Interface crack problem in elastic dielectric/piezoelectric bimaterials

By considering an isotropic elastic dielectric material as a transversely isotropic piezoelectric material with little piezoelectricity, the interface crack problem in elastic/piezoelectric bimaterials is treated in this paper based on Stroh's complex potential theory (1958) with the impermeable crack model. In order to obtain universal results, Numerical results of the near tip stress field and the electric field for 35 kinds of dissimilar bimaterials constructed by five kinds of elastic dielectric materials, namely Epoxy, Polymer, Al₂O₃, SiC and Si₃N₄, and seven kinds of piezoelectric ceramics, namely PZT-4, BaTiO₃, PZT-5H, PZT-6B, PZT-7A, P-7, and PZT-PIC151, are presented. It is concluded that all the combinations lead to the same results: in which the first crack tip singularity parameter does not vanish whereas the second parameter always vanishes. From the physical point of view, an interface crack in such a dissimilar material shows a similar oscillating singularity as that in dissimilar elastic bimaterials. Moreover, by using a maximization value technique, the regular inverse square root singularity r ^–1/2 of the stress and the electric field at the crack tip can be realized, although, theoretically, an interface crack in such bimaterials possesses the well-known oscillating singularity r ^{–1/2± i}. Of great significance is that, in the absence of mechanical loadings, a purely electric loading can induce relative large model I or II stress intensity factor for a interface crack in some elastic/piezoelectric bimaterials, which implies that the electric-induced failure may be realized in such bimaterials.     

Singular fields of a mode III interface crack in a power law hardening bimaterial

A high order of asymptotic solution of the singular fields near the tip of a mode III interface crack for pure power law hardening bimaterials is obtained by using the hodograph transformation. It is found that the zero order of the asymptotic solution corresponds to the assumption of a rigid substrate at the interface, and the first order of it is deduced in order to satisfy completely two continuity conditions of the stress and displacement across the interface in the asymptotic sense. The singularities of stress and strain of the zero order asymptotic solutions are –1/(n ₁+1) and –n/(n ₁+1) respectively (n=n ₁, n ₂ is the hardening exponent of the bimaterials). The applicability conditions of the asymptotic solutions are determined for both zero and first orders. It is proved that the Guo-Keer solution [23] is limited in some conditions. The angular functions of the singular fields for this interface crack problem are first expressed by closed form.     

Crack path selection in piezoelectric bimaterials

The problem of crack path selection in piezoelectric bimaterials is considered in this paper. Based on the Stroh formulation for anisotropic material and Green’s functions for piezoelectric bimaterials, the crack problem is expressed in terms of coupled singular integral equations at first, and then the equations are used to solve for stress and electric displacement fields numerically. A crack impinging an interface joining two dissimilar materials may arrest or may advance by either penetrating the interface or deflecting into it. The competition between deflection and penetration is investigated using the maximum energy release rate criterion. Numerical results are presented to study the role of remote electroelastic loads on the path selection of crack extension.     

Boundary element analysis of fracture mechanics in anisotropic bimaterials

This paper presents a boundary element formulation for the analysis of linear elastic fracture mechanics problems involving anisotropic bimaterials. The most important feature associated with the present formulation is that it is a single domain method, and yet it is accurate, efficient and versatile. In this formulation, the displacement integral equation is collocated on the uncracked boundary only, and the traction integral equation is collocated on one side of the crack surface only. The complete Green's functions for anisotropic bimaterials are also derived and implemented into the boundary integral formulation so that discretization along the interface can be avoided except for the interfacial crack part. A special crack-tip element is introduced to capture exactly the crack-tip behavior.Numerical examples are presented for the calculations of stress intensity factors for a straight crack with various locations in infinite bimaterials. It is found that very accurate results can be obtained by the proposed method even with relatively coarse discretization. Numerical results also show that material anisotropy can greatly affect the stress intensity factor.     

Elastic plastic mode III crack under internal shear

A complete solution is presented for the problem of a mode III crack in an infinite elastic perfectly-plastic solid under internal shear stress. This problem is the anti-plane strain equivalent of a mode I crack with internal pressure. The problem is transformed into a boundary value problem for a potential function. The particular case when the applied stress σA is equal to the yield stress σ0 is solved analytically, and the distance to the elastic-plastic boundary is obtained in closed form. The general case when σA σ0 is solved numerically by using the Boundary Element Method for potential problems. Numerical results are given for the distance to the elastic-plastic boundary and the crack tip opening displacement. The extent of the plastic zone ahead of the crack tip is shown to vary linearly with the ratio σA/σ0) when 0.5 ≤ (σA/σ0) ≤ 1. This revised version was published online in August 2006 with corrections to the Cover Date.     

Analysis of sharp angular notches in anisotropic materials

The singular stresses at the tip of a sharp angular notch are analysed for the most general case of elastic anisotropy. The problem is stated in relation with the kinked crack and is modelled by means of continuous distributions of dislocations which are assumed to be singular at the notch vertex, the kind of the main singularity λ being unknown and weaker than at the crack tip. The Mellin transform is applied to obtain a system of simultaneous functional equations that enables one to find the parameter λ. The reciprocal theorem is used to compute the generalised stress intensity factor which characterises the singular stresses in a neighbourhood of the notch tip. This revised version was published online in July 2006 with corrections to the Cover Date.     

Propagation of a Mode-II crack in a viscoelastic medium with different bulk and shear relaxation

A solution for a crack propagating under shear-loading in an isotropic viscoelastic medium with different relaxation under volume and shear deformations is presented. The medium is infinite and the semi-infinite crack propagates along the x ₁-axis at constant speed V, which may take any value up to the speed of dilatational waves. The requisite Riemann–Hilbert problem for the steady-state case has been solved and the asymptotics of the stress component σ₁₂ directly ahead of the crack and at infinity have been obtained.     

Analytical evaluation of J-integral for elliptical and parabolic notches under mode I and mode II loading

In the present work the J-integral (indicated here as J_Vρ because two parallel flanks are not present) was calculated by using, along the free border, the exact analytical stress distribution for the ellipse and the asymptotic one for parabolic notches. The material was assumed as homogeneous isotropic and linear elastic. First, for an ellipse under remote tensile loading, the expression of J_Vρ has been analytically calculated on the basis of Inglis’ equations. The equations have been used to prove that, in terms of J-integral, the crack is the limit case of an equivalent elliptic notch. Furthermore, by distinguishing the symmetric and skew-symmetric terms, the well-known Stress Intensity Factors (SIF) of mode I and II for a crack in a wide plate under tension are obtained by adding a limiting condition. Second, by means of Creager–Paris’ equations, J_Vρ has been analytically calculated for a parabolic notch of assigned tip notch radius ρ. The asymptotic value of J_Vρ and the relationship between the peak stress and the relative SIF are the same as the ellipse. Finally, as an engineering application, we provide an accurate formula for the evaluation of the Notch Stress Intensity Factors of a crack, mainly subjected to tensile stress, from the peak stress of the equivalent ellipse under the same loading.     

Prediction of fracture toughness for heat-resistant steels considering specimen dimensions. Report 2. Ductile fracture

A model of ductile failure of a body with a crack has been developed which enables predicting fracture toughness on the upper shelf of the fracture toughness temperature dependence taking into account the influence of the stress state. The model is based on the physical-mechanical model of ductile failure which is controlled by the critical value εf reached by plastic strain at the crack tip ε _i ^ρ . In this case it is assumed that both the ε _i ^ρ value, which precedes the crack growth onset by the mechanism of pore coalescence, and the critical strain εf are functions of specific stress state parameters, namely: the critical strain is a function of the stress state triaxiality σ _m /σ _n (σ _m is the hydrostatic stress, σ _i is the stress intensity), and ε _i ^ρ is a function of the parameter χ introduced, which is an explicit function of all three principal local stresses in the process zone at the crack tip and which defines the degree to which the stress state approaches the plane strain conditions for a body of specified thickness. The model developed has two modifications one of which enables predicting fracture toughness of large-size bodies from the results of testing only small cylindrical specimens without cracks (smooth and with a circular recess) and the other from the results of testing small cylindrical specimens and small specimens with a crack. Translated from Problemy Prochnosti, No. 2, pp. 5–19, March–April, 1997.     

Parallel crack near the interface of magnetoelectroelastic bimaterials

A parallel crack near the interface of magnetoelectroelastic bimaterials is considered. The crack is modelled by using the continuously distributed edge dislocations, which are described by the density functions defined on the crack line. With the aid of the fundamental solution for the edge dislocation, the present problem is reduced to a system of singular integral equations, which can be numerically solved by using the Chebyshev numerical integration technique. Then, the stress intensity factor (SIF), the magnetic induction intensity factor (MIIF) and the electric displacement intensity factor (EDIF) at the crack tips are evaluated. Using these fracture criteria, the cracking behaviour of magnetoelectroelastic bimaterials is investigated. Numerical examples demonstrate that the interface, mechanical load, magnetic load and material mismatch condition are all important factors affecting the fracture toughness of the magnetoelectroelastic bimaterials.     

Quasi-static crack tip fields in rate-sensitive FCC single crystals

In this work, the effects of loading rate, material rate sensitivity and constraint level on quasi-static crack tip fields in a FCC single crystal are studied. Finite element simulations are performed within a mode I, plane strain modified boundary layer framework by prescribing the two term (K − T) elastic crack tip field as remote boundary conditions. The material is assumed to obey a rate-dependent crystal plasticity theory. The orientation of the single crystal is chosen so that the crack surface coincides with the crystallographic (010) plane and the crack front lies along [10[`1]][10\overline 1] direction. Solutions corresponding to different stress intensity rates [(K)\dot]\dot{{K}}, T-stress values and strain rate exponents m are obtained. The results show that the stress levels ahead of the crack tip increase with [(K)\dot]\dot{{K}} which is accompanied by gradual shrinking of the plastic zone size. However, the nature of the shear band patterns around the crack tip is not affected by the loading rate. Further, it is found that while positive T-stress enhances the opening and hydrostatic stress levels ahead of crack tip, they are considerably reduced with imposition of negative T-stress. Also, negative T-stress promotes formation of shear bands in the forward sector ahead of the crack tip and suppresses them behind the tip.     

Simulation of Fatigue Crack Propagation in Ductile Metals by Blunting and Re-sharpening

Laird and Smith [(1962). Philosophical Magazine 8, 847–857] proposed a plastic sliding-off mechanism for the stage II fatigue crack growth via striation formation. In their view, the fatigue crack extension results solely from the changing character of deformation at the crack tip during loading and unloading. In particular, the crack tip blunts during the loading stage and folds into a double notch during the unloading stage, resulting in striation formation. In order to verify Laird’s plastic blunting mechanism for ductile polycrystals as well as for ductile fcc single crystals, FE calculations were performed for a rectangular plate with an initially sharp crack under plane strain conditions. The plate was subjected to a fully reversed tension-to-pressure cyclic load perpendicular to the crack plane (Mode 1). In the single crystal case the crack propagation simulations were carried out for cracks with crack plane (001) for two different crack growth orientations [110] and [100]. No initial radius for the crack tip was assumed. The actual shape of the crack tip followed from an initially sharp crack by repeated remeshing. To model the constitutive behavior typical for polycrystalline ductile metals, J₂ hypo-elasto-plasticity model with Armstrong–Frederick kinematic hardening was used. To model the constitutive behavior typical for ductile fcc single crystals, a geometrically nonlinear version of Cailletaud’s model based on the multiplicative elasto-plastic decomposition of the deformation gradient was implemented into the FE program ABAQUS. For simplicity, only octahedral slip systems were considered. Using repeated remeshing for severely distorted elements at the advancing crack tip, deformation patterns in the sense of Laird’s mechanism for fatigue crack propagation with striation formation were obtained in the case of the polycrystal simulation as well as in the case of the single crystal simulation for [110] crack growth direction. The simulation for [100] crack growth direction with the same stress level as for [110] direction also yielded crack extension by progressive large deformations but without striation formation. The dependence of the fatigue striation formation on the crack growth direction as predicted by the simulation of crack propagation in single crystals is verified by the experimental results of Neumann [(1974). Acta Metallurgica 22, 1155–1165] on pure copper single crystals.     

A Note for the Crack Problem of Functionally Graded Materials

Employing the power-type function of material properties, a crack lying between the functionally graded materials (FGMs) and homogeneous substrate is studied by an asymptotic analysis from that of bimaterials, J-integral and the numerical calculations. The present results show that when the curve of the material property is concave, i.e. the power (m) of function of material property is great than 1, the stress distribution near the crack-tip is the same as that of homogeneous materials, which is in agreement with previous findings. However, if the curve of the material property is convex corresponding to 0< m <1, our results show that the stress distribution is strongly affected by m and it can be obtained asymptotically from that of bimaterials.     

A unified and integrated numerical‐analytical approach for evaluation of Jk‐integrals of an interfacial crack in orthotropic and isotropic bimaterials

A new unified and integrated method for the numerical‐analytical calculation of J_k‐integrals of an in‐plane traction free interfacial crack in homogeneous orthotropic and isotropic bimaterials is presented. The numerical algorithm, based on the boundary element crack shape sensitivities, is generic and flexible. It applies to both straight and curved interfacial cracks in anisotropic and/or isotropic bimaterials. The shape functions of semidiscontinuous quadratic quarter point crack tip elements are correctly scaled to adapt the singular oscillatory near tip field of tractions. The length of crack is designated as the design variable to compute the strain energy release rate precisely. Although an analytical equation relating J₁ and stress intensity factors (SIFs) exists, a similar relation for J₂ in debonded anisotropic solids for decoupling SIFs is not available. An analytical expression was recently derived by this author for J₂ in aligned orthotropic/orthotropic bimaterials with a straight interface crack. Using this new relation and the present computed J_k values, the SIFs can be decoupled without the need for an auxiliary equation. Here, the aforementioned analytical relation is reconstructed for cubic symmetry/isotropic bimaterials and used with the present numerical algorithm. An example with known analytical SIFs is presented. The numerical and analytical magnitudes of J_k for an interface crack in orthotropic/orthotropic and cubic symmetry/isotropic bimaterials show an excellent agreement.     

Cylindrical crack <Emphasis Type="Italic">vis a vis</Emphasis> parallel crack pair: shielding,energetics and asymptotics

Fibrous materials often contain cylindrical cracks due to delamination along the matrix-fiber interface. It is instructive to analyse a cylindrical crack of length 2a and diameter 2h in a homogeneous medium and compare the results with those for a pair of parallel cracks of length 2a and spacing 2h. The pair of parallel cracks mutually shielding each other is examined here with regard to the variation of stress intensity factors and energetics including the asymptotic limit of a pair of nearly coalescing parallel cracks. A unified formulation for parallel cracks/cylindrical crack based on crack opening displacement (COD) in terms of Chebyshev polynomials is developed. The characteristic variation of stress intensity factors as the cracks approach each other (h → 0) shows that the stress intensity factors vanish for the case of a vanishingly small cylindrical crack but not for the 2D parallel pair of cracks. The 2D case of a pair of collapsing parallel cracks ensures a finite energy release rate asymptoting to that of a single crack. Further research is needed to establish definitive asymptotic bounds for the case of extremely closely spaced cracks on the lines of Hutchinson and Suo (Adv Appl Mech 29:377–384, 1992), Kachanov (1993) and Gorbatikh et al. (Int J Fract (Lett Fract Micromech) 143:377–384, 2007). Results are presented for different values of Poisson’s ratio.     

Fracture mechanics of piezoelectric materials

This paper presents an analysis of crack problems in homogeneous piezoelectrics or on the interfaces between two dissimilar piezoelectric materials based on the continuity of normal electric displacement and electric potential across the crack faces. The explicit analytic solutions are obtained for a single crack in an infinite piezoelectric or on the interface of piezoelectric bimaterials. For homogeneous materials it is found that the normal electric displacement D₂, induced by the crack, is constant along the crack faces which depends only on the remote applied stress fields. Within the crack slit, the perturbed electric fields induced by the crack are also constant and not affected by the applied electric displacement fields. For bimaterials, generally speaking, an interface crack exhibits oscillatory behavior and the normal electric displacement D₂ is a complex function along the crack faces. However, for bimaterials, having certain symmetry, in which an interface crack displays no oscillatory behavior, it is observed that the normal electric displacement D₂ is also constant along the crack faces and the electric field E₂ has the singularity ahead of the crack tip and has a jump across the interface. Energy release rates are established for homogeneous materials and bimaterials having certain symmetry. Both the crack front parallel to the poling axis and perpendicular to the poling axis are discussed. It is revealed that the energy release rates are always positive for stable materials and the applied electric displacements have no contribution to the energy release rates. This revised version was published online in July 2006 with corrections to the Cover Date.     

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.     

The Mixed Mode Crack Problem in an Inhomogeneous Orthotropic Medium

The mixed mode crack problem in plane elasticity for a graded and oriented material is considered. The material property grading is intentional, whereas the property orientation or orthotropy is usually the consequence of material processing. It is assumed that the crack is located in a plane perpendicular to the direction of property grading and the principal axes of orthotropy are parallel and perpendicular to the crack. The four independent engineering constants E₁₁, E₂₂, G₁₂, and ν₁₂ are replaced by a stiffness parameter, E = √E₁₁ E₂₂, a stiffness ratio, δ = (E₁₁/E₂₂)^1/4, a Poisson's ratio, ν = √ν₁₂ ν₂₁, and a shear parameter κ₀ = (E/2G₁₂) - ν. The corresponding mixed boundary value problem is reduced to a system of integral equations which is solved for various loading conditions and material parameters. The results presented consist of the strain energy release rate, the stress intensity factors and the crack opening displacements. It is found that generally the stress intensity factors increase with increasing material inhomogeneity parameter and shear parameter and with decreasing stiffness ratio. This revised version was published online in July 2006 with corrections to the Cover Date.     

Screw dislocation interacting with interfacial and interface cracks in piezoelectric bimaterials

Interface and interfacial cracks interacting with screw dislocations in piezoelectric bimaterials subjected to antiplane mechanical and in-plane electrical loadings are studied within the framework of linear piezoelectricity theory. Straight dislocations with the Burgers vector normal to the isotropic basal plane near the interface or interfacial crack are considered. The dislocations are characterized by a discontinuous electric potential across the slip plane and are subjected to a line-force and a line-charge at the core. An explicit solution for the screw dislocation in piezoelectric bimaterial with straight interface is found based on the solution of a similar problem for infinite homogenous medium. The obtained relation is independent of the nature of singularity. This fundamental result is used to analyze dislocation interacting with a set of collinear interfacial cracks in piezoelectric bimaterials. Three solutions for the screw dislocation interacting with a semi-infinite crack, finite crack, and edge crack between two bonded dissimilar piezoelectric materials are obtained in closed-form. These solutions can be used as Green’s functions for the analyses of interfacial cracks in piezoelectric bimaterials.     

Dynamic crack growth along an elastoplastic bimaterial interface

Summary The near-tip stress and deformation rate fields of a crack dynamically propagating along an interface between dissimilar elastic-plastic bimaterials are presented in this paper. The elastic-plastic materials are characterised by theJ ₂-flow theory with linear plastic hardening. The solutions are assumed to be of variable-separable form with a power-law singularity in the radial direction. Two distinct solutions corresponding to the tensile and shear solutions exist with slightly different singularity strengths and very different mixities at the crack tip. The phenomenon of discrete and determinate mixities at the interfacial crack tip is confirmed in dynamic crack growth. This is not an artifact of the variable-separable solution assumption, arising from the linear-hardening material model. The dynamic crack analysis shows that the mixity of the near-tip field is mainly determined by the given material parameters and affected slightly by the crack propagation velocity. A significant variation of the mixity is observed near to the coalescing point of the tensile and shear solutions. The strength of the singularity is almost determined by the smaller strain-hardening alone, and dynamic inertia decreases the stress intensity. The asymptotic solutions reveal that the crack propagation velocity changes only the stress field of the tensile mode significantly. With increasing the crack propagation velocity, the stress singularity of the tensile solutions decreases obviously and the stress triaxiality at the tip (=0) falls considerably at the unity effective stress. These observations imply that the fracture toughness of the interface crack under tensile mode may be significantly higher than that under quasi-static conditions.