Abstract
Depending on composition, processing, microstructure, and loading mode, steels may demonstrate various inelastic deformation mechanisms, notably dislocation glide, deformation twinning, and solid-solid phase changes. Damage in the form of voids, often leading to failure by macro-cracking, is also of current interest. A finite-deformation constitutive model is constructed in order to address such mechanisms in isotropic polycrystals under static and dynamic loading, where the latter encompasses high pressures and extreme strain rates pertinent to ballistic penetration. Slip and twinning are isochoric, and their relative contributions to kinematics are not explicitly distinguished in the present application. Phase changes among, e.g., face-centered-cubic (FCC), body-centered-cubic (BCC), body-centered-tetragonal (BCT), and hexagonal (HCP) structures are admitted, with corresponding deviatoric and volumetric strains. Porosity from voids contributes to volumetric strain. A new consistent thermodynamic framework incorporating an Eulerian strain tensor and internal state variables is developed, whereby kinetic equations for slip–twinning, phase changes, and damage evolution result in contributions to dissipation. An objective rate form of the model, derived assuming small deviatoric elastic strain, is implemented numerically. The model is applied to three different primarily austenitic, medium-high Mn steels. Specifically, representations of a slip-dominated (SLIP), a TRIP (transformation-induced plasticity) and a TWIP (twinning-induced plasticity) steel are calibrated to quasi-static tension, quasi-static compression, and dynamic compression data, at both room and high temperatures. A novel functional form of material strength distinguishes hardening profiles of the different alloys. Experimental data are reasonably well represented. Model extrapolations for dynamic strength and pressure in regimes pertinent to shock compression are analyzed. Predictions for multi-axial loading of shear with simultaneous expansion or contraction demonstrate competing physical mechanisms among the alloys that could be leveraged for optimal ballistic performance.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
Introduction
Steels and iron alloys are perhaps the most widely used metallic materials in structural engineering applications. Composition, processing, and microstructure lead to vastly different performance measures, e.g., yield strength, strain hardening, ductility, and fracture resistance. Dynamic performance metrics include resistance to adiabatic shear localization, spall fracture, and fragmentation [1]. Analysis of structure–property–performance relationships, especially at high loading rates, is complicated by the possible presence of multiple phases and strongly temperature-dependent mechanisms: dislocation motion, generation, and annihilation; deformation twinning; transformations among phases; and flow instabilities. An improved understanding of effects of such mechanisms, some complementary and some competing, on dynamic failure resistance is sought to enable design and optimization of multiphase steels, including TWIP [2] and TRIP [3] steels, for static [4] and ballistic [1, 5] applications.
Dynamic failure is of keen interest in this work, notably shear localization [6] and fracture. Failure in ductile steels is often associated with void nucleation, growth, and coalescence, where instability occurs by shear banding accelerated by linkages between voids or second-phase particles [7]. Brittle fracture (cracks versus voids) is logically more prevalent in steels of higher hardness, and catastrophic dynamic fracture may be preceded by adiabatic shear localization, for example plugging failure in ballistic penetration [1, 8,9,10]. Explicit studies of localization and failure in steel microstructures, wherein secondary phases and voids or cracks are resolved individually, include [7, 11, 12].
The model developed in the present work is intended for numerical studies of localization and failure in steels across a range of loading rates, temperatures, and pressures. Effects of rate and temperature dependence have been studied often before for metals, as has plasticity–damage incurring local strain softening from voids [13, 14]. However, no systematic study of combined effects of inelastic flow from slip and twinning, deviatoric and volumetric strains from phase changes, softening and dilatation from porosity, and/or failure by cracking, for different imposed loading rates from static to highly dynamic (e.g., ballistic) at low and high temperatures seems to exist.
In this work, a generic continuum constitutive model capable of addressing such mechanisms in a simple yet physically credible way is constructed. This model conforms to accepted thermo-mechanical laws of nonlinear solid mechanics of crystalline solids (e.g., [10, 15,16,17,18]), here directed toward macroscopically isotropic polycrystals with evolving microstructure. Results of numerical simulations can then guide design of steels for optimum performance. For example, in TWIP steels competition among slip, twinning, and phase change depends on intrinsic stacking fault energy, SFE [2, 19], which is strongly temperature-dependent and can be tuned via chemistry and processing [3, 5]. Mechanical and thermal treatments can be employed to advantageously affect hardness, ductility, and anisotropy.
No prior model known to the authors contains all above-mentioned physical ingredients; published models may resolve fewer mechanisms with more or less fidelity than pursued here. For example, a homogenization approach was used to study quasi-static stress–strain behavior and localization in TRIP multiphase steels [4], but this approach did not account for twinning, voids, evolving temperature, or strain-rate dependence. Single-crystal models have been used to understand twinning and phase changes in austenitic steels at different ambient temperatures [3, 20,21,22,23,24], but these did not consider failure mechanisms (voids or cracks) or adiabatic shear localization under dynamic conditions. Experiments suggest importance of temperature-dependent phase transformations inside adiabatic shear bands observed in austenite–ferrite duplex steels [25], but modeling of such complex behaviors remains to be undertaken. Models incorporating homogenized flow rules for slip and phase changes in polycrystalline TRIP steels include [26,27,28,29]. These models can account for rate and temperature effects on transformations and flow stress, but they do not address nonlinear pressure–volume behavior arising in impact events, nor ductile failure mechanisms. Kinetic models for shock-induced phase changes have been researched for decades [30]—perhaps most notably the \(\alpha \rightarrow \epsilon\) (i.e., BCC \(\rightarrow\) HCP) contractive transformation in iron that has been described in sophisticated crystal-level simulations [31]—but much less attention has been devoted to modeling this and other transformations in alloys with different compositions and more complex microstructures in dynamic loading regimes. The expansive transformations of \(\gamma\) (FCC austenite) and \(\epsilon\) phases to \(\alpha\)-martensite are of primary interest for compositions and loading protocols examined in this work.
This paper is organized as follows. Constitutive theory is developed in “Constitutive Model” section. Since this is the first of what should be a series of works utilizing this model framework, stepwise derivations from fundamental principles and basic physical arguments are included; these will be abbreviated or omitted in future works. A number of more lengthy derivations are deferred to “Appendix 1”. Governing equations and numerical algorithms are summarized in “Numerical Implementation, Materials, and Parameters” section, where material properties and parameters are tabulated for three different steel alloys tested experimentally. Model results for quasi-static (tension and compression) and dynamic (Kolsky-bar, compression only) loading are reported and compared with test data in “Static and Dynamic Loading” section. Results for uniaxial strain compression (akin to adiabatic ramp loading) are compared among the three alloys in “Dynamic High-Pressure Response” section, including assessment of predicted quasi-static and dynamic high pressure responses in the context of available shock Hugoniot data on other steels. Predictions for dynamic, multi-axial pressure–shear loading are described in “Dynamic Volumetric-Simple Shear Response” section. Conclusions follow thereafter in “Conclusions” section. Standard notation of modern continuum mechanics is used. Vectors and tensors are written in bold, with scalar components, referred to a fixed Cartesian coordinate frame, written in plain italics. When index notation is used, summation applies over repeated indices.
Constitutive Model
A finite-deformation, thermoelastic–viscoplastic model is constructed, capable of representing prominent physical mechanisms in polycrystalline multiphase SLIP, TRIP, or TWIP steels at different loading rates and temperatures. A local volume element is assumed to contain a statistically representative microstructure, with a sufficient number of grains of each phase as well as defects (dislocations, twin and phase boundaries, pores, micro-cracks, etc.), to adequately reflect the response of a material point via continuum mechanical theory. Individual microscopic features are not resolved explicitly. Rather, effects of microstructure are captured by evolving internal state variables that quantify defect densities and volume fractions of phases within each local volume element. Local deformations within each phase, or within (un)twinned regions, are not distinguished explicitly.
Kinematics
Let \({{{\varvec{x}}}} = {{{\varvec{x}}}}({{{\varvec{X}}}},t)\) denote the time-dependent spatial position of a material particle referentially located at \({{{\varvec{X}}}}.\) Let \(\nabla _{0}(\cdot ) = \frac{\partial }{\partial {{{\varvec{X}}}}} (\cdot )\) be the material gradient. The deformation gradient \({{{\varvec{F}}}}\) is decomposed multiplicatively as
where \({{{\varvec{F}}}}^{E}\) and \({{{\varvec{F}}}}^{P}\) are the reversible thermoelastic deformation and the total residual “plastic” deformation that persists upon local unloading. The latter is further deconstructed as
with \({{{\varvec{F}}}}^{pl}\) due to dislocation glide, \({{{\varvec{F}}}}^{tw}\) due to twinning, \({{{\varvec{F}}}}^{tr}\) due to phase transformations, and \({{{\varvec{F}}}}^d\) due to damage mechanisms. Each term in (1) and (2) is presumed invertible with positive determinant. Similar decompositions have been used elsewhere for twinning and phase changing materials [3, 24, 32], but without the residual deformation contribution from damage.
Later in “Plastic Flow” section and thereafter, contributions from plastic slip and deformation twinning to yielding and flow resistance are combined into functional kinematic and strength quantities that do not explicitly delineate between the two mechanisms. This is a standard approach [2, 33, 34] for modeling of TWIP- and TRIP-type alloys that simplifies parameterization of the model when insufficient data are available to distinguish effects of slip and twinning and assign model parameters accordingly. However, separation of mechanisms is maintained in the theoretical presentation until “Plastic Flow” section, which should facilitate extension of the thermo-mechanical theory to alternative implementations in which distinct slip and twinning kinetics are resolved explicitly, for example as in single crystal formulations [3, 24].
Let \(\rho\) and \(\rho _{0}\) denote spatial and referential mass densities for an element of the whole (composite) microstructure. Then conservation of mass dictates
Slip and twinning are isochoric, following standard kinematic arguments for crystalline solids [15, 17, 18]. Volume change from phase transformations may be dilatational or contractive depending on the direction of transformation and the structure of each phase. Damage from voids or open cracks is assumed to be dilatational only, i.e.,
Let \({\bar{{{\varvec{F}}}}} = J^{-1/3} {{{\varvec{F}}}}\) denote the isochoric part of the deformation gradient, with similar notation used for other terms in (2). Then, assuming that the contribution from damage is spherical, (2) becomes
Accordingly, residual shape changes are due to slip, twinning, and transformations, while residual volume changes are due to transformations and porosity or damage. Dilatation from dislocation cores, stacking faults, and twin boundaries is presumed small enough to be neglected, a reasonable assumption when defect densities do not approach their theoretical maximum limits [18, 35, 36].
Denote the material time derivative of field quantity \(f({{{\varvec{X}}}},t)\) by \({\dot{f}} ({{{\varvec{X}}}},t)= \frac{\partial }{\partial t} f ({{{\varvec{X}}}},t),\) and denote \(\nabla (\cdot ) = \frac{\partial }{\partial {{{\varvec{x}}}}} (\cdot )\) the spatial gradient. Particle velocity is \(\dot{{{\varvec{x}}}}.\) The spatial velocity gradient is
where \({{{\varvec{d}}}}\) and \(\pmb {\omega },\) with corresponding superscripts, are symmetric and skew parts of \({{{\varvec{l}}}}\) and its thermoelastic and inelastic parts. The total plastic velocity gradient pushed forward to the spatial frame is then, using (2) and (5),
Total plastic strain rate \({{{\varvec{d}}}}^{P}\) and total plastic spin \(\pmb {\omega }^{P}\) both have deviatoric contributions from rates of slip, twinning, and phase changes embedded in \({\bar{{{\varvec{l}}}}}^{P}.\) Volumetric contributions to the rate of \(J^{P}\) arise from phase changes and damage; these affect \({{{\varvec{d}}}}^{P}\) but not \(\pmb {\omega }^{P}.\) Later in this work, the usual continuum plasticity assumption \(\pmb {\omega }^{P} = \mathbf{0 }\) [10, 15, 17] is imposed for isotropic polycrystals. The total Eulerian deviatoric plastic strain rate is decomposed as follows, where contributions to each term associated with its corresponding rate of inelastic deformation gradient, omitted here for brevity, can be calculated readily from time differentiation of (5):
An Eulerian thermoelastic strain \({{{\varvec{D}}}}^{E}\) is used, justified by prior analysis on different strain measures for crystalline solids under shock compression [37,38,39,40]:
This tensor, though constructed from “Eulerian” field variable \({{{\varvec{F}}}}^{E-1}({{{\varvec{x}}}},t),\) is referred to intermediate configuration coordinates. Therefore it is naturally invariant under changes of spatial observer and suitable as a state variable for isotropic and anisotropic materials [18, 37]. Its material time derivative is
Thermoelastic volume change is
The polar decompositions \({{{\varvec{F}}}}^{E} = {{{\varvec{V}}}}^{E} {{{\varvec{R}}}}^{E} = {{{\varvec{R}}}}^{E} {{{\varvec{U}}}}^{E}\) are introduced, along with the following unimodular thermoelastic stretches and traceless thermoelastic strain tensor, respectively:
An assumption made in some later derivations is that deviatoric thermoelastic strains are “small”; specifically, letting \(||{{{\varvec{A}}}}|| = ({{{\varvec{A}}}}:{{{\varvec{A}}}})^{1/2}\) for symmetric second-order tensor \({{{\varvec{A}}}},\) this entails
Thermomechanical Balance Laws
Local conservation laws for mass, linear momentum, and angular momentum are of the standard forms
Denoted by \(\pmb {\sigma }\) and \({{{\varvec{P}}}}\) are the Cauchy and first Piola–Kirchhoff stress tensors, respectively, related by \(\pmb {\sigma } = J^{-1} {{{\varvec{P}}}} {{{\varvec{F}}}}^{\text {T}}.\) Denoted by \({{\varvec{b}}}\) is the body force per unit initial volume.
Thermodynamic potentials are expressed on a per unit initial volume basis, converted to a per unit mass basis via division by \(\rho _{0}.\) Three such potentials—internal energy density U, Helmholtz free energy density \({\varPsi },\) and entropy density \(\eta\)—are classically related by
Absolute temperature is \(T > 0.\) Denote the heat flux in the reference configuration by the vector \({{{\varvec{Q}}}}\) and a scalar heat source per unit reference volume by R. The local balance of energy is
A local entropy production inequality and an alternative expression following from (16) and (17) are, respectively,
Let \(p = -\frac{1}{3} {\text {tr}} \pmb {\sigma }\) be Cauchy pressure and \(\pmb {\sigma }' = \pmb {\sigma } + p \mathbf{1 }\) the stress deviator. Stress power per unit reference volume can be decomposed as follows using (6)–(8) and (11) [see (97) of “Appendix 1” for details]:
Constitutive Assumptions and Implications
Thermoelastic potentials depend on thermoelastic strain \({{{\varvec{D}}}}^{E},\) temperature T or entropy \(\eta ,\) and three dimensionless scalar internal state variables \(\upsilon ,\chi ,d,\) grouped into the vector \(\pmb {\xi }\) with components \(\xi ^{i}\) (\(i=1,2,3)\):
Here, \(\upsilon \in [0,1-\upsilon _{0}]\) is the volume fraction of the second phase of microstructure that has undergone transformation after the initial time. The volume fraction of the second phase in the initial state is \(\upsilon _{0} \in [0,1].\) Attention is restricted to two-phase materials such that a single scalar variable \(\upsilon\) suffices to quantify local changes in solid phase content. Attention is also restricted to isotropic damage such that a single scalar variable d suffices, noting \(d \in [0,1)\) is the damage variable. The state variable later associated primarily with strain hardening behavior is \(\chi ;\) its energetic contributions account for residual elastic stress fields from dislocations and surface energy of twin boundaries. Twinned volume fractions and dislocation densities are not tracked explicitly. A potentially more rigorous approach would consider energy functions measured per unit volume in the thermoelastically unloaded intermediate configuration [17, 18, 35, 41]; such refinement is not pursued here to maintain simplicity.
Phase volume fractions are measured per unit volume of the solid material, i.e., this solid volume does not include the void volume. The total volume fraction of the first phase is \(1-(\upsilon +\upsilon _{0}).\) In two-phase steels, for example, undergoing a transformation from \(\gamma\)-austenite to \(\alpha\)-martensite, \(\upsilon _{0}\) would be the fraction of \(\alpha\) phase present initially with \(1-\upsilon _{0}\) the initial remainder consisting of \(\gamma\) phase. Slip and/or twinning may occur within all phases, and \(\chi\) is representative of dislocation and deformation–twin boundary content in the entire microstructure. This is a notably relevant assumption for martensitic transformations in steels whereby transformed regions could inherit the dislocation density of the parent phase [26]. Variable \(\chi\) does not account for twin boundaries that form spontaneously upon transformation, e.g., finely twinned martensitic variants.
Define the conjugate stress to \({{{\varvec{D}}}}^{E}\) and (negative) conjugate forces to \(\xi ^{i}\) as
The rate of free energy in (20) becomes
Substitution of (19) and (22) into the second of (18) produces the thermoelastic equations
and the dissipation inequality
The contribution of heat conduction to entropy production is presumed non-negative, which can be ensured by proper choice of constitutive model for \({{{\varvec{Q}}}}\) [17, 18], to be shown later for Fourier conduction. This leaves the condition \({\mathfrak {D}}^{mech} \ge 0,\) which states that the rate of inelastic working should equal or exceed any rate of free energy increase associated with internal state variables.
From (20), (23), and letting \(T = T({{{\varvec{D}}}}^{E},\eta ,\pmb {\xi }),\) thermoelastic relations in terms of derivatives of internal energy are
Let \(c^{E}({{{\varvec{D}}}}^{E},T,\pmb {\xi })\) denote specific heat per unit reference volume at constant elastic strain; from (23) and (25),
The rate of internal energy is
Substituting (19), (26), and (27) into (17) produces a temperature evolution equation:
where the total rate of plastic work and thermal stress coefficients are, respectively,
The Grüniesen tensor is \(\pmb {{\varGamma }}^{E}.\) Isothermal and isentropic, respectively, second-order thermodynamic elastic coefficients (not necessarily constants) are
Thermal expansion coefficients are \(\pmb {\alpha }^{E} = \partial {{{\varvec{D}}}}^{E} / \partial T,\) with partial differentiation taken at constant stress \({{{\varvec{S}}}}\) and constant internal state \(\pmb {\xi }.\) Specific heat at constant stress is \(c^{S}.\) Then [18, 42]
Define \(\beta ^{P}\) as an instantaneous dissipation ratio, not a constant in this theory, similar to a Taylor–Quinney parameter:
Substitution into (28) gives the energy balance in a more compact form:
The transformation \(\nabla _{0} \cdot {{{\varvec{Q}}}} = J \nabla \cdot {{{\varvec{q}}}}\) used in (33) follows from the Piola identity \(\nabla \cdot (J^{-1} {{{\varvec{F}}}}) = \mathbf{0 }\) [43], where the spatial heat flux vector \({{{\varvec{q}}}}\) is the Piola transform of \({{{\varvec{Q}}}},\) i.e., \({{{\varvec{q}}}} = J^{-1} {{{\varvec{F}}}} {{{\varvec{Q}}}}\) [17, 18].
While models based on internal energy are often more convenient for analysis of adiabatic and wave propagation problems [18, 38], the present theory will hereafter focus on free energy. The reason is that energy potentials associated with phase transformations are more often parameterized in terms of temperature rather than entropy [2, 3, 26].
Total free energy per unit reference volume \({\varPsi }\) of (20) is deconstructed into three scalar functions as
where the latter two are further composed of the sums
Thermoelastic strain energy density is W, phase energy with decoupled specific heat energy (\(g_{T}\)) and latent heat energy (\(g_{\upsilon }\)) is g. Total defect energy is h. Stored energy of dislocation structure and twin boundaries is \(h_{\chi },\) and stored energy of phase boundaries and twin boundaries that form within the second phase, simultaneously upon phase changes, is included in \(h_{\upsilon }\) along with other athermal microstructure effects. The magnitude of \(h_{\chi }\) will later be linked to cumulative plastic strain (and thus, strain hardening and slip barriers), while \(h_{\upsilon }\) will provide an intrinsic athermal energy barrier to phase transformations. Specific forms applicable to steels are forthcoming in the remainder of “Constitutive Model” section, along with kinetic equations for rates of internal state variables and inelastic deformations.
Thermoelasticity
Strain Energy
Isotropic symmetry is assumed for local thermoelastic response of a polycrystalline material element. Total strain energy—including dilatational and deviatoric elastic strain energies and thermoelastic coupling—is the following function measured per unit reference volume:
Energy function (36) is an assumed form, not derived from prior fundamental principles. The second and third terms in (36) account, respectively, for nonlinear pressure–volume response and thermoelastic coupling. Initial temperature is \(T_{0}.\) Damage-dependent bulk and shear moduli are expressed as follows, in terms of isothermal second-order elastic constants \(B_{0}\) and \(G_{0}\) of the solid without damage:
Specifically, \(B_{0}\) is the isothermal bulk modulus of the material without damage, and \(G_{0}\) is the shear modulus of the material without damage. Degradation functions \(\zeta ^{B} \in [0,1]\) and \(\zeta ^{G} \in [0,1]\) are defined later for solids with voids in “Damage” section. Thermal stress coefficients are constant for simplicity:
Denoted by \(A_{0}\) is the volumetric coefficient of thermal expansion, and specific heat at constant volume is \(c_{V} = c^{E}\) for isotropic symmetry [18, 42].
The isothermal elastic modulus tensor of (30) is, in component form,
Higher-order thermoelastic coupling, e.g., temperature dependence of elastic moduli, is omitted. Furthermore, differences in G and B among different crystal phases are not addressed here; this simplifying assumption has been invoked often elsewhere in the context of limited available properties for individual phases comprising TRIP steels [26,27,28,29].
Thermodynamic stress is, from the first of (21) and noting that only W depends on \({{{\varvec{D}}}}^{E}\) in (34),
Cauchy stress is
Pressure
The trace of (41) is [see (98) of “Appendix 1” for details]
where small deviatoric thermoelastic strain approximation (14) is used. Cauchy pressure is then, using (14) with (37) [see (99)],
When \(J^{P} = 1\) and \(B = B_{0} \leftrightarrow \zeta ^{B} = 1,\) this becomes the Birch–Murnaghan equation-of-state that is accurate for many polycrystalline solids at least up to moderate shock pressures [18, 44, 45]. Effects of phase change and damage are implicitly included via dependence of volume on both mechanisms, recalling that \(J = J^{E} J^{P}\) with \(J^{P} = J^{tr} J^{d},\) and via dependence of B on damage through \(\zeta ^{B}\) of (87), at least for tensile stress states.
Deviatoric Stress Rate
The deviatoric Cauchy stress is, from (41), and invoking (14) [see (100)],
where \({\bar{{{\varvec{d}}}}}^{E} = {{{\varvec{d}}}}^{E} - \frac{1}{3} {\text {tr}} ( {{{\varvec{D}}}}^{E}) \mathbf{1 } = {\bar{{{\varvec{d}}}}} - {\bar{{{\varvec{d}}}}}^{P}\) is the deviatoric Eulerian thermoelastic strain rate. The material time derivative of (44) is calculated as follows, noting that \({\dot{F}}^{E \, -1}_{\alpha k} = -{F}^{E \, -1}_{\alpha j} {\dot{F}}^{E}_{j \beta } {F}^{E \, -1}_{\beta k}\) [17] and using the second of (37) [see (102)]:
An objective rate form of deviatoric Cauchy stress under the assumption (14) is then
with \(\zeta ^{G}\) prescribed later in (87) and \({\bar{G}}\) the tangent shear modulus:
Now assume that, for an isotropic material, \(\pmb {\omega } = \pmb {\omega }^{E}\) is the total spin tensor, equal to the elastic spin. Further assume that the rate of deviatoric elastic stretching is small relative to rates of elastic rotation and dilatation, such that convective products \(\pmb {\sigma }' {\bar{{{\varvec{d}}}}}^{E}\) and \({\bar{{{\varvec{d}}}}}^{E} \pmb {\sigma }'\) can be omitted in (47). In this case,
The first three terms following the approximation sign comprise the Jaumann rate of deviatoric Cauchy stress. The rightmost term in square braces depends on total and plastic volume change rates and the damage rate. Factors of \(\frac{5}{3}\) and \(-\frac{2}{3}\) arise since the volumetric elastic strain rate is maintained in the convective terms carried forward from (47).
Plastic Flow
A composite yield stress \(K^{P}\) is used for computation of the total deviatoric inelastic strain rate \({\bar{{{\varvec{d}}}}}^{P}\) in the two-phase material, as in, e.g., [28, 29, 33]. The total deviatoric inelastic strain rate tensor is coaxial with the deviatoric Cauchy stress tensor [26,27,28,29, 33] and can be derived from a \(J_{2}\)-based flow potential \({\varOmega }^{P}\):
The magnitude of the deviatoric stress is \({\bar{\sigma }},\) related to the Von Mises stress \(\sigma _{v},\) the second invariant \(J_{2},\) and the triaxiality \({\varSigma }\) as
The deviatoric flow direction is normal to \({\varOmega }^{P}\):
The associated deviatoric flow rule is
The rate of inelastic working is
The Kuhn–Tucker conditions and consistency condition are [46]
Inelastic volumetric deformation is non-associative.
The total flow resistance of the composite \(K^{P}\) depends on the total flow resistance of each phase, as well as damage incurred in the solid. The flow resistance of each phase, in turn, depends on resistance to mechanisms active in each phase, namely dislocation glide, deformation twinning, and phase transformations. These mechanistic resistances depend on strain history, strain rate, and temperature. Let subscript \(i=1,2\) distinguish the two phases, where the relevant transformation direction, e.g., austenite to martensite in the TRIP steel studied later, is \(1 \rightarrow 2.\) Total volume fractions of each crystallographic phase are, recalling nomenclature of “Constitutive Assumptions and Implications” section,
This notation is convenient for labeling of properties and parameters that depend on crystallographic phase measured absolutely by \((\upsilon _{1},\upsilon _{2})\) rather than initial and transient transformed fractions measured by \((\upsilon _{0},\upsilon ).\) Let \(K^{P}_{0}\) denote the total composite flow resistance in the absence of damage. A rule of mixtures is applied as in [28, 29, 33]:
The degradation function \(\zeta ^{P}(d)\) is defined explicitly later in “Damage” section for a material with voids and acts uniformly over the composite microstructure. Since slip, twinning, and phase transformations may operate in parallel, their individual contributions combine according to [33]
For one-way transformations, \(K^{tr}_{2} \rightarrow \infty\) if there is no effect of transformation of the first phase on strength of the second phase; however, \(K^{tr}_{2}\) can be finite to account for localized softening of initial material corresponding to \(\upsilon _{0}\) under stress-assisted transformations, for example. Martensite often consists of finely-twinned layers arranged in a herringbone pattern [21, 22]; effects of these twins that form during the phase change are captured by \(K^{tr}_{i}\) and not \(K^{pl{-}tw}_{i}.\) The latter quantifies resistance to plastic slip and deformation twinning in phase 1 for \(i=1,\) and for resistance to plastic slip and twinning that occurs after transformation in phase 2 for \(i=2.\)
In principle, the \(K^{tr}_{i}\) contribution should occur only when the phase transformation rate is nonzero. However, use of a function \(K^{tr}_{i}\) depending explicitly on \({\dot{\upsilon }}\) proved too sensitive to slight variations in stress state and temperature, leading to unrealistic oscillations and instabilities in numerical predictions, especially near the transformation threshold. Later in (82), a smooth double-well function of \(\upsilon\) itself is used to modulate \(K^{tr}_{i},\) with zeroes at \(\upsilon = 0,1.\) The end result is a capability to produce sigmoidal stress–strain curves witnessed experimentally in some TRIP steels [26, 27].
As in [33, 34], plastic slip and deformation twinning are not distinguished explicitly in the flow resistance. This assumption is justified since the volume fraction of twins is relatively low in austenetic TWIP steels, and since effects of twinning manifest primarily as barriers to slip, reducing dislocation mean free paths, accelerating pile-ups, and increasing overall plastic hardening [2]. More intricate models exist wherein twin volume fractions and twinning resistances are tracked explicitly, but these models require calibration of more parameters—a problematic issue since effects of twinning and slip on overall strength are usually difficult to distinguish in conventional test data—and are more computationally cumbersome. The transformation resistance \(K^{tr}_{i}\) is addressed later in “Phase Transformations” section. The remainder of “Plastic Flow” section addresses \(K^{pl{-}tw}_{i}\) and the contribution of defects associated with slip and twinning to free energy, \(h_{\chi }(\chi ).\)
Let \({\dot{e}}_{0}\) be a reference strain rate, and \(m_{i},p_{i}\) be material constants for each phase \(i=1,2.\) Denote by \(k_{i}\) the slip–twinning resistance at an applied plastic strain rate \({\dot{e}}^{P} = {\dot{e}}_{0}\) and a reference (room) temperature \(T=T_{R}.\) Let \({\bar{\sigma }}_{i}\) be the magnitude of deviatoric stress in phase i. When plastic flow is occurring, a local viscoplastic flow rule is operative, where \({\dot{e}}^{P} = \sqrt{\frac{2}{3}} {\dot{\epsilon }}^{P}\) is coarsely approximated as the same in each phase:
Thermal softening due to increasing dislocation mobility at higher temperatures, as well as decreased flow resistance and hardening scaling with the shear modulus, is measured by \(p_{i} \ge 0\) [47]. The melt temperature at ambient pressure is \(T_{M}.\) Angled brackets \(\langle x \rangle = \frac{1}{2}(x + |x|)\) parse positive values, preventing hardening at low temperatures, enforcing null strength for \(T \ge T_{M},\) and rendering non-integer values of \(p^{i}\) allowable for \(T < T_{R}.\) Inversion of (59) gives, at local yield \({\bar{\sigma }}_{i} = \sqrt{\frac{2}{3}} K_{i}^{pl{-}tw},\)
Relation (60) for flow stress could be prescribed directly. However, the same result is derived here from the more physically appealing kinetic law (59), following the viewpoint in [15]. The strain rate sensitivity for plastic flow in phase i is \(m_{i} ={ \partial \ln {\bar{\sigma }}_{i} }/{ \partial \ln {\dot{e}}^{P}}.\)
Mises-equivalent function \(k_{i}\) depends on cumulative conjugate inelastic strain \(e^{P} = \int _{0}^{t} {\dot{e}}^{P}(\tau ) \text {d}\tau \ge 0\) as follows [48]:
The initial yield stress at \(T=T_{R}\) and \({\dot{e}}^{P} = {\dot{e}}_{0},\) a constant for a given material phase of specific chemical composition and processing treatment, is \(\sigma _{0i}.\) The strain hardening coefficient with dimensions of stress is \(\theta _{0i},\) and \(\delta _{i}\) and \(\epsilon _{i}\) are fitting parameters that control the hardening profile [48]:
Herein, \(k_{i}\) is a function that accounts for, in addition to usual dislocation-mediated hardening, the effects of twinning on flow resistance (twins acting as slip barriers [2]) and phase transformations on flow resistance of the first phase (e.g., the Greenwood–Johnson effect [49]). This is simply a stated assertion; its functional form is not derived from micro-mechanics or other fundamental physical arguments. Since activity of the latter two mechanisms is strongly dependent on SFE [2, 3, 24], \(\delta _{i}\) and \(\epsilon _{i}\) are likely correlated to SFE [48], though possible dependence of these parameters on temperature is omitted for simplicity. It will later be shown that SLIP steels with high SFE demonstrate a relatively low value of \(\delta _{i},\) whereas TWIP and TRIP steels (phase 1) demonstrate higher values of \(\delta _{i}.\) Such findings are consistent with trends reported for these and other steel compositions in [48]. Success of (61) for fitting tensile stress–strain data of a large number of medium to high-Mn steels (16 datasets) with different SFEs that collectively undergo combinations of slip, twinning, and phase transformations is reported in Supplementary Material of [48].
Let \({\mathfrak {D}}^{pl{-}tw}\) be the net dissipation due to plastic slip and twinning, where the sum of their rates is also assumed coaxial with the Cauchy stress deviator:
The internal variable \(\chi\) associated with defects accumulated in conjunction with slip and twinning (distinct from phase changes addressed by \(\upsilon\)) is the corresponding cumulative inelastic strain due to these processes:
The stored defect energy \(h_{\chi },\) which may not be analytic in \(\chi ,\) obeys the following differential equation, where \(\beta ^{pl{-}tw} \in [0,1]\) is the prescribed Taylor–Quinney factor for slip and twinning processes:
The effective deviatoric strain rate from transformations is \({\dot{\epsilon }}^{tr},\) to be defined in the next section. With regard to (33), \(\beta ^{P} \rightarrow \beta ^{pl{-}tw}\) when \({\dot{\upsilon }}\rightarrow 0\) and \({\dot{d}} \rightarrow 0,\) but in general, \(\beta ^{P} \ne \beta ^{pl{-}tw}\) since \(\beta ^{P}\) accounts for working from all inelastic deformation mechanisms and all internal variables (\(\chi ,\upsilon ,d\)), while \(\beta ^{pl{-}tw}\) accounts only for slip–twinning and \(\chi .\)
Delineation between stored energy changes due to slip and twinning processes (e.g., contributions from microscopic elastic fields of dislocations and twin boundary surfaces) from free energy changes due to phase transformations is essential. The latter correspond to athermal barriers and latent heat energy of phase changes, both of which explicitly contribute to transformation kinetics in the framework of “Phase Transformations” section. The choice of \(\beta ^{pl{-}tw},\) in contrast, does not directly enter transformation kinetics, though transformation rates are secondarily affected since \(\beta ^{pl{-}tw}\) can influence evolution of temperature when plastic straining occurs.
Phase Transformations
Detailed kinematics for the transformation process, free energy associated with phase content, kinetics for the transformation rate, and the contribution of phase changes to the effective yield stress are discussed in this section. The total deformation rate associated with phase transformation consists of a deviatoric part coaxial with the deviatoric Cauchy stress [26,27,28,29, 33] and a volumetric (i.e., spherically symmetric) part:
First consider the volume change, which, recalling \(\upsilon = \upsilon _{2} - \upsilon _{0}\) is the volume fraction of the transformed material acquired after the initial time, follows as
The stress-free mass density of phase i at \(T_{0}\) is \((\rho _{0})_{i}.\) When \(\upsilon _{0} = 0,\) reference mass density is simply \(\rho _{0} = (\rho _{0})_{1}.\)
Now consider the shape change. Let \(\gamma ^{tr} \ge 0\) be the maximum (simple) shear strain incurred by an element of crystalline material under total transformation [3, 24]; this depends on the crystal structure and lattice parameters of each phase. Let \(A^{tr} \in [0,1]\) be a scalar function describing the net contribution of each transforming crystal to the total shape change. Its value may potentially evolve with deformation and is typically less than unity since local shape changes not aligned with \({{{\varvec{N}}}}^{P}\) tend to cancel themselves [26, 33]. Then
Since \({\dot{\epsilon }}^{tr} \ge 0\) by definition, validity of (68) requires \({\dot{\upsilon }} \ge 0,\) meaning the present kinematic treatment is restricted to one-way (irreversible) transformations.
Next consider the contribution of phase change to the total free energy in (34) and (35). The temperature dependent energy function is \(g(T,\upsilon ) = g_{T} (T) + g_{\upsilon }(T,\upsilon ).\) A uniform and constant specific heat \(c_{V}\) is used, identical in each phase for simplicity, giving the standard free energy contribution [18]
Latent heat per unit initial volume upon transformation from phase 1 to 2 is contained in function \({\varLambda }(T),\) such that
Finally, the athermal contribution of surface energy of phase boundaries and other defects induced during the transformation process is quantified by \(h_{\upsilon }\):
Here, \({\varUpsilon }_{0}\) is an energy barrier per unit area (assumed constant for simplicity) and \(l_{0}\) is a characteristic spacing between planar defects such as phase lamellae and phase boundaries. More intricate models for contributions of latent heat and microstructure that exceed the present level of fidelity are available elsewhere, e.g., [22].
The net dissipation from phase changes is now addressed, influencing a thermodynamically consistent kinetic law for transformation. From (66)–(71) [see (103)],
where, with \(\alpha ^{tr}\) given in (104), the mechanical driving force, thermal barrier, and athermal (microstructure-dependent) energy barriers to transformation are, respectively,
The kinetic law for transformation per unit volume of the initial phase ensuring that \({\mathfrak {D}}^{tr} \ge 0\) is motivated by [21, 22]:
where \({\dot{\upsilon }}_{max}\) is the maximum transformation rate, \(\mu\) is a dimensionless viscosity, and recalling the usual definitions
The quotient \([1-(\upsilon + \upsilon _{0})]\) in (74) accounts for reduction in maximum transformation rate per unit reference volume as the volume fraction of second phase increases [26, 27]. For example, as the total volume of phase 1 shrinks to zero, the corresponding maximum transformation rate approaches zero. The Heaviside (unit) step function is \({\text {H}}(x) = 0 \forall x < 0\) and \({\text {H}}(x) = 1 \forall x \ge 0.\) Denoted by \({\dot{e}}_{1}\) is reference strain rate above which the time scale for transformation kinetics limits the maximum effective transformation rate relative to the total plastic strain rate \({\dot{e}}^{P}.\) Dimensionless parameter \(r \ge 0\) controls the magnitude of this rate-limiting effect.
According to (74), the mechanical driving force must exceed the thermal and athermal energy barriers for transformation to proceed. Note that \(f^{ath}\) is always positive by definition, but \(f^{mech}\) and \(f^{th}\) are unrestricted in sign. The maximum transformation rate is not a constant; it occurs when the transformation strain rate approaches the total inelastic strain rate, i.e., \({\dot{\epsilon }}^{tr} \rightarrow {\dot{\epsilon }}^{P}\) as \({\dot{\upsilon }} \rightarrow {\dot{\upsilon }}_{max}\):
The total effective inelastic strain rate \({\dot{\epsilon }}^{P}\) potentially consists of contributions from slip, twinning, and phase transformations (deviatoric part only). Definition (76) is a necessary condition that prohibits the transformation contribution from exceeding the total, since \({\dot{\epsilon }}^{pl{-}tw}\) and \({\dot{\epsilon }}^{tr}\) are both always non-negative. The product of other terms multiplying \({\dot{\upsilon }}_{max}\) in (74) is always less than or equal to unity according to definitions of each variable.
Denote by \(T_{T}\) the “transformation temperature”, a theoretical temperature at which phase 1 can instantly transform into phase 2 at zero stress and without an energy barrier (\(f^{mech} = f^{ath} = 0\)). Define latent heat constant \(\lambda _{T}\) at the transformation temperature \(T_{T}\) and zero stress as the internal energy per unit reference volume required for complete transformation \(\upsilon = 0 \rightarrow \upsilon =1\) in the theoretical limit that \({\mathfrak {D}}^{tr} = 0.\) The entropy change associated with this transformation is \(\lambda _{T}/T_{T}.\) Then, as in [21, 22],
The latent heat constant \(\lambda _{T} = \lambda (T_{T})\) is related to latent heat measured at the “martensite start” temperature \(M_{s},\) when differences in specific heats between the two phases are neglected, via
For usual transformation from a higher- to lower-entropy phase, \(\lambda _{T} < 0.\) In that case, the thermal barrier \(f^{th}\) will be positive for \(T > T_{T}\) and will decrease linearly with decreasing temperature T. From (70) and (77), the following reduced form of a contribution to internal energy rate in (27) is derived [see (105)]:
In the usual case when \(\lambda (T) = \lambda _{T} \cdot (T/T_{T}) < 0,\) the contribution of (79) to the internal energy rate tends to require heat energy, consistent with the definition of \(\lambda _{T}.\)
Finally consider the effect of transformation on the overall inelastic strain resistance \(K^{P}_{i}\) in (57), which enters through the transformation resistance \(K^{tr}_{i}\) in (58). The latter term accounts for the Magee effect, that is, the direct contribution of the transformation strain to the overall flow stress [49]. Typically this mechanism, in isolation, induces softening as deformation is accommodated directly by the transformation process. A phenomenological approach, requiring a scalar function \(z^{tr}_{i}\) that can presumably be fitted to stress–strain and transformation fraction data, is used here to incorporate \(K^{tr}_{i};\) related models addressing similar physics include [26, 27, 33].
Assume that \(K^{P}_{i} \rightarrow (1 - z^{tr}_{i}) K^{pl{-}tw}_{i}\) as the transformation process affects the overall flow stress of phase i, where \(z^{tr}_{i}\) is a scalar function, likely state- and history-dependent, that when positive denotes the fractional decrease in strength due to accommodation from the transformation strain rate. The unlikely case \(z^{tr}_{i} < 0\) would imply an increase in strength (distinct from apparent composite hardening due to an increase in volume fraction of stiffer martensite) and is still admissible in this model, if deemed physical. The direct transformation contribution to flow resistance of phase i is defined as
Notice \(|K^{tr}_{1}| \rightarrow \infty\) as \(z^{tr}_{i} \rightarrow 0,\) so \(K^{tr}_{i}\) does not affect \(K^{P}_{i}\) when \(z^{tr} = 0.\) With this definition, (58) gives, simply,
Even if \(z^{tr}_{i} > 0,\) the overall flow stress of the composite, \(K^{P}_{0}\) in (57), will likely increase as the phase change proceeds if \(K_{2}^{pl{-}tw} > K^{pl{-}tw}_{1},\) i.e., if the second phase is harder than the first, which is typical of austenite-to-martensite transformations in steels [4]. A smooth, analytic functional form of \(z^{tr}\) that depends only on transformed volume fraction \(\upsilon\) to be used later is
Parameter \(\iota ^{tr}_{i} \in (0,1]\) when transformation strain accommodation induces softening in phase i.
An equation such as (81) could simply be introduced directly to account for combined resistance of all deviatoric inelastic deformation mechanisms. However, preceding derivations demonstrate how to obtain (81) from a composite yield function (58) with a physical origin [33], where slip–twinning and transformation mechanisms operate in parallel. Similarly, contributions to dislocation mean free paths from slip, twinning, and phase changes combine in parallel in [24]. The composite mean free path, in turn, affects dislocation density kinetics and hardening rates in this single crystal model [3, 24].
When (55) and (76) are enforced, the following restrictions on transformation rate apply:
This model predicts strain-assisted transformation when \({\dot{\upsilon }} > 0\) occurs first in a strain history if \({\bar{\sigma }}\) entering \(f^{mech}\) exceeds \(\sqrt{\frac{2}{3}}\) times the temperature- and rate-dependent initial yield stress of phase 1. Depending on temperature and loading rate, strain-assisted transformation may occur for \(z^{tr} > 0,\) but it is the only admissible kind of transformation when \(z^{tr} < 0.\) Stress-assisted transformation occurs when \({\dot{\upsilon }} > 0\) and \({\bar{\sigma }}\) is less than the above-mentioned threshold. Stress-assisted transformation is possible only if \(z^{tr} > 0.\) Implementation of the theory for strain-assisted transformation is straightforward and follows the approach outlined in “Algorithms” section, whereby the transformation rate is modulated by the total inelastic strain rate. An approximate method of implementing stress-assisted transformation is discussed in “TRIP Steel” section.
Conditions in (83) depict phase transformations as phenomena that can be modeled with plasticity-like constructions. Such conditions are most appropriate for TRIP steels in cases where dislocation glide precedes and/or accompanies phase changes. Similar approaches, where a phase transformation rate ultimately manifests from a deviatoric “plastic” strain rate, include those for strain-assisted transformations in [26, 27, 29] and stress-assisted transformations in [33]. Martensitic transformations in materials so modeled involve coordinated movements of partial dislocations similar to deformation twinning [24], shear/slip band intersections as initiation sites [26], or interactions of martensitic nuclei with super-dislocation linear defects [33]. In such cases, the analogy to plasticity theory appears most relevant.
However, martensitic transformations can also be driven purely by changes in temperature and/or pressure, and no deviatoric deformation need occur at all. In the present framework, when \(A^{tr} = 0\) or \({\bar{\sigma }} = 0,\) no deviatoric deformation arises from phase transformations (\({\dot{\epsilon }}^{tr} = 0\)), so \({\dot{\upsilon }}\) is unconstrained by the total deviatoric plastic flow rate. The plasticity-like conditions in (83) would no longer apply to the transformation rate, and (76) would become ineffective since \({\dot{\epsilon }}^{P}\) contains no direct contributions from phase changes. In that case, (76) could be replaced by \({\dot{\upsilon }}_{max} = {\dot{\upsilon }}_{0} =\) constant as in the model of [22], where \({\dot{\upsilon }}_{0}\) would limit the rate of transformation under hydrostatic or stress-free (purely thermal) conditions; however, this feature is not relevant to deformation paths of the TRIP steel studied in the present work so is not developed further.
Damage
A degradation model rooted in micromechanics explicitly defines d as the void volume fraction \({\phi }\):
Here \({\tilde{\rho }} > 0\) is the mass density of the composite material in the unloaded, but possibly phase-transformed, state without any voids, and \({\hat{\rho }}\) is the mass density of the composite material, externally unloaded at the reference temperature, with voids. Since \(0 < {\hat{\rho }} \le {\tilde{\rho }},\) bounds \(d \in [0,1)\) are fulfilled by this physical definition.
Define the Poisson’s ratio of the undamaged material at its initial state by the usual isotropic linear elastic relation:
Dilatational and deviatoric strain energy densities now are affected by voids using coefficients obtained from the isotropic elastic analysis of [50]:
Note that \(\kappa ^{B}\) is not a constant since it depends on \(d = {\phi }.\) The following degradation functions are then used for \(0 \le d < 1,\) based on [50, 51]:
The value of
indicates elastic compression (\(q<0\)) versus tension (\(q>0\)) or neutral loading (\(q=0\)). Under volumetric compression, the volumetric strain energy density in W is not degraded. Under compressive pressure, voids collapse and are presumed to leave the bulk modulus unaltered [51,52,53]. Interpenetration of matter that might occur were the bulk modulus reduced to zero in compression is thereby avoided. If \(\kappa ^{B} \ge 1\) or \(\kappa ^{G} \ge 1,\) the corresponding minimum of \(\zeta ^{B}\) or \(\zeta ^{G}\) is set to zero when \(d \ge 1 / \kappa ^{B}\) or \(d \ge 1 / \kappa ^{G},\) respectively.
Relations from [50] for \(\kappa ^{B}\) and \(\kappa ^{G}\) are accurate to \({\text {O}}(d^{3})\) and \({\text {O}}(d^{2}),\) respectively, in linear isotropic elastic solids. They are extrapolations for nonlinear elastic solids like the one considered here, since analogous analytical formulae do not exist. See also [54] for interpretation of these equations in linear elasticity. A material element for which \(d=1\) is completely voided, with no solid material. Although this extreme limiting condition can never be fully realized physically, the tangent bulk modulus degrades to zero by default for \(q \ge 0\) in (87), and vanishing of the tangent shear modulus results automatically as \(d \rightarrow 1\) since \(\kappa ^{G} > 1\) for \(-1< \nu _{0} < \frac{1}{2}\): \(\zeta ^{B}(1,q) = \zeta ^{G}(1) = 0.\)
The kinetic equation and initial condition used for the damage variable, i.e., the void volume fraction \({\phi } = d,\) is adapted from works of Cocks and Ashby [55,56,57]:
This model, sometimes with augmentations [58], has been widely used for representing both static and dynamic ductile failure behavior in metals, including steels [57, 59]. This model explicitly tracks the void fraction and thus the dilatation from damage, and it contains only two parameters: \({\hat{m}} \ge \frac{1}{2}\) related to viscoplastic rate sensitivity, and the positive constant \(c_{0}.\) The latter is needed to enable the possibility of damage growth without assignment of an initial pore fraction that would induce a spurious initial softening and initial volume change. In other words, \(c_{0},\) which does not affect stiffness or volume, is used in place of \(\phi _{0} > 0.\) A similar, but not identical, form is given in [60], attributed to [55]. Other models for steels use an effective phenomenological damage variable d not necessarily linked to the void fraction [9, 61,62,63,64], and most of these require more fitting parameters. According to (89), damage can only increase under conditions in which triaxiality is positive (tension) and plastic flow is simultaneously taking place.
It can be verified using (36), (86), and (89) that damage evolution is dissipative for \({\dot{d}} \ge 0\) [see (106)]:
Since \({\dot{\epsilon }}^{P} \ge 0\) by definition and damage cannot decrease—notice \({\dot{d}} \ge 0\) according to the conditions on \({\varSigma }\) in (89) and voids can nucleate or grow only when pressure is tensile—it follows that (90) is always satisfied.
Porosity affects slip resistance \(K^{P} = K^{P}_{0} \cdot \zeta ^{P} ({\phi })\) of (57) in addition to tangent elastic moduli. Following prior theories of ductile failure [13, 57, 60, 65], the total plastic work rate times total specific volume of the continuum with voids equals the rate of working of the non-degraded yield stress \(\sqrt{\frac{2}{3}} K^{P}_{0}\) times specific volume of the solid (matrix) phase:
Thus, the total dynamic yield stress of the composite is degraded linearly by the usual factor [13, 60, 65] of \(\zeta ^{P} = 1-d.\)
Thermodynamic Summary
For an isotropic material, let \(\pmb {\kappa } = \kappa _{T} \mathbf{1 }\) be the symmetric spatial conductivity tensor, where for simplicity, \(\kappa _{T} > 0\) is a constant. The conduction equation and thermal dissipation in (24) reduce to
The total dissipation inequality (24) is then
Each term in \({\mathfrak {D}}^{mech},\) the total (mechanical) dissipation under adiabatic conditions, has been verified as non-negative: \({\mathfrak {D}}^{pl{-}tw}\) in (65) with (63), \({\mathfrak {D}}^{tr}\) in (72) with (74), and \({\mathfrak {D}}^d\) in (90) with (89). The energy balance in the form of temperature rate in (28) becomes, applying the time derivative of the last approximation in (14) and applying (79), in the absence of point heat sources (\(R = 0\)):
The first term on the right is the net dissipation from plasticity, twinning, phase changes, and damage kinetics. The second accounts for thermoelastic coupling, the third for latent heat of transformation, and the last for heat conduction.
Numerical Implementation, Materials, and Parameters
The theory of “Constitutive Model” section is implemented in a user-material subroutine compatible with ABAQUS and ale3d [66] simulation packages. Numerical algorithms, properties, and parameters for three steel alloys are described subsequently.
Algorithms
A staggered semi-implicit scheme is used to integrate the constitutive equations over the time domain. Let subscripts n and \(n+1\) correspond to values of variables at successive time increments, where \(n \in [0,N] .\) Rates (e.g., deformation rate, spin, temperature rate, transformation rate) are approximated as fixed over any single window \(t \in [t_{n}, t_{n+1}].\) Analysis begins at \(n = 0\) when \(t = t_{0} = 0\) and concludes when \(t = t_{N}.\) The algorithm proceeds step-wise as follows:
-
(1)
Initialize (if \(n = 0\)) or receive deformation, stress, and state variables: \({{{\varvec{d}}}},\pmb {\omega },{{{\varvec{F}}}}_{n},{{{\varvec{F}}}}_{n+1},\pmb {\sigma }_{n},T_{n},e^{P}_{n},\upsilon _{n},\phi _{n},{\dot{\upsilon }},{\dot{\phi }}.\)
-
(2)
Update deviatoric stress \(\pmb {\sigma }'_{n+1}\) and plastic strain rate \({\dot{e}}^{P}\) using radial return [46] with state values \(T_{n},\) \(\upsilon _{n},\) \(\phi _{n}.\)
-
(3)
Calculate temperature contribution \({\varDelta }{T}^{P}\) from plastic work over time increment \({\varDelta }t = t_{n+1} - t_{n}.\)
-
(4)
Calculate pressure \(p_{n+1}\) and temperature change \({\varDelta }T^{E}\) from EOSFootnote 1 using \(J_{n+1},\) \(T_{n} + {\varDelta }T^{P},\) \(\upsilon _{n}\), and \(\phi _{n}.\)
-
(5)
Update deviatoric stress tensor \(\pmb {\sigma }'_{n+1}\) by integrating the objective rate form (49).
-
(6)
Calculate transformation rate \({\dot{\upsilon }}\) and fraction \(\upsilon _{n+1}\) using \({\dot{e}}^{P},\) stress \(\pmb {\sigma }_{n+1},\) and temperature \(T_{n} + {\varDelta }T^{P} + {\varDelta }T^{E}.\)
-
(7)
Calculate temperature change \({\varDelta }T^{\upsilon }\) due to mechanical and latent heat effects from phase transformation.
-
(8)
Calculate damage (porosity) rate \({\dot{\phi }},\) damage \(\phi _{n+1},\) temperature change from damage \({\varDelta }T^{\phi }\) using \(\pmb {\sigma }_{n+1}\) and \({\dot{e}}^{P}.\)
-
(9)
Store \(T_{n+1} = T_{n} + {\varDelta }T^{P} + {\varDelta }T^{E} + {\varDelta }T^{\upsilon } + {\varDelta }T^{\phi },\) \(e^P_{n+1} = e^{P}_{n} + {\dot{e}}^{P} {\varDelta }t,\) \(\upsilon _{n+1},\) \(\phi _{n+1},\) \({\dot{\upsilon }},\) and \({\dot{\phi }}\) for next increment.
-
(10)
Set \(n \rightarrow n + 1\) and \(t \rightarrow t + {\varDelta }t.\) Return to step 1 iff updated \(n< N \leftrightarrow t < t_{N}.\)
Accuracy of steps 4 and 6 in particular is contingent on reasonably small changes of porosity and phase volume fraction over any time increment \({\varDelta }t.\) The model is intended for use in explicit dynamic finite element simulations, so a fully updated elastic–plastic Jacobian tensor (i.e., tangent modulus tensor) is neither required nor computed. In step 2, a cut-off minimum plastic strain rate is enforced in the algorithm such that viscoplastic yield strength remains finite at very low (static) strain rates. If the material is melted (\(T \ge T_{M}\)) or fully damaged (\(\phi\) attains a threshold \(\phi _{F} \le 1\)), then \(\pmb {\sigma }' = \mathbf{0 }\) and steps 2, 3, and 5–8 are bypassed. Crucial model equations are summarized in Box 1 for ease of reference.
Box 1. Constitutive equations implemented numerically
Kinematics | |
\({{{\varvec{d}}}} = {{{\varvec{d}}}}^{E} + {{{\varvec{d}}}}^{P} = {\bar{{{\varvec{d}}}}}^{E} + {\bar{{{\varvec{d}}}}}^{P} + {\textstyle {\frac{1}{3}}}({\dot{J}}^{E} J^{E \,-1} + {\dot{J}}^{P} J^{P \,-1}) \mathbf{1 }\) | |
\(J^{P} = J^{E \,-1} J = J^{tr} J^{d} = (1 + \delta ^{tr} \upsilon ) \cdot (1 - {\phi })^{-1}\) | |
\({\bar{{{\varvec{d}}}}}^{P} = {\dot{\epsilon }}^{P} {{{\varvec{N}}}}^{P} = ({\dot{\epsilon }}^{pl{-}tw} + {\dot{\epsilon }}^{tr}) {{{\varvec{N}}}}^{P}\) | |
\({\dot{\epsilon }}^{tr} = {\textstyle {\frac{1}{2}}} A^{tr} \gamma ^{tr} {\dot{\upsilon }}\) | |
Stress and stress rate | |
\(\pmb {\sigma } = \pmb {\sigma }' - p \mathbf{1 } = {\bar{\sigma }} \left( {{{\varvec{N}}}}^{P} + \sqrt{ \textstyle {\frac{3}{2}}} {\varSigma }\mathbf{1 }\right)\) | |
\(p = J^{P \, -1} B_{0} \left[ {\textstyle {\frac{3}{2}}} (J^{E \, -7/3}-J^{E \, -5/3}) \{ \zeta ^{B} - {\textstyle {\frac{3}{4}}} (B'_{0} -4)(1-J^{E \, -2/3} ) \} + J^{E \, -5/3} A_{0} (T-T_{0}) \right]\) | |
\(\overset{\triangledown }{\pmb {\sigma }'} = 2 G_{0} \frac{\zeta ^{G}}{J^{E \, 7/3} J^{P} } ({\bar{{{\varvec{d}}}}} - {\bar{{{\varvec{d}}}}}^{P}) = {\dot{\pmb {\sigma }}}' + \pmb {\sigma }' \pmb {\omega } - \pmb {\omega } \pmb {\sigma }' + \pmb {\sigma }' \left[ {\textstyle {\frac{1}{3}}} ( 5 {\text {tr}} {{{\varvec{d}}}} - 2 {\text {tr}} {{{\varvec{d}}}}^{P}) - \frac{ \text {d}\zeta ^{G} }{\text {d}{\phi } } \frac{\dot{{\phi }}}{ \zeta ^{G}} \right]\) | |
Yield condition and flow strength | |
\({\bar{\sigma }} \le \sqrt{ \textstyle {\frac{2}{3}}} [ \{1-(\upsilon + \upsilon _{0}) \} K^{P}_{1} + (\upsilon +\upsilon _{0}) K^{P}_{2}](1-{\phi })\) | |
\(K^{P}_{i} = k_{i} \left( \frac{{\dot{\epsilon }}^{P}}{{\dot{\epsilon }}_{0}} \right) ^{m_{i}} \langle 1- {\bar{T}}^{p_{i}} \rangle \left[ 1 - z^{tr}_{i} \right] \left[ 1 + k_{s} \left\langle \ln \frac{{\dot{e}}^{P} }{{\dot{e}}_{s}} \right\rangle \right]\) | |
\(k_{i} = \sigma _{0i} + \frac{\theta _{0i}}{2} \left[ \epsilon _{i} - \frac{1}{\delta _{i}} \ln \{ \cosh [ \delta _{i} (e^{P} - \epsilon _{i})] \} \right] + \frac{\theta _{0i}}{2 \delta _{i}} \ln \{ \cosh ( - \delta _{i} \epsilon _{i} ) \}\) | |
Transformation kinetics and damage rate | |
\({{\dot{\upsilon }}} = \frac{2 [1-(\upsilon + \upsilon _{0})] }{A^{tr} \gamma ^{tr}} \tanh \left\langle \frac{ {\bar{\sigma }} \alpha ^{tr} - ( {\varLambda }+ {\varUpsilon }_{0} / l_{0} )}{\mu {\varUpsilon }_{0} / l_{0} }\right\rangle \cdot {\text {H}}({\dot{e}}^{P} - {\dot{e}}_{1}) \left( \frac{{\dot{e}}^{P}}{{\dot{e}}_{1}} \right) ^{-r} {\dot{\epsilon }}^{P}\) | |
\(\alpha ^{tr} = \frac{1}{2} A^{tr} \gamma ^{tr} + \sqrt{\frac{3}{2}} \frac{{\varSigma }\delta ^{tr}}{1+\delta ^{tr} \upsilon }\) | |
\(\dot{{\phi }} = \sinh \left[ \frac{ 2( 2 {\hat{m}} - 1) }{2 {\hat{m}} + 1} {\varSigma }\right] \left[ \frac{1}{(1-{\phi })^{{\hat{m}}} } - \{ 1- (\phi +c_{0}) \} \right] {\dot{\epsilon }}^{P} \quad ({\varSigma }\ge 0)\) | |
Temperature rate and driving forces | |
\({\dot{T}} = \frac{1}{c_{V}} \left[ {\mathfrak {D}}^{mech} - \frac{A_{0} B_{0} T }{(J^{E})^{2/3}} {\text {tr}} ({{{\varvec{d}}}}^{E}) + \frac{\lambda _{T} T }{T_{T}} {\dot{\upsilon }} + J \kappa _{T} \nabla ^{2} T \right]\) | |
\({\mathfrak {D}}^{mech} = {\beta }^{pl{-}tw} {\bar{\sigma }} ({\dot{\epsilon }}^{P} - {\dot{\epsilon }}^{tr}) + \langle f^{mech} - f^{th} - f^{ath} \rangle {\dot{\upsilon }} - \left[ \frac{p}{1-{\phi }} + \frac{\partial W}{\partial \phi} \right] \dot{{\phi }}\) | |
\(f^{mech} = {\bar{\sigma }} \alpha ^{tr}, \quad f^{th} = {\varLambda }= -(\lambda _{T}/T_{T}) (T-T_{T}), \quad f^{ath} = {\varUpsilon }_{0} / l_{0}\) |
Materials
Three alloys are labeled according to dominant inelastic deformation mechanisms for loading protocols of present interest. The SLIP alloy undergoes dislocation glide, with little or no evidence of twinning or phase changes. The TWIP alloy undergoes dislocation glide and deformation twinning, with negligible phase change. The TRIP alloy may demonstrate all three mechanisms, but it is most distinguished by transformation behavior when deformed at room temperature. Stacking fault energy (SFE) decreases from SLIP to TWIP to TRIP alloys. Chemical compositions, calculated intrinsic SFE, and calculated martensite start temperatures (assuming uniform \(\gamma\)-austenite initially) are listed in Table 1, where the latter are obtained using thermodynamic methods discussed in [67]. More details regarding material compositions, microstructures, and processing histories can be found in [68] (SLIP steel only) and several forthcoming publications [69, 70].
Static tensile experiments on the SLIP alloy are described in more detail in [68]. A comprehensive account of static and dynamic tension and compression tests, at different temperatures, on all three alloys will be reported in forthcoming publications [69, 70]. Data may be more abundant for other, common steels, which could conceivably be used to demonstrate model calibration and performance in lieu of results shown in “Static and Dynamic Loading” section. For example, torsion data is available for rolled homogeneous armor (RHA) steel at different loading rates, in addition to tension and compression data [59]. However, a goal of this work is comparison of trends among three ductile medium–high Mn steels with the same alloying elements and very similar Mn content (see Table 1), but demonstrating different dominant deformation and strain hardening mechanisms, i.e., slip, twinning, or phase changes, due primarily to large differences in SFE. Constitutive models and parameter sets already exist for other common steels, but not for the present ones, which demonstrate promising mechanical behavior among this class of alloys, for example, high ductility and ultimate tensile strength [68]. The current model is intended for future use in ballistic simulations, and no complete models and parameters presently exist elsewhere for the alloys of Table 1 spanning loading regimes of interest. Many, if not most, constitutive models developed elsewhere for steels and similar alloys likewise do not consider all three stress states for calibration or validation, with torsion most frequently omitted. For example, the model in [33] considers tension and compression, those in [24, 27, 62] consider only tension, and those in [71, 72] consider only compression.
Analogous remarks apply for the high-pressure (e.g., shock compression) analysis and predicted dynamic strength in “Dynamic High-Pressure Response” section, as well as the dynamic simple shear analysis in “Dynamic Volumetric-Simple Shear Response” section. Experimental data for calibration and validation of functional forms and parameters in these regimes are missing for the three alloys considered herein. The value of these analyses, and the credibility of model predictions, would be greatly enhanced with this data. Certain information may be available for other steels. For example, strength data exist for other alloys under dynamic torsion [59] and plate impact with time-resolved lateral stress measurements [73,74,75]. But calibration of the present model to data for other steels of different compositions, microstructures, and/or mechanisms would not fulfill the goals of the current comparative study on promising new steels emphasized above, nor facilitate ballistic simulations of these advanced materials. Other models developed for ballistic impact of steels also frequently omit calibration to multiple stress states, notably dynamic torsion. For example, the model used in [9] is calibrated only to tensile data on smooth and notched specimens, and the plasticity components of the model used in [6] are calibrated only to compression data.
Parameters
The constitutive model implemented for SLIP and TWIP steels requires around 20 parameters; the TRIP steel model requires around 10 additional parameters to address phase changes. The numbers of parameters is modest considering the scope of physics addressed—nonlinear thermoelasticity, thermo–visco-plasticity, deformation twinning, phase transformations, and damage evolution—over a wide range of loading rates, stress states, and temperatures. Some parameters can be obtained from basic physical arguments or estimated from those of iron or other steels, e.g., certain thermoelastic properties. Others are calibrated, as reported in “Static and Dynamic Loading” section, to test data on the specific alloys of Table 1. A default mass density typical of steel or iron alloys is listed; density variations among the three alloys are ignored in Table 2 since subsequent material-point calculations (no wave propagation) do not use this information.
Complete sets of material parameters are listed in Tables 2 and 3, where the latter contains those exclusive to phase-transforming materials (i.e., TRIP steel). When no reference is listed for a particular value, then that parameter is calibrated to test data that follows later, rather than prescribed a priori. Most parameter values are shared among all three (SLIP, TWIP, TRIP) alloys. Those unique to one alloy or phase are so labeled in the second column of Table 2.
Static and Dynamic Loading
Model results for quasi-static and dynamic compression and quasi-static tension loading are compared with experimental data on SLIP, TWIP, and TRIP alloys listed in Table 1. All of these loading protocols correspond to uniaxial stress conditions. Let the loading direction be along \(X_{1},\) with the magnitude of the only nonzero Cauchy stress component \(\sigma = |\sigma _{11}|.\) The magnitude of natural (logarithmic) strain is denoted by \(\epsilon = | \ln F_{11} |.\) In the loading direction, deformation is imposed in calculations as \(F_{11} = 1 \pm {\dot{\epsilon }} t,\) where \({\dot{\epsilon }}\) is a constant nominal strain rate, not strictly equal to the material time derivative of \(\epsilon .\) Lateral deformations are adjusted incrementally to maintain equilibrium and a uniaxial stress state. For quasi-static loading, \({\dot{\epsilon }} = {\dot{e}}_{0} = 10^{-3}\)/s. For dynamic loading, \({\dot{\epsilon }} = 2500\)/s. These strain rates are representative of quasi-static (Instron) and dynamic [Kolsky bar or split-Hopkinson-pressure bar (SHPB)] experiments. Initial temperatures considered here are \(T = T_{0} = T_{R} = 293\) K (room) and \(T = T_{0} = 473\) K (elevated). For quasi-static loading, isothermal conditions are assumed, and both tension and compression data are available from experiments. For dynamic loading, adiabatic conditions are assumed, and only compression data are studied. In either case, thermal conductivity \(\kappa _{T}\) is irrelevant. Phase volume fractions are measured in situ using diffraction techniques [68].
SLIP Steel
Stress–strain data from model and experiment are compared for the SLIP alloy in Fig. 1. Parameters \(\sigma _{0},\) \(\theta _{0},\) \(\delta ,\) and \(\epsilon\) listed in Table 2 (subscript \(i=1\)) are calibrated to fit the quasi-static, room temperature tensile data in Fig. 1a. Thermal softening parameter p is then chosen to match the elevated temperature data in Fig. 1a. Results for quasi-static compression in Fig. 1b are predictions obtained using these same parameters, unadjusted. Finally, rate sensitivity m is chosen to best match the dynamic compression data in Fig. 1c, along with an appropriate value of \(\beta ^{pl{-}tw}.\) Agreement between model and experiment is respectable considering only six parameters have been adjusted. A quantitative assessment of model accuracy, along with data on experimental variability, is included in Table 5 of “Appendix 2”.
Uniaxial stress response, SLIP steel: a quasi-static tension, b quasi-static compression and c dynamic compression
TWIP Steel
Stress–strain data from model and experiment are compared for the TWIP alloy in Fig. 2. The same calibration procedure followed in “SLIP Steel” section is repeated here. First, parameters \(\sigma _{0},\) \(\theta _{0},\) \(\delta ,\) and \(\epsilon\) are selected to fit the quasi-static, room temperature tensile data in Fig. 2a. Thermal softening parameter p is chosen to match the high temperature data in Fig. 2a. Results for quasi-static compression in Fig. 2b are predictions with these same parameters. Rate sensitivity m is chosen to best match the dynamic compression data in Fig. 2c in conjunction with Taylor–Quinney factor \(\beta ^{pl{-}tw}.\) Agreement between model and experiment is excellent for quasi-static tension and static compression at elevated temperature. Agreement is also respectable for dynamic room temperature compression. Agreement is favorable for dynamic elevated temperature compression and quasi-static room temperature compression only at \(\epsilon \lesssim 0.2.\) At higher compressive strains, the model under-predicts room-temperature static strength and over-predicts high-temperature dynamic strength. This alloy apparently demonstrates tension–compression asymmetry and more complex thermally activated deformation mechanisms that are not captured by the present constitutive equations at large strains. More sophisticated fitting functions, e.g., as in [77], could be added to likely improve agreement, albeit at the expense of more than six adjusted parameters.
Uniaxial stress response, TWIP steel: a quasi-static tension, b quasi-static compression and c dynamic compression
Possible sources of stress-state dependence of plastic flow behavior (e.g., different flow strengths among tension, compression, and torsional loading to the same equivalent strain) in nominally isotropic polycrystals are discussed in [77]: deformation-induced texturing (textural hardening/softening), evolution of dense dislocation walls and micro-bands differently depending on stress state, deformation twinning (including interactions of twins with dislocations and dislocation structures), and phase transformations, especially those incurring a volume change for which pressure contributes to driving force. A direct comparison of model results for tension, compression, and torsion is provided in Fig. 13 of “Appendix 2”, with complementary discussion on trends observed in several other steels.
TRIP Steel
Stress–strain data from model and experiment are compared for the TRIP alloy in Fig. 3. Calibration is more involved than for cases addressed in “SLIP Steel” and “TWIP Steel” sections, since the phase transformation model requires adjustment of several additional parameters as marked in the rightmost column of Table 3. A simple and sufficient assumption made first is that the \(\alpha\)-martensite phase is plastically stiff but non-hardening, with a static room-temperature strength of \(\sigma _{02} \gg \sigma _{01}.\) Experiments show negligible transformation behavior at elevated temperature. Thus, high-temperature quasi-static data in Fig. 2a, b are first used to calibrate \(\sigma _{01},\) \(\theta _{01},\) \(\delta _{01},\) and \(\epsilon _{01},\) where the latter two are simply chosen as the same as for the TWIP alloy, which appears sufficient for the present data fitting. Rate sensitivity m is next chosen to best match the elevated-temperature dynamic compression data in Fig. 3c, along with \(\beta ^{pl{-}tw}.\) The following six parameters are then adjusted simultaneously to fit the room temperature data in Fig. 3 for quasi-static tension, quasi-static compression, and dynamic compression: thermal softening parameter p, shear product \(A^{tr} \gamma ^{tr},\) and phase kinetic constants \(\mu ,\) \({\dot{e}}_{1},\) r, and \(\iota ^{tr}.\) Also considered in this calibration procedure is the evolution of phase volume fraction reported in Fig. 4. In total, 12 parameters have been systematically selected to obtain model results in Figs. 3 and 4.
Uniaxial stress response, TRIP steel: a quasi-static tension, b quasi-static compression and c dynamic compression
Volume fraction of \(\alpha\)-martensite in TRIP steel. All tests shown start at room temperature; experimental data for static compression unavailable
Agreement of the model with high-temperature data is excellent except for the small amount of softening not captured at \(\epsilon \gtrsim 0.45\) in Fig. 3c. Now consider room temperature loading, for which martensitic phase changes occur. Under static and dynamic compression, there is no drastic drop in initial yield stress relative to the high-temperature cases. Phase transformations are deemed strain-assisted, that is, \({\dot{\upsilon }}\) becomes nonzero after a nominal yield stress for plastic flow is attained. The knee in each model stress–strain curve near the initiation point in Fig. 3b, c is unrealistic, but experimentally observed behavior for \(\epsilon \gtrsim 0.05\) is well-represented. A more sophisticated function \(z^{tr}_{i}\) than that introduced in (82) is likely needed to address subtle behaviors near the transformation initiation point. Similar to results shown for compression in Fig. 4, a relative reduction in martensitic volume fraction with increasing applied strain rate has been noted elsewhere for a different TRIP steel [29].
Under quasi-static tensile loading at room temperature, the model represents gross stress–strain response reasonably well in Fig. 3a, although it omits high-frequency serrations in the test data corresponding to the Portevin–Le Chatelier effect most likely attributed to dynamic strain aging. The initial yield stress is lowered by the onset of a phase change, meaning this case corresponds to stress-assisted transformation. In principle, the theory of “Phase Transformations” section can address such behavior, given an appropriate function \(z^{tr}_{i}.\) However, a fully implicit numerical integration scheme appears needed to properly account for the drop in initial yield stress without overshooting a transformation-reduced yield stress in the elastic regime. The latter problematic issue emerges in the staggered semi-implicit implementation of “Algorithms” section: a very small elastic strain increment can produce a deviatoric stress that exceeds the anticipated transformation-reduced yield stress but that remains below the nominal initial flow stress for conditions under which stress-assisted transformation would not take place. An approximate resolution used in the present calculations is assignment of different values of \(\sigma _{01}\) (reduced), \(\theta _{01}\) (increased), and \(\iota ^{tr}_{i}\) (increased) for quasi-static tensile deformation (see values in parentheses in Tables 2, 3), benefiting from the a priori knowledge that stress-assisted behavior is expected. For general loading protocols in which the transformation mechanism, i.e., strain- or stress-assisted, is unknown, a more robust numerical algorithm is required. However, parameters for the two cases can still presumably be used in distinct calculations with the present implementation to suggest bounds on the expected response.
Summary
The following trends among parameters, test data, and model results are inferred from comparison of findings in “SLIP Steel”, “TWIP Steel”, and “TRIP Steel” sections:
-
Initial static, room-temperature yield stress \(\sigma _{01}\) decreases from SLIP (600 MPa) to TWIP (470) to TRIP (450) alloys, as does initial hardening coefficient \(\theta _{01}\): 5500, 2250, and 2200 MPa.
-
TWIP steels (all temperatures) and TRIP steels (elevated temperatures, no phase change) demonstrate relatively linear strain hardening over ranges examined, which is well-captured by high values of \(\delta _{i}\) (20) and \(\epsilon _{i}\) (0.6).
-
SLIP steels demonstrate concave-down hardening behavior, well-captured by low values of \(\delta _{i}\) (5) and \(\epsilon _{i}\) (0.09).
-
Strain rate sensitivity m is the same (0.015) for SLIP and TWIP steels, and much lower (0.003) for TRIP steel.
-
Thermal softening p decreases from SLIP (1.0) to TWIP (0.9) to TRIP steel (0.6), while Taylor–Quinney factor \(\beta ^{pl{-}tw}\) varies from SLIP (0.6) to TWIP (0.9) to TRIP steel (0.8).
Predicted temperature rises for all three alloys under dynamic compression are compared in Fig. 5a, for starting temperatures of \(T_{0} = 293\) K (room) and \(T_{0}=473\) K (elevated). Temperature rise is greatest for the TWIP steel in each case, due to its combination of high \(\beta ^{pl{-}tw} = 0.9\) and relatively high stiffness, especially at large strain \(\epsilon \gtrsim 0.4.\)
Model results for a temperature under dynamic compression and b porosity under quasi-static tension
Predicted porosity, i.e., damage, is reported in Fig. 5b. Quantitative experimental data on porosity are not available for these alloys. However, qualitative observations of fractured surfaces of tensile SLIP specimens revealed dimples and an approximate \(\phi\) on the order of 1% at logarithmic strains on the order of \(\epsilon \approx 0.3.\) The value of \(c_{0} = 0.01\) listed in Table 2 has been chosen to reflect this behavior. Parameter \({\hat{m}} = 10\) entering the Cocks–Ashby void growth law of (89) is assigned uniformly among the three alloys. This value falls within the range advocated in [58] for tantalum. Use of \({\hat{m}} = 1/m_{i}\) produced much larger values of m, leading to unrealistically rapid void growth. More brittle behavior was observed for tensile failure of the TWIP and TRIP steels in experiments, corresponding to lower \(\phi\) at fracture. The value of \(c_{0}\) was adjusted downwards, by an order of magnitude accordingly, to \(c_{0} = 0.001\) for TWIP and TRIP alloys, producing porosity on the order of 0.6% and 0.01% respectively at strains corresponding to failure in corresponding quasi-static tensile experiments. Evolution of porosity under quasi-static tension at high temperatures, not shown in Fig. 5b, is nearly identical to corresponding room-temperature results since (89) does not contain any explicit temperature dependence. In all cases (SLIP, TWIP, and TRIP alloys at room and elevated temperatures), porosity values are low enough that overall stress–strain behavior is relatively unaffected over regimes reported in Figs. 1, 2, and 3. No pore growth is incurred for compressive loading whereby \({\varSigma }< 0.\) See Table 6 in “Appendix 2” for a quantitative evaluation of tensile failure data and model results.
Dynamic High-Pressure Response
The intent of the present section is evaluation and comparison of model predictions at strain rates and pressures comparable to those attained during shock compression and ballistic events, for the specific alloys of Table 1. Adiabatic uniaxial strain loading is imposed uniformly in the \(X_{1}\)-direction at a constant rate \({\dot{\epsilon }}.\) This loading program can be achieved immediately (and trivially) using a single material point calculation, without need for equilibrium iterations. More realistic approaches for rate-dependent plasticity that would resolve a shock waveform explicitly include steady wave methods [78, 79] and finite difference or finite element methods [38, 66, 80, 81], often with artificial viscosity. The latter, more computationally intensive, methods may be considered further in future work. The present simplified homogeneous boundary conditions are sufficient for the present purpose of verification of the high-pressure EOS and comparison of extrapolated dynamic strengths among the three alloys. Experimental techniques for measuring dynamic strength at similarly extreme conditions are discussed in [82, 83].
The deformation gradient is \({{{\varvec{F}}}} = \mathbf{1 } - {\dot{\epsilon }} t {{{\varvec{e}}}}_{1} \otimes {{{\varvec{e}}}}_{1}.\) Nominal strain is \(\epsilon = {\dot{\epsilon }} t = 1 - J,\) with \(J = \det {{{\varvec{F}}}} = F_{11} = V/V_{0}.\) The Cauchy stress tensor is diagonal with longitudinal stress \(P = - \sigma _{11} \ge 0\) and lateral stresses \(\sigma _{22} = \sigma _{33},\) the latter equal from symmetry of loading mode and material isotropy. Maximum shear stress is defined as \(\tau = -\frac{1}{2}(\sigma _{11}-\sigma _{22}).\) Cauchy pressure is \(p = -\frac{1}{3} \sigma _{kk} = P - \frac{4}{3} \tau .\) When the material has no “strength”, \(\tau \rightarrow 0\) and \(P \rightarrow p.\) Three high strain rates are applied in subsequent calculations: \({\dot{\epsilon }} = 10^{4}\)/s, \(10^{6}\)/s, and \(10^{8}\)/s. The rate of \(10^{4}\)/s is lower than what would expectedly be incurred within a steady structured plastic wave in steel but is four times larger than that achieved in SHPB experiments reported in “Static and Dynamic Loading” section. The rate of \(10^{6}\)/s is physically representative of that incurred by a steady shock of normal stress magnitude \(P \approx 10\) GPa in iron, while the rate of \(10^{8}\)/s is representative of a stronger shock of magnitude \(P \approx 30\) GPa if the Swegle–Grady scaling law is accurate [84]. Also considered in some subsequent calculations is adiabatic quasi-static uniaxial strain, with \({\dot{\epsilon }} = 10^{-3}\)/s. These conditions are used to acquire an approximation of the strength contribution to the “equilibrium stress” on the Hugoniot, i.e., the static yield stress in the wake of the plastic rise when the strain rate has decayed to a negligible magnitude [84].
An enhancement to the baseline viscoplastic model of “Plastic Flow” section, applicable only at very high strain rates, is now explained. At low to moderate strain rates, flow resistance is controlled mainly by local obstacles, with dislocation motion thermally activated. At higher rates as occurring in shock compression, for example, resistance becomes more dependent on viscous and phonon drag, where relativistic effects limit maximum glide velocity. Homogeneous dislocation nucleation also becomes more important at very high strain rates. Constitutive models that seek to address thermally activated and drag regimes include [85,86,87,88]. Apparent strength increases can also be due to enhanced strain hardening at high rates from increases in dislocation density production and dynamic changes in dislocation structures [73], which in turn can result in strain-rate-history effects. To induce increased strength at very high strain rates in a simple yet effective way, the viscoplastic yield function in (60) is augmented via the final term in square braces below:
This full equation is already given in the phase i flow strength \(K^{P}_{i}\) of Box 1. Values of \(k_{s} = 0.25\) and \({\dot{e}}_{s} = 10^{4}\)/s are as listed in Table 2, where \(k_{s}\) is chosen—for demonstration purposes in the absence of very high rate data on the three steels of present interest—to give an approximate doubling of flow strength for each hundredfold increase in strain rate above \({\dot{e}}_{s}.\) Results presented already in “Static and Dynamic Loading” section are not affected by this modification for shock hardening since \({\dot{e}}^{P} < {\dot{e}}_{s}\) for quasi-static loading and for dynamic compression at 2500/s pertinent to SHPB tests. The enhanced rate hardening term in (95), which contains no memory of loading history, only affects instantaneous strength.
Reported in Fig. 6a is the predicted longitudinal stress P for all three alloys deformed adiabatically at a quasi-static rate of \({\dot{\epsilon }} = 10^{-3}\)/s, along with the predicted hydrostat. The latter (isothermal hydrostat) is calculated under spherical deformation of the form \({{{\varvec{F}}}} = (1 - \epsilon )^{1/3} \mathbf{1 }\) with \(T = T_{R} =\) constant and \(P = p.\) Since no deviatoric deformation, slip, twinning, or phase changes are predicted for spherical compression, this curve is identical among the three alloys and is independent of loading rate. Differences in P among alloys, due only to strength and related temperature effects, are nearly indiscernible in Fig. 6a since \(P \approx p.\) Model results compare favorably with shock compression data on other steels [89]. However, possible effects of strain rate history on material strength and plastic dissipation, for example as a real material is loaded along the Raleigh line through a steady structured shockwave to each terminal Hugoniot state, are omitted. Furthermore, shock dissipation due to curvature of the pressure–volume Hugoniot is omitted in the present homogeneous uniaxial strain calculations. The leading terms in this contribution to entropy production, and pressure rise, are third-order in volume change across the shock front, and these can become significant in metals for compression exceeding 25%; for weak shocks this volumetric dissipation is usually safely neglected [16]. At very high shock stresses (\(P \gtrsim 120\) GPa), higher-order entropic effects omitted by the constant strain-rate approximation could contribute to under-prediction of P relative to shock Hugoniot data in Fig. 6a.
Adiabatic uniaxial strain and isothermal hydrostatic predictions: a longitudinal stress (with quasi-static strength) compared with shock compression Hugoniot data on other steel alloys [89] and b temperature assuming constant strain rate of \(10^{6}\)/s during the plastic rise
Temperatures for all three alloys are reported versus compression in Fig. 6b. Differences among alloys are very small, with temperature rise greatest in the TWIP steel, in agreement with trends at lower strain rates in Fig. 5a. Temperature rise is much larger here for shock-type than for SHPB-type loading, with values in excess of 1000 K achieved at 35% compression. However, melting does not occur: \(T < T_{M} = 1800\) K in all cases.
Transformation to \(\alpha\)-martensite is expansive and thus strongly inhibited by high-pressure–loading. Accordingly, negligible transformation is predicted for the present loading protocol by the kinetic model of “Phase Transformations” section with parameters of Table 3. More conceivable is the reverse transformation of any initial \(\alpha\) phase (recall \(\upsilon _{0} = 0.1\) for TRIP alloy) to \(\epsilon\) as observed in iron and some other steels depending on composition [30]. While the TWIP steel consists entirely of \(\gamma\) austenite, the SLIP steel does contain a small, but non-negligible, initial fraction of \(\alpha\) that could likewise transform to \(\epsilon\) under high pressure–loading. The present model could conceivably be re-parameterized and implemented to address the \(\alpha \rightarrow \epsilon\) transformation if it is discovered, in future experiments, to occur in these particular alloys.
Results in Fig. 7 focus on the SLIP alloy. As evident in Fig. 7a, longitudinal stress does not vary substantially with loading rate over the range from quasi-static (i.e., equilibrium) to what might be incurred within a stronger shock at \({\dot{\epsilon }} = 10^{8}\)/s. Temperature rise increases moderately with increasing strain rate over this domain since material strength and plastic dissipation are enhanced at higher rates according to (95). Shear stress \(\tau\) is predicted versus P for quasi-static conditions and three dynamic loading rates in Fig. 7c. Hugoniot data from lateral stress measurements on other steel alloys [73,74,75] show strength similar to the equilibrium strength of the SLIP alloy at low shock pressures, but exceeding the quasi-static strength of the model by around a factor of two at compressive stresses in excess of a few GPa. Discrepancies could be due to differences in material properties, differences in strain rate history incurred in the present ramp loading versus shock compression loading leading to different transient rates and end states, and potential deficiencies in the present strength model that does not account for effects of strain rate history on flow resistance.
Adiabatic uniaxial strain response, SLIP steel: a longitudinal stress, b temperature and c shear stress
Strength extrapolations at high loading rates and high pressures for all three alloys are shown in Fig. 8. As intended by (95), \(\tau\) increases markedly with each hundredfold increase in strain rate for the choice \(k_{s} = 0.25.\) When \(k_{s} = 0\) (e.g., omission of enhanced drag), on the other hand, increases in \(\tau\) with rate are much less substantial. Peak strengths achieved at \({\dot{\epsilon }} = 10^{6}\)/s are predicted at 1.2 GPa for the SLIP alloy, 0.9 GPa for the TWIP steel, and 0.7 GPa for the TRIP steel. These projections assume \(k_{s}\) is identical among the three materials, an assumption that would benefit from experimental confirmation. Thermal softening overrides strain hardening at larger compressions. In these calculations, \(0.7 \lesssim e^{P}/ \epsilon \lesssim 0.8.\)
Predicted shear stress under uniaxial strain at \({\dot{\epsilon }} = 10^{6}\)/s: a SLIP steel, b TWIP steel and c TRIP steel
Dynamic Volumetric-Simple Shear Response
Homogeneous deformation without localization is fully admissible in rate-dependent solids that strain soften [90]. However, when a local perturbation in the deformation field is introduced, for example geometrically or through a physical property variation, localization into a band or region of concentrated deformation is likely when a material region is deformed to or beyond its point of maximum load-bearing capacity [10, 91]. Let \(\sigma _{C}\) denote the maximum stress measure corresponding to this critical loading point beyond which instability can ensue, and \(\epsilon _{C}\) the corresponding strain measure. By these broad definitions, for the same loading protocol and initial conditions, a material with larger \(\sigma _{C}\) should support a larger stress prior to failure by strain localization, while a material with a larger \(\epsilon _{C}\) should accommodate more strain prior to instability. Localization tendencies for classes of idealized elastic–viscoplastic solids are analyzed in [10, 91]. More robust measures of localization tendency for a given material require further details regarding the initial-boundary value problem under consideration, e.g., boundary conditions and the nature of any perturbation.
In what follows next, a multi-axial loading protocol is imposed on a material element to obtain results that should provide some insight into dynamic shear localization behavior on the three steel alloys of present interest whose properties are compiled in “Numerical Implementation, Materials, and Parameters” section. Localization is not simulated explicitly; homogeneous deformation is imposed in order to obtain \(\sigma _{C},\) \(\epsilon _{C},\) and related quantities expected to correlate with conditions for shear band formation in numerical solutions of initial-boundary value problems wherein perturbations in field variables are admitted. A deformation gradient \({{{\varvec{F}}}}\) combining simple shear in the 1–2 plane with spherical expansion/contraction is, in matrix form,
Simple shear strain is \(\gamma = \int {\dot{\gamma }} \, \text {d}t.\) The rate of volume change is controlled by \({\dot{e}},\) where \({\dot{e}} > 0\) for tension (\(J > 1\)), \({\dot{e}} = 0\) for “neutral” loading (\(J=1\)), and \({\dot{e}} < 0\) for compression (\(J < 1\)). A dynamic shear strain rate of \({\dot{\gamma }} = 10^{3}\)/s is imposed in subsequent calculations. Adiabatic conditions are assumed consistent with high strain rates. For non-neutral loading, dynamic spherical deformation rates of \({\dot{e}} = \pm 200\)/s are imposed simultaneously. Note that p is nonzero under neutral loading (\({\dot{e}} = 0\)) due to changes in temperature, porosity, and/or phase content. Under loading conditions of (96), \(\sigma _{13} = \sigma _{23} = 0,\) and \(\tau = \sigma _{12} \ge 0\) is the dominant shear stress component that emerges from nonzero \({\dot{\gamma }}.\) Here \(\tau\) is strictly work conjugate to \({\dot{\gamma }}\) only for neutral loading. Normal stresses are generally nonzero, where \(\sigma _{11} \approx \sigma _{22} \approx \sigma _{33}\) and may differ slightly, even in these isotropic solids, due to geometric and material nonlinearities. For comparison of results with those from uniaxial normal stress loading, effective shear stress and strain measures are defined as \(\sigma = \sqrt{3} \tau\) and \(\epsilon = \gamma / \sqrt{3}.\)
Shown in Fig. 9 are model predictions for effective shear stress versus effective shear strain for all three alloys and all three loading protocols. Corresponding pressure and temperature histories are reported in respective Figs. 10 and 11. Phase transformation in the TRIP alloy and porosity evolution for all three steels under shear + tension loading are given in Fig. 12a and b, respectively. Phase transformations are not enabled in the calculations for SLIP or TWIP alloys, and voids do not manifest in any of the alloys for compressive or neutral loading.
Combined dynamic simple shear- and pressure–loading predictions for effective shear stress: a SLIP alloy, b TWIP alloy and c TRIP alloy
Combined dynamic simple shear- and pressure–loading predictions for pressure: a SLIP alloy, b TWIP alloy and c TRIP alloy
Combined dynamic simple shear- and pressure–loading predictions for temperature: a SLIP alloy, b TWIP alloy and c TRIP alloy
Combined dynamic simple shear- and pressure–loading predictions for a transformation to \(\alpha\) phase in TRIP alloy (null transformation for shear + compression) and b porosity under shear + tension loading for all three alloys
Trends in results for the SLIP and TWIP alloys are similar, though the strain \(\epsilon _{C}\) at peak load is significantly larger in the latter. Shear stress in Fig. 9 increases with strain due to strain hardening, then plateaus and decreases due to thermal softening, and due to damage softening in the case of tensile loading. As shown in Fig. 10, pressure rise is large in compression and smaller but positive in neutral shear due to thermoelastic coupling. Tensile pressure is mitigated under expansion from porosity which simultaneously accommodates dilatation and softens the elastic moduli. As shown in Fig. 11, temperature rise from plasticity–damage dissipation and thermoelastic coupling is largest for compression, lowest for tension, and intermediate for neutral loading. Pore growth (Fig. 12b) correlates with lower stresses obtained in Fig. 9a, b for shear + tension loading.
Trends for the TRIP alloy differ somewhat from those of the other two steels. As shown in Fig. 12a, transformation to the stiff \(\alpha\) phase is substantial for tensile loading. For neutral loading, transformation is moderate, with \(\upsilon\) reaching a plateau of 0.138 at \(\epsilon \approx 0.08.\) At larger applied strains, further transformation is suppressed by the combination of compressive pressure and temperature rise. Transformation does not occur at all for shear + compression loading due to the high compressive stress and temperature rise. Effects of phase change, or lack thereof, are reflected in Fig. 9c, wherein strength under shear + tension loading increases rapidly due to martensitic transformation, then drops due to thermal softening and, to a lesser extent, ductile damage evolution. Loading under shear + compression demonstrates notably lower shear stress, since transformation to the harder phase does not occur at all. Perhaps the most promising result for the TRIP alloy is evident in Fig. 12b. Porosity is significantly lower in the TRIP steel than the TWIP steel at the same strain level, even though the kinetic law and material parameters for damage initiation and growth are assumed identical for both alloys (Table 2). Damage evolves more slowly in the TRIP steel than the other alloys since dilatation is accommodated by the expansive martensitic transformation rather than void growth. As a result, the material is able to sustain a larger tensile pressure under shear + tension loading than the other two alloys since the former maintains a larger tangent bulk modulus. In a real material, rupture by void coalescence or fracture, which is not addressed in the present calculations, would be expected to ensue before the void volume fraction becomes very large, though the maximum sustainable porosity under combined tensile-shear loading is unknown for these steels.
Data for key solution variables at the point of peak load are reported in Table 4, where state \((\cdot )_{C}\) corresponds to \(\text {d}\tau / \text {d}\gamma = 0 \Leftrightarrow \text {d}\sigma / \text {d}\epsilon = 0.\) These data substantiate trends discussed already in the context of Figs. 9, 10, 11, and 12. The following findings are noteworthy for each of the three loading protocols:
-
Shear + tension: TRIP steel has largest \(\sigma _{C},\) highest tensile pressure \(-p_{C},\) low porosity \(\phi _{C},\) and largest \(\epsilon _{F};\)
Table 4 Critical predictions for dynamic (\(10^{3}\)/s), multi-axial pressure–shear loading: peak stress \(\sigma _{C}\) and corresponding ductility \(\epsilon _{C},\) pressure \(p_{C},\) temperature \(T_{C},\) porosity \(\phi _{C},\) and transformed phase fraction \(\upsilon _{C}\) -
Shear neutral: TWIP steel has largest \(\sigma _{C}\) and corresponding strain \(\epsilon _{C},\) despite largest \(T_{C};\)
-
Shear + compression: TWIP steel has largest \(\sigma _{C},\) compressive pressure \(p_{C},\) and temperature \(T_{C}.\)
These results suggest that the TWIP alloy would perform best for ballistic applications dominated by shear and compressive strain states (e.g., thick-target penetration), while the TRIP alloy would perform well in ballistic applications wherein tensile strain dominates (e.g., spall [92, 93]). If a failure porosity of \(\phi _{F} = 0.01\) is assumed for all three materials, then corresponding equivalent shear strain to failure \(\epsilon _{F}\) under dynamic shear + tension loading is approximately 50% larger in the TRIP steel than the SLIP and TWIP steels, as shown in the rightmost column of Table 4. By tuning material chemistry and initial microstructure, it should be possible to engineer an alloy with optimal properties for a given application guided by model predictions, once sufficiently validated by ballistic experiments.
Conclusions
A thermodynamically consistent nonlinear constitutive model for steels has been formulated with comprehensive features thought unique among models of this class. Encompassed are nonlinear thermoelasticity relevant for high pressure/shock loading, viscoplasticity for slip and twinning across a wide range of temperatures and strain rates, ductile damage growth, and martensitic phase transformations affected by stress state, temperature, and loading rate. Volume changes associated with porosity and phase changes are included. The model has been demonstrated as capable of representing data on three different alloys with very different hardening behaviors and deformation mechanisms. Extrapolations of the model to conditions pertinent to the shock regime appear physically reasonable, though validations of strength predictions in this regime await experiments on these particular alloys. Model predictions for simultaneous simple shear with tension or compression suggest how different physical mechanisms in the three steels should affect their tendency for localization and dynamic failure. Future work will seek to apply the model to finite element simulations of tensile and shear instabilities and ballistic penetration.
Notes
EOS is an acronym for equations of state herein relating pressure, specific volume, temperature, and thermoelastic free energy of “Thermoelasticity” section.
References
Crouch IG, Cimpoeru SJ, Li H, Shanmugam D (2017) Armour steels. In: Crouch IG (ed) Theoretical and applied mechanics. Woodhead, Amsterdam, pp 55–115
De Cooman BC, Estrin Y, Kim SK (2018) Twinning-induced plasticity (TWIP) steels. Acta Mater 142:283–362
Madilava M, Schwedt A, Wong SL, Roters F, Prahl U, Bleck W (2018) Temperature dependent strain hardening and fracture behavior of TWIP steel. Int J Plast 104:80–103
Lani F, Furnemont Q, Van Rompaey T, Delannay F, Jacques PJ, Pardoen T (2007) Multiscale mechanics of TRIP-assisted multiphase steels: II. Micromechanical modelling. Acta Mater 55:3695–3705
Wengrenovich NJ, Olson GB (2015) Optimization of a TRIP steel for adiabatic fragment protection. Mater Today 2S:639–642
Bronkhorst CA, Cerreta EK, Xue Q, Maudlin PJ, Mason TA, Gray GT (2006) An experimental and numerical study of the localization behavior of tantalum and stainless steel. Int J Plast 22:1304–1335
Vernerey FJ, McVeigh C, Liu WK, Moran B, Tewari D, Parks DM, Olson GB (2006) The 3-D computational modeling of shear-dominated ductile failure in steel. JOM 58:45–51
Zukas JA, Nicholas T, Swift HF, Greszczuk LB, Curran DR (1992) Impact dynamics. Krieger, Malabar
Børvik T, Langseth M, Hopperstad OS, Malo KA (1999) Ballistic penetration of steel plates. Int J Impact Eng 22:855–886
Wright TW (2002) The physics and mathematics of adiabatic shear bands. Cambridge University Press, Cambridge
McVeigh C, Vernerey F, Liu WK, Moran B, Olson G (2007) An interactive micro-void shear localization mechanism in high strength steels. J Mech Phys Solids 55:225–244
Kadkhodapour J, Anbarlooie B, Hosseini-Toudeshky H, Schmauder S (2014) Simulation of shear failure in dual phase steels using localization criteria and experimental observation. Comput Mater Sci 94:106–113
Gurson AL (1977) Continuum theory of ductile rupture by void nucleation and growth: Part I—yield criteria and flow rules for porous ductile media. J Eng Mater Technol 99:2–15
Needleman A, Tvergaard V (1992) Analyses of plastic flow localization in metals. Appl Mech Rev 45:S3–S18
Nemat-Nasser S (2004) Plasticity: a treatise on finite deformation of heterogeneous inelastic materials. Cambridge University Press, Cambridge
Davison L (2008) Fundamentals of shock wave propagation in solids. Springer, Berlin
Clayton JD (2011) Nonlinear mechanics of crystals. Springer, Dordrecht
Clayton JD (2019) Nonlinear elastic and inelastic models for shock compression of crystalline solids. Springer, Cham
Curtze S, Kuokkala VT (2010) Dependence of tensile deformation behavior of TWIP steels on stacking fault energy, temperature and strain rate. Acta Mater 58:5129–5141
Cherkaoui M, Berveiller M, Lemoine X (2000) Couplings between plasticity and martensitic phase transformation: overall behavior of polycrystalline TRIP steels. Int J Plast 16:1215–1241
Turteltaub S, Suiker ASJ (2005) Transformation-induced plasticity in ferrous alloys. J Mech Phys Solids 53:1747–1788
Turteltaub S, Suiker ASJ (2006) A multiscale thermomechanical model for cubic to tetragonal martensitic phase transformations. Int J Solids Struct 43:4509–4545
Tjahjanto DD, Turteltaub S, Suiker ASJ (2008) Crystallographically based model for transformation-induced plasticity in multiphase carbon steels. Contin Mech Thermodyn 19:399–422
Wong SL, Madilava M, Prahl U, Roters F, Raabe D (2016) A crystal plasticity model for twinning- and transformation-induced plasticity. Acta Mater 118:140–151
Yuan F, Bian X, Jiang P, Yang M, Wu X (2015) Dynamic shear response and evolution mechanisms of adiabatic shear band in an ultrafine-grained austenite–ferrite duplex steel. Mech Mater 89:47–58
Stringfellow RG, Parks DM, Olson GB (1992) A constitutive model for transformation plasticity accompanying strain-induced martensitic transformations in metastable austenitic steels. Acta Metall Mater 40:1703–1716
Tomita Y, Iwamoto T (1995) Constitutive modeling of TRIP steel and its application to the improvement of mechanical properties. Int J Mech Sci 37:1295–1305
Serri J, Martiny M, Ferron G (2005) Finite element analysis of the effects of martensitic phase transformation in TRIP steel sheet forming. Int J Mech Sci 47:884–901
Kohar CP, Cherkaoui M, El Kadiri H, Inal K (2016) Numerical modeling of TRIP steel in axial crashworthiness. Int J Plast 84:224–254
Duvall GE, Graham RA (1977) Phase transitions under shock-wave loading. Rev Mod Phys 49:523–579
Barton NR, Benson DJ, Becker R (2005) Crystal level continuum modelling of phase transformations: the \(\alpha \leftrightarrow \epsilon\) transformation in iron. Model Simul Mater Sci Eng 13:707–731
Levitas VI (1998) Thermomechanical theory of martensitic phase transformations in inelastic materials. Int J Solids Struct 35:889–940
Grujicic M, Sankaran N (1997) Dispersed-phase martensitic transformation controlled deformation behavior of two-phase metallic materials. Int J Solids Struct 34:4421–4446
Lloyd JT, Williams CL, Clayton JD (2020a) Dynamic strength of AZ31B-4E and AMX602 magnesium alloys under shock loading. J Dyn Behav Mater. https://doi.org/10.1007/s40870-019-00230-x
Clayton JD (2009) A continuum description of nonlinear elasticity, slip and twinning, with application to sapphire. Proc R Soc Lond A 465:307–334
Clayton JD (2014a) An alternative three-term decomposition for single crystal deformation motivated by non-linear elastic dislocation solutions. Q J Mech Appl Math 67:127–158
Clayton JD (2013) Nonlinear Eulerian thermoelasticity for anisotropic crystals. J Mech Phys Solids 61:1983–2014
Lloyd JT, Clayton JD, Becker R, McDowell DL (2014a) Simulation of shock wave propagation in single crystal and polycrystalline aluminum. Int J Plast 60:118–144
Clayton JD (2014b) Shock compression of metal crystals: a comparison of Eulerian and Lagrangian elastic–plastic theories. Int J Appl Mech 6:1450048
Clayton JD (2015) Crystal thermoelasticity at extreme loading rates and pressures: analysis of higher-order energy potentials. Extreme Mech Lett 3:113–122
Bennett KC, Regueiro RA, Borja RI (2016) Finite strain elastoplasticity considering the Eshelby stress for materials undergoing plastic volume change. Int J Plast 77:214–245
Thurston RN (1974) Waves in solids. In: Truesdell C (ed) Handbuch der Physik, vol VI. Springer, Berlin, pp 109–308
Clayton JD (2014c) Differential geometry and kinematics of continua. World Scientific, Singapore
Birch F (1947) Finite elastic strain of cubic crystals. Phys Rev 71:809–824
Jeanloz R (1989) Shock wave equation of state and finite strain theory. J Geophys Res 94:5873–5886
Simo J, Hughes TJR (1998) Computational inelasticity. Springer, New York
Johnson GR, Cook WH (1983) A constitutive model and data for materials subjected to large strains, high strain rates, and high temperatures. In: Proceedings of the 7th international symposium on ballistics, The Hague, Netherlands, pp 541–547
Lloyd JT, Field DM, Limmer KR (2020b) A four parameter hardening model for TWIP and TRIP steels. Mater Des 194:108878
Leblond JB, Mottet G, Devaux JC (1986) A theoretical and numerical approach to the plastic behaviour of steels during phase transformations—I. Derivation of general relations. J Mech Phys Solids 34:395–409
Mackenzie JK (1950) The elastic constants of a solid containing spherical holes. Proc Phys Soc B 63:2–11
Clayton JD, Freed AD (2020) A constitutive framework for finite viscoelasticity and damage based on the Gram–Schmidt decomposition. Acta Mech 231:3319–3362
Clayton JD, Knap J (2014) A geometrically nonlinear phase field theory of brittle fracture. Int J Fract 189:139–148
Clayton JD, Knap J (2016) Phase field modeling of coupled fracture and twinning in single crystals and polycrystals. Comput Methods Appl Mech Eng 312:447–467
Kachanov M, Tsukrov I, Shafiro B (1994) Effective moduli of solids with cavities of various shapes. Appl Mech Rev 47:S151–S174
Cocks ACF, Ashby MF (1980) Intergranular fracture during power-law creep under multiaxial stresses. Met Sci 14:395–402
Cocks ACF, Ashby MF (1982) On creep fracture by void growth. Prog Mater Sci 27:189–244
Bammann DJ, Chiesa ML, Horstemeyer MF, Weingarten LI (1993) Failure in ductile materials using finite element methods. In: Jones N, Wierzbicki T (eds) Structural crashworthiness and failure. Elsevier Applied Science, The Universities Press, Belfast, pp 1–54
Barton NR (2015) Results from a new Cocks–Ashby style porosity model. AIP Conf Proc 1793:100029
Whittington WR, Oppedal AL, Turnage S, Hammi Y, Rhee H, Allison PG, Crane CK, Horstemeyer MF (2014) Capturing the effect of temperature, strain rate, and stress state on the plasticity and fracture of rolled homogeneous armor (RHA) steel. Mater Sci Eng A 594:82–88
Marin EB, McDowell DL (1996) Associative versus non-associative porous viscoplasticity based on internal state variable concepts. Int J Plast 12:629–669
Johnson GR, Cook WH (1985) Fracture characteristics of three metals subjected to various strains, strain rates, temperatures and pressures. Eng Fract Mech 21:31–48
Celentano DJ, Chaboche JL (2007) Experimental and numerical characterization of damage evolution in steels. Int J Plast 23:1739–1762
Chung K, Ma N, Park T, Kim D, Yoo D, Kim C (2011) A modified damage model for advanced high strength steel sheets. Int J Plast 27:1485–1511
Abu Al-Rub RK, Ettehad M, Palazotto AN (2015) Microstructural modeling of dual phase steel using a higher-order gradient plasticity-damage model. Int J Solids Struct 58:178–189
Becker R, Needleman A, Richmond O, Tvergaard V (1988) Void growth and failure in notched bars. J Mech Phys Solids 36:317–351
Noble CR, Anderson AT, Barton NR et al (2017) ALE3D: an arbitrary Lagrangian–Eulerian multi-physics code. Technical Report LLNL-TR-732040. Lawrence Livermore National Laboratory (LLNL), Livermore
Field DM, Baker DS, Van Aken DC (2017) On the prediction of \(\alpha\)-martensite temperatures in medium manganese steels. Metall Mater Trans A 48:2150–2163
Magagnosc DJ, Field DM, Meredith CS, Walter TR, Limmer KR, Lloyd JT (2021) Superior strength and ductility in a low density duplex steel studied by in-situ neutron diffraction. Mater Sci Eng A 799:140252
Field DM et al (2020) Variable properties in a two-stage TRIP Mn steel. Mater Des (in preparation)
Meredith CS et al (2020) Mechanical behavior and microstructural evolution of TRIP, TWIP, and SLIP multi-phase steels at dynamic strain rates (in preparation)
Roters F, Raabe D, Gottstein G (2000) Work hardening in heterogeneous alloys—a microstructural approach based on three internal state variables. Acta Mater 48:4181–4189
Steinmetz DR, Jäpel T, Wietbrock B, Eisenlohr P, Gutierrez-Urrutia I, Saeed-Akbari A, Hickel T, Roters F, Raabe D (2013) Revealing the strain-hardening behavior of twinning-induced plasticity steels: theory, simulations, experiments. Acta Mater 61:494–510
Millett JCF, Bourne NK, Rosenberg Z (1997) Shear stress measurements in copper, iron, and mild steel under shock loading conditions. J Appl Phys 81:2579–2583
Whiteman G, Millett JCF, Bourne NK (2007) Shear stress measurements in copper, iron, and mild steel under shock loading conditions. AIP Conf Proc 955:673–676
Whiteman G, Millett JCF (2012) Shear stress measurements in stainless steel 2169 under 1D shock loading. Exp Mech 52:265–273
Guinan MW, Steinberg DJ (1974) Pressure and temperature derivatives of the isotropic polycrystalline shear modulus for 65 elements. J Phys Chem Solids 35:1501–1512
Miller MP, McDowell DL (1996) Modeling large strain multiaxial effects in FCC polycrystals. Int J Plast 12:875–902
Lloyd JT, Clayton JD, Austin RA, McDowell DL (2014b) Plane wave simulation of elastic–viscoplastic single crystals. J Mech Phys Solids 69:14–32
Lloyd JT, Clayton JD, Austin RA, McDowell DL (2015) Shock compression modeling of metallic single crystals: comparison of finite difference, steady wave, and analytical solutions. Adv Model Simul Eng Sci 2:14
Winey JM, Gupta YM (2006) Nonlinear anisotropic description for the thermomechanical response of shocked single crystals: inelastic deformation. J Appl Phys 99:023510
Vogler TJ, Clayton JD (2008) Heterogeneous deformation and spall of an extruded tungsten alloy: plate impact experiments and crystal plasticity modeling. J Mech Phys Solids 56:297–335
Vogler TJ, Chhabildas LC (2006) Strength behavior of materials at high pressures. Int J Impact Eng 33:812–825
Brown JL, Alexander CS, Asay JR, Vogler TJ, Ding JL (2013) Extracting strength from high pressure ramp-release experiments. J Appl Phys 114:223518
Swegle JW, Grady DE (1985) Shock viscosity and the prediction of shock wave rise times. J Appl Phys 58:692–701
Johnson JN, Tonks DL (1992) Dynamic plasticity in transition from thermal activation to viscous drag. In: Shock compression of condensed matter-1991. Elsevier, Amsterdam, pp 371–378
Preston DL, Tonks DL, Wallace DC (2003) Model of plastic deformation for extreme loading conditions. J Appl Phys 93:211–220
Austin RA, McDowell DL (2011) A dislocation-based constitutive model for viscoplastic deformation of FCC metals at very high strain rates. Int J Plast 27:1–24
Austin RA, McDowell DL (2012) Parameterization of a rate-dependent model of shock-induced plasticity for copper, nickel, and aluminum. Int J Plast 32:134–154
Marsh S (ed) (1980) LASL shock Hugoniot data. University of California Press, Berkeley
Needleman A (1988) Material rate dependence and mesh sensitivity in localization problems. Comput Methods Appl Mech Eng 67:69–85
Wright TW, Batra RC (1985) The initiation and growth of adiabatic shear bands. Int J Plast 1:205–212
Savinykh AS, Garkushin GV, Razorenvo SV, Wolf S, Kruger L (2015) Influence of the temperature-induced martensitic-austenitic transformation on the strength properties of high-alloy steels under dynamic loading. Combust Explos Shock Waves 51:124–129
Eckner R, Kruger L, Motylenko M, Savinykh AS, Razorenov SV, Garkushin GV (2018) Deformation mechanisms and microplasticity of austenitic TRIP/TWIP steel under flyer plate impact. EPJ Web Conf 183:03007
Acknowledgements
J.D.C. and J.T.L. acknowledge support of the CCDC Army Research Laboratory. The authors thank D.M. Field, D.J. Magagnosc, C.S. Meredith, and T.R. Walter for sharing various data on steel alloys prior to external publication.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendices
Appendix 1: Theoretical Derivations
Equation (19) is derived as
Equation (42) is derived as
Equation (43) is derived as
Equation (44) is derived as
Equation (45) is derived as
Equation (46) is derived as
Equation (72) is derived as
where
Equation (79) is derived as
Equation (90) is derived as
Appendix 2: Data Analysis
In most cases, 2–3 experiments were performed for each alloy at each initial temperature and loading rate. For every loading protocol, the number of experiments is shown in Table 5 along with experimental variation in effective stress, model prediction of effective stress, and the error in this prediction relative to the experimental mean. Comparisons are made at two discrete strain levels \(\epsilon\) in each case, corresponding to availability of test data (e.g., some compression experiments were unloaded prior to attainment of very large deformations). Stress variability is defined as \({\varDelta }\sigma = {\text {max}}_{i} | {\bar{\sigma }} - \sigma _{i} |,\) where \({\bar{\sigma }}\) is the mean among tests and \(\sigma _{i}\) is the measured value for experiment i, where i runs from 1 to the number of tests shown in corresponding rows of the table. Modeling error in the rightmost column is defined, at each \(\epsilon ,\) as \(|\sigma - {\bar{\sigma }} | / {\bar{\sigma }} ,\) with the model result denoted simply by \(\sigma .\) Agreement with tensile data is generally superior to that with compression data since the calibration procedure of “Static and Dynamic Loading” section focuses on the tensile response. Of the 36 cases quantified in Table 5, modeling error exceeds 4% only 11 times. The most inaccurate results, with error exceeding 13%, correspond to static compression of the TWIP steel at room temperature (noting that the model, calibrated to tensile response, fails to capture tension–compression asymmetry in the TWIP alloy), and to dynamic high-temperature compression of the TWIP steel, for which thermal softening is apparently not well represented.
Since experimental data on the three alloys of Table 1 for compression and tension are available, but torsion data are notably absent, model predictions for all three stress states and all three steels are compared versus one another and versus trends for several other steels reported elsewhere. Results for the SLIP and TWIP steels are shown in Fig. 13a. These show negligible effect of triaxiality on stress–strain behavior, with the exception of tensile softening due to porosity that initiates at \(\epsilon \gtrsim 0.35\) in the SLIP steel. In each case, effective stress \(\sigma\) is von Mises stress, and effective strain is \(\epsilon = \sqrt{2/3} \int {{{\varvec{d}}}}: {{{\varvec{d}}}} \text {d}t.\) For tension and compression, these definitions are consistent with true stress and logarithmic strain measures used in “Static and Dynamic Loading” section. For free-end torsion, \(\sigma = \sqrt{3} \tau\) and \(\epsilon = \gamma / \sqrt{3},\) with \(\tau\) and \(\gamma\) the usual engineering shear stress and strain measures.
Model predictions of effect of loading mode on effective stress \(\sigma\) under quasi-static, room temperature conditions a SLIP and TWIP alloys and b TRIP alloy
Results in Fig. 13b demonstrate a marked effect of stress state on strength behavior of the TRIP steel. For strain-assisted transformation, stress in tension very slightly exceeds that in torsion, since the contribution of tensile pressure is not enough to drastically accelerate phase changes in the former relative to the latter. In contrast, martensitic transformation is impeded by compressive stress, leading to a softer response. Stress-assisted transformation corresponds to the results shown in Fig. 3a, and is thought more realistic than strain-assisted transformation for static tensile loading. In this case, strain accommodation by phase changes initiated prior to plastic yield induces a load reduction, which in turn retards the transformation rate at low applied strains.
Trends predicted in Fig. 13 are similar, but not identical, to those reported for RHA steel in [59]. Quasi-static room temperature data for RHA show very similar plastic strength behavior for tension, compression, and shear, at least until damage accumulates. At larger strain levels, porosity nucleated at hard particles was more severe in torsion than in tension, leading to softer torsional behavior and earlier failure [59]. This behavior cannot be captured by the traditional Cocks–Ashby type damage model (89), used in the present work, which requires positive mean stress for porosity increases.
Data and/or modeling on 304L stainless steel in [77] show tensile strength exceeding compressive strength, with torsional hardening the lowest among the three stress states. Expansive martensitic transformation from the \(\gamma\) to \(\alpha '\) phase was reported to be greater in compression loading than torsional loading, despite the negative contribution to mechanical driving force induced by pressure in the former. Less localized slip and different dislocation structures observed in torsion than compression were suggested as responsible for the observed stress-state dependencies [77]. If relative torsional softening occurs in any of the three alloys of present study—which must be confirmed or refuted by future torsion experiments—adjustments to the present model framework are necessary to capture such behavior. For example, a \(J_{3}\)-based yield function is advocated in [29, 77], which provides satisfactory description of their experimental results.
Table 6 compares experimental data and model calibrations for quasi-static tensile failure behavior of the three alloys in Table 1, at room and elevated temperatures. The logarithmic tensile strain to failure is \(\epsilon _{F}.\) This is the average ductility among experimental tests, while it is the exact failure strain imposed in calibration of porosity \(\phi _{F}\) for the damage model. The experimental variation of ductility is \({\varDelta }\epsilon _{F} = {\text {max}}_{i} | \epsilon _{F} - \epsilon _{i} |\) with \(\epsilon _{i}\) the strain to failure of each experiment i at the given temperature for the alloy under consideration. Similarly, experimental variation of tensile strength at failure is \({\varDelta }\sigma _{F} = {\text {max}}_{i} | {\bar{\sigma }}_{F} - \sigma _{i} |,\) where \({\bar{\sigma }}_{F}\) is the average over experiments i with individual tensile strengths \(\sigma _{i}.\) The predicted tensile strength of the model at \(\epsilon _{F}\) is \(\sigma _{F},\) and the error in this prediction is \(|\sigma _{F} - {\bar{\sigma }}_{F} | / {\bar{\sigma }}_{F}.\) In order to achieve perfect agreement in \(\epsilon _{F}\) between model and experiment, a composition- and temperature-dependent failure porosity \(\phi _{F}\) must be assigned in each case. Determination of whether or not such values are physically descriptive requires microscopy of fractured surfaces as in [59], for example. Agreement between model and experiment for tensile strength at \(\epsilon _{F}\) is within 8% error, and within 4.5% when high temperature data for TWIP and TRIP steels are excluded.
Rights and permissions
About this article
Cite this article
Clayton, J.D., Lloyd, J.T. A Dynamic Finite-Deformation Constitutive Model for Steels Undergoing Slip, Twinning, and Phase Changes. J. dynamic behavior mater. 7, 217–247 (2021). https://doi.org/10.1007/s40870-020-00279-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40870-020-00279-z