Evaluation of the Elastic T-stress in Three-dimensional Crack Problems Using an Integral Formula

An integral formulation for computation of T-stresses along a three-dimensional crack front is presented. The mutual M-integral expressed through J-integrals (on analysed and auxiliary fields) provides sufficient information for determining T-stresses on the base of found relation between the M-integral and T-stresses. The auxiliary fields are selected to eliminate the singular terms in the asymptotic expansion of the stresses near the crack tip. Variation of the T-stress along the crack front of an edge-cracked plate is presented.     

Extraction of stress intensity factors from generalized finite element solutions

The performance of several superconvergent techniques to extract stress intensity factors (SIFs) from numerical solutions computed with the generalized finite element method is investigated. The contour integral, the cutoff function and the J-integral methods are considered. An implementation of the extraction techniques based on a sequence of mappings that are independent of the underlying solution method or discretization is proposed. It is shown that this approach is suitable for virtually any mesh-free or mesh-based solution method. Several numerical examples demonstrating the convergence of the computed SIF and the flexibility of the proposed implementation are presented. The path independence of the extraction methods is also investigated. Numerical experiments demonstrate that the contour integral and the cutoff function methods are more robust than the J–integral method with the CFM being the most accurate.     

Evaluation of fracture parameters in continuously nonhomogeneous piezoelectric solids

A contour integral method is developed for computation of stress intensity and electric intensity factors for cracks in continuously nonhomogeneous piezoelectric body under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium is the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for computation of physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LBIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LBIEs. The accuracy of the present method for computing the stress intensity factors (SIF) and electrical displacement intensity factors (EDIF) are discussed by comparison with available analytical or numerical solutions.     

Application of the Jk integral to mixed mode crack problems for anisotropic composite laminates

The J_k integral method for determining mixed mode stress intensity factors separately in the cracked anisotropic plate is developed. Stress intensity factors are indirectly determined from the value of J₁ and J₂. The J₂ integral can be evaluated efficiently from a finite element solution, neglecting the contribution from the portion of the integration contour along the crack faces, by selecting the integration contour in the vicinity of the crack tip. Using functions of a complex variable, the complete relations between J₁, J₂ and K_I, K_II for anisotropic materials are derived conveniently by selecting narrow rectangular contours shrinking to the crack tip. Compared to the existing path independent integral methods, the present method does not involve calculating the auxiliary solution and hence numerical procedures become quite simple. Numerical results to various propblems are given and demonstrate the accuracy, stability and versatility of the method.     

An Interaction Integral Method for Computing Fracture Parameters in Functionally Graded Magnetoelectroelastic Composites

A contour integral method is developed for the computation of stress intensity, electric and magnetic intensity factors for cracks in continuously nonhomogeneous magnetoelectroelastic solids under a transient dynamic load. It is shown that the asymptotic fields in the crack-tip vicinity in a continuously nonhomogeneos medium are the same as in a homogeneous one. A meshless method based on the local Petrov-Galerkin approach is applied for the computation of the physical fields occurring in the contour integral expressions of intensity factors. A unit step function is used as the test functions in the local weak-form. This leads to local integral equations (LIEs) involving only contour-integrals on the surfaces of subdomains. The moving least-squares (MLS) method is adopted for approximating the physical quantities in the LIEs. The accuracy of the present method for computing the stress intensity factors (SIF), electrical displacement intensity factors (EDIF) and magnetic induction intensity factors (MIIF) are discussed by comparison with numerical solutions for homogeneous materials.     

T-stress in the Zener-Stroh arc crack problem in plane elasticity

A Zener-Stroh curved crack is defined such that the crack undergoes an initial displacement discontinuity. A singular integral equation is suggested to solve the Zener-Stroh curved crack problem. General formulation for evaluating the stress intensity factors and the T-stresses at the crack tips of a Zener-Stroh curved crack is carried out. For the Zener-Stroh arc crack, T-stresses at the crack tips can be evaluated in a closed form.     

Evaluation of the stress intensity factors and the T-stress in periodic crack problem

This paper investigates the T-stress at crack tips in the periodic crack problem. Remote tension in the y-direction is applied to cracks with an arbitrary inclined angle. The original stress field can be considered a superposition of a uniform stress field and a perturbation stress field. The problem of evaluating the stresses in the perturbation field can be considered a superposition of many single crack problems. A Fredholm integral equation is suggested for the solution of the perturbation stress field. In the equation, the loading on the crack face is chosen as unknown quantity. Once the integral equation is solved, the stress intensity factors and the T-stress at the crack tip can be evaluated immediately. For solving the integral equation and evaluating stresses in the perturbation field, the remainder estimation technique is suggested for evaluating the influences on the central crack from infinite cracks. The technique can considerably improve convergence in computation. Many results for the stress intensity factors and the T-stresses in periodic cracks are presented. It is shown that the interaction is significant for the closer cracks.     

Interaction integrals for fracture analysis of functionally graded magnetoelectroelastic materials

This paper presents the domain form of interaction integrals based on three independent formulations for computation of stress intensity factors, electric displacement intensity factors and magnetic induction intensity factors for cracks in functionally graded magnetoelectroelastic materials. Conservation integrals of J-type are derived based on the governing equations for magnetoelectroelastic media and the crack tip asymptotic fields of homogeneous magnetoelectroelastic medium as auxiliary fields. Each of the formulations differs in the way auxiliary fields are imposed in the evaluation of interaction integrals and each of them results in a consistent form of the interaction integral in the sense that extra terms naturally appear in their derivation to compensate for the difference in the chosen crack tip asymptotic fields of homogeneous and functionally graded magnetoelectroelastic medium. The additional terms play an important role of ensuring domain independence of the presented interaction integrals. Comparison of numerically evaluated intensity factors through the three consistent formulations with those obtained using displacement extrapolation method is presented by means of two examples.     

Evaluating J‐integral from Displacement Fields Measured by Digital Image Correlation

Path integral, domain integral and least squares methods for evaluating J‐integral from measured displacement fields for a power‐law hardening material are described in this paper. The values of the J‐integral are evaluated by applying the path and domain integral methods to the displacement fields obtained by elastoplastic finite element analysis and the displacement fields obtained through the measurement using digital image correlation. Results show that the values obtained by the domain integral method are slightly better than those by the path integral method, because the domain integral method efficiently uses the full‐field measurement data. The values of the J‐integral are also evaluated by the least squares method with the Hutchinson, Rice, and, Rosengren displacement fields. Results show that the J‐integral can be obtained by the least squares method simply and easily without any calculation of the integration. The J‐integral values obtained by the least squares method agree well with the values obtained using other methods. Because J‐integral can be evaluated easily by any method described in this paper, it is expected that these methods are applicable to various fracture problems during experimental evaluation of structural components.     

An interaction energy integral method for computation of mixed-mode stress intensity factors along non-planar crack fronts in three dimensions

A new interaction energy integral method for extracting mixed-mode stress intensity factors along the fronts of non-planar, three-dimensional cracks is described. In the method, interaction energy contour integrals are defined and expressed in domain form. The mixed-mode stress intensity factors are obtained by evaluating the domain integrals as a post-processing step in the finite element method. To assess the accuracy of the method, two benchmark problems are considered. The first problem is that of an arc crack in an infinitely extended solid subjected to equibiaxial tension. The second is that of a lens-shaped crack embedded in an infinite solid subjected to hydrostatic tension. Excellent agreement is obtained between the numerical and corresponding analytical results obtained from the literature.     

An extended equivalent domain integral method for mixed mode fracture problems by the p-version of FEM

A new path-independent contour integral formula is presented to estimate the crack-tip integral parameter, J-value, for two-dimensional cracked elastic bodies which may quantify the severity of the crack-tip stress fields. The conventional J-integral method based on a line integral has been converted to an equivalent area or domain integral (EDI) by the divergence theorem. It is noted that the EDI method is very attractive because all the quantities necessary for computation of domain integrals are readily available in a finite element analysis. The details and its implementation are extended to the p-version FE model with hierarchic elements using integrals of Legendre polynomials. By decomposing the displacement field obtained from the p-version finite element analysis into symmetric and antisymmetric displacement fields with respect to the crack line, the Mode-I and Mode-II non-dimensional stress intensity factors can be determined by using the decomposition method. The example problems for validating the proposed techniques are centrally oblique cracked plates under tensile loading. The numerical results associated with the variation of oblique angles show very good agreement with the existing solutions. Also, the selective distribution of polynomial orders and the corner elements for automatic mesh generation are applied to improve the numerical solution in this paper. © 1998 John Wiley & Sons, Ltd.     

A thermomechanical fracture modeling and simulation for functionally graded solids using a residual-strain formulation

This paper addresses mixed-mode crack growth in two-dimensional functionally graded solids under thermomechanical loads, and investigates the effect of mechanical and thermal loads as well as the T-stress on their crack growth behavior. A novel residual strain-based formulation in the interaction integral method is developed and used for the accurate evaluation of mixed-mode stress intensity factors and/or the T-stress. Simulation of mixed-mode crack propagation in functionally graded materials including solid oxide fuel cells under thermomechanical loads is performed by means of the finite element method and the generalized interaction integrals in conjunction with a remeshing algorithm. An iterative procedure is used for crack growth simulation including the calculation of mixed-mode stress intensity factors and/or the T-stress by means of the generalized interaction integral method, determination of crack growth direction and crack initiation condition based on selected fracture criteria, and local automatic remeshing along the crack path. The present approach employs a user-defined crack increment at the beginning of the simulation. Crack trajectories and fracture parameters obtained by the present simulation for thermomechanical loads are assessed for some numerical examples in comparison with those for mechanical loads.     

A comparison of stress intensity factors obtained through crack closure integral and other approaches using eXtended element-free Galerkin method

In this paper, a new approach for extracting stress intensity factors (SIFs) by the extended element-free Galerkin method, through a crack closure integral (CCI) scheme, is proposed. The CCI calculation is used in conjunction with a local smoothing technique to improve the accuracy of the computed SIFs in a number of case studies of linear elastic fracture mechanics. The cases involve problems of mixed-mode, curved crack and thermo-mechanical loading. The SIFs by CCI, displacement and stress methods are compared with those based on the M-integral technique reported in the literature. The proposed CCI method involves very simple relations, and still gives good accuracy. The convergence of the results is also examined.     

Mixed-mode separation in dynamic fracture mechanics: New path independent integrals

This paper presents a theoretical and numerical analysis of mixed-mode separation in fracture dynamics, based on new path independent integrals. The M-integral method proposed by Chen and Shield is generalized for dynamic fracture applications. Strain energy density is expressed as a function of the actual displacement field and of an auxiliary kinematically admissible field. On the other hand, the concept of the G-integral developed by Destuynder using the Rice J-integral is extended to dynamic problems. Introducing the same concept in the M-integral formulation leads to the new path-independent integral M in elastodynamics. Numerical tests give us accurate results of separated mode.     

A new variable‐order singular boundary element for two‐dimensional stress analysis

A new variable‐order singular boundary element for two‐dimensional stress analysis is developed. This element is an extension of the basic three‐node quadratic boundary element with the shape functions enriched with variable‐order singular displacement and traction fields which are obtained from an asymptotic singularity analysis. Both the variable order of the singularity and the polar profile of the singular fields are incorporated into the singular element to enhance its accuracy. The enriched shape functions are also formulated such that the stress intensity factors appear as nodal unknowns at the singular node thereby enabling direct calculation instead of through indirect extrapolation or contour‐integral methods. Numerical examples involving crack, notch and corner problems in homogeneous materials and bimaterial systems show the singular element's great versatility and accuracy in solving a wide range of problems with various orders of singularities. The stress intensity factors which are obtained agree very well with those reported in the literature. Copyright © 2002 John Wiley & Sons, Ltd.     

Fracture Mechanics Parameters of Crack Surface Zones Under Volumetric Strains

This report presents the fracture mechanics parameters that enable a theoretical treatment of strained crack surface zones. Green's functions are given for the computation of mode-I and mode-II stress intensity factors as well as T-stresses. Application is shown for strained zones of constant thickness and square-root shaped profile.     

Interaction integrals for thermal fracture of functionally graded materials

This paper addresses finite element evaluation of the non-singular T-stress and mixed-mode stress intensity factors in functionally graded materials (FGMs) under steady-state thermal loads by means of interaction integral. Interaction integral provides an accurate and efficient numerical framework in evaluating these fracture parameters in FGMs under thermal as well as mechanical loads. We use a non-equilibrium formulation and the corresponding auxiliary (secondary) fields tailored for FGMs. Graded finite elements have been developed to account for the spatial gradation of thermomechanical properties. This paper presents various numerical examples in which the accuracy of the present method is verified.     

Contour integral method for stress intensity factors of interface crack

A general Betti's reciprocal work theorem with interface cracks of a bimaterial is established in this paper, and a path independent contour integral method for the stress intensity factor (SIF) of the interface crack was obtained. When the stress and displacement fields in a specimen are calculated by the finite element method, the SIF K _I and K _II of interface cracks can be obtained immediately by a contour integral. Some solutions of interesting examples, such as two collinear interface cracks, are also given.Presented at the Far East Fracture Group (FEFG) International Symposium on Fracture and Strength of Solids, 4–7 July 1994 in Xi'an China.     

Computing stress intensity factors for curvilinear cracks

The use of the interaction integral to compute stress intensity factors around a crack tip requires selecting an auxiliary field and a material variation field. We formulate a family of these fields accounting for the curvilinear nature of cracks that, in conjunction with a discrete formulation of the interaction integral, yield optimally convergent stress intensity factors. In particular, we formulate three pairs of auxiliary and material variation fields chosen to yield a simple expression of the interaction integral for different classes of problems. The formulation accounts for crack face tractions and body forces. Distinct features of the fields are their ease of construction and implementation. The resulting stress intensity factors are observed converging at a rate that doubles that of the stress field. We provide a sketch of the theoretical justification for the observed convergence rates and discuss issues such as quadratures and domain approximations needed to attain such convergent behavior. Through two representative examples, a circular arc crack and a loaded power function crack, we illustrate the convergence rates of the computed stress intensity factors. The numerical results also show the independence of the method from the size of the domain of integration. Copyright © 2015 John Wiley & Sons, Ltd.