A Semi-analytical Model of Maxienmal First Principal Stress at Mode I Crack Tip

Abstract

The first principal stress plays a key role in ductile fracture processes. Investigation of the distribution and evolution of the first principal stress at the crack tip is essential for exploring elastoplastic fracture behaviors. A semi-analytical model was developed in this study to determine the maximal first principal stress at the mode I crack tip with 3D constraints for materials following the Ramberg–Osgood law. The model, based on energy density equivalence and dimensional analysis, was validated through finite element analysis (FEA) of various materials and geometric dimensions of specimens with mode I cracks, under over 100 different types of working conditions. The dimensionless curves of maximal first principal stress versus load, as predicted by the model, agreed well with the FEA results, demonstrating the accuracy and applicability of the model. This research can provide a basis for future theoretical predictions of crack initiation and propagation.

1 Introduction

Under the effects of manufacturing methods, damage, and other factors, structures in aerospace, nuclear power, chemical, and other engineering sectors frequently exhibit defects and cracks that can significantly impact the safety of systems. Investigation of fracture mechanical behaviors of cracked structures plays a key role in structural integrity analysis. To comprehend structural fractures, first crack tip stress field and deformation distribution should be understood.

Characterization of crack tip stress distribution has been an active research topic. In 1939, Westergaard [1] first described biaxial stress fields in internal cracks of a pressurized cylinder, which is considered as one of the earliest and most well-known methods to comprehend stress fields in cracked structures. In 1957, Williams [2] demonstrated that stress fields around crack tips in isotropic elastic materials could be represented as an infinite series. Since then, the Williams series expansion has been the most popular analytical technique for modeling the mechanical field around crack tip in a planar domain. The first two components of the Williams series expansion denote stress intensity [3, 4] and T-stress [5] factors, respectively. For each cracked structure, the coefficient sequence in the Williams series expansion depends on the existing loading system and geometry. In addition, higher-order stress terms significantly affect the engineering description of stress fields for different fractured specimens [6]. In 2013, Berto and Lazzarin [7] derived an equation series to precisely define crack tip stress components, while the accuracy of complete stress fields in front of crack tips was unsatisfactory. In 2016, Larisa and Pavel [8] analytically determined stress expansion coefficients at crack tips for two collinear finite cracks with similar lengths in an infinite plane medium. Their study particularly focused on the effect of varying numbers of series expansion terms on stress distribution. The primary application of the Williams series expansion is in linear elastic fracture mechanics, and the analytical calculation of crack tip stress expansion coefficients remains an area deserving of in-depth studies.

Cherepanov [9] and Rice [10] independently proposed the J-integral for elastic–plastic fracture mechanics, an energy line integral demonstrating path independence for every contour surrounding the crack tip to characterize the crack-tip stress field intensity. In order to tackle the power-law plasticity problem of mode I cracks under plane conditions, Hutchinson [11] and Rice and Rosengren [12] presented the crack tip stress and strain field (HRR field) characterized by the J-integral in 1968. It has been proved that the J-integral uniquely determines crack tip stress and strain field strength. The proposed HRR field is a milestone in investigating crack tip stress and strain fields. Recent theoretical, experimental, and numerical analyses have revealed that the HRR field could accurately capture the inherent stress and strain properties in an annular region at the crack tip. The control region of the HRR field varies with the geometric configuration and size of the specimen. In general, the stress and strain field with high constraints under plane strain conditions can be effectively described by the HRR field.

For low-constrained cracked specimens under plane strain conditions, J–Q [13,14,15], J–A₂ [16, 17], and J–A [18,19,20] theories have been developed to describe the crack tip field. According to O’Dowd and Shih et al. [13,14,15], the J–Q theory offers a two-term crack tip field solution capable of accurately predicting stress fields under a wide spectrum of conditions ranging from small yield to large yield or pure plasticity. However, as an empirical parameter, Q depends on finite element calculations for specific materials or configurations, as well as the distance and angle between a chosen reference point and the crack tip. Yang and Chao et al. [16, 17] developed the J–A₂ theory on the basis of three asymptotic expansions of the crack tip stress field. Later, using a similar method, Nikishkov and Matvienko [18,19,20] developed a new version of the J–A theory, which was almost identical to the J–A₂ theory. Under some circumstances, constraint parameters such as Q, A₂, A, or others could be transformed into one another, and their descriptive capabilities are comparable. For cracked specimens with different constraints and materials, the constraint parameters had to be separately determined by finite element calculations and could not be expressed analytically.

In recent years, many researchers have explored new methods to characterize crack tip stress fields. Withers [21] found that tomographic imaging and synchrotron X-ray diffraction could capture crack initiation and growth at 3D crack tips. The quantitative information of fracture mechanics regarding the area around the crack tip, such as fracture strength factor and stress field, could be retrieved to monitor crack tip fracture behavior. Sutton et al. [22] applied digital image correlation (DIC) to examine 3D effects near crack tips. Srilakshmi et al. [23] investigated crack growth in composite patch-repaired aluminum panels under fatigue loading using finite element analysis (FEA) and DIC. However, further development is needed to fully leverage these technologies for studying stress fields at crack tips.

Existing equations for stress and strain fields mainly focus on plane conditions, while cracked problems under 3D constraints have rarely been studied. The paucity of comprehensive research on complex constraint conditions at crack tips is also a significant factor limiting our ability to predict crack initiation. It is essential to develop methods for analytically characterizing crack tip stress under 3D constraint conditions, which can not only depict stress characteristics but also effectively combine ductile fracture criteria with crack initiation studies, to lay a theoretical foundation for predicting crack structure and fracture behavior.

Previous research on critical ductile fracture criteria [24, 25] considered the critical first principal stress as the primary control parameter for ductile fractures. In this research, a semi-analytical model of critical first principal stress at the crack tip was developed using energy density equivalence and dimensional analysis, and the model’s accuracy was validated through FEA of different materials and geometric dimensions of specimens with mode I cracks.

2 Load–Displacement Model for Specimens with Mode I Cracks

2.1 Energy Density Equivalence

Uniaxial equivalent stress–strain relationships of representative volume elements (RVEs) are fundamental in continuum mechanics. In elastic–plastic fracture mechanics, the Ramberg–Osgood (R–O) law is commonly used to characterize constitutive relationships of materials under quasi-static conditions, which is stated as

$$ \left\{ {\begin{array}{*{20}l} {\varepsilon _{{{\text{eq}}}} = \varepsilon _{{{\text{e-eq}}}} + \varepsilon _{{{\text{p-eq}}}} } \hfill \\ {\varepsilon _{{{\text{e-eq}}}} = \frac{{\sigma _{{{\text{eq}}}} }}{E}} \hfill \\ {\varepsilon _{{{\text{p-eq}}}} = (\frac{{\sigma _{{{\text{eq}}}} }}{K})^{n} } \hfill \\ \end{array} } \right. $$

(1)

where ε_p-eq and ε_e-eq are the equivalent plastic and elastic strains, respectively; σ_eq and ε_eq are total equivalent stress and strain, respectively; K is stress strength coefficient in terms of MPa; E is elastic modulus in terms of GPa; and n is strain hardening exponent, which is a dimensionless parameter.

According to Chen and Cai [26,27,28], for any loaded specimens, the effective deformation region has a median energy density point, and the RVE strain energy density at this point is equal to the ratio of the specimen’s total strain energy to the effective deformation volume, i.e., the average strain energy. Based on the von Mises equivalence principle, the RVE strain energy density at a medium point subjected to complex stress states is also equal to that of the RVE under uniaxial equivalent stress states. This allows for a semi-analytical link between strain energy and material constitutive relation parameters, geometric sizes, and the load or displacement of loaded specimens.

In the case of homogeneous and isotropic materials, the strain energy U of a loaded specimen can be stated as

$$ U = \iiint\limits_{\Omega } {\mu (x,{\kern 1pt} {\kern 1pt} {\kern 1pt} y,{\kern 1pt} {\kern 1pt} {\kern 1pt} z{\text{)d}}V} $$

(2)

where Ω is the effective deformation region of the loaded specimen, and μ(x, y, z) is the RVE strain energy density at any point in Ω.

On the basis of integral mean value theorem, there must be a point M(x_M, y_M, z_M) in Ω at which the relationship between strain energy density μ_M and total strain energy U can be expressed as

$$ \mu_{M} (x_{M} ,y_{M} ,z_{M} ){ = }\frac{U}{{V_{{{\text{eff}}}} }} $$

(3)

where V_eff is the volume of effective deformation region of the specimen.

For the RVE at the medium point, there is

$$ \mu_{M} = \int\limits_{0}^{{\varepsilon_{ij - M} }} {\sigma_{ij} {\text{d}}\varepsilon_{ij} } = \int\limits_{0}^{{\varepsilon_{{{\text{eq-}}M}} }} {\sigma_{{{\text{eq}}}} {\text{d}}\varepsilon_{{{\text{eq}}}} } $$

(4)

where σ_ij and ε_ij are stress and strain tensors under complex stress states, and ε_eq-M and ε_ij-M are equivalent strain and strain tensor under certain deformation conditions, respectively.

According to Eqs. (2) and (4), the effective deformation region strain energy for a loaded specimen is stated as

$$ U = \mu_{M} V_{{{\text{eff}}}} { = }V_{{{\text{eff}}}} \int\limits_{0}^{{\varepsilon_{{{\text{eq-}}M}} }} {\sigma_{{{\text{eq}}}} {\text{d}}\varepsilon_{{{\text{eq}}}} } $$

(5)

If the uniaxial equivalent constitutive relation of the material can be described as σ_eq = f (ε_eq, E, K, n), the relationship between ε_eq-M and dimensionless displacement ${\frac{v}{{v^{*} }}}$ on the loaded line (with ${v^{*}}$ being the characteristic displacement of the specimen) can be stated as ε_eq-M = ϕ $\frac{v}{v^{*}}$ , and that between $\frac{v}{v^{*}}$ and the volume of dimensionless effective deformation region V_eff/V^* (with V^* being the characteristic volume of the specimen) can be expressed as V_eff/V^* = φ $\frac{v}{v^{*}}$ , the analytical description of the strain energy U of the loaded specimen can be obtained based on Eq. (5) as

$$ U{ = }V^{*} \varphi \left( {\frac{v}{{v^{*} }}} \right)\int\limits_{0}^{{\phi \left( \frac{v}{v^{*}} \right)}} {f(\varepsilon_{{{\text{eq}}}} ,E,K,n){\text{d}}\varepsilon_{{{\text{eq}}}} } $$

(6)

The link between strain energy, material constitutive parameters, geometric sizes, and load or displacement on the load line can be precisely calculated after identifying the specific forms of functions f, ϕ, and φ.

2.2 3D Constraint Function of Specimens with Mode I Cracks

For specimens with mode I cracks under 3D constraints, the geometric dimension {L} consists of the width W, crack length a, and thickness B; denoted as {L} = {W, a, B}. Chen and Cai [26,27,28] developed an independent power law function relationship for dimensionless displacement v/v^* on the loading line and dimensionless effective deformation volume $\overline{V}_{{{\text{eff}}}}$ for specimens with different geometric configurations. Therefore, the dimensionless effective deformation volume $\overline{V}_{{{\text{eff}}}}$ for specimens with mode I cracks can be described as a separation relationship between dimensionless displacement function and dimensionless geometric function. This separation relationship is expressed as

$$ \overline{V}_{{{\text{eff}}}} { = }\varphi \left( {\left\{ {\overline{L}} \right\},\overline{v}} \right){ = }G\left( {\frac{a}{W},\;\frac{B}{W}} \right)\left[ {k_{01} \left( {\frac{v}{{v^{*} }}} \right)^{{k_{2} }} } \right] $$

(7)

where v^* is the characteristic displacement of the specimen (equal to W for specimens with mode I cracks), k₀₁ is the effective volume coefficient associated with geometric sizes, and k₂ is the effective volume exponent associated with specimen deformation.

${(1} - a/W{)}^{{m_{1} }}$ parameters suggested by Landes et al. [29] were utilized to further break down the dimensionless geometric function G(a/W, B/W) as

$$ G\left( {\frac{a}{W},\frac{B}{W}} \right){ = }k_{02} \left( {{1} - \frac{a}{W}} \right)^{{m_{1} }} g\left( \frac{B}{W} \right) $$

(8)

where g(B/W) is the dimensionless thickness separation function, and k₀₂ and m₁ are the effective volume reduction coefficient and index, respectively.

To address the effective deformation volume analysis of specimens with mode I cracks across a broad thickness range, g(B/W) is anticipated as

$$ g\left( \frac{B}{W} \right){ = }k_{03} \left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{m_{2} }} $$

(9)

where b₁ and b₂ are thickness coefficients, and k₀₃ and m₂ are the effective volume reduction coefficient and index, respectively.

Substitution of Eqs. (8) and (9) into Eq. (7) yields the dimensionless effective deformation volume $\overline{V}_{{{\text{eff}}}}$ for specimens with mode I cracks as

$$ \overline{V}_{{{\text{eff}}}} { = }\frac{{V_{{{\text{eff}}}} }}{{W^{2} B}}{ = }k_{1} \left( {{1} - \frac{a}{W}} \right)^{{m_{1} }} \left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{m_{2} }} \left( {\frac{v}{{v^{*} }}} \right)^{{k_{2} }} $$

(10)

where k₁ is a constant only associated with specimen geometry, and k₁ = k₀₁k₀₂k₀₃.

The characteristic volume V^* is expressed as

$$ V^{*} { = }W^{2} B\left( {{1} - \frac{a}{W}} \right)^{{m_{1} }} \left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{m_{2} }} $$

(11)

Thus, from Eqs. (10) and (11), it can be inferred that

$$ \varphi \left( {\frac{v}{{v^{*} }}} \right) = \frac{{V_{{{\text{eff}}}} }}{{V^{*} }}{ = }k_{1} \left( {\frac{v}{{v^{*} }}} \right)^{{k_{2} }} $$

(12)

where all undetermined constants k₁, k₂, m₁, m₂, b₁, and b₂ can be calculated using a small amount of FEA.

Based on Eqs. (6) and (12), the analytical function of strain energy for specimens with mode I cracks under 3D constraints can be written as

$$ \left\{ {\begin{array}{*{20}l} {U{\text{ = }}V^{{\text{*}}} k_{1} \left( {\frac{v}{{v^{*} }}} \right)^{{k_{2} }} \int\limits_{0}^{{\phi \left( {\frac{v}{{v^{*} }}} \right)}} {f(\varepsilon _{{{\text{eq}}}} ,E,K,N){\text{d}}\varepsilon _{{{\text{eq}}}} } } \hfill \\ {V^{{\text{*}}} {\text{ = }}W^{2} B\left( {{\text{1}} - \frac{a}{W}} \right)^{{m_{1} }} \left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{m_{2} }} } \hfill \\ \end{array} } \right. $$

(13)

2.3 Plastic Load–Displacement Semi-analytical Model for Specimens with Mode I Cracks

In specimens with mode I cracks under 3D constraints, only pure plastic deformation was considered. According to the R–O law, the material constitutive function f in Eq. (6) can be expressed as

$$ \sigma_{{{\text{eq}}}} = f\left( {\varepsilon_{{{\text{eq}}}} ,\;E,\;K,\;n} \right) = K\varepsilon_{{\text{p-eq}}}^{1/n} $$

(14)

Substitution of Eq. (14) into Eq. (13) yields the plastic strain energy U_p for specimens with mode I cracks as

$$ U_{{\text{p}}} = \frac{{nKV^{*} }}{n + 1}k_{1} \left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)^{{k_{2} }} \left[ {\phi \left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)} \right]^{1 + 1/n} $$

(15)

where $v_{{\text{p}}}$ is the plastic displacement on the loading line. Referring to [26,27,28], the function ϕ is defined as

$$ \phi \left( {\frac{v}{{v^{*} }}} \right) = k_{3} \left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)^{{k_{4} }} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} $$

(16)

where k₃ and k₄ are the effective strain coefficient and index under pure plastic conditions, respectively.

Substitution of Eq. (16) into Eq. (15) yields

$$ \left\{ \begin{gathered} \frac{{U_{{_{\text{p}} }}^{{}} }}{{U_{{_{{}} }}^{*} }} = \xi_{{\text{p}}} \left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)^{{m_{{\text{p}}} + 1}} \hfill \\ \left\{ \begin{array}{l} \displaystyle U^{*} = KV^{*} { = }KW^{2} B\left( {{1} - \frac{a}{W}} \right)^{{m_{1} }} \\ \displaystyle \qquad\quad\left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{m_{2} }} \hfill \\ \displaystyle v^{*} { = }W \hfill \\ \displaystyle \xi_{{\text{p}}} = \frac{{nk_{{1}} k_{{3}}^{{{1}/n + {1}}} }}{{n + {1}}} \hfill \\ \displaystyle m_{{\text{p}}} = \frac{{k_{{4}} }}{n} + k_{{4}} + k_{{2}} - {1} \hfill \\ \end{array} \right. \hfill \\ \end{gathered} \right. $$

(17)

where U^* is the characteristic strain energy function associated with specimen geometry, ξ_p is the curvature of the plastic energy-displacement curve, and m_p is the plastic deformation index.

Under quasi-static plastic conditions and following the functional principle, U_p is determined as

$$ U_{{\text{p}}} = W = \int_{0}^{{v_{{\text{p}}} }} {P{\text{d}}v} $$

(18)

Substituting Eq. (17) into Eq. (18), the derivatives of Eq. (18) with respect to $v_{{\text{p}}}$ are obtained as

$$ \left\{ \begin{gathered} \frac{P}{{P_{{}}^{*} }} = \left( {1 + m_{{\text{p}}} } \right)\xi_{{\text{p}}} \left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)^{{m_{{\text{p}}} }} \hfill \\ \left\{ \begin{gathered} P^{*} = KA^{*} { = }KWB\left( {{1} - \frac{a}{W}} \right)^{{m_{1} }} \left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{m_{2} }} \hfill \\ v^{*} { = }W \hfill \\ \xi_{{\text{p}}} = \frac{{nk_{{1}} k_{{3}}^{{{1}/n + {1}}} }}{{n + {1}}} \hfill \\ m_{{\text{p}}} = \frac{{k_{{4}} }}{n} + k_{{4}} + k_{{2}} - {1} \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} \right. $$

(19)

where A^* is the characteristic area, given by A^* = V^*/$v^{*}$, and P^* is the characteristic load associated with geometric dimensions. Equation (19) outlines the dimensionless plastic load–displacement model for specimens with mode I cracks under 3D constraints (abbreviated as the CS-pv model).

3 Maximal Principal Stress Model at Mode I Crack Tip

3.1 FEA of Specimens with Mode I Cracks

Compact tension (CT) and single-edge notch-bending (SEB) specimens were selected as typical mode-I-cracked specimens to investigate crack tip stress fields. Their deformation processes were analyzed under 3D large deformation conditions using the finite element software ANSYS 14.5. Solid 185 elements were utilized to develop quarter models of the CT and SEB specimens. The meshes at crack tips were locally refined, and the selected mesh density was not sensitive to the calculated results. Table 1 summarizes the details of CT and SEB models, with red shaded faces in both models indicating symmetric constraints applied during calculations.

Table 1 Details of the finite element model for CT and SEB specimens

3.2 Distribution and Evolution of Crack-Tip First Principal Stress

Studies on ductile material critical fracture behaviors in [24, 25] have identified the primary controlling parameter of ductile fracture as the first principal stress. Accurate description of the maximal crack tip first principal stress is essential for predicting the onset of mode I cracks using the stress fracture criterion. Numerous experiments [30,31,32] have demonstrated that crack initiation occurs at the crack tip on a surface with symmetric thickness. FEA was used to determine the distribution of the first principal stress in the crack tip region where the crack surface and symmetric thickness surface meet.

A CT specimen was employed as an example to show the distribution of the first principal stress at the intersection line between the crack surface and the symmetric thickness surface under different loads. The constitutive parameters of the CT specimen were K = 800 MPa, E = 210 GPa, and n = 6, and its geometric parameters were a/W = 0.5 and B/W = 0.5. The obtained results are displayed in Fig. 1. The material yield stress, denoted as σ₀, was obtained from stress–strain curves to normalize the first principal stress. It was observed that the first principal stress distribution first increased and then decreased in the crack tip region, with the maximal first principal stress point gradually moving away from the crack tip with increasing load. In Fig. 2, the first principal stress at the maximal point was denoted as σ_1max^T. When the relationship between the first principal stress and stress triaxiality at the maximal principal stress point satisfied the ductile material critical fracture criterion [24, 25], crack initiation occurred.

Fig. 1

Distribution of crack tip first principal stress

Fig. 2

Schematic diagram of maximal first principal stress σ_1max^T

Establishing a dimensionless relationship between the maximal first principal stress and the load (σ_1max^T/σ₀-P/P^*) during the loading process is crucial to pinpoint mode I crack initiation under 3D constraints. Large-deformation FEA was applied to obtain σ_1max^T/σ₀-P/P^* curves of CT and SEB specimens with various materials and geometric dimensions. The results are illustrated in Figs. 3–4. The σ_1max^T/σ₀-P/P^* curves reveal the following rules:

• (1) For all specimens with various materials and geometries, the trend of σ_1max^T/σ₀-P/P^* curves evolved in the following manner: Initially, there was a significant increase in σ_1max^T/σ₀ as P/P^* rose, marking the ascending period. Once the specimens underwent a certain degree of elastic–plastic deformation, σ_1max^T/σ₀ stabilized, showing no significant change with further increases in P/P^*, indicating the onset of the stable period.

• (2) Figs. 3a and 4a indicated that as crack length a/W increased from 0.5 to 0.8, the value of σ_1max^T/σ₀ under the same P/P^* significantly decreased during the ascending period, while remaining relatively constant during the stable period.

• (3) Figs. 3b and 4b demonstrated a slight change in σ_1max^T/σ₀-P/P^* curves for specimens with mode I cracks of varying B/W ratios. This slight change must be carefully investigated because thickness parameters significantly influenced crack initiation. Among specimens with different B/W ratios, those with values of 0.1 and 6 determined the minimal and maximal σ_1max^T/σ₀ values under the same P/P^*, respectively.

• (4) Figs. 3c, d and 4c, d showed the influences of material parameters K, E, and n on σ_1max^T/σ₀-P/P^* curves. The ascending periods of σ_1max^T/σ₀-P/P^* curves exhibited clear correlations with K, E, and n, while the stable periods were only closely related to n.

Fig. 3

σ_1max^T/σ₀-P/P^* curves of CT specimens

Fig. 4

σ_1max^T/σ₀-P/P^* curves of SEB specimens

3.3 Description Model for Crack-Tip Maximal First Principal Stress

To accurately describe the dimensionless relationship between the crack-tip maximal first principal stress and the load for specimens with mode I cracks under 3D constraints, all influencing factors, including geometric dimensions a/W, B/W, and material parameters K, E, and n, needed consideration.

The crack-tip maximal first principal stress σ_1max^T exhibited a positive relationship with the load and displacement of specimens with mode I cracks. Following the principle of dimensional analysis, the stress similarity criterion π₁, load similarity criterion π₂, and pure plastic displacement similarity criterion π₃ are defined as

$$ \left\{ \begin{gathered} \pi_{1} = \frac{{\sigma_{1\max }^{{\text{T}}} }}{{\sigma_{0} }} \hfill \\ \pi_{2} = \frac{P}{{P^{*} }} \hfill \\ \pi_{3} = \frac{{v_{{\text{p}}} }}{{v^{*} }} \hfill \\ \end{gathered} \right. $$

(20)

The relationship between π₁, π₂, and π₃ is stated as

$$ \pi_{1} = k_{5} \pi_{2} \pi_{3}^{{k_{6} }} + k_{7} \pi_{2} \pi_{3}^{{k_{8} }} $$

(21)

where k₅ and k₆ are parameters in the ascending periods of σ_1max^T/σ₀-P/P^* curves, associated with a/W, B/W, K, E, and n, while k₇ and k₈ pertain to the stable period of σ_1max^T/σ₀-P/P^* curves only associated with n.

From Eqs. (20) and (21), it can be deduced that

$$ \frac{{\sigma_{1\max }^{{\text{T}}} }}{{\sigma_{0} }} = k_{5} \frac{P}{{P^{*} }}\left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)^{{k_{6} }} + k_{7} \frac{P}{{P^{*} }}\left( {\frac{{v_{{\text{p}}} }}{{v^{*} }}} \right)^{{k_{8} }} $$

(22)

The parameters k₅ to k₈ were defined through the large-deformation FEA of various geometries and materials of specimens with mode I cracks.

By substituting the CS-pv model from Eq. (19) into Eq. (22), the dimensionless relationship between the crack-tip maximal principal stress and the load can be obtained as

$$ \left\{ \displaystyle\begin{array}{l} \frac{{\sigma_{1\max }^{{\text{T}}} }}{{\sigma_{0} }} = k_{5} \left[ {\frac{1}{{\left( {1 + m_{{\text{p}}} } \right)\xi_{{\text{p}}} }}} \right]^{{\frac{{k_{6} }}{{m_{{\text{p}}} }}}} \left( {\frac{P}{{P^{*} }}} \right)^{{\frac{{k_{6} }}{{m_{{\text{p}}} }} + 1}} \\ \qquad\qquad+ k_{7} \left[ {\frac{1}{{\left( {1 + m_{{\text{p}}} } \right)\xi_{{\text{p}}} }}} \right]^{{\frac{{k_{8} }}{{m_{{\text{p}}} }}}} \left( {\frac{P}{{P^{*} }}} \right)^{{\frac{{k_{8} }}{{m_{{\text{p}}} }} + 1}} \hfill \\ \left\{ \begin{array}{l} k_{5} = p_{1} e^{{p_{2} (\frac{1}{n + 1} + 1)}} (\frac{K}{E})^{{p_{3} (\frac{1}{n + 1} + 1)}} (1 - \frac{a}{W})^{{p_{4} }} \\ \qquad\qquad\left( {\left[ {\arctan \left( {b_{1} \frac{B}{W}} \right) + b_{2} } \right]^{{p_{5} }} + p_{6} } \right) \hfill \\ k_{6} = p_{7} (\frac{K}{E}) + \frac{{p_{8} }}{n + 1} \hfill \\ k_{7} = p_{9} \left( {1 + \frac{1}{n + 1}} \right)^{{p_{10} }} \hfill \\ k_{8} = 1 + \frac{{p_{11} }}{n + 1} \hfill \\ \end{array} \right. \hfill \\ \end{array} \right. $$

(23)

Equation (23) is referred to as the description model for the maximal first principal stress in specimens with mode I cracks (abbreviated as the CS-σ_1max^T model). The parameters p₁ to p₁₁ can be determined through fitting calculations. Moreover, the CS-σ_1max^T model can be universally applied across various materials and geometric dimensions.

4 Validation of the Description Model for Maximal Principal Stress

4.1 Model Parameter Determination

The CS-σ_1max^T model is a mathematical equation with universal applicability and adheres to dimensional analysis. Model parameters, including k₁ to k₄, m₁, m₂, b₁, and b₂, can be obtained through FEA, while p₁ to p₁₁ are determined by fitting simulated data for different materials and geometries. Figure 5 illustrates the accurate parameter determination process. Table 2 outlines the parameter values for CT and SEB specimens.

Fig. 5

Flow diagram of model parameter determination

Table 2 Parameters of the description model for maximal first principal stress

4.2 Validation of CS-σ _1max ^T Model

CT and SEB specimens of various materials and geometric dimensions were tested. For CT specimens, material parameters were K = 200, 400, 600, and 800 MPa, E = 70, 110, and 210 GPa, and n = 4, 6, 8, 10, and 12, and geometric parameters were a/W = 0.5, 0.6, 0.7, and 0.8 and B/W = 0.1, 0.3, 0.5, 0.7, 1.5, 3, and 6. For SEB specimens, material parameters were K = 300, 600, 900, and 1200 MPa, E = 70, 110, and 210 GPa, and n = 3, 5, 7, 9, and 11, and geometric parameters were a/W = 0.5, 0.6, 0.7, and 0.8 and B/W = 0.1, 0.25, 0.5, 0.75, 1.5, 3, and 6. The calculations encompassed over 100 working conditions.

The σ _1max^T/ σ₀-P/P^* curves corresponding to these working conditions were obtained through FEA and compared with the prediction results from the CS-σ_1max^T model, as presented in Figs. 6, 7, 8, 9, 10. The σ_1max^T/σ₀-P/P^* curves presented good consistency with the results derived using the CS-σ_1max^T model under all conditions for both CT and SEB specimens, irrespective of variations in a/W, B/W, K, E, or n. This validated the ability of the CS-σ_1max^T model to accurately describe the dimensionless relationship between crack-tip maximal first principal stress and load across diverse materials and geometries.

Fig. 6

σ_1max^T/σ₀-P/P^* curves with different a/W

Fig. 7

σ_1max^T/σ₀-P/P^* curves with different B/W

Fig. 8

σ_1max^T/σ₀-P/P^* curves with different K

Fig. 9

σ_1max^T/σ₀-P/P^* curves with different E

Fig. 10

σ_1max^T/σ₀-P/P^* curves with different n

5 Conclusions

The research examined the correlations between load, displacement, and maximal first principal stress at crack tips of specimens with mode I cracks under 3D constraints, with constitutive relationships conforming to the R–O law. The following conclusions were drawn:

• (1) According to energy density equivalence and dimensional analysis, a CS-pv model was developed to characterize the dimensionless relationship between plastic displacement and load for specimens with mode I cracks under 3D constraints, considering the universality of materials and geometric dimensions.

• (2) The effects of constitutive and geometric parameters on dimensional relationships between the maximal first principal stress and the load were analyzed through large-deformation finite element analyses. The ascending periods of σ_1max^T/σ₀—P/P^* curves exhibited obvious correlations with all parameters a/W, B/W, K, E, and n. In contrast, the stable periods of the curves were only closely related to n.

• (3) A CS-σ_1max^T model was developed to describe the dimensionless relationship between the crack-tip maximal first principal stress and the load. The effects of geometric parameters a/W and B/W, as well as constitutive parameters K, E, and n on σ_1max^T/σ₀—P/P^* curves were all taken into consideration.

• (4) The CS-σ_1max^T model was validated through large-deformation finite element analyses under over 100 different working conditions. The predicted σ_1max^T/σ₀—P/P curves using the developed model agreed well with finite element results for CT and SEB specimens comprising various materials and geometries.