The stress distribution around a crack perpendicular to an interface between materials

The plane strain problem of a crack terminating perpendicular to a planar interface between two isotropic half spaces with different elastic constants is solved to obtain the distribution of stress in the vicinity of the crack tip. The relative elastic constants are shown to strongly affect the relative magnitudes of the various stress components as well as their radial drop off with distance from the crack tip. The implications of the results with regard to failure modes in composite materials are discussed.     

Elastic and plastic fracture analysis of a crack perpendicular to an interface between dissimilar materials

Elastic and plastic fracture analysis of a Mode I crack perpendicular to an interface between dissimilar materials is carried out. Continuously distributed dislocations are used to simulate the crack. The simulation will cause singular integral equations with Cauchy kernel. By solving the singular integral equations numerically, the effects of crack depth (distance from the interface to the crack middle point) and Dundurs’ parameters on the Mode I stress intensity factor are investigated systematically. Then, based on the Dugdale model, the plastic zone size, and the crack tip opening displacement of the crack under uniform loadings are investigated. The effects of uniform loadings, crack depth, and Dundurs’ parameters on the plastic zone size and the crack tip opening displacement are examined. Numerical results show that when the crack is embedded in a stiffer material, the values of both the normalized plastic zone size and the normalized crack tip opening displacement are larger than 1. On the contrary, if the crack is embedded in a softer material, the values of both the normalized plastic zone size and the crack tip opening displacement are less than 1.     

Dominance of asymptotic crack tip fields in elastic functionally graded materials

The stress field surrounding an edge crack in an elastic functionally graded plate is calculated using two dimensional finite element analysis. The property gradient direction is parallel to the crack line and loading is constrained to be symmetric such that a pure mode I situation is achieved. The extent of dominance of asymptotic fields is evaluated by comparing the stress field calculated from the finite element analysis to that calculated by asymptotic equations. Two separate forms of the asymptotic stress fields, one for homogeneous materials and another for continuously nonhomogeneous materials are used. The shape and extent of the dominance regions of each asymptotic field and their dependence on crack length and material nonhomogeneity is also presented. Under the pure mode I conditions considered here, it is seen that both asymptotic fields exist around the crack tip with the one for homogeneous materials in general being embedded in the one for continuously nonhomogeneous materials. The ligament length is seen to primarily control the extent of development of the asymptotic stress field for nonhomogeneous materials. The steepness of the material gradient affects the relationship between the two asymptotic stress fields and therefore the extent of their dominance.     

Two-parameter characterization of the near-tip stress fields for a bi-material elastic-plastic interface crack

A particular case of interface cracks is considered. The materials at each side of the interface are assumed to have different yield strength and plastic strain hardening exponent, while elastic properties are identical. The problem is considered to be a relevant idealization of a crack at the fusion line in a weldment. A systematic investigation of the mismatch effect in this bi-material plane strain mode I dominating interface crack has been performed by finite strain finite element analyses. Results for loading causing small scale yielding at the crack tip are described. It is concluded that the near-tip stress field in the forward sector can be separated, at least approximately, into two parts. The first part is characterized by the homogeneous small scale yielding field controlled by J for one of the interface materials, the reference material. The second part which influences the absolute value of stresses at the crack tip and measures the deviation of the fields from the first part can be characterized by a mismatch constraint parameter M. Results have indicated that the second part is a very weak function of distance from the crack tip in the forward sector, and the angular distribution of the second part is only a function of the plastic hardening property of the reference material.     

Determination of coefficients of crack tip asymptotic fields using the scaled boundary finite element method

Coefficients of the Williams expansion of the linear elastic crack tip asymptotic field can only be evaluated analytically for a few simple cases. Numerical solution is necessary in the general case, and the presence of the singular term presents numerical difficulties. The scaled boundary finite element method is a new semi-analytical approach to computational mechanics developed by Wolf and Song. This paper shows that when the scaling centre is located at the crack tip, the scaled boundary finite element solution converges to the Williams expansion. Consequently the coefficients of the Williams expansion, including the stress intensity factor and the T-stress, can be determined directly without further processing. The technique is applied to several problems for which coefficients of the Williams expansion are available, and close agreement with existing results is obtained with very few degrees of freedom.     

Mixed mode asymptotic crack tip fields in orthotropic materials: Derivation and range of dominance

In the first part of this work the asymptotic stress field distribution surrounding a crack in a generally orthotropic solid (i.e. one in which the material and loading axes do not coincide) is derived. The resulting crack tip stress intensity factors are also related to the energy release rate of the cracked solid. In any physical situation, however, the range of dominance of these asymptotic fields will be limited, and will depend on specific geometry and loading parameters. Thus, the second part of this work deals with the range of dominance of the derived stress fields in edge cracked bending loaded fiber reinforced composite plates. The lower and upper limits of the range of dominance of the solution are respectively determined from three-dimensional and two-dimensional full field finite element solutions. The comparison of full field solutions with the asymptotic result provides information on the latter's range of dominance. In all cases the effects of mixed mode loading are considered.     

Stress intensity factor for a crack normal to an interface between two orthotropic materials

In this paper, the problem of a crack normal to an interface in two joined orthotropic plates is studied as a plane problem. Body force method is used to investigate dependence of the stress intensity factor on the elastic constants: E _x1, E _y1, G _xy1, V _xy1 for material 1 and E _x2, E _y2, G _xy2, V _xy2 for material 2. A particular attention is paid to simplifying kernel functions, which is used in the body force method, so that all the elastic constants involved can be represented by three new parameters: H ₁, H _2I, H ₃ for the mode I deformation and H ₁, H _2II, H ₃ for the mode II deformation. From the kernel function so obtained it is found that the effects of the eight elastic constants on the stress intensity factors can be expressed by the three material parameters, H ₁, H _2I, H ₃ and H ₁, H _2II, H ₃, respectively for K _I and K _II. Furthermore, it is also found that the dependence of K _I on H ₁, H _2I, H ₃ is exactly the same as the dependence of K _II on H ₁, H _2II, H ₃. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.

     

Analytical investigation of crack tip fields in viscoplastic materials

A macroscopic stationary crack in viscoplastic materials is considered under mode I creep loading conditions. Typical representations of constitutive laws with internal variables. (back stress) which can be derived from a scalar potential function are used to model the inelastic material behaviour. It is shown that, in the limit of high stresses, the constitutive equation adopt the form of Norton's law thus leading to a singular field of the HRR-type at the crack tip, if the material functions for the constitutive equations are represented by power laws. The amplitude of the crack tip field can be evaluated using a crack tip integral. It reduces to the well-known C ^* expression at the crack tip and includes additonal domain integrals which ensure the independence of the choice of the integration contour and the area enclosed around the crack tip in regions where primary creep and linear eleastic effects cannot be neglected. The paper concentrates on results characterising the crack tip fields which can be derived analytically. Numerical aspects focused upon the right modelling of the crack tip zone, the range of validity of the crack tip field and the calculation of the crack tip parameter and the creep zones will be discussed in a subsequent paper.     

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.

     

Virtual crack closure technique for an interface crack between two transversely isotropic materials

The virtual crack closure technique makes use of the forces ahead of the crack tip and the displacement jumps on the crack faces directly behind the crack tip to obtain the energy release rates \({{\mathcal {G}}}_I\) and \({\mathcal {G}}_{II}\). The method was initially developed for cracks in linear elastic, homogeneous and isotropic material and for four noded elements. The method was extended to eight noded and quarter-point elements, as well as bimaterial cracks. For bimaterial cracks, it was shown that \({\mathcal {G}}_I\) and \({\mathcal {G}}_{II}\) depend upon the virtual crack extension \(\varDelta a\). Recently, equations were redeveloped for a crack along an interface between two dissimilar linear elastic, homogeneous and isotropic materials. The stress intensity factors were shown to be independent of \(\varDelta a\). For a better approximation of the Irwin crack closure integral, use of many small elements as part of the virtual crack extension was suggested. In this investigation, the equations for an interface crack between two dissimilar linear elastic, homogeneous and transversely isotropic materials are derived. Auxiliary parameters are used to prescribe an optimal number of elements to be included in the virtual crack extension. In addition, in previous papers, use of elements smaller than the interpenetration zone were rejected. In this study, it is shown that these elements may, indeed, be used.     

Plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant materials

The plane strain asymptotic fields for cracks terminating at the interface between elastic and pressure-sensitive dilatant material are investigated in this paper. Applying the stress-strain relation for the pressure-sensitive dilatant material, we have obtained an exact asymptotic solution for the plane strain tip fields for two types of cracks, one of which lies in the pressure-sensitive dilatant material and the other in the elastic material and their tips touch both the bimaterial interface. In cases, numerical results show that the singularity and the angular variations of the fields obtained depend on the material hardening exponent n, the pressure sensitivity parameter μ and geometrical parameter λ. This revised version was published online in August 2006 with corrections to the Cover Date.     

Dynamic asymptotic mode III crack tip field in rate dependent materials

The asymptotic field at a dynamically growing crack tip in strain-rate sensitive elastic-plastic materials is investigated under anti-plane shear loading conditions. In the conventional viscoplasticity theory, the rate sensitivity is included only in the flow stress. However, it is often found that the yield strength is also affected by previous strain rates. The strain rate history effects in metallic solids are observed in strain rate change tests in which the flow stress decreases gradually after a rapid drop in strain rate. This material behavior may be explained by introducing the rate sensitivity in the hardening rule in addition to the flow rule. The strain-rate history effect is pronounced near the propagating crack where the change of strain rates take place. Effects of the rate dependency in the flow rule and the hardening rule on the crack propagation are analyzed. The order of the stress singularity in the asymptotic field is determined in terms of material parameters which characterize the rate sensitivity of the material. The results show that an elastic sector is present in the wake zone when the rate-dependency is considered only in the hardening rule. Terminal crack propagation speed is determined by applying the critical stress fracture criterion and the critical strain criterion to the asymptotic fields under the small scale yielding condition.     

Boundary element analysis for an interface crack between dissimilar elastoplastic materials

The boundary element method (BEM) is presented for elastoplastic analysis of cracks between two dissimilar materials. The boundary integral equations and integral representation of stress rates are written in such a form that all integrals can be evaluated by the regular Gaussian quadrature rule. An advanced multidomain BEM formulation is suggested for the solution of analysed problems where the substantial reduction of stiffness matrix is observed. The elastoplastic behaviour is modelled through the use of an approximation for the plastic component of the stresses. The boundary and the yielding zone are discretized by elements with quadratic approximations. In numerical examples the path independence of the J- and L-integrals for a straight interface crack and a circular arc-shaped interface crack are investigated, respectively. The influence of the different values of Young's modulus on the J-integral, shape and size of plastic zones is treated too.     

Families of asymptotic tensile crack tip stress fields in elastic-perfectly plastic single crystals

In this work, two families of asymptotic near-tip stress fields are constructed in an elastic-ideally plastic FCC single crystal under mode I plane strain conditions. A crack is taken to lie on the (010) plane and its front is aligned along the direction. Finite element analysis is first used to systematically examine the stress distributions corresponding to different constraint levels. The general framework developed by Rice (Mech Mater 6:317–335, 1987) and Drugan (J Mech Phys Solids 49:2155–2176, 2001) is then adopted to generate low triaxiality solutions by introducing an elastic sector near the crack tip. The two families of stress fields are parameterized by the normalized opening stress prevailing in the plastic sector in front of the tip and by the coordinates of a point where elastic unloading commences in stress space. It is found that the angular stress variations obtained from the analytical solutions show good agreement with finite element analysis.     

Large strain field near an interface crack tip

In this paper the elastostatic field near the tip of an interface crack between two materials is analyzed with the fully nonlinear theory. By dividing the crack tip field into narrowing sectors and an expanding sector, the asymptotic equations for the crack tip field are derived and solved. The singular characters of stress and strain near the crack tip are revealed. The crack opening shape is discussed for various cases. It is shown that when large deformation is taken into account the oscillatory singularity does not occur.