A finite element investigation of quasi-static and dynamic asymptotic crack-tip fields in hardening elastic-plastic solids under plane stress

The asymptotic structures of crack-tip stress and deformation fields are investigated numerically for quasi-static and dynamic crack growth in isotropic linear hardening elastic-plastic solids under mode I, plane stress, and small-scale yielding conditions. An Eulerian type finite element scheme is employed. The materials are assumed to obey the von Mises yield criterion and the associated flow rule. The ratio of the crack-tip plastic zone size to that of the element nearest to the crack tip is of the order of 1.6 × 10⁴. The results of this study strongly suggest the existence of crack-tip stress and strain singularities of the type r ^s (s < 0) at r=0, where r is the distance to the crack tip, which confirms the asymptotic solutions of variable-separable type by Amazigo and Hutchinson [1] and Ponte Castañeda [2] for quasi-static crack growth, and by Achenbach, Kanninen and Popelar [3] for dynamic crack propagation. Both the values of the parameter s and the angular stress and velocity field variations from the present full-field finite element analysis agree very well with those from the analytical solutions. It is found that the dominance zone of the r ^s-singularity is quite large compared to the size of the crack-tip active plastic zone. The effects of hardening and inertia on the crack-tip fields as well as on the shape and size of the crack-tip active plastic zone are also studied in detail. It is discovered that as the level of hardening decreases and the crack propagation speed increases, a secondary yield zone emerges along the crack flank, and kinks in stress and velocity angular variations begin to develop. This dynamic phenomenon observable only for rapid crack growth and for low hardening materials may explain the numerical difficulties, in obtaining solutions for such cases, encountered by Achenbach et al. who, in their asymptotic analysis, neglected the existence of reverse yielding zones along the crack surfaces.     

Analytical forms of higher-order asymptotic elastic-plastic crack-tip fields in a linear hardening material under antiplane shear

An analytical study of the higher-order asymptotic solutions of the stress and strain fields near the traction-free crack tip under antiplane shear in a linear hardening material is investigated. The results show that every term of the asymptotic fields is controlled by both elasticity and plasticity and all the higher-order asymptotic fields are governed by linear nonhomogeneous equations. The first four term solutions are presented analytically and the first four terms are described by two independent parameters J and K ₂. The amplitude of the second order term solution is only dependent on the material properties, but independent of loading and geometry. This paper focuses on the case with traction-free crack surface boundary conditions. The effects of different crack surface boundary conditions, such as clamped and mixed surfaces, on the crack-tip fields are also presented. Comparison of multi-term solution with leading term solution, and finite element solution in an infinite strip with semi-infinite crack under constant displacements along the edges is provided.     

Mode I plane stress crack-tip constraint for elastic-plastic materials with different strain hardening

Finite element analyses and simulations have been undertaken to investigate the triaxial constraint in the crack-tip regions of a stationary crack and a steady-state growing crack under mode I plane stress for elastic-plastic materials with different strain hardening. The results show that the triaxial constraint in the crack-tip region is independent of specimen geometry, and material strain hardening, both for a stationary and an extending crack quasi-statically. The triaxial constraints for the various configurations examined are in better accordance with those required by the HRR solution for a stationary crack, which defines the low and similar constraints in crack-tip regions for different material strain hardening in the plane stress case. Along the entire ligament ahead of a crack tip, the constraint level transites gradually from that defined by the HRR solution within the near tip zone to that characterized by the stress intensity factor K _I in the far field.     

Crack-tip fields in elastic-plastic material under plane stress mode I loading

New results on the crack-tip fields in an elastic power-law hardening material under plane stress mode I loading are presented. Using a generalized asymptotic expansion of the stress function, higher-order terms are found which have newly-discovered characteristics. A series solution is obtained for the elastic-plastic crack-tip fields. The expansion of stress fields contains both the and terms where t_i is real and t_k is complex; the terms σ⁽ⁱ⁾ _pq(θt_i) and σ^(k) _rsθt_k) are real and complex functions of θ respectively. Comparing the results with that for the plane strain mode I loading shows that: (1) the effect of higher-order solutions on the crack-tip fields is much smaller; and (2) the path-independent integral J also controls the second-order or third-order term in the asymptotic solutions of the crack-tip fields for most of the engineering materials (1 < n < 11) in plane stress, while the J-integral does not control the second and the third-order terms for the plane strain mode I case for n > 3. These theoretical results imply that the crack-tip fields can be well characterized by the J-integral, and can be used as a criterion for fracture initiation under plane stress mode I loading. This is in agreement with existing full-field solutions and experimental data that J at crack growth initiation is essentially independent of in-plane specimen geometry. The comparison confirms the theoretical asymptotic solutions developed in this study. This revised version was published online in August 2006 with corrections to the Cover Date.     

An elastic-plastic finite element analysis of crack tip fields under biaxial loading conditions

A square plate containing a central crack and subjected to biaxial stresses has been studied by a finite element analysis. An elastic analysis shows that the crack opening displacement and stress of separation ahead of the crack tip are not affected by the mode of biaxial loading and therefore the stress intensity factor adequately describes the crack tip states in an elastic continuum.

An elastic-plastic analysis involving more than localized yielding at the crack tip provides different solutions of crack tip stress fields and crack face displacements for the different modes of biaxial loading.

The equi-biaxial loading mode causes the greatest separation stress but the smallest plastic shear ear and crack displacement. The shear loading system induces the maximum size of shear ear and crack displacement but the smallest value of crack tip separation stress.

     

A new finite element for treating plane thermomechanical heterogeneous solids

This study is concerned with the development and implementation of a new finite element which is capable of treating the problem of interacting circular inhomogeneities in heterogeneous solids under mechanical and thermal loadings. The general form of the element, which is constructed from a cell containing a single circular inhomogeneity in a surrounding matrix, is derived explicitly using the complex potentials of Muskhelishvili and the Laurent series expansion method. The newly proposed eight‐noded plane element can be used to treat quite readily the two‐dimensional steady‐state heat conduction and thermoelastic problems of an elastic circular inclusion embedded in an elastic matrix with different thermomechanical properties. Moreover, the devised element may be applied to deal with arbitrarily and periodically located multiple inhomogeneities under general mechanical and thermal loading conditions using a very limited number of elements. The current element also enables the determination of the local and effective thermoelastic properties of composite materials with relative ease. Three numerical examples are given to demonstrate its versatility, accuracy and efficiency. Copyright © 1999 John Wiley & Sons. Ltd.     

Influence of isotropic strain hardening on plane strain dynamic growth in elastic-plastic materials

In this paper, dynamic crack growth in an elastic-plastic material is analysed under mode I, plane strain, small-scale yielding conditions using a finite element procedure. The material is assumed to obey J₂ incremental theory of plasticity with isotropic strain hardening which is of the power-law type under uniaxial tension. The influence of material inertia and strain hardening on the stress and deformation fields near the crack tip is investigated. The results demonstrate that strain hardening tends to oppose the role of inertia in decreasing plastic strains and stresses near the crack tip. The length scale near the crack tip over which inertia effects are dominant also diminishes with increase in strain hardening. A ductile crack growth criterion based on the attainment of a critical crack tip opening displacement is used to obtain the dependence of the theoretical dynamic fracture toughness on crack speed. It is found that the resistance offered by the elastic-plastic material to high speed crack propagation may be considerably reduced when it possesses some strain hardening.     

A hybrid adaptive finite element phase-field method for quasi-static and dynamic brittle fracture

The phase-field approach has unique advantages in describing fracture phenomena, which has received extensive attention in the past decade. Nevertheless, the phase-field modeling of fracture is computationally demanding, due to the high temporal-spatial resolution required for crack tracking. In this contribution, a novel hybrid adaptive finite element phase-field method (ha-PFM) is developed to solve brittle fracture problems under quasi-static and dynamic loading. ha-PFM can dynamically track the propagation of the cracks and adaptively refine the meshes based on a novel crack tip identification strategy. Afterward, the refined meshes in the noncrack progression region are reconverted into coarse meshes. This scheme prominently reduces the computational cost, eg, CPU time and memory usage. Unlike the previous adaptive phase-field method, multilevel hybrid triangular and quadrilateral elements were developed to discretize the computational domain, which eliminates hanging nodes and ensures that the meshes in the vicinity of the crack tip are highly isotropic. Several representative benchmarks containing quasi-static and dynamic fracture were reinvestigated with ha-PFM, and its excellent performance is substantiated by comparison with the standard phase-field method and literature results.     

Higher-order crack-tip creep stress fields in materials under biaxial loading

We have developed and implemented a method for calculating the fields of parameters of the crack-tip creep stress-strain state by taking direct account of the higher-order terms. The paper presents some calculated data on the fields of crack-tip stresses, creep strain rates, and amplitude ratios in the creep case. The influence of loading biaxiality on redistribution of stresses between the creep stages and on the constraint parameters in failure is assessed. Translated from Problemy Prochnosti, No. 6, pp. 25–43, November–December, 2008.     

Finite deformation finite element analyses of tensile growing crack fields in elastic-plastic material

Finite deformation finite element analyses of plane strain stationary and quasi-statically growing crack fields in fully incompressible elastic-ideally plastic material are reported for small-scale yielding conditions. A principal goal is to determine the differences between solutions of rigorous finite deformation formulation and those of the usual small-displacement-gradient formulation, and thereby assess the validity of the (nearly all) extant studies of ductile crack growth that are based on a small-displacement-gradient formulation. The stationary crack case with a significantly blunted tip is studied first; excellent agreement in stress characteristics at all angles about the crack tip and up to a radius of about three times the crack tip opening displacement is shown between Rice and Johnson's [1] approximate analytical solution and our numerical solution. Outside this radius, the numerical results agree very well with Drugan and Chen's [2] small-displacement-gradient analytical characteristics solution in the region of principal plastic deformation. Thus we identify accurate analytical representations for the stress field throughout the plastic zone of a blunted stationary crack. For the growing crack case, the macroscopic difference in crack tip opening profiles between previous small-displacement-gradient solutions and the present results is shown to be negligible, as is the difference in the stress fields in plastic regions. The stress characteristics again agree very well with analytical results of [2]. The numerical results suggest—in agreement with a recent analytical finite deformation study by Reid and Drugan [3]—that it is the finite geometry changes rather than the additional spin terms in the objective constitutive equation that cause any differences between the small-displacement-gradient and the finite deformation solutions, and that such differences are nearly indistinguishable for growing cracks.     

Quasi-statically growing crack-tip fields in elastic perfectly plastic pressure-sensitive materials under plane strain conditions

Quasi-statically growing crack-tip fields in elastic perfectly plastic pressure-sensitive materials under plane strain conditions are investigated in this paper. The materials are assumed to follow the Drucker-Prager yield criterion and the normality flow rule. The asymptotic mode I crack-tip fields are assumed to follow the five-sector assembly of Drugan et al. (1982) for Mises materials. The crack-tip sectors, in turns, from the front of the crack tip are a constant stress sector, a centered fan sector, a non-singular plastic sector, an elastic sector and finally a trailing non-singular plastic sector bordering the crack face. The results of the asymptotic analysis show that as the pressure sensitivity increases, the plastic deformation shifts to the front of the tip, the angular span of the elastic unloading sector increases, and the angular span of the trailing non-singular plastic sector bordering the crack surface decreases. As the pressure sensitivity increases to about 0.6, the angular span of the trailing non-singular plastic sector almost vanishes. The effects of the border conditions between the centered fan sector and the first non-singular plastic sector on the solutions of the crack-tip fields for both Mises and pressure-sensitive materials are investigated in details. This revised version was published online in August 2006 with corrections to the Cover Date.     

A finite element procedure for axisymmetric elastomeric solids under general loading

A finite element procedure, which utilizes Fourier series, is developed to compute the large deformations and deflections in axisymmetric elastomeric solids subjected to non-axisymmetric loading. Nearly incompressible and incompressible, isotropic strain energy density functions describe the elastic properties. The nearly incompressible functions become compatible with an analytical integration scheme by using a novel variable to approximate the dilatation. A large spherical elastomeric bearing is analysed with this procedure.     

A finite element method for determining the angular variation of asymptotic crack tip fields

A finite element method for computing the angular variation of asymptotic singular solutions is presented. For the method to be applicable, the asymptotic fields must admit a separable form in polar coordinates. The radial dependence of the fields is assumed known. We provide details of the application of the method to the problem of a stationary semi-infinite crack in a Ramberg-Osgood material subjected to in-plane remote mixed mode elastic fields. This example demonstrates the primary strengths of the method: the material model is easily implemented and accurate solutions are obtained using coarse meshes.

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Résumé On présente une méthode par éléments finis pour le calcul de la variation angulaire des solutions singulières asymptotiques. Pour que la méthode soit applicable, il faut que les champs asymptotiques admettent une forme séparable en coordonnées polaires. On suppose connue la dépendance radiale du champs. Des détails sont fournis sur l'application de la méthode au problème d'une fissure semi-infinie stationnaire dans un matériau de Ramberg-Osgood soumis à des charges éloignées et dans le même plan, Cet exemple démontre les principaux points forts de la méthode: on peut aisément y introduire un modèle du matériau et obtenir des solutions précises tout en utilisant des mailles larges.

     

A finite element approach to optimal design of plastic structures in plane stress

Plane stress structures of any shape and boundary conditions are simulated by finite element models with homogeneous stress and strain fields in each element. Besides the given live loads, dead loads (such as self-weight) depending on the unknown thickness distribution are allowed for. Possible practical requirements on geometry (minimum thickness, areas of equal thickness, areas of thickness variation in a prescribed way) are taken into account. Yield surfaces of the materials are piecewise linearized. On this basis the minimum weight design problem is formulated in terms of a linear program. This program is dualized and the pair of dual programming problems is discussed. The mechanical interpretation leads to the generalization of some known limit design theory results. Some numerical examples are given at the end.     

Representation of singular fields without asymptotic enrichment in the extended finite element method

In this paper, we replace the asymptotic enrichments around the crack tip in the extended finite element method (XFEM) with the semi‐analytical solution obtained by the scaled boundary finite element method (SBFEM). The proposed method does not require special numerical integration technique to compute the stiffness matrix, and it improves the capability of the XFEM to model cracks in homogeneous and/or heterogeneous materials without a priori knowledge of the asymptotic solutions. A Heaviside enrichment is used to represent the jump across the discontinuity surface. We call the method as the extended SBFEM. Numerical results presented for a few benchmark problems in the context of linear elastic fracture mechanics show that the proposed method yields accurate results with improved condition number. A simple code is annexed to compute the terms in the stiffness matrix, which can easily be integrated in any existing FEM/XFEM code. Copyright © 2013 John Wiley & Sons, Ltd.     

Stress‐hybrid quadrilateral finite element with embedded strong discontinuity for failure analysis of plane stress solids

A formulation of a quadrilateral finite element with embedded strong discontinuity, suitable for the material failure numerical analysis of plane stress solids, is presented. The kinematics of standard finite element is enhanced by displacement jumps that vary linearly along the embedded discontinuity line. They are described by four kinematic parameters that are related to four element separation modes. The modes are designed for no stress transfer over the discontinuity line at its fully softened (opened) state. As for the material, the bulk of the element is assumed to be elastic, and the softening plasticity, in terms of discontinuity tractions and displacement jumps, is assumed along the discontinuity line. The bulk stresses are described by the optimal five‐parameter interpolation. The combination of stress interpolation and enhanced kinematics yields simple form of the element stiffness matrix. To achieve efficient implementation, the stiffness matrix is statically condensed for both the enhanced kinematic parameters and the stress parameters. In a set of numerical examples, the performance of the derived element is illustrated. Obtained results are compared with some other representative embedded discontinuity quadrilateral elements (displacement‐based and enhanced assumed strain based). It turns out that the element performs very well. Copyright © 2013 John Wiley & Sons, Ltd.     

The asymptotic structure of small-scale yielding interfacial free-edge joint and crack-tip fields

Summary The problem of the small-scale yielding (SSY) plane-strain asymptotic fields for the interfacial free-edge joint singularity is examined in detail, and comparisons are made with the interfacial crack tip. The geometries are idealized as isotropic elasto-plastic materials with Ramberg-Osgood power-law hardening properties bonded to a rigid elastic substrate. The resulting fields are shown to be singular and are presented in terms of radial and angular distributions of stress and displacement, and as idealized plastic slip-line sectors. A fourth-order Runge-Kutta numerical method provides solutions to fundamental equations of equilibrium and compatibility that are verified with those of a highly focused finite element (FE) analysis. It is shown that, as in the case of the crack, the asymptotic singular fields are only dependent on the hardening parameter and only a small range of interfacial mode-mix ratios are permitted. The order for the stress singularity may be formulated in terms of the hardening parameter and the elastic solution for incompressible material. The rigid-slip-line field for the interfacial free-edge joint is presented, and it is shown that there is some significant similarity between the asymptotic fields of the deviatoric polar stresses for the joint and the crack-tip having an elastic wedge sector.