Angular Stress Distributions Research Articles

Theoretical solutions for 2D mode-I crack-tip stress fields in power-law plastic materials based on the stress factor derived from the developed median-energy–density equivalence method

This study presents the energy density equivalence equation, demonstrating that the strain energy density of an arbitrary representative volume element (RVE) under a complex stress state equals to that under a uniaxial stress state when volume change is neglected. Based on the developed median-energy–density equivalence method, the analytical solution for equivalent stress of the RVE at the energy density equivalence point in power-law plastic materials is derived in a new method and the concept of stress factor is proposed. Additionally, novel special functions are introduced to accurately describe the angular distributions of stresses in this study. By combining the stress factor, integral theoretical solutions for mode-I crack tip stress fields in power-law plastic materials under finite plane conditions are proposed. Finally, parameterized studies are conducted on bending and tension specimens under plane-strain and plane-stress conditions. The results of stress distributions and contour lines predicted by the integral theoretical solutions are compared with the finite element analysis and HRR solutions for five mode-I plane specimens, providing verification of the solutions’ applicability.

Steady-State Crack Growth in Nanostructured Quasi-Brittle Materials Governed by Second Gradient Elastodynamics

The elastodynamic stress field near a crack tip propagating at a constant speed in isotropic quasi-brittle material was investigated, taking into account the strain gradient and inertia gradient effects. An asymptotic solution for a steady-state Mode-I crack was developed within the simplified strain gradient elasticity by using a representation of the general solution in terms of Lamé potentials in the moving framework. It was shown that the derived solution predicts the nonsingular stress state and smooth opening profile for the growing cracks that can be related to the presence of the fracture process zone in the micro-/nanostructured quasi-brittle materials. Note that similar asymptotic solutions have been derived previously only for Mode-III cracks (under antiplane shear loading). Thus, the aim of this study is to show the possibility of analytical assessments on the elastodynamic crack tip fields for in-plane loading within gradient theories. By using the derived solution, we also performed analysis of the angular distribution of stresses and tractions for the moderate speed of cracks. It was shown that the usage of the maximum principal stress criterion within second gradient elastodynamics allows us to describe a directional stability of Mode-I crack growth and an increase in the dynamic fracture toughness with the crack propagation speed that were observed in the experiments with quasi-brittle materials. Therefore, the possibility of the effective application of regularized solutions of strain gradient elasticity for the refined analysis of dynamic fracture processes in the quasi-brittle materials with phenomenological assessments on the cohesive zone effects is shown.

Unified theoretical solutions for describing the crack-tip stress fields of finite specimens with mode-I crack under fully plastic conditions

The analytical equation to express the equivalent stress of the representative volume element (RVE) is derived from the energy density equivalence. Based on the analytical expression for equivalent stress of the RVE, extensive analyses are conducted for the stress fields in the forward sector near the crack-tip of finite specimens with mode-I crack for pure power-law hardening materials. Further, unified theoretical solutions with various special functions for describing the stress fields are proposed under fully plastic and plane-strain conditions. Comparisons of radial stress distributions in various angles, angular stress distributions, and contour lines of the equivalent stress between the results predicted by the finite element analysis, HRR (Hutchinson and Rice and Rosengren) solutions, and the proposed unified theoretical solutions are performed. The excellent capability of the proposed unified theoretical solutions for predicting the crack-tip stress fields is verified.

Asymptotic stress fields near the crack tip in perfect plastic materials under mixed mode loading (plane strain conditions)

In the paper the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities are presented. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities for plane strain conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions all values of the mixity parameter to within 0.1 were considered and analysed. The asymptotic solution to the statically determinate problem is obtained by the eigenfunction expansion method. Steady-state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher 0.89 respectively for plane strain conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain.

Mixed mode fracture in perfect plastic materials for plane stress conditions

In the study the asymptotic stress fields in the neighbourhood of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities are presented. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities for plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions all values of the mixity parameter to within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained by the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the asymptotic solution for the mixity parameter less than 0.39 and five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter.

Asymptotic stress fields in the vicinity of the crack in perfectly plastic solids under mixed mode loading

In the paper presents the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for a full range of the mode mixities. This objective is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions, all values of the mixity parameter within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained using the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher 0.89 respectively for plane strain conditions and seven sectors determine the asymptotic solution for the mixity parameter less than 0.39, while five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

Asymptotic crack-tip fields in perfect plasticity under mixed mode loading

In the paper the refined presentation of the asymptotic stress fields in the vicinity of the crack tip in perfectly plastic Mises materials under mixed mode loading for the full range of the mode mixities is given. This objective of the paper is engendered by the necessity of considering all the values of the mixity parameter for the full range of the mode mixities both for plane strain and plane stress conditions to grasp stress tensor components behaviour in the vicinity of the crack tip as the mixity parameter is changing from 0 to 1. To gain a better understanding of the stress distributions all values of the mixity parameter to within 0.1 were considered and analyzed. The asymptotic solution to the statically determinate problem is obtained by the eigenfunction expansion method. Steady - state stress distributions for the full range of the mode mixities are found. The type of the mixed mode loading is controlled by the mixity parameter changing from zero for pure mode II loading to 1 for pure mode I loading. It is shown that the analytical solution is described by different relations in different sectors, the value of which is changing from 7 sectors to 5 sectors. At loadings close to pure mode II, seven sectors determine the solution whereas six and five sectors define the solution for the mixity parameter higher 0.33 and less than 0.89 and higher than 0.89 respectively for plane strain conditions and seven sector determine the asymptotic solution for the mixity parameter less than 0.39 and five sectors determine the solution for other values of the mixity parameter for plane stress conditions. The number of sectors depends on the mixity parameter. The angular stress distributions are not fully continuous and radial stresses are discontinuous for some values of the mixity parameter. It is interesting to note that the characteristic feature of the asymptotic solution obtained is the presence of a segment of values of the mixity parameter for which the solution does not depend on the mixity parameter (the solution does not depend on the mixity parameter for the mixity parameter from 0.89 to 1 and the solution coincides with the solution for mode I crack in perfect plastic materials for plane strain conditions). Thus, the salient point of the study is that the asymptotic solution is described by the same formulae for all values of the mixity parameter from 0.89 to 1 for plane strain. For plane stress conditions this segment can’t be observed. The solution in each sector corresponds to the certain value of the mixity parameter. The obtained solutions for plane strain and plane stress conditions can be considered as the limit solution for power law hardening materials and creeping power law materials.

Fundamental solutions and dual boundary element methods for fracture in plane Cosserat elasticity.

In this paper, both singular and hypersingular fundamental solutions of plane Cosserat elasticity are derived and given in a ready-to-use form. The hypersingular fundamental solutions allow to formulate the analogue of Somigliana stress identity, which can be used to obtain the stress and couple-stress fields inside the domain from the boundary values of the displacements, microrotation and stress and couple-stress tractions. Using these newly derived fundamental solutions, the boundary integral equations of both types are formulated and solved by the boundary element method. Simultaneous use of both types of equations (approach known as the dual boundary element method (BEM)) allows problems where parts of the boundary are overlapping, such as crack problems, to be treated and to do this for general geometry and loading conditions. The high accuracy of the boundary element method for both types of equations is demonstrated for a number of benchmark problems, including a Griffith crack problem and a plate with an edge crack. The detailed comparison of the BEM results and the analytical solution for a Griffith crack and an edge crack is given, particularly in terms of stress and couple-stress intensity factors, as well as the crack opening displacements and microrotations on the crack faces and the angular distributions of stresses and couple-stresses around the crack tip.

Biaxial Loading Effect on Higher-Order Crack Tip Parameters

Abstract The higher-order asymptotic crack-tip fields are considered for a Mode I in power-law plastic and creeping materials under plane strain conditions. Using an asymptotic expansion and separation of variables for the stress function, a series solution is obtained for stress components at a crack tip. In addition, full-field finite element analysis based on a modified layer approach is employed to model the effects of biaxial loading on nonlinear behavior. Loadings were applied related to a range of biaxial stress ratio (−2, +2). The radial and angular stress distributions and higher-order amplitude coefficients for plastic materials are obtained. Crack tip higher-order fields under various conditions of biaxial loading and spanning the range of times from small scale creep to extensive creep are presented. Account is taken of the radial distance accompanying crack tip blunting. The phenomenon of stress redistributions along crack plane on creep time as a function of biaxial stress ratio is stated. The regions of dominance of the HRR-type field under various biaxial stress ratio, crack distance, hardening exponent, and creep time are found. Good agreement analytical findings with finite element results conforms that HRR-solution corresponds only to the equi-biaxial tension which is a particular case of biaxial loading. It is further demonstrated that the higher-order both for plastic and creeping fields are controlled through constraint parameter A2 by biaxial stress ratio. By fitting numerical results for plastic and creeping materials, two empirical formulas were obtained to describe the higher-order terms amplitude coefficients A2 distributions depending on biaxial stress ratio, crack distance, hardening exponent, and creep time.

Elastic perfectly-plastic asymptotic mixed mode crack tip fields in plane stress

Elastic perfectly-plastic asymptotic plane stress crack tip fields have been constructed by assembling elastic, constant stress and fan sectors under a complete range of mixed mode I/II states of loading. The angular stress distributions are fully continuous, and do not contain the stress discontinuities which have been a feature of many previously proposed solutions. The analytic solutions are verified by finite element solutions under contained yielding conditions. The structure of the elastic perfectly-plastic fields is compared to the structure of the asymptotic strain hardening fields.

Evaluation of power-logarithmic singularities, T-stresses and higher order terms of in-plane singular stress fields at cracks and multi-material corners

The scaled boundary finite-element method is extended to analyze the in-plane singular stress fields at cracks and multi-material corners. A complete singular stress field is represented semi-analytically as a series of matrix power functions of the radial coordinate originating from the singular point. This method is capable of directly evaluating orders of singularity, stress intensity factors, T-stresses, higher order terms, angular distributions of stresses for each term and, if present, power-logarithmic singularities. In this method, the singular functions are represented analytically and are not evaluated close to the singular point. Numerical examples are calculated to demonstrate the simplicity and to evaluate the accuracy of the scaled boundary finite-element method.

Asymptotic fields for dynamic crack growth in non-associative pressure sensitive materials

Asymptotic near-tip fields are analyzed for a plane strain Mode I crack propagating dynamically in non-associative elastic–plastic solids of the Drucker–Prager type with an isotropic linear strain hardening response. Eigen solutions are obtained over a range of material parameters and crack speeds, based on the assumption that asymptotic solutions are variable-separable and fully continuous. A limiting speed, beyond which a tendency to slope discontinuity in angular distributions of stresses and velocities is detected, is found to deviate from the associative models. At low strain-hardening rates, the onset of the plastic potential corner zone ahead of the crack-tip imposes another limit to the crack speed. Correspondingly, those limits imply the limits to the degree of non-associativity at a given crack speed. In addition, a tendency to slope discontinuity in the angular radial stress distribution sets another limit on the non-associativity at vanishing hardening rates.

Crack-tip field for fast fracture of an elastic–plastic–viscoplastic material incorporated with quasi-brittle damage. Part 1. Large damage regime

Abstract An asymptotic analysis of the near-tip field is presented in terms of the coordinate perturbation technique for fast crack propagation in an elastic–plastic–viscoplastic material with damage. A damage variable is incorporated in the constitutive relation based upon the strain-equivalence principle of damage mechanics. The damage evolution law used is a quasi-brittle type, in which both equivalent and hydrostatic stresses are involved. A non-singular stress field is obtained, as the damage has such a substantial influence on the material behaviour that the high stresses are relaxed at the crack tip. An analytical expression is obtained which explicitly shows the variation of stresses approaching the crack tip, and numerical computations of the angular distributions of stresses and strains are also presented.

Mode I and mode II crack tip asymptotic fields with strain gradient effects

The strain gradient effect becomes significant when the size of fracture process zone around a crack tip is comparable to the intrinsic material lengthl, typically of the order of microns. Using the new strain gradient deformation theory given by Chen and Wang, the asymptotic fields near a crack tip in an elastic-plastic material with strain gradient effects are investigated. It is established that the dominant strain field is irrotational. For mode I plane stress crack tip asymptotic field, the stress asymptotic field and the couple stress asymptotic field can not exist simultaneously. In the stress dominated asymptotic field, the angular distributions of stresses are consistent with the classical plane stress HRR field; In the couple stress dominated asymptotic field, the angular distributions of couple stresses are consistent with that obtained by Huang et al. For mode II plane stress and plane strain crack tip asymptotic fields, only the stress-dominated asymptotic fields exist. The couple stress asymptotic field is less singular than the stress asymptotic fields. The stress asymptotic fields are the same as mode II plane stress and plane strain HRR fields, respectively. The increase in stresses is not observed in strain gradient plasticity for mode I and mode II, because the present theory is based only on the rotational gradient of deformation and the crack tip asymptotic fields are irrotational and dominated by the stretching gradient.

Asymptotic fields for dynamic crack growth in pressure-sensitive elastic–plastic materials

Asymptotic analyses for dynamic propagation of mode I planar cracks in pressure-sensitive elastic–plastic materials have been carried out. The material model adopted is based on the Drucker–Prager yield surface obeying the associate flow rule with linear isotropic hardening. The asymptotic solution is assumed to be of the variable-separable form with a power singularity in the radial coordinate from the crack tip. Attention is focused on the inertia effect on some features of the asymptotic solutions. It is found that for plane-strain cases, the range of pressure sensitivity can be expanded by increasing the crack speed due to a delay in the occurrence of a hydrostatic stress state ahead of the crack tip. An increase in crack speed produces a corresponding change in the characteristics of the governing equations because of a tendency to produce strain and stress ‘jumps’. Material and loading mode-dependent speed limits have also been studied under plane-strain and plane-stress conditions. In addition, numerical results are presented for the strength of singularity, the angular distributions of stresses and velocities, and the crack-tip constraint.

Eigenspectra and Orders of Singularity at a Crack Tip for a Power-Law Creeping Medium

The objective of this paper is to study the eigenspectrum of a creeping body with a static mode I crack. The coordinate perturbation technique is employed to pose the required asymptotic equations. To attain eigensolutions a numerical scheme is worked out and the results obtained provide the information including the number of singularities, and their orders, as well as the angular distributions of stresses. In particular, additional eigenvalues of the HRR/RR problem that have not been reported previously are presented in the context, indicating the fact that other singular fields may exist for the problem. The discussion is also performed to the higher order asymptotic solutions that differ from those resulting from the eigen expansion treatment.

Mixed mode near-tip fields for cracks in materials with strain-gradient effects

Large strain gradients exist near the tip of a crack due to stress singularity. The strain-gradient effect becomes significant when the size of the fracture process zone around a crack tip is comparable to the intrinsic material length, l, typically on the order of microns. Fleck and Hutchinson's [(1993) A phenomenological theory for strain-gradient effects in plasticity. J. Mech. Phys. Solid 41, 1825–1857], strain-gradient plasticity theory is applied to investigate the asymptotic field near a mixed mode crack tip in elastic as well as elastic-plastic materials with strain-gradient effects. It is established that the dominant strain field is irrotational. For an elastic material, stresses and couple stresses have the square-root singularity near the crack tip, and are governed by three variables (two for mode I and II stress fields, and the third, resulting from higher order stresses, for the couple stress field). Stresses ahead of a crack tip in elastic materials with strain-gradient effects could be more than 50% higher than their counterparts in materials without strain-gradient effects. For an elastic-power law hardening strain-gradient material, an analytical solution is obtained. The mixed mode stress field in strain-gradient plasticity is the linear superposition of their counterparts in mode I and II. The angular distribution of stresses and couple stresses for several near-tip mode mixities clearly indicate that the new near-tip field in strain-gradient plasticity differs significantly from the HRR field. Stresses ahead of a crack tip in elastic-plastic solids with strain-gradient effects could be more than 2.5 times their counterparts in the HRR field. The asymptotic analysis compares favorably with available finite element results. The relevance of this solution to materials, in particular the size of the dominant zone, is discussed.

Singular stress fields at V-notch tips in elastoplastic pressure-sensitive materials

Elastoplastic solutions with the higher-order terms for V-notches in materials exhibiting pressure-sensitive yielding and plastic volumetric deformations are presented. It is shown that under plane strain conditions the variable-separable solution exists within some limited pressure-sensitivities. The limit values grow significantly with increasing notch angle. The leading singularity is a decreasing function of notch angle. The small notch angle can hardly affect the singularity. The plane stress fields are generally more singular than the plane strain ones under the same conditions. The pressure-sensitivity does not affect the plane strain field singularity, but the angular stress distributions. The plane stress singularity is slightly increased by the high pressure-sensitivities at the large notch angles. The second-order exponent grows significantly with increasing notch angle. At a notch angle greater than 60°, the elasticity enters the second-order terms in all materials under plane strain conditions, while the plane stress second-order solutions contain the elasticity effects for all notches. It implies that the second-order terms in the notch analysis may not give a significant improvement in characterising the full stress fields. For an apex notch bounded to a rigid substrate, the leading-order singularity falls with increasing notch angle more slowly than that in the homogeneous pressure-sensitive materials. It vanishes at a notch angle of about 125° for all strain-hardening exponents. The elasticity affects the second-order solutions when notch angle becomes large. Whereas the stress fields are dominated by the hoop stress under assumptions of the traction-free crack surfaces, the shear stress is significant for large angle notches.

Singular stresses at the tip of a sharp notch in power-law-hardening materials under antisymmetric loading

We studied stresses near the tip of a sharp V-notch in a material with power-law hardening under the action of antisymmetric loading. The problem is reduced to the solution of a fourth-order differential eigenvalue equation. We present the results of numerical analysis of the order of singularity of displacements and the angular distribution of stresses. We also construct the plastic zone near the tip of the notch for various values of the opening angle of the notch and the hardening index of the material. In the case where the material is linearly elastic, our numerical results are in good agreement with known results.

Mechanics of transonic debonding of a bimaterial interface: The in-plane case

The report of a recent experiment by Liu, Lambros and Rosakis (1993) has focused attention on several issues concerning the process of transonic debonding of a bimaterial interface. As a sequel to our investigation on the anti-plane shear case (Yu and Yang, 1994), the present paper focuses on the in-plane transonic debonding case. We present complete solutions under conditions of both plane strain and plane stress for an interfacial crack advancing within various velocity regimes below the longitudinal wave speed of the stiffer material component. The stress singularities and the angular stress distributions in different transonic regimes are determined in closed form. The coefficients representing stress intensities are deter mined explicitly for the case of point forces acting on the crack faces and translating with the crack tip. The boundedness of energy flow across the interface precludes debonding beyond the higher Rayleigh wave speed.