Weight functions for bimaterial interface cracks

An efficient approach using the analytically decoupled near-tip displacement solution for bimaterial interface cracks presented in this paper involves: (1) the calculation of the decoupled strain energy release rates G _I and G _II associated respectively with the decoupled stress intensity factors K _I and K _II and (2) the extension of Rice's displacement derivative representation of Bueckner's weight function vectors beyond the homogeneous media. It is shown that the stress intensity factors for a bimaterial interface crack predicted by the present approach agree very well with those solutions available in the literature. The computational efficiency is enhanced through the use of singular elements in the crack-tip neighborhood.As reported in the homogeneous case, the calculated weight function for a bimaterial interface crack is load-independent but depends strongly on geometry and constraint conditions. Due to the coupling nature of the stress intensity factors of a bimaterial interface crack, the invariant characteristics of the dimensionless weight function vectors are different from those of a crack in homogeneous material. In addition, the elastic constants of two constituents can significantly alter the weight function behavior for a cracked bimaterial medium.Due to the load-independent characteristic of the weight functions, the stress intensity factors for a bimaterial interface crack can be obtained accurately and inexpensively by performing the sum of worklike products between the applied loads and the weight functions for the cracked bimaterial body under any loading conditions once the weight functions are explicitly predetermined. The same calculation can also be applied for the identical cracked bimaterial medium with different constraint conditions by including the self-equilibrium forces that contain both the external loads and the reaction forces induced at the constraint locations. Moreover, the physical interpretation of the weight functions can provide a guidance for damage tolerant design application.     

Stress intensity factor analyses of interface cracks between dissimilar anisotropic materials using the finite element method

Delamination along an interface between dissimilar materials is the primary cause of failure in microstructures like electronic packages, micro-electro-mechanical systems (MEMS), and so on. Fracture mechanics is a powerful tool for the evaluation of delamination. However, many materials used in microstructures such as composite materials and single crystals are anisotropic materials. Stress intensity factors of an interface crack between dissimilar anisotropic materials, which were proposed by Hwu, are useful for evaluating the reliability of microstructures. However, numerical methods that can analyze the stress intensity factors of an interface crack between anisotropic materials have not been developed. We propose herein a new numerical method for the analysis of an interface crack between dissimilar anisotropic materials. The stress intensity factors of an interface crack are based on the generalized plane strain condition. The energy release rate is obtained by the virtual crack extension method in conjunction with the finite element method for the generalized plane strain condition. The energy release rate is separated into individual modes of the stress intensity factors K_I, K_II, and K_III, using the principal of superposition. The target problem to be solved is superposed on the asymptotic solution of displacement in the vicinity of an interface crack tip, which is described using the Stroh formalism. Analyses of the stress intensity factors of center interface cracks between semi-infinite dissimilar anisotropic media subjected to concentrated self-balanced loads on the center of crack surfaces and to uniform loads are demonstrated. The present method accurately provides mode-separated stress intensity factors using relatively coarse meshes for the finite element method.     

A numerical model for calculation of stress intensity factors in particle-reinforced metal–matrix composites

A two-dimensional finite element model was developed to study effects of particle diameter, mechanical properties of the fiber and matrix materials and loading conditions (Mode 1 and Mixed-Mode). A theoretical model was proposed to calculate the stress intensity factor for an interface crack in Particle-Reinforced Metal–Matrix Composites. The displacement Correlation Method was used to calculate the stress intensity factors K ₁ and K ₂. In the present model the fiber and matrix materials were modeled in linear elastic conditions. The interface crack was considered between the fiber and matrix, without the presence of the interphase. Obtained results show that the key role on the stress intensity factors played by the relative elastic properties of the fiber and matrix. The results also show that K ₁ and absolute K ₂ values increase for both Mode 1 and mixed-mode loading condition once Young’s modulus of the fiber material increases. The values of K ₁ and K ₂ stress intensity factors decrease when the fiber volume fraction increases for Mode 1 loading.     

Elastodynamic Response of Cracked Orthotropic Strip Under Impact Loading

The paper presents the elastodynamic response of an infinite orthotropic strip of finite width containing a central crack opened by suddenly applied stresses. Integral transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane solved by iterations in the low frequency case. Analytic expressions for the dynamic stress intensity factors are obtained. Numerical results for two particular materials are given. The influence of different orthotropic constants on the magnitude of the overshoots in the stress intensity factors K ₁(t) and K ₂(t) are found.     

Fracture toughness of the + 45° / – 45° interface of a laminate composite

Experiments are carried out to determine the delamination toughness for a crack along the interface between two transversely isotropic materials. The material chosen for study consists of carbon fibers embedded within an epoxy matrix. A crack is introduced between two layers of this material, with fibers in the upper layer along the + 45°-direction and those in the lower layer along the − 45°-direction both with respect to the crack plane. The Brazilian disk specimen is employed in the testing. To calibrate the specimens, stress intensity factors are obtained which result from the applied load, as well as residual curing stresses. It may be noted that all three modes are coupled, leading to a three-dimensional problem. The finite element method and a mechanical M-integral are employed to determine the stress intensity factors arising from the applied load. For the residual stresses, a three-dimensional conservative thermal M-integral is presented for stress intensity factor determination. The stress intensity factors found for the applied load and residual stresses are superposed to obtain a local energy release rate, together with two phase angles. From the load at fracture, the critical interface energy release rate or interface toughness as a function of phase angles ψ and ϕ is determined. Results are compared to a fracture criterion.     

Complex variable method of computing Jk for bi-material interface cracks

A method using functions of a complex variable is developed for evaluation of J₁ and a modified J₂ integrals for bi-material interface cracks. This method, used in conjunction with the finite element method, would be useful in the prediction of stress intensity factors for cracks lying between the interface of two dissimilar materials. Since the direct evaluation of J₂ poses difficulties in modeling the singular behavior in the near vicinity around the crack tip for bi-material crack problems, it is modified by evaluating it around a contour path of small radius from the crack tip within the singularity dominated zone. It is shown that the stress intensity factors for a bi-material interface crack can be accurately evaluated using these j_k integrals.     

An integral representation of the stress intensity factors for three-dimensional static problems

For finding suitable expressions for the stress intensity factors (SIFs) under a general three-dimensional condition, the first stress invariant and the displacement tangent to a crack edge are analyzed. By using Green's theorem, the SIFs are expressed by integrals for the most general situations. K _I and K _II are expressed by integrals of the first stress invariant and its partial derivative. K _III is expressed by an integral of the displacement tangent to the crack edge and its partial derivative. The integrals include a surface integral on a smooth surface of arbitrary shape, and a line integral along part of the surface's boundaries. The expressions are valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. The expressions provide a new approach for the determination of the SIFs.     

Representation of stress intensity factors by path integrals of the first stress invariant or anti-plane displacement for general two-dimensional static problems

Determining stress intensity factors (SIFs) is a difficult task either analytically or experimentally. The difficulty arises from the fact that there is no simple and accurate expression for the SIFs under general circumstances. As a result, the determination of the SIFs is usually a complex process. For finding a suitable expression for the SIFs, the first stress invariant and anti-plane displacement are analyzed, and Green's theorem is used. It is found that the stress intensity factors can be represented by path integrals involving only the first stress invariant or anti-plane displacement for general two-dimensional static problems. K _I and K _II are represented by path integrals of the first stress invariant and its partial derivative. K _III is represented by a path integral of the anti-plane displacement as well as its partial derivative. The integrals are path-independent and valid for an arbitrarily shaped elastic medium with stationary cracks of arbitrary shape. They are also valid for a body containing isolated inhomogeneities such as holes and inclusions. If a crack is straight near its tip, and if the straight portion of the crack can be treated as a cut along the radius of a simply connected circular disk, there exists another kind of integrals representation that does not include the partial derivative terms in the representation for K _I. The representation by these integrals provides a new approach to determine the SIFs experimentally, which is simpler and more accurate. This is because the integrals are exact expressions for the SIFs and involve only the first stress invariant or anti-plane displacement.     

State‐of‐the‐art review on extended stress intensity factor concepts

The stress intensity factor concept for describing the stress field at pointed crack or slit tips is well known from fracture mechanics. It has been substantially extended since Williams' basic contribution (1952) on stress fields at angular corners. One extension refers to pointed V‐notches with stress intensities depending on the notch opening angle. The loading‐mode‐related simple notch stress intensity factors K₁, K₂ and K₃ are introduced. Another extension refers to rounded notches with crack shape or V‐notch shape in two variants: parabolic, elliptic or hyperbolic notches (‘blunt notches’) on the one hand and root hole notches (‘keyholes’ when considering crack shapes) on the other hand. Here, the loading‐mode‐related generalised notch stress intensity factors K_1ρ, K_2ρ and K_3ρ are defined. The concepts of elastic stress intensity factor, notch stress intensity factor and generalised notch stress intensity factor are extended into the range of elastic–plastic (work‐hardening) or perfectly plastic notch tip or notch root behaviour. Here, the plastic notch stress intensity factors K_1p, K_2p and K_3p are of relevance. The elastic notch stress intensity factors are used to describe the fatigue strength of fillet‐welded attachment joints. The fracture toughness of brittle materials may also be evaluated on this basis. The plastic notch stress intensity factors characterise the stress and strain field at pointed V‐notch tips. A new version of the Neuber rule accounting for the influence of the notch opening angle is presented.     

Double crack problem in nonlocal elasticity

The singular stress field around a sharp notch tip is expressed as a sum of two independent fields: a symmetric field with a stress singularity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+% cacaWGYbWaaWbaaSqabeaacaaIXaGaeyOeI0Iaeq4UdW2aaSbaaWqa% aiaaigdaaeqaaaaaaaa!3CC3!\[1/r^{1 - \lambda _1 } \]and a skew-symmetric field with a stress singularity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaac+% cacaWGYbWaaWbaaSqabeaacaaIXaGaeyOeI0Iaeq4UdW2aaSbaaWqa% aiaaikdaaeqaaaaaaaa!3CC4!\[1/r^{1 - \lambda _2 } \]. The intensities of the symmetric and skew-symmetric singular stress fields are defined in terms of constants K _I and K _II, respectively. In this study, a plane problem of a strip with single or double edge notches under tension or in-plane bending is considered. The bisector of the notch may be inclined to the edge, so that the two singular stress fields with different singularities may be created simultaneously at the notch tip. The body force method is used to calculate the stress intensity factors K _I and K _II. In numerical analysis, basic density functions of the body forces are introduced to characterize the stress singularity at the notch tip. The advantages of this technique are the high accuracy of results, due to the smoothness of the unknown weight functions, and the presence of the direct relation between the values of K _I and K _II and the values of unknown weight functions. The stress intensity factors are systematically calculated for the various geometrical conditions.     

Particle-size dependence of thermoelastic stress intensity factor in two-phase materials

The stress intensity factor is estimated for an annular crack originating from the particle-matrix interface in residual stress fields associated with a spherical particle of lower thermal expansion than that of the matrix. It is shown that the stress intensity factor is a function of particle size and pre-existing crack length. Spontaneous matrix cracking will occur when the particle size exceeds a critical value,R _c. Close agreement between the calculated and experimental values forR _c is obtained. The analysis is applicable to all particulate composites where there is volume increase of a particle induced either by phase transformation or thermal expansion mismatch.     

Boundary integral equation fracture mechanics analysis of plane orthotropic bodies

In this paper, a quick and efficient means of determining stress intensity factors, K _I and K _II, for cracks in generally orthotropic elastic bodies is presented using the numerical boundary integral equation (BIE) method. It is based on the use of quarter-point singular crack-tip elements in the quadratic isoparametric element formulation, similar to those commonly employed in the BIE fracture mechanics studies in isotropic elasticity. Analytical expressions which enable K _Iand K _II to be obtained directly from the BIE computed crack-tip nodal traction, or from the computed nodal displacements, of these elements are derived. Numerical results for a number of test problems are compared with those established in the literature. They are accurate even when only a very modest number of boundary elements are used.     

Interface fracture analysis of joints with a ductile interlayer

Plane strain problem of an interface crack with two interface shear yield zones and one crack-face contact zone is studied. The plastic yielding of the interlayer is stimulated by the interface shear yield zones and contact zone is included near one tip of the interface crack. An interesting and important phenomenon found in this analysis is that for such an interface crack the applied compressive normal stress can increase the stress intensity factor K ₁ at one of the two crack tips, the size of the crack-face contact zone, and the maximum value of the crack opening, in a combined normal and shear stress field. Examples are given for two pairs of materials used in ceramic-metal brazed joints, Si₃N₄/Ni and Si₃N₄/Incoloy 909.     

BEM for crack‐inclusion problems of plane thermopiezoelectric solids

The problem of interactions between an inclusion and multiple cracks in a thermopiezoelectric solid is considered by boundary element method (BEM) in this paper. First of all, a BEM for the crack–inclusion problem is developed by way of potential variational principle, the concept of dislocation, and Green's function. In the BE model, the continuity condition of the interface between inclusion and matrix is satisfied, a priori, by the Green's function, and not involved in the boundary element equations. This is then followed by expressing the stress and electric displacement (SED) and elastic displacements and electric potential (EDEP) in terms of polynomials of complex variables ξ_t and ξ_k in the transformed ξ‐plane in order to simulate SED intensity factors by the BEM. The least‐squares method incorporating the BE formulation can, then, be used to calculate SED intensity factors directly. Numerical results for a piezoelectric plate with one inclusion and a crack are presented to illustrate the application of the proposed formulation. Copyright © 2000 John Wiley & Sons, Ltd.     

A methodology for measuring interface fracture properties of composite materials

A methodology is presented for measuring interface fracture properties of composite materials. A bimaterial Brazilian disk specimen with a crack along the interface is employed. The specimen is analyzed by means of the finite element method and a conservative integral to determine stress intensity factors as a function of loading angle and crack length. A weight function is employed to determine the effect of residual curing stresses on the stress intensity factors. These are combined to determine the critical interface energy release rate _ic as a function of stress intensity phase angle for tests carried out on a glass/epoxy material pair.     

A fractographic criterion for subcritical crack-growth boundaries in hot-pressed alumina

The percent intergranular fracture (PIF) was measured along radii extending from fracture origins in hot-pressed alumina specimens, fractured at various loading rates and temperatures, and plotted versus estimates of stress intensity factors (K _I) at the various crack lengths. Minima in PIF occur at values ofK _I that are close to the critical stress intensity factors (K _IC) for cleavage on various crystal lattice planes in sapphire. The subcritical crack-growth boundary (K _I=K _IC of the polycrystalline material) occurs near the primary minimum in PIF suggesting that this minimum can be used as a criterion for locating this boundary. In addition, it was noted that the polycrystallineK _IC (4.2 MPa m^1/2) is very close to theK _IC for fracture on {¯1 ¯1 2 6} planes which is 4.3 MPa m^1/2. These observations suggest that critical crack growth begins when increased fracture energy can no longer be absorbed by cleavage on these planes. There is a secondary minimum atK _I>K _IC that appears to be associated with theK _IC necessary for fracture on combinations of planes selected by the fracture as alternatives to the high fracture-toughness basal plane.     

Stress intensity factor analysis of an interface crack between dissimilar anisotropic materials under thermal stress using the finite element analysis

New numerical methods were presented for stress intensity factor analyses of two-dimensional interfacial crack between dissimilar anisotropic materials subjected to thermal stress. The virtual crack extension method and the thermal M-integral method for a crack along the interface between two different materials were applied to the thermoelastic interfacial crack in anisotropic bimaterials. The moving least-squares approximation was used to calculate the value of the thermal M-integral. The thermal M-integral in conjunction with the moving least-squares approximation can calculate the stress intensity factors from only nodal displacements obtained by the finite element analysis. The stress intensity factors analyses of double edge cracks in jointed dissimilar isotropic semi-infinite plates subjected to thermal load were demonstrated. Excellent agreement was achieved between the numerical results obtained by the present methods and the exact solution. In addition, the stress intensity factors of double edge cracks in jointed dissimilar anisotropic semi-infinite plates subjected to thermal loads were analyzed. Their results appear reasonable.     

Conducting cracks in dissimilar piezoelectric media

Complete stress and electric fields near the tip of a conducting crack between two dissimilar anisotropic piezoelectric media, are obtained in terms of two generalized bimaterial matrices proposed in this paper. It is shown that the general interfacial crack-tip field consists of two pairs of oscillatory singularities. New definitions of real-valued stress and electric field intensity factors are proposed. Exact solutions of the stress and electric fields for basic interface crack problems are obtained. An alternate form of the J integral is derived, and the mutual integral associated with the J integral is proposed. Closed form solutions of the stress and electric field intensity factors due to electromechanical loading and the singularities for a semi-infinite crack as well as for a finite crack at the interface between two dissimilar piezoelectric media, are also obtained by using the mutual integral.     

Moving crack at the interface between two dissimilar magnetoelectroelastic materials

Summary Analytical solutions for an anti-plane Griffith crack moving at the interface between two dissimilar magnetoelectroelastic media under the conditions of permeable crack faces are formulated using the integral transform method. The far-field anti-plane mechanical shear and in-plane electrical and magnetic loadings are applied to the magnetoelectroelastic materials. Expressions for stresses, electric displacements, and magnetic inductions in the vicinity of the crack tip are derived. Field intensity factors for magnetoelectroelastic material are obtained. The stresses, electric displacements and magnetic inductions at the crack tip show inverse square root singularities, and it is found that the dynamic stress intensity factor (DSIF), the dynamic electric displacement intensity factor (DEDIF) and the dynamic magnetic induction intensity factor (DMIIF) are independent of the remote electromagnetic loads. The moving speed of the crack has influence on the DEDIF and the DMIIF. When the crack is moving at lower speeds 0 ≤ M≤ M_c1 or higher speeds M_c2 < M < 1, the crack will propagate along its original plane, while in the range of M_c1 < M < M_c2 , the propagation of the crack possibly brings about the branch phenomena in magnetoelectroelastic media.