Abstract
This book is focused on constitutive description of mechanical behavior of engineering materials: both conventional (e.g., polycrystalline homogeneous isotropic or anisotropic metallic materials) and nonconventional ones (e.g., heterogeneous multicomponent usually anisotropic composite materials) fabricated by modern material engineering. Effective material properties at the macrolevel depend on both the material microstructure (isotropic or originally anisotropic in general case) and on dissipative phenomena occurred on fabrication and consecutive loading phase resulting in irreversible microstructure changes (acquired anisotropy). The material symmetry is a background and anisotropy is a core around which the book is formed. In this way a revision of classical rules of enhanced constitutive description of materials is required. The aim of this introductory chapter lies in providing, apart from classical definitions of tensor single invariants, also the choice of state variables necessary to describe irreversible microstructure changes accompanying coupled dissipative phenomena, and basic definitions of common invariants of either two second-order tensors (e.g., stress/strain and damage tensors) or two different-order tensors (e.g., stress/strain and fourth-order structural tensors). Concise classification of anisotropic materials with respect to symmetry of elastic matrices as referred to the crystal lattice symmetry is given, and extended analogy between symmetries of constitutive material matrices (elastic and yield/failure) is also discussed. Next, strain and complementary energy as function of either stress/strain invariants (initial elastic isotropy) or common stress/strain—damage invariants (damage acquired anisotropy) are mentioned. Constitutive equation of linear elasticity in terms of common invariants of strain and structural orthotropic tensors is given. The scope of this chapter provides necessary tools for more extended constitutive description of materials which exhibit either virgin anisotropy or damage or phase change acquired anisotropy following microstructure changes.
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Keywords
- Single or common tensor invariants
- Material symmetry and constitutive matrices
- Virgin or acquired anisotropy
- Shear and volumetric change coupling
- Strain energy of anisotropic materials
- Damage and phase change state variables
- Constitutive tensors analogy
1.1 Second-Order Tensors
1.1.1 Stress Tensor and Stress Tensor Invariants
Stress tensor \(\varvec{\sigma }\), when mathematical \(\sigma _{ij} \quad i,j=1,2,3\), or \(i,j=x,y,z\), and engineering notations are used is furnished as
where \(x,y,z\) denote cartesian coordinate system.
When symmetry of the stress tensor \(\sigma _{ij}=\sigma _{ji}\) is assumed, the stress tensor can be represented as columnar stress vector as follows:
When the definition of stress deviator is assumed as
where \(\sigma _\mathrm{h}=\frac{1}{3}\sigma _{kk}\) denotes either hydrostatic or mean stress, while \(\delta _{ij}=\left\{ \begin{array}{ll} 1 &{} i=j\\ 0 &{} i\ne j \end{array}\right. \) denotes Kronecker’s symbol, decomposition of the stress tensor into the stress axiator and the stress deviator takes the following form:
where absolute notation \(\sigma _\mathrm{h}\mathbf{1}\) and \({\varvec{s}}\) are used for the stress axiator and the stress deviator, respectively
Classical stress transformation rule from \(i,j\) to \(k,l\) directions is
where second-order tensor transformation rule is applied and \(a_{ki},a_{lj}\) denote direction cosines of the transformation from the original frame \(i,j=x,y,z\) in the new reference frame \(k,l=\xi ,\eta ,\zeta \). It is possible to distinguish the specific transformation into eigendirections (principal directions) for which the corresponding stress tensor takes the diagonal representation
Three principal stresses are determined as real roots of the cubic equation, being solution of eigenproblem for the stress tensor \(\varvec{\sigma }\)
where \(\lambda _i=\sigma _1,\sigma _2,\sigma _3\) stand for eigenvalues. These principal stresses are real roots of the characteristic equation of stress tensor \(\lambda _i=\sigma _i\)
which can be rewritten in the equivalent fashion
Three coefficients of the characteristic equation (1.10) \(I_{1\sigma },I_{2\sigma },I_{3\sigma }\) are called the principal invariants of the stress tensor and may be defined in terms of stress components
Apart from the principal invariants, the basic stress invariants also called the generic stress invariants are of particular importance, namely
It is seen that the basic stress invariants can be interpreted as traces of subsequent powers of stress tensor \(\varvec{\sigma }\), \(\varvec{\sigma }^2=\varvec{\sigma }\cdot \varvec{\sigma }\), \(\varvec{\sigma }^3=\varvec{\sigma }\cdot \varvec{\sigma }\cdot \varvec{\sigma }\), if appropriate coefficients 1, 1/2, 1/3 are used. Note that the basic invariants differ from the principal invariants, which are coefficients of the characteristic equation (1.10).
The basic stress invariants \(J_{1\sigma },J_{2\sigma },J_{3\sigma }\) are expressed in terms of the principal stress invariants \(I_{1\sigma },I_{2\sigma },I_{3\sigma }\) as follows:
The reciprocal relations are
Decomposition of the stress tensor into the stress axiator (spherical tensor) and the stress deviator (1.3 \(-\) 1.5) leads to the following system of principal or generic invariants of the stress deviator
where, in similar fashion as in Eq. (1.12), subsequent powers of the stress deviator \({\varvec{s}}\), \({\varvec{s}}^2={\varvec{s}}\cdot {\varvec{s}}\), \({\varvec{s}}^3={\varvec{s}}\cdot {\varvec{s}}\cdot {\varvec{s}}\) are used. Note that the first basic deviatoric stress invariant \(J_{1s}\) is equal to zero according to definition (1.3).
Additionally, some engineering tensor stress invariants characterized by the stress dimension homogeneity [MPa], by contrast to the above defined basic invariants of different dimensions [MPa], [MPa\(^2\)], [MPa\(^3\)] are frequently used as
The first of them \(\sigma _\mathrm{h}\) is easily recognized as the mean stress and the second \(\sigma _\mathrm{eq}\) represents the commonly used stress intensity also called the effective stress .
1.1.2 Strain Tensor and Strain Tensor Invariants
Strain tensor \({\varvec{\varepsilon }}=\varepsilon _{ij}\) when uniform mathematical notation \(i,j=1,2,3\) or \(i,j=x,y,z\), and the engineering notation are used, is furnished as
where \(x,y,z\) denote cartesian coordinate frame.
Transformation of the strain tensor is described in a similar fashion as the stress tensor transformation (1.6), namely
Similarly, the principal strains can be obtained by solution of the eigenproblem of the tensor \({\varvec{\varepsilon }}\)
or equivalently as solution of characteristic equation of strain tensor
Coefficients of the above equation \(I_{1\varepsilon },I_{2\varepsilon },I_{3\varepsilon }\) denote the principal invariants of the small (linearized) strain tensor and are defined as the homogeneous, scalar functions of the strain components
If symmetry of the strain tensor is assumed equivalent representation of the strain tensor in the form of columnar strain vector may be applied as
When the definition of the strain deviator, analogous to the stress deviator (1.3), is used, we arrive at
where \(\varepsilon _\mathrm{m}\) denotes mean (volumetric) strain. Decomposition of the strain tensor into the strain axiator and the strain deviator is given according to the scheme
when the absolute notation was used, where \(\varepsilon _\mathrm{m}\mathbf{1}\) and \({\varvec{e}}\) denote the strain axiator and the strain deviator, respectively
The basic or the generic strain tensor invariants are defined in analogous fashion as in Eq. (1.12)
The basic strain tensor invariants \(J_{1\varepsilon },J_{2\varepsilon },J_{3\varepsilon }\) are expressed in terms of the principal strain invariants \(I_{1\varepsilon },I_{2\varepsilon }, I_{3\varepsilon }\) as
The reciprocal relationships are
The principal invariants of the strain deviator may be determined in an analogous way as the principal invariants of the stress deviator (1.15), namely
1.1.3 Matrix Representation of Stress and Strain Tensors
Stress \(\sigma _{ij}\) and strain \(\varepsilon _{ij}\) are the second-rank tensors having in general \(3^2=9\) components, since each of indices \(i,j\) runs from 1 to 3. Each of them can be interpreted as linear transformation of a certain vector to another vector. In case of the stress tensor, linear transformation of direction cosines \(n_j\) into a traction vector \(p_i\) according to rule
is written down or
when the expanded form is used. Applying the matrix-vector notation to the above formulae the equivalent form is reached
In Eq. (1.32) the second-rank stress tensor is represented by the \(3\times 3\) tensor representation matrix and analogously, the strain tensor representation matrix
Due to symmetry conditions of both the stress \(\sigma _{ij}=\sigma _{ji}\) and the strain \(\varepsilon _{ij}=\varepsilon _{ji}\) tensors, both representation matrices are symmetric, comprising 6 independent components each. When engineering notation is used, replacing \(1,2,3\) frame by \(x,y,z\) cartesian coordinate frame, and introducing appropriate notation \(\sigma _{ij}=\tau _{ij}\) and \(\varepsilon _{ij}=\frac{1}{2}\gamma _{ij}\) for \(i\ne j\), we arrive at
1.1.4 Decomposition of Strains
In the case of infinitesimal deformation the total strain \(\varepsilon _{ij}\) can be expressed as the sum of the elastic (reversible) strain \(\varepsilon _{ij}^\mathrm{e}\), inelastic (irreversible) strain \(\varepsilon _{ij}^\mathrm{I}\), and thermal strain \(\varepsilon _{ij}^\mathrm{T}\):
In the process of deformation, various microstructural rearrangements of material structure may take place, for example, the changes in density and configuration of dislocations, the development of microscopic cavities, changes from primary to secondary phase, etc. All these rearrangements may contribute to both reversible and irreversible strains (cf. Abu Al-Rub and Voyiadjis [1]), therefore:
where \(\varepsilon _{ rs}^\mathrm{E}\) is a “pure” elastic strain, and \(\varepsilon _{ rs}^\mathrm{ed},\ldots ,\varepsilon _{ rs}^\mathrm{p},\ldots \) are respectively the reversible and irreversible components of the total strain induced by dissipative phenomenon (see Fig. 1.1), e.g., plastic flow , damage , phase transformation , etc. For example, in the case of thermo-elastic-plastic-damage material the total strain tensor \(\varepsilon _{ij}\) is expressed as
while its damage-induced component, \(\varepsilon _{ rs}^\mathrm{d}\), consists of both reversible (\(\mathrm{ed}\)) and irreversible (\(\mathrm{Id}\)) damage strain terms:
Components of the strain tensor \(\varepsilon _{ij}^{k}\) induced by \(k\)th dissipative phenomenon
1.2 Fourth-Order Tensors and Matrix Representation
1.2.1 Stiffness and Compliance Matrices—Voigt’s Notation
General linear elasticity equation for anisotropic material, frequently called the generalized Hooke law, takes the forms
where the fourth-rank elasticity tensors, stiffness \(E_{ijkl}\) or compliance \(E_{ijkl}^{-1}\), are defined, in general by \(3^4=81\) components, since each of indices \(i,j,k,l\) runs through 1, 2, 3. Because of the symmetry of the stress \(\sigma _{kl}=\sigma _{lk}\) and the strain \(\varepsilon _{ij}=\varepsilon _{ji}\) tensors, both the stiffness and compliance tensors are symmetric with respect to change of indices in pairs \(i\leftrightarrow j\) and \(k\leftrightarrow l\)
Additionally, because of property of positive definiteness of strain energy or complementary energy the symmetry with respect to change of indices between pairs \(ij\leftrightarrow kl\) must also hold
Because of symmetry conditions (1.40) and (1.41) from among 81 components of stiffness or compliance tensors, only 21 are independent. In order to describe the generalized Hooke’s law (1.39) by use of vector-matrix Voigt’s notation, stress and strain tensors are written as columnar stress and strain vectors, if the following scheme of change between tensor \(i,j=1,2,3\) and vectors \(k=1,2,\ldots ,6\) indices holds:
From the above scheme we obtain the following representations of stress and strain tensors :
Analogous scheme is applied to the first and second pairs of indices of stiffness and compliance tensors
where appropriate factors 2 or 4 are applied.
For instance, if the axial strain \(\varepsilon _{11}\) is considered the transformation scheme is as follows:
In case the shear strain \(\varepsilon _{23}\) is considered, the following are furnished:
Finally, the generalized Hooke’s law (1.39) is represented in vector-matrix notation as follows:
or
or equivalently
where \([{\mathbb E}]\) or \([{\mathbb E}^{-1}]\) denote representation matrices of elastic stiffness or compliance tensors, whereas \(\left\{ {\varvec{\varepsilon }}\right\} \) and \(\left\{ \varvec{\sigma }\right\} \) denote the columnar vectors of strain and stress, respectively. When columnar vectors of stress and strain are used as well as elasticity matrices are explicitly written down, Hooke’s law is furnished as
or
where engineering notation for shear stress \(\tau _j=\sigma _j\) \((j=4,5,6)\) is used.
It should be mentioned that symmetric stiffness \([E_{ij}]=[E_{ji}]\) and symmetric compliance \([E^{-1}_{ij}]=[E^{-1}_{ji}]\) matrices, both having dimension \(6\times 6\), are representation matrices of fourth-rank elasticity tensors \(E_{ijkl}\) or compliance \(E_{ijkl}^{-1}\). Transformation of each matrix to another coordinate frame can be performed if the matrix notation (1.51) is replaced by the tensor notation (1.39), or by use of the appropriate transformation matrix \([{\mathbb Q}]\)
For instance, if the stiffness matrix is considered the transformation matrix takes the form
the coefficients of which are scalar products of corresponding direction cosines \(q_{ij}=n_i n_j\) between both coordinate frames.
1.2.2 The Choice of State Variables
The irreversible rearrangements of the internal structure can be represented by a group of variables describing the current state of material microstructure:
where \(V^{k}\) may be scalars, vectors, or even rank tensors. For damage description, in the case where the damaged material remains isotropic, the current state of damage is often represented by the scalar variable \(V^\mathrm{d}=D\) denoting the volume fraction of cracks and voids \(\mathrm{d}V^\mathrm{d}\) in the total volume \(\mathrm{d}V^{0}\). Damage acquired orthotropy requires a second-order tensor, for example, the classical Murakami–Ohno [38] tensor \(V_{ij}^\mathrm{d}=D_{ij}\), see Eq. (1.61). In the most general case of anisotropy the description of damage needs to be embodied in an eight-order tensor (cf. Cauvin and Testa [6]), while the principle of strain equivalence allows using fourth-order tensors, see Sect. 1.2.3. For phase transformation analysis the scalar variable \(V^\mathrm{ph}=\xi \) is commonly adopted (cf. Egner and Skoczeń [14]), which denotes the volume fraction of the secondary phase in the total volume of the two-phase Representative Volume Element. However, a scalar variable is not capable of describing the acquired anisotropy due to partially directional nature of the secondary inclusions in the primary matrix. Therefore, instead of scalar variable a second-order phase change tensor can be defined in analogy to the damage tensor:
where \(\xi _{i}\) describes the ratio of the secondary phase area \(\mathrm{d}A_{i}^\mathrm{ph}\) to the total area \(\mathrm{d}A_{i}^{0}\) on the principal plane of normal unit vector \({\varvec{n}}_{i}\) (cf. Egner [13]). Another group of state variables consists of internal (hidden) variables corresponding to the modifications of loading surfaces:
where \(r^\mathrm{p},r^\mathrm{d}\) correspond to isotropic expansion of the loading surface, \(\alpha _{ij}^\mathrm{p},\alpha _{ij}^\mathrm{d}\) affect loading surface translatoric displacements, \(l_{ijkl}^\mathrm{p},l_{ijkl}^\mathrm{d}\) are hardening tensors of the fourth order which includes varying lengths of axes and rotation of the loading surface, and \(g_{ijklmn}^\mathrm{p},g_{ijklmn}^\mathrm{d}\) describe changes of the curvature of the loading surface (distortion) related to appropriate dissipative phenomenon (cf. Kowalsky et al. [27], see Fig. 1.2). The complete set of state variables \(\{V_\mathrm{st}\}\) reflecting the current state of the thermodynamic system consists of observable variables: elastic (or total) strain tensor \(\varepsilon _{ij}^\mathrm{e}\) and absolute temperature \(T\), and two groups of microstructural \(\{V^{k}\}\) and hardening \(\{h^{k}\}\) state variables:
Modifications of the loading surface in the space of thermodynamic conjugate forces \(\{J^{k}\}\) (after [13])
When thermo-elastic-plastic-damage two-phase material is considered, the exemplary set of state variables for a general case of hardening/softening effects induced by different dissipative phenomena is further listed in Table 7.1.
When the material is subjected to reverse tension-compression cycles, the unsymmetrical behavior in tension and compression is observed as the unilateral response due to partial crack closure effect. To describe the phenomenon of the unilateral damage, also called the damage deactivation or the crack closure/opening effect, a decomposition of the stress or strain tensors into the positive or negative projection is usually introduced using the fourth-rank projection operators (cf. Krajcinovic [30]; Bielski et al. [4]):
where the fourth-rank tensor \(B_{ijkl}^{(\varepsilon )}\) is built of directional cosines between the principal and the current spatial systems, \(n_{iI}^{(\varepsilon )}\) and \(\kappa (\varepsilon _{I})=\mathrm{H}(a)+\zeta \mathrm{H}(-a)\), \(\mathrm{H}\) is a Heaviside function and \(\zeta \) is a material constant.
1.2.3 Damage and Damage Effect Tensors
So far constitutive description of material has not accounted for influence of damage. Damage means existence of microvoids and microcracks in the material that result in essential deterioration of mechanical properties at the macroscale, such as strength and stiffness or compliance.
Three configurations used in CDM: a initial, b damaged, c effective pseudo-undamaged
In the simplest case when microvoids are spherical and homogeneously distributed in material, damage is described by the scalar damage variable \(D\), usually called the damage parameter, Fig. 1.3
Scalar damage variable \(D\) , introduced by Kachanov [24] and Rabotnov [46], represents the loss of effective area from the initial \(A\) to the damaged \(A^\mathrm{d}\) states. In order to generalize the scalar damage variable to the case when microvoids exhibit clearly directional nature, the vector damage variable \(D_{i}\), is proposed by Davison and Stevens [11], Kachanov [25], Krajcinovic and Fonseka [28]
Murakami and Ohno [39] introduced more general damage variable defined by the symmetric second-rank damage tensor \({\varvec{D}}\), capable of capturing an orthotropic damage nature
Recently, researches aimed towards correct description of damage mechanism in elastic-brittle rock-like materials, ceramics or concrete led to definition of the fourth-rank damage tensors, e.g., Chaboche [8], Krajcinovic [29] or Lubarda and Krajcinovic [35]. Apart from the above-mentioned damage variables possessing clear geometric interpretation other damage variables referring to physical planes, described in details e.g., by Gambarotta and Lagomarsino [15], Seweryn and Mróz [48] should also be mentioned. More general classifications of damage variables were listed in following subject monographs by Krajcinovic [29, 30], Skrzypek and Ganczarski [51], Betten [3] or Murakami [40].
In the frame of continuum damage mechanics (CDM), three configurations are considered: initial configuration \(\mathcal{A}\) that describes material in undamaged state \(D(\mathcal{A})=0\), physical configuration \(\mathcal{B}\) referring to the damaged state \(D(\mathcal{B})\ne 0\), and the equivalent, fictitious pseudo-undamaged configuration \(\mathcal{C}\) in which real heterogeneous material is substituted by a homogeneous material, free of damage \(D(\mathcal{C})=0\), as schematically is shown in Fig. 1.3.
The physical (damaged) configuration \(\mathcal{B}\) is equivalent to the effective (pseudo-undamaged) configuration \(\mathcal{C}\) in a certain sense, for instance, of strain equivalence Chaboche [7], stress equivalence Taher et al. [54], or elastic strain energy equivalence Cordebois and Sidoroff [10]. In physical configuration \(\mathcal{B}\) damage state manifests through the effective elasticity modulus \(\widetilde{E}\), for instance,
where the hypotheses of strain or stress equivalence (first formula) or elastic energy equivalence are used. Contrarily, in the effective configuration \(\mathcal{C}\) damage state manifests by the definition of the effective variables \(\widetilde{\sigma },\widetilde{\varepsilon }\), respectively
or equivalently
The damage effect matrix, being matrix representation of the damage effect tensor
is expressed in terms of the damage parameter \(D\) as follows:
where the diagonal form is applicable.
Damage effect matrix plays an essential role in definitions of the damage effective stress tensor \(\widetilde{\varvec{\sigma }}\)
and the damage effective compliance (stiffness) matrix \([\widetilde{{\mathbb E}}^{-1}]\)
For brevity, in all the above equations (1.62–1.68) the assumption of material isotropy in undamaged state (\(\mathcal{A}\)) is applied.
Assumption of the isotropic damage nature is too strong a simplification since usually microvoids or microcracks are of oval or directional shapes. A proper damage description requires application of orthotropic damage representation (1.61), which under the assumption of the principal damage frame reduces to the diagonal form, where \(D_1,D_2,D_3\) components may be interpreted by reduction of effective areas 1, 2, 3 (1.60), hence
Chosen representations of the damage effect matrix based on various hypotheses, after Chen and Chow [9], Skrzypek [49], Murakami [40], can be defined as follows:
or
or
The damage effective stress can be defined, for instance in the following two ways, both satisfying symmetry of the effective stress \(\widetilde{\varvec{\sigma }}({\varvec{D}})\):
or
Exemplary effective compliance matrices take the following representations, Skrzypek and Ganczarski [51]:
or
In both cases, for the sake of brevity, material isotropy at the undamaged state was assumed.
In a more general case of full damage anisotropy the fourth-rank damage tensor \(D_{ijkl}\), built of 21 independent components, should be used.
Following Cauvin and Testa [6] the effective stiffness tensor is defined as
where fourth-rank tensors \({\mathbb R}\) and \({\mathbb D}\) stand for damage effect and damage tensors, respectively. In general case of full damage anisotropy the \(6\times 6\) matrix representation of the fourth-rank damage tensor is as follows:
As a particular case the orthotropic damage is considered as example for which the unsymmetric orthotropic damage matrix reduces to
In the particular case when the orthotropic symmetry of damaged material is considered, the damage tensor takes the following matrix representation, after Cauvin and Testa [6], also Ganczarski [17]:
defined by 12 independent elements, in general nonsymmetric because three elements under diagonal \(D_{2211},D_{3311},D_{3322}\) are truly independent.
In the narrower case of transverse isotropy (in the plane 2, 3), number of independent elements of the tensor \(D_{ijkl}\) reduces to 5, namely \(D_1,D_2,D_3,D_4,D_5\)
Two components \(D_{2211}=D_{3311}=D_2\prime \) are dependent, and expressed as
This kind of transverse isotropy will further be classified as transverse isotropy case of hexagonal symmetry (5 independent components in contrast to another transverse isotropy of tetragonal symmetry where all 6 components are truly independent, see Table 1.1).
The \(6\times 6\) transversely isotropic compliance matrix is of the following form:
in which damage affected modules expressed in terms of damage variables are
More accurate description of anisotropic damage may be provided by use of fabric tensors, see Murakami [40], Voyiadjis and Kattan [55], Yun-bing and Xing-fu [56], Lubarda and Krajcinovic [35]. For this reason a unit spherical surface around a given point \(P(x)\) in the RVE is considered (see Fig. 1.4), and the directional distribution \(\xi ({\varvec{n}})\) of the microvoid density on the unit sphere is defined as a polynomial function of the direction vector \({\varvec{n}}\)
Expression (1.85) is a generalized Fourier series with respect to the irreducible tensor bases \(f_{ij}({\varvec{n}})\), \(f_{ijkl}({\varvec{n}})\), ...
The tensor bases \(f_{ij}({\varvec{n}})\), \(f_{ijkl}({\varvec{n}})\), ... are symmetric with respect to the indices, consist of even-order tensor components, and have vanishing trace.
Unit spherical surface to represent directional void distribution
The tensors \(D_{0}\), \(D_{ij}\), \(D_{ijkl}\), ... characterize the directional distribution of damage, and are called fabric tensors. For given \(\xi ({\varvec{n}})\) they can be derived by calculating the following integrals (cf. Murakami [40]):
The even-order tensors \(D_{0}\), \(D_{ij}\), \(D_{ijkl}\), ... represent completely the damage state of the materials, and have been used as the internal state variables in thermodynamic modeling of creep and brittle damage, see Onat and Leckie [43], Lacy et al. [32].
Concluding, it is worth to mention that virgin material anisotropy may either manifest from the very beginning of the elastic response when appropriate anisotropic formulation of Hooke’s law is required or at damage initiation phase when damage acquired anisotropy appears as shown above. In the last case the elasticity matrix at the virgin state may have isotropic nature, whereas after some dissipative process initiates it changes to anisotropic form.
1.3 Common Invariants of the Second-Order and Fourth-Order Tensors
1.3.1 Common Invariants of Two Second-Order Tensors: The Stress/Strain and the Damage Tensors
A fundamental (irreducible) set of common invariants of two second-order tensors comprises 10 invariants. In a particular case when the common strain-damage space \(({\varvec{\varepsilon }},{\varvec{D}})\) is considered they are furnished as
When another stress-damage commonly used space \((\varvec{\sigma },{\varvec{D}})\) is considered the following holds:
1.3.2 Common Invariants of Two Different-Order Tensors: The Second Stress/Strain and the Fourth-Order Structural Tensors
The orthotropic material is characterized by three mutually perpendicular symmetry planes determined by three second-rank tensors called the structural tensors in terms of which the elastic strain energy \(\mathcal{W}\) can be represented as
When axes of material orthotropy coincide with axes of reference frame the structural tensors take the simplified forms
for which the following holds:
Condition (1.92) means that the structural tensors are mutually dependent. Hence, elastic strain energy (1.90) can be represented in terms of two structural tensors chosen as independent, e.g., \({\varvec{M}}^{(1)}\) and \({\varvec{M}}^{(2)}\)
Analogously, strain tensor can be written as \({\varvec{\varepsilon }}={\varvec{1}}\cdot {\varvec{\varepsilon }}= {\varvec{\varepsilon }}\cdot {\varvec{1}}\), which finally leads to
Summing up, the above equations assure symmetry of the strain tensor \({\varvec{\varepsilon }}\)
The following representation of elastic strain energy in terms of 7 invariants can be obtained:
comprising both 3 single strain invariants and 4 common strain and structural tensor invariants. However, based on (1.95) two first-strain invariants can be represented as
whereas the third strain invariant \(\frac{1}{3}\mathrm{tr}({\varvec{\varepsilon }}\cdot {\varvec{\varepsilon }}\cdot {\varvec{\varepsilon }})\) is ignored because strain energy must be a quadratic function of strain \({\varvec{\varepsilon }}\). For further details see Sect. 1.7.3.
1.4 Classification of Elastic Materials with Respect to Symmetry Groups and Classes
For further considerations, analogy between the
crystal lattice symmetry
groups and classes and corresponding symmetry of the
stiffness matrices
defined for crystalline materials might be useful (cf. e.g. Nye [42]). Unit cells of the eight conventional crystal lattices are demonstrated based on Love [34] and Jastrzebski [23], whereas corresponding
constitutive elasticity matrices
are schematically sketched applying Nye’s graphics (symbol 
refers to independent element, symbol 
refers to dependent element, symbols 
or 
represent pairs of identical matrix elements, symbols 
stand for pairs of elements in which one is doubled (effect of engineering notation applied to shear strain \(\gamma _{ij}=2\varepsilon _{ij}\)), whereas symbols 
denote pairs of elements of the same absolute value but opposite signs, respectively.
1.4.1 Triclinic Hooke’s Anisotropy (21 Constants)
Deformation of representative cube taken of the generally anisotropic material of triclinic symmetry subjected to exemplary axial tension along three axes is fully anisotropic. This means that it comprises both anisotropic axial strains (transformation of the cube to a rectangular prism) and anisotropic shear strains (transformation of the rectangular prism to a parallelepiped), as schematically sketched in Fig. 1.5. In such a case of general deformation the elastic compliance matrix is fully populated. In other words, all components of the columnar stress vector depend on all six components of the columnar strain vector (36 combinations). Final representation of compliance matrix for fully anisotropic (triclinic) material is as follows:
Schematic deformation of representative cube of anisotropic triclinic material under uniaxial tension along three axes
Symmetry of the elastic compliance matrix (1.98) results from symmetry of both stress and strain tensors, namely
In should be pointed out that the symmetry \({\mathbb E}_{ij}^{-1}={\mathbb E}_{ji}^{-1}\) holds for elements of compliance matrix but not for corresponding engineering material constants \(E_{ii},\nu _{ij},G_{ij},\eta _{(i)jk}\) , \(\mu _{ij(ki)}\) as shown in (1.100) versus (1.98) (Table 1.2)
Elastic engineering modules of five types can be sorted in the following way, after Lekhnitskii [33]:
-
\(E_{ii}\)—axial modules (3 generalized Young’s modules)
-
\(G_{ij}\)—shear modules for planes parallel to the coordinate planes (3 generalized Kirchhoff’s modules)
-
\(\nu _{ij}\)—Poisson’s ratios characterizing the contraction in the direction of one axis when tension is applied in the direction of another axis (3 generalized Poisson’s coefficients)
-
\(\mu _{ij(kl)}\)—coefficients characterizing shears in planes parallel to the coordinate planes resulting from shear stresses acting in other planes parallel to the coordinate planes (3 Chencov’s modules)
-
\(\eta _{i(jk)}\)—mutual influence coefficients characterizing extensions in the directions of the coordinate axes resulting from shear stresses acting in the coordinate planes (9 Rabinovich’s modules)
The aforementioned modules are listed in Table 1.3. In case of full anisotropy the shear stress acting in one plane results in a shear strain appearing in another plane. This effect is described by the three Chencov modules. Hence, the bottom right-hand side block of the compliance matrix (1.100) is fully populated, in contrast to the case of isotropy where shear stress acting in one plane results in shear strain in the same plane exclusively. This means that in case of isotropy the considered block of compliance matrix must have the diagonal form.
In order to describe effect of axial stresses on shear strains (upper right-hand side block), as well as effect of shear stresses on axial strains (lower left-hand side block), it is necessary to define 9 additional modules \(\eta _{(i)jk}\), called Rabinovich’s modules where the appropriate symmetry conditions hold (1.99). The total number of discussed modules is equal to 21. However, only 18 of them are truly independent because the compliance matrix \([{\mathbb E}^{-1}]\) has to obey transformation with respect to three Euler angles. It should be pointed out that in general case of anisotropy it is not possible to find any reference frame for which any element of the compliance matrix can be equal to zero. The general case of anisotropy corresponds to the triclinic symmetry lattice cell in which all three edges differ from each other and all three angles between them differ from each other and none of them is equal to \(90^{\circ }\), as shown in item 1 of Table 1.1.
1.4.2 Monoclinic Hooke’s Anisotropy (13 Constants)
Among anisotropic materials the narrower group called monoclinic symmetry can be distinguished. Monoclinic or oblique symmetry corresponds to monoclinic space lattice cell symmetry in which all three edges differ from each other, whereas two angles are equal to \(90^{\circ }\) and one is different, as shown in item 2 of Table 1.1. The corresponding stiffness matrix symmetry characterizes through incomplete population in which only 13 elements are not equal to zero, as shown below.
In other words, in case of monoclinic symmetry only three of the Rabinovich modules and only one of the Chencov modules are different from zero.
1.4.3 Trigonal/Rhombohedral Hooke’s Anisotropy (6 Constants)
Another important narrower case of material anisotropy called trigonal anisotropy can be distinguished. The trigonal anisotropy corresponds to the rhombohedral cell lattice in which all three edges are equal to each other and all three angles are equal but different from \(90^{\circ }\), as shown in item 3 of Table 1.1. The corresponding compliance matrix takes the following representation:
It is seen that in case of trigonal symmetry among Rabinovich’s modules only two are nonzeroth but in fact only one of them is independent because they only differ in sign. Additionally, only one Chencov’s modulus is different from zero but in fact it is the dependent modulus due to the specific coupling between components \(2E_{14}^{-1}=E_{56}^{-1}\) and \(E_{24}^{-1}=-E_{14}^{-1}\) as well as \(E_{11}^{-1}=E_{22}^{-1}\), \(E_{44}^{-1}=E_{55}^{-1}\), \(E_{13}^{-1}=E_{23}^{-1}\) whereas \(E_{66}^{-1}=(E_{11}^{-1}-E_{12}^{-1})/2\) must hold. Finally for trigonal symmetry only 6 elements of the compliance matrix are independent, see Berryman [2].
1.4.4 Orthorhombic Hooke’s Orthotropy (9 Constants)
The majority of engineering materials exhibit a specific symmetry property, which may result in reduction of the number of nonzeroth elastic modules. It can be done when, for chosen symmetry group or class, some particular material directions are defined in such a way that transformation of the compliance matrix from an arbitrary coordinate frame to the given structural symmetry frame leads to the zeroth population of the top right-hand side and the bottom left-hand side blocks of the compliance matrix (1.98), and additionally the bottom right-hand side block possesses a diagonal form. In such practically important cases both the nine Rabinovich \(\eta _{(i)jk}\) and the three Chencov \(\mu _{ij(kl)}\) modules are equal to zero, and consequently, coupling between the shear stresses and elongations does not exist such that shear strains are produced exclusively by the action of stresses at the same planes. In this particular symmetry, called orthotropy, there exist three mutually perpendicular axes (1, 2, 3) that determine the three material orthotropy planes. The orthotropy symmetry case corresponds to the orthorhombic lattice in which all three edges differ each from other but all angles are equal to \(90^{\circ }\), as presented in item 4 of Table 1.1.
The following conditions must hold to assure matrix symmetry:
Finally, in case of orthotropy the number of independent material constants is nine, that is, three generalized Hooke’s modules \(E_{11},E_{22},E_{33}\), 3 generalized Kirchhoff’s modules \(G_{12},G_{23},G_{31}\) and three generalized Poisson’s ratios \(\nu _{21},\nu _{23},\nu _{31}\).
1.4.5 Tetragonal Hooke’s Transverse Isotropy (6 Constants)
For several engineering applications the general orthotropic symmetry model seems too complicated, since additional symmetry conditions frequently appear. Particularly, when conditions of isotropy hold in selected orthotropy plane the so-called transverse isotropy obeys.
In case of so-called tetragonal symmetry material properties in the plane (1, 2) satisfy condition of cubic symmetry, see item 5 of Table 1.1
Hence, in case of transverse isotropy of tetragonal symmetry the number of independent material constants is equal to 6: \(E_{11}\), \(E_{33}\), \(G_{23}\), \(G_{12}\), \(\nu _{21}\), \(\nu _{31}\). Corresponding crystal lattice is sketched in item 5 of Table 1.1, where tetragonal lattice being special case of the orthorhombic lattice with \(a=b\ne c\) obeys.
When the constraints (1.105) are applied to compliance matrix (1.103) the transverse isotropy tetragonal symmetry case yields
It follows from the constraints (1.105) that six independent material constants define the tetragonal symmetry matrix:
-
\(E_{11},E_{33}\)—two Young’s modulus in the plane of isotropy and direction perpendicular to this plane,
-
\(\nu _{21},\nu _{31}\)—two Poisson’s ratios referring to transverse contraction or swelling caused by tension or compression in direction perpendicular to isotropy plane,
-
\(G_{12},G_{23}\)—two different Kirchhoff’s modules in the isotropy or orthotropy planes.
1.4.6 Hexagonal Hooke’s Transverse Isotropy (5 Constants)
In special case of the transverse isotropy called hexagonal symmetry the additional constraint must obey for the shear modulus in the isotropy plane
where modulus \(G_{12}\) is expressed in terms of the transverse Young modulus \(E_{11}\) and transverse Poisson’s ratio \(\nu _{21}\). Hence, in case of the transverse isotropy of hexagonal symmetry the number of independent constants is equal to 5: \(E_{11}\), \(E_{33}\), \(G_{23}\), \(\nu _{21}\), \(\nu _{31}\). A choice of the five independent material constants from among six can be performed in an optional way, for instance
Rolled metals, some multi-phase composite materials, basalt, or columnar ice are examples of transversely isotropic materials, however, precise distinction between the tetragonal or hexagonal symmetry classes is often difficult (see for example Gan et al. [16]).
1.4.7 Cubic Hooke’s Symmetry (3 Constants)
Further reduction in the number of independent constants leads to cubic symmetry for which the compliance matrix is characterized by three independent material constants \(E_{11}=E_{22}=E_{33}=E\), \(G_{23}=G_{31}=G_{12}=G\) and \(\nu _{21}=\nu _{31}=\nu _{32}=\nu \). Hence, the following form of the compliance matrix is furnished:
Note that in case of cubic symmetry the condition (1.107) does not hold. The corresponding cubic or regular lattice is shown in item 7 of Table 1.1. A particular example of the cubic symmetry material is nickel-based single crystal superalloy widely used in aircraft engines, especially for turbine blades as discussed by Desmorat and Marull [12]. The cubic symmetry is the narrower symmetry case known from crystallography, see Jastrzebski [23], since fully isotropic crystal lattices are unknown.
1.4.8 Isotropic Hooke’s Symmetry (2 Constants)
All the aforementioned symmetry groups have equivalences in existing crystal lattice systems. Nevertheless, even narrower than the cubic symmetry called isotropy is frequently used. The isotropy requires the infinite symmetry group which means that all material directions are equivalent in terms of mechanical, thermal, electric, optical, and magnetic properties. In other words it is not possible to distinguish any specific direction. The isotropy is helpful when describing the majority of polycrystalline materials either in a virgin state or artificially fabricated as particulate composites, nano-composites , etc., see item 8 of Table 1.1.
In an isotropic material physical properties are independent of the reference frame. Hence, any optional reference frame \(x,y,z\) is sufficient for unique definition of material properties. In order to derive mathematical form of the Hooke law of isotropic material it is most convenient to apply superposition of strain components \(\left\{ {\varvec{\varepsilon }}\right\} = \left\{ \varepsilon _x, \varepsilon _y, \varepsilon _z, \gamma _{yz}, \gamma _{zx}, \gamma _{xy}\right\} \) caused by subsequent stress components \(\left\{ {\varvec{\sigma }}\right\} = \left\{ \sigma _x, \sigma _y, \sigma _z, \tau _{yz},\right. \) \(\left. \tau _{zx}, \tau _{xy}\right\} \) (see Table 1.2). Applying vector-matrix notation the isotropic Hooke law takes the form
where the isotropic compliance matrix \([{\mathbb E}^{-1}]\) takes the following representation.
It is clear that the elastic isotropic material is uniquely defined by two independent material constants, the choice of which from among \(E,G,\nu \) is optional. In the above representation diagonal modules \(E\) and \(G\) are chosen as independent. Hooke’s law can also be transformed to the following inverse relation, (see Ottosen and Ristinmaa [44]):
where the isotropic stiffness matrix \([{\mathbb E}]\) is defined as
Format of elastic stiffness matrix (1.113) involves elements all dependent on both \(E\) and \(\nu \) such that the format equivalent to (1.111) cannot be achieved. Explicit separation of the diagonal matrix elements related to shear deformation and the off-diagonal matrix elements related to extension is possible by use of the format expressed in terms of Lamé’s constants \(\lambda \) and \(\mu \)
where the classical definitions of Lamé’s constants hold
It is worth to mention that the last format (1.114) can be interpreted by use of Nye graphics (
or 
) where three off-diagonal first quarter elements and three diagonal third quarter elements are considered as independent.
The considered case of elastic isotropy is the only symmetry case for which it is possible to separate effects of shape and volume changes when decomposition of strain and stress tensors into deviators and axiators (1.5), (1.25) is used as
Two modules in the above pair of relations called the bulk modulus \(K\) and the Kirchhoff modulus \(G\) can be expressed in terms of the Young modulus \(E\) and the Poisson ratio \(\nu \)
However, in all other cases of material anisotropy (items 1 to 7 in Table 1.1) aforementioned separation of volumetric from shear effects is impossible.
In the particular case of plane stress state in the \(x,y\) plane strain component \(\varepsilon _z\) can be expressed in terms of strain components in \(x,y\) plane as follows:
Finally, plane stress stiffness matrix \({\mathbb E}\) can be reduced to the \(3\times 3\) matrix
1.5 Analogy Between Constitutive Fourth-Order Tensors: The Elastic (Hooke’s) and the Yield/Failure (von Mises’) of the Same Symmetry
Identification of material symmetry in elastic range of deformation ( anisotropy, orthotropy, transverse isotropy, isotropy, etc.) is a starting point to appropriate description of both the limit criteria that control transition from the elastic range into the state connected with energy dissipation ( material damage, plastic yield, phase change, etc.) as well as correct constitutive description of deformation processes in nonelastic range. It can be expected that if material in the elastic range exhibits isotropic behavior, then at least in the initial phase of plastic yielding it will approximately save properties of isotropy. The nature of elastic deformation resulting from interatomic distances change in crystal lattice is qualitatively different from the nature of plastic deformation commonly interpreted as plastic microslips considered usually as slips and dislocations between atom layers inside lattice. However, it can be expected that during more advanced plastic deformation certain orientation of plastic slip systems in the particular grains leading to appearance of a material texture characterized by an acquired anisotropy is observed (metal forming processes like rolling, drawing and press forming, see Mróz and Maciejewski [37]).
On the other hand if material even in elastic range is characterized by an anisotropy (e.g., long fiber reinforced composites, wood, biological tissues) it can be expected that in nonelastic range it will also exhibit anisotropy. However, it will be possible decrease of a symmetry class toward more general plastic anisotropy, for instance due to gradual evolution of elastic orthotropy. It can be however noticed that in case of dissipative processes different from plasticity (e.g., material damage or failure) loss of isotropy may be expected just in the elastic range, as observed in elastic brittle materials e.g., ceramics, composites, concrete , etc. Additionally, initiation and growth of other dissipative processes connected with plastic yielding, phase change or other structural changes may result in change in the initial symmetry class. For example, in case of spheroidal graphite cast iron which generally exhibits brittle-ductile behavior a gradual transition from elastic anisotropy caused by directional damage to a state close to isotropy may be observed.
It can be assumed that features of anisotropy present in the elastic range are in general inherited in nonelastic range if some dissipative processes like plastic yield, damage, failure are present. Notice however that even, in the case when in inelastic range material behaves as isotropic, initiation of inelastic range (plasticity, damage, or failure) may provoke a material symmetry change. It was previously discussed that in case of damage evolution the fourth-rank damage effect tensor \([\mathbb {M}({\varvec{D}})]\) may be used to describe degeneration of the elasticity tensor \([\widetilde{\mathbb {E}}]=[\mathbb {M}({\varvec{D}})]^\mathrm{T}[\mathbb {E}][\mathbb {M}({\varvec{D}})]\), in a similar fashion effect of other dissipative phenomena such as plastic yield, structural change due to phase transformation may result in anisotropy nucleation and growth.
Analogy between
crystal unit cells
of
space lattices
and constitutive matrices of
elasticity
and
initiation of plasticity
is presented in Table 1.4. In the fundamental book by Love [34] the analogy between crystal symmetry classes and groups from one side and appropriate forms of elastic strain energy function \(\mathcal{W}=\frac{1}{2}\left\{ {\varvec{\varepsilon }}\right\} ^{\mathrm {T}} [{\mathbb {E}}]\left\{ {\varvec{\varepsilon }}\right\} \) from the other, is demonstrated. In this book an extension of the aforementioned analogy also for symmetry of constitutive matrix of plastic yield initiation 
appearing in the
von Mises criterion
is proposed. Unit cells of the four chosen space lattices have been presented following Jastrzebski [23], whereas corresponding constitutive elasticity matrices have schematically been presented applying Nye [42] graphics (symbol 
refers to independent element, symbol 
refers to dependent element, whereas symbols 
or 
represent pairs of identical matrix elements).
In case of full anisotropy the complete analogy between the Hooke matrix and the von Mises plasticity matrix holds (21 independent matrix elements in both classes). However, when narrower symmetry groups are considered: orthotropic, transversely isotropic of tetragonal or hexagonal classes, it is necessary to notice that elastic matrices are usually defined in stress tensor coordinates, whereas plastic constitutive matrices are often defined in the narrower stress deviator coordinates.
Reduction of the tensorial space to the deviatoric one is always equivalent to imposing additional constraints, hence the number of independent elements of plasticity matrix is always lower than the corresponding number of independent elements of elasticity matrix. Namely, it is clear that the 6-element orthotropic deviatoric Hill’s matrix corresponds to the 9-element orthotropic Hooke’s matrix. Similarly, the 4-element transversely isotropic tetragonal class Hill’s matrix corresponds to the 6-element Hooke’s matrix, when the independence of Hill’s matrix of hydrostatic stress is imposed. Finally, the 3-element transversely isotropic hexagonal class Hu–Marin matrix corresponds to the 5-element transversely isotropic hexagonal class Hooke matrix. Let us note that pairs of identical matrix elements are arranged in the same way in both matrices of elasticity and plasticity.
Nevertheless, some dependent elements in the plasticity matrix (as represented by symbol 
) correspond to independent elements of elasticity matrix (sketched by symbol 
), but general population of both matrices remains unchanged.
The commonly used term “transversely isotropic criterion” may be misleading as long as an additional distinction between the tetragonal and the hexagonal symmetry is not introduced. The aforementioned distinction is known from the literature dealing with prediction of composite behavior in elastic range and its validation by experiments. For example, Sun and Vaidya [53] examined two types of materials: Boron/Al composite and Graphite/Epoxy composite, and found that some of them exhibit tetragonal while others hexagonal symmetry classes. However, even this distinction between tetragonal and hexagonal symmetry classes may be insufficient to describe some composite materials, for example, SiC/Ti unidirectional lamina examined by Herakovich and Aboudi [19]. This is basically caused by residual stresses that appear after cooling-down during fabrication process .
The above considerations are limited to the description of initial yield surface only. Generally, it is assumed that during plastic hardening the initial yield surface possessing certain symmetry is rebuilt in an isotropic way, which is generally not true. This question was discussed, e.g., by Malinin and Rżysko [36], who invoked Mursa [41] results for OTCz Titanium Alloy that confirms assumption of isotropic nature of plastic hardening. However, Hu and Marin’s [22] findings for Aluminum Alloy showed anisotropic nature of plastic hardening rather than isotropic.
Nevertheless, the plastic hardening theory is usually taken in an isotropic fashion, e.g., Malinin and Rżysko [36], Ottosen and Ristinmaa [44], Hill [20, 21]. Such approach, although commonly used, may be questionable in light of the aforementioned experimental testing, some of which confirm such assumption, cf. Mursa [41] ( Titanium alloy) but others contradict it cf. Hu and Marin [22] (Aluminum alloy), Kowalewski and Śliwowski [26] (influence of first common invariant).
1.6 Strain Energy and Complementary Energy—The State Potentials for Isotropic or Anisotropic Materials
Material is called elastic if its response (deformation) is independent of loading history (Fig. 1.6), which means that stress is determined to be strain
or vice versa
After the fully closed loading–unloading cycle (A-B-A), the initial material state A is recovered, independent of the loading–unloading path, Fig. 1.6b.
Schematic illustration of elastic material response: a strain energy and complementary energy, b independence of final state of loading history
When the concept of strain energy per unit volume \(\mathcal{W}\) [Nm/m\(^3\)] is introduced, the following definitions hold:
in case of the uniaxial or the multiaxial loadings, respectively. In the following fashion the complementary energy per unit volume \(\mathcal{C}\) [Nm/m\(^3\)] is defined as
It is seen from Fig. 1.6 that the following is true:
It should be emphasized that in the considered case of pure elastic material both the strain energy \(\mathcal{W}\) and complementary energy \(\mathcal{C}\) are independent of loading path but depend on the current state exclusively.
In a more general case, when the deformation process is accompanied by permanent (irreversible) changes in material microstructure, for instance, resulting from plastic yielding, damage growth , or phase transformation during martensitic change or other irreversible phenomena, the strain energy and the complementary energy depend on loading history.
In the elastic material for which strain energy depends on the current state only \(\mathcal{W}(\varepsilon _{ij})\) but does not depend on strain path
the strain energy can be used as an invariant potential function for the stresses
In a similar fashion the complementary energy that depends on the current state only \(\mathcal{C}(\sigma _{ij})\) but does not depend on strain path
can be used as an invariant potential function for the strain as follows:
In a general case of nonlinear elastic material the strain energy and the complementary energy are not equal to each other, \(\mathcal{W}\ne \mathcal{C}\), whereas only in the case of linear elastic material the equality \(\mathcal{W}=\mathcal{C}\) holds.
In the above considerations the initial state was treated as stress and strain free, point A (\(\sigma =0,\varepsilon =0\)) in Fig. 1.6. In the more general case a residual stress and/or strain are built-in A\(^\mathrm{res}\) (\(\sigma =\sigma ^\mathrm{res}, \varepsilon =\varepsilon ^\mathrm{res}\)). This residual state may result from fabrication process or prior loading history in which some irreversible changes of material structure have occurred (e.g., cyclic plasticity) or certain residual stresses or strains have been built-in (e.g., after cooling-down of long fiber reinforced composite characterized by different thermal properties of fiber and matrix). Note that in general this residual state is unknown since the whole history of material, which contains complete information about fabrication, its initial machining, as well as concerning unloading process prior to the appearance of this self-balanced residual stress, is unknown.
Consider the process of elastic deformation of material starting from the residual state A\(^\mathrm{res}(\sigma ^\mathrm{res},\varepsilon ^\mathrm{res})\) toward the final state B\((\varDelta \sigma ,\varDelta \varepsilon )\), assuming at the beginning uniaxial tension (see Fig. 1.7).
Process of elastic deformation of material with prior residual state included
The increment of elastic strain energy of material corresponding to applied strain \(\varDelta \mathcal{W}(\varDelta \varepsilon )\) in the presence of residual stress \(\varepsilon ^\mathrm{res}\) is equal to
In the particular case of linear Hooke’s law for isotropic material it yields
where
obey. In the more general case of multiaxial deformation state the strain energy per unit volume of elastic material in the presence of residual stress may be written as
whereas in case, if linear elastic material is assumed, the linear relation combining stress and strain increments is furnished as
Equation (1.132) represents the increment of elastic energy \(\varDelta \mathcal{W}\) in the presence of the residual state \(\varepsilon _{ij}=\varepsilon _{ij}^\mathrm{res}+\varDelta \varepsilon _{ij}\), hence
where the fourth-rank stiffness tensor \(E_{ijkl}\) is used. Note that Eq. (1.134) is true both for isotropic and anisotropic materials of optional class of symmetry. The stiffness tensor \(E_{ijkl}\) comprises complete information defining the elastic material response.
In a similar way, the complementary energy increment \(\varDelta \mathcal{C}\) of elastic material in the presence of residual stress can be written as
If the linear elastic material is assumed we arrive at
where \(E_{klmn}^{-1}\) stands for the compliance tensor of elastic material
The above Eq. (1.137) is an extension of the law of linear elastic material to the case of existence of a nonzeroth residual stress and strain \(\varDelta \sigma _{ij} = \sigma _{ij} - \sigma _{ij}^\mathrm{res}\), \(\varDelta \varepsilon _{kl} = \varepsilon _{kl} - \varepsilon _{kl}^\mathrm{res}\)
or
When the vector-matrix notation is used the fourth-rank elastic tensors \(E_{ijkl}\) or \(E_{klmn}^{-1}\) can be represented by the symmetric \(6\times 6\) matrices: \([{\mathbb E}]\) or \([{\mathbb E}^{-1}]\) called the stiffness or the compliance matrices, respectively, whereas the tensors \(\sigma _{ij} - \sigma _{ij}^\mathrm{res}\) or \(\varepsilon _{kl} - \varepsilon _{kl}^\mathrm{res}\) take the format of columnar vectors of overstress or overstrain, respectively,
Hence, when the Voigt notation is used Eqs. (1.138) and (1.139) can be written in equivalent fashion
or
1.7 Elastic Strain Energy as Function of Invariants
The stress and the strain invariants are presented in Sect. 1.1. In the present section the elastic strain energy per unit volume \(\mathcal{W}\) expressed as the scalar product of both these tensors
will also be presented in terms of invariants. In the case of isotropic material three basic invariants of the strain tensor are sufficient for unique representation of the strain energy, whereas in case of elastic material comprising damage the use of common invariants defining internal material microstructure is necessary (see Sect. 1.3).
1.7.1 Elastic Strain Energy of Isotropic Materials
The simplest example of the scalar function of tensorial argument is the elastic strain energy \(\mathcal{W}({\varvec{\varepsilon }})\). In the case of isotropic material the strain tensor is uniquely determined in terms of three basic or generic strain invariants (1.26) as follows:
Constitutive law of elastic material (1.126) can be written as follows:
where
hence,
Introducing the Lamé elastic constants \(\lambda =\frac{\nu E}{(1+\nu )(1-2\nu )}\) and \(\mu =\frac{E}{2(1+\nu )}\) with
we arrive at the classical Hooke law of the isotropic material
Summing up, the isotropic elastic Hooke material is uniquely defined by the strain energy which depends on the first and the second basic invariants of the strain tensor
but does not depend on the third invariant \(J_{3\varepsilon }\).
1.7.2 Strain or Complementary Energy of Elastic-Damage Material—Common Strain-Damage and Stress-Damage Invariants; the Helmholtz or the Gibbs State Potentials
Theory of invariants allows to determine minimal number the basic invariants from which all other tensorial invariants necessary to obtain a sufficiently general representation of the state equations can be built (cf. e.g., Spencer [52], Rymarz [47]). Usually the strain energy per unit volume \(\mathcal{W}(\varepsilon _{ij})\) or the complementary energy per unit volume \(\mathcal{C}(\sigma _{ij})\) is taken as the state potential of elasticity (see Sect. 1.7.1). As shown in Sect. 1.7.1, in case of elastic isotropy three invariants sufficiently determine both types of energy \(\mathcal{W}(J_{i\varepsilon })\) or \(\mathcal{C}(J_{i\sigma })\), \(i=1,2,3\).
A scalar function dependent on a pair of tensorial arguments, each of them being the symmetric second-rank tensor, is a more complex case. The representative example of such a case is the strain energy of damaged material \(\mathcal{W}({\varvec{\varepsilon }},{\varvec{D}})\) . Analogous to the isotropic material (1.144), both tensors \({\varvec{\varepsilon }}\) or \({\varvec{D}}\) are determined by their single basic invariants \(J_{i\varepsilon }\) or \(J_{iD}, \ i=1,2,3\). However, the scalar function dependent on both arguments \(\mathcal{W}({\varvec{\varepsilon }},{\varvec{D}})\) has to be uniquely defined not only by single invariants \(J_{i\varepsilon }\) and \(J_{iD}\) but also by the common invariants \(J_{j\varepsilon D}, \ j=1,2,3,4\). This leads to the format dependent on six single and four common invariants (total 10)
In addition, the strain energy \(\mathcal{W}\) has to be a decreasing function with damage growth since energy is released during the damage nucleation and growth, so it has to be linear with respect to \({\varvec{D}}\). Hence, the strain energy cannot depend either on the third strain invariant \(J_{3\varepsilon }\) and on the two single damage invariants \(J_{2D}\), \(J_{3D}\) and also on the two common invariants \(J_{3\varepsilon D}\), \(J_{4\varepsilon D}\) (underlined arguments in Eq. (1.151)). Based on the above physical reasons the strain energy of elastic damaged material can completely be represented in terms of a combination of five invariants (three single and two common)
In this way an invariant representation of the Helmholtz free energy per unit mass is furnished and finally applied as the state potential that determines the stress state in a unique fashion
Note also that when the representation (1.152) is specified, some combinations of invariants are allowed for which the scalar function \(\psi ({\varvec{\varepsilon }},{\varvec{D}})\) remains quadratic with respect to \({\varvec{\varepsilon }}\). Hence, following Murakami and Kamiya [38] the free energy function \(\rho \psi ({\varvec{\varepsilon }}, {\varvec{D}})\) per unit mass is furnished as
or
when the equivalent representation is used, e.g., Skrzypek et al. [50].
Remember that the above formulas (1.154) and (1.155) for the Holmholtz free energy refer to the specific case of elastic anisotropy which is acquired as the result of damage nucleation and growth. Hence, in a virgin state where damage does not exist the energy representation of the isotropic elastic material has to be recovered, such that symbol \({\varvec{\varepsilon }}\) has to be referred to the elastic strain \({\varvec{\varepsilon }}^\mathrm{e}\).
In a general 3D case the following matrix representation of the constitutive equation with total formulation holds:
where \(^{\mathrm {s}}\widetilde{E}_{ij}\) represents effective elastic-damage secant stiffness matrix. The damage acquired anisotropy is described by the 6\(\times \)6 symmetric secant stiffness matrix as follows (cf. Skrzypek et al. [50]):
The alternative formulation based on a concept of the complementary energy \(\mathcal{C}\) represented by a scalar function of the two tensorial arguments \(\varvec{\sigma }\) and \({\varvec{D}}\), namely \(\mathcal{C}(\varvec{\sigma },{\varvec{D}})\), leads to the Gibbs potential function per unit mass \(\mathcal{G}\) as follows (cf. Hayakawa–Murakami [18], Murakami [40]):
where the crack closure effect due to compressive stress, originally introduced in Hayakawa–Murakami [18], is omitted.
Repeating the above reasoning for physical nature of the Gibbs complementary energy \(\mathcal{C}(\varvec{\sigma },{\varvec{D}})\), only five of the above aforementioned ten (1.158) common stress and damage invariants can be admitted, namely
Hence, in case of the elastic isotropic material in a virgin state which changes to anisotropic material due to damage evolution, the Gibbs state potential takes the following format (cf. Hayakawa and Murakami [18]):
which is complementary to (1.155). The matrix representation of secant compliance matrix referring to Hayakawa–Murakami type elastic-plastic-damage material is as follows:
where
Note that the Gibbs complementary energy per unit mass refers to elastic strains \({\varvec{\varepsilon }}^\mathrm{e}\) and is represented in the stress space by the quadratic function of \(\varvec{\sigma }\) linear with respect to \({\varvec{D}}\), in a similar way as the Helmholtz free energy \(\rho \psi ({\varvec{\varepsilon }},{\varvec{D}})\) but defined in the strain space .
Four material constants \(\eta _i\) appearing in the Helmholtz state potential (1.155) as well as four constants \(\vartheta _i\) appearing in the Gibbs state potential (1.160) (\(i=1,2,3,4\)) act as additional material constants to the elastic constants of the virgin elastic isotropic material: \(\lambda , \mu \) or \(E,\nu \), defining effect of damage on the state equation. Namely, when the Helmholtz potential function \(\mathcal{W}=\rho \psi ({\varvec{\varepsilon }},{\varvec{D}})\) is used as the stress potential we arrive at the state equation \(\varvec{\sigma }={\mathbb E}({\varvec{D}}):{\varvec{\varepsilon }}\)
On the other hand, when the formulation based on the Gibbs potential function is used as the strain potential \(\mathcal{C}=\rho \mathcal{G}(\varvec{\sigma },{\varvec{D}})\) we obtain the state equation in equivalent form \({\varvec{\varepsilon }}={\mathbb E}^{-1}({\varvec{D}}):\varvec{\sigma }\)
Note however that in the case of elastic damaged material constitutive matrices stiffness \([{\mathbb E}({\varvec{D}})]\) and compliance \([{\mathbb E}^{-1}({\varvec{D}})]\) are rebuilt following damage evolution such that originally isotropic elastic material acquires an anisotropy.
The state equation of elastic damaged material (1.155) was calibrated for the high strength concrete by Murakami and Kamiya [38], see also Skrzypek [49] as shown in Table 1.5.
Apart from the constants of isotropic elasticity \(E,\nu \) (\(\lambda ,\mu \)) additional four constants \(\eta _i\) (\(i=1,2,3,4\)) are shown in Table 1.5.
The state equation of elastic moderate ductility with damage (1.164) was calibrated for spheroidal graphite cast iron FCD400 by Hayakawa and Murakami [18], see also Skrzypek [49] as shown in Table 1.6.
Apart from the constants of isotropic elasticity \(E,\nu \) (\(\lambda ,\mu \)) additional four constants \(\vartheta _i\) (\(i=1,2,3,4\)) are shown Table 1.6.
The more extended analysis including: crack closure effect under compressive stress, the initial damage threshold, and the subsequent damage growth during the hardening phases can be found in Murakami and Kamiya [38], Hayakawa and Murakami [18], Skrzypek et al. [50], Bielski et al. [4], Kuna-Ciskał and Skrzypek [31].
1.7.3 Strain Energy of the Elastic Orthotropic Materials—The Structural Tensors
So far the case of scalar function of second-order tensors expressed in terms of invariants has been discussed. The more general case of a scalar function of a pair of tensorial arguments being the second-order and the structural tensors is considered in this section. The strain energy of orthotropic material \(\mathcal{W}=\mathcal{W}({\varvec{\varepsilon }},{\varvec{M}}^{(i)})\) is the representative example of such a case.
The constitutive equation of orthotropic hyperelastic material is obtained by differentiation of the strain energy function, cf. Boehler [5]
where the following definitions of common invariants are used:
and definitions (1.91) hold. Following Boehler [5], in order to determine the constitutive equation of linear orthotropic material we choose, (see also Ottosen and Ristinmaa [44])
The coefficients \(\beta _1,\beta _2,\beta _3\) can be substituted by corresponding coefficients \(\alpha _2,\alpha _4, \alpha _5\) in order to satisfy symmetry of orthotropic stiffness matrix
The above yields the constitutive equation of linear orthotropic material by use of common invariants of strain and structural tensors
Equation (1.169) can be rewritten in the classical form at \(\varvec{\sigma }={\mathbb E}:{\varvec{\varepsilon }}\) when the consecutive tensor products \({\varvec{\varepsilon }}\cdot {\varvec{M}}^{(i)}\) and their traces are defined. For instance,
from where one finds
and
When the remaining products \({\varvec{\varepsilon }}\cdot {\varvec{M}}^{(2)}\) and \({\varvec{\varepsilon }}\cdot {\varvec{M}}^{(3)}\) are calculated analogously, the coefficients preceding the components of the strain tensor are grouped, and when the engineering notation is consequently used the state equation (1.169) can finally be furnished in the following form:
Subsequent elements of stiffness matrix of the orthotropic elastic material \([{\mathbb E}]\) are expressed in terms of coefficients \(\alpha _i\) as follows:
Note that the above described procedure of linear orthotropic elasticity derivation is based on the theory of invariant representation which differs from the conventional approach (1.103). More detailed distinction between different ways of formulating the linear elasticity constitutive laws will be presented in Sect. 1.9.
1.8 Remarks on Irreducible Coupling of Volumetric and Shear Response in Anisotropic Materials
In the general case of full material anisotropy complete mutual coupling between all stress and strain components holds. In fact, the generalized Hooke law (1.39) with the compliance matrix for general anisotropy taken in the form (1.98) leads to (after Rabinovich [45])
Note that in the above equations elastic extensions \(\varepsilon _{11},\) \(\varepsilon _{22},\varepsilon _{33}\) depend not only on all normal stresses \(\sigma _{11},\sigma _{22},\sigma _{33}\) but also on all shear stresses \(\tau _{23},\tau _{31},\tau _{12}\) (through the generalized Young modules \(E_{ii}\) and the Rabinovich modules \(\eta _{ij(k)}\)), resulting in nonzeroth elements of symmetric constitutive matrix of elasticity in its right top block. Moreover, the shear strains \(\gamma _{23}\), \(\gamma _{31},\gamma _{12}\) depend on all shear stresses \(\tau _{23},\tau _{31},\tau _{12}\) (through the generalized Kirchhoff modules \(G_{ij}\) and the Chencov coefficients \(\mu _{ij(kl)}\)) as well as on all normal stresses \(\sigma _{11},\sigma _{22},\sigma _{33}\) such that the left bottom and the right bottom blocks of the elasticity matrix are fully populated. The above remarks lead in consequence to the conclusion that, in all cases different from isotropy, pure volumetric deformation is inseparable from pure shear deformation. In other words, irreducibility of elasticity equations (1.175) into uncoupled law of volume change and law of shape change holds when the decomposition of strain and stress tensors into axiator and deviator \({\varvec{\varepsilon }}=\varepsilon _\mathrm{m}\mathbf{1}+{\varvec{e}}\) and \({\varvec{\sigma }}=\sigma _\mathrm{m}\mathbf{1}+{\varvec{s}}\) is used.
This impossibility is inevitable even in a narrower case of orthotropy (1.103) in spite of the fact that shear stresses are uncoupled to the extensions and, vice versa, normal stresses do not result in shear strains. In order to trace this let us rewrite (1.103) as
Calculating the unit volume change called dilatation \(\varTheta =\varepsilon _{11}+\varepsilon _{22}+\varepsilon _{33}\) we obtain
or recalling the symmetry of elasticity matrix (1.104) the equivalent form is furnished
Note that in case of orthotropy dilatation is expressed not only in terms of the hydrostatic stress \(\varTheta =\varTheta ^\mathrm{izo}(\sigma _\mathrm{h})\) but by the more general function \(\varTheta =\varTheta ^\mathrm{orto}(\sigma _{11},\sigma _{22},\sigma _{33}; E_{ij},\nu _{ij})\) or \(\varTheta =\varTheta ^\mathrm{orto}(\sigma _{kk};{\mathbb E}_{ijkl}^{-1})\).
In the particular case of isotropy when \(E_{ij}=E\), \(\nu _{ij}=\nu \) the above equations reduce to the classical form
or
in which dilatation or mean strain \(\varepsilon _\mathrm{m}\) depends on hydrostatic stress \(\sigma _\mathrm{h}\) exclusively.
Contrary to the previous case for material orthotropy by use of the following definition of deviatoric strain:
we obtain
In other words, the pure shear deformation obtained by subtracting of the dilatation from the full deformation depends also on \(\varTheta ^\mathrm{orto}\), so separation of these two effects is impossible.
1.9 Cauchy’s Elasticity, Hyperelasticity, or Hypoelasticity
In the theory of linear elasticity in case of infinitesimal deformations occurring in isothermal or adiabatic conditions the constitutive relations linking tensors of stress and strain can be defined in three equivalent ways:
-
According to the Cauchy formulation it is assumed that there exists an equilibrium state, called natural state, for which all components of the stress and strain tensors are equal to zero and to which material returns after removing loadings. An environment of natural state obeys unique value relation between stress and strain as
$$\begin{aligned} \sigma _{ij} = E_{ijkl} \varepsilon _{kl} \end{aligned}$$(1.183) -
According to the Green formulation, also called hyperelasticity, it is postulated an existence of function of elastic strain energy per unit volume \(\mathcal{W}\) which is equal to zero in an environment of natural state and such that an increment of work done by stress is equal to an increment of strain energy
$$\begin{aligned} \sigma _{ij} = \frac{\partial \mathcal{W}}{\partial \varepsilon _{ij}} \qquad \mathcal{W} = \frac{1}{2}\sigma _{ij}\varepsilon _{kl} = \frac{1}{2} E_{ijkl}\varepsilon _{ij}\varepsilon _{kl} \end{aligned}$$(1.184) -
According to the third formulation, called hypoelasticity, it is postulated an incremental relation of the following form:
$$\begin{aligned} \mathrm{d}\sigma _{ij} = E_{ijkl}\mathrm{d}\varepsilon _{kl} \qquad \text{ or } \qquad \frac{\partial \sigma _{ij}}{\partial t} = E_{ijkl} \frac{\partial \varepsilon _{kl}}{\partial t} \end{aligned}$$(1.185)
For all three cases: Cauchy’s, hyper- and hypoelasticity tensor \(E_{ijkl}\) may depend on temperature but is independent of stress and strain tensors.
Note however that in the general case of nonlinearity constitutive tensors of elasticity or hyperelasticity (1.183) and (1.184) may differ from constitutive tensor of hypoelasticity (1.185). In the first case tensor representative matrix \({\mathbb E}\) is the secant matrix \([{\mathbb E}]=[_\mathrm{sec}{\mathbb E}]\), whereas in the other case it is the tangent matrix \([{\mathbb E}]=[_\mathrm{tan}{\mathbb E}]\).
It is worth to mention that although Cauchy, hyper- and hypoformulations of elasticity are alternative in case of theory of infinitesimal deformations, they may lead to essentially different results after entering the finite deformation range. Namely, introducing definitions of finite strains
where \(\epsilon _{ij}\) and \(\in _{ij}\) stand for Green’s and Almansi’s strain tensors, respectively, and corresponding stress tensors
where \(\sigma _{ij}\) and \({\varSigma }_{ij}\) denote the Lagrange and the second Piola–Kirchhoff stress tensors instead of formulations (1.183–1.185) we arrive at mutually different formulations
In case of hypoelastic material subjected to finite deformation appropriate constitutive equation (1.188)\(_3\) comprises both the symbol of objective derivative of the stress tensor \(\mathrm{D}{\sigma _{ij}}/\mathrm{D}t\) and an effect of change of stress tensor resulting from rigid rotation which is described by skew-symmetric spin tensor
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Ganczarski, A.W., Egner, H., Skrzypek, J.J. (2015). Introduction to Mechanics of Anisotropic Materials. In: Skrzypek, J., Ganczarski, A. (eds) Mechanics of Anisotropic Materials. Engineering Materials. Springer, Cham. https://doi.org/10.1007/978-3-319-17160-9_1
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