Dissimilar Elastic Materials Research Articles

The polarization vectors at the interface and the secular equation for Stoneley waves in monoclinic bimaterials

Stoneley waves propagating in the direction of the x 1 –axis in a bimaterial consisting of two half–spaces x 2 ⩾ 0 and x 2 ⩽ 0 of dissimilar anisotropic elastic materials are considered. Several invariants, independent of x 2 , that relate the displacement and the stress are obtained for a general anisotropic elastic bimaterial. We then study the case when both materials have the symmetry plane at x 2 = 0 (or x 1 = 0). The displacement and the traction at the interface x 2 = 0 describe two separate elliptic paths. They are polarized on planes that contain the x 2 –axis (or x 1 –axis). Moreover, the x 2 –axis (or x 1 –axis) is one of the principal axes of the ellipse. In the rest of the paper we consider special monoclinic bimaterials with the symmetry plane at x 2 = 0, x 1 = 0 or x 3 = 0. The elastic constants for the two monoclinic materials are identical except that those elastic constants that would vanish if the material were orthotropic have the opposite sign for the two materials. It is shown that one of the ellipses degenerates into a line along a coordinate axis while the other ellipse is on a coordinate plane normal to this coordinate axis. When the symmetry plane is at x 3 = 0, both ellipses degenerate into lines along the x 1 – and x 2 –axes. Explicit expressions of the polarization vectors and the secular equation are presented for all three monoclinic bimaterials.

Cohesive zone modeling of interface fracture in elastic bi-materials

Some basic issues regarding the cohesive zone modeling of interface fracture between two dissimilar elastic materials are studied. The dependence of the cohesive energy density on the phase angle is first discussed under small scale cohesive zone conditions. It is then shown that in general the stress singularities in tension and shear cannot be simultaneously removed at the cohesive zone tip if a single cohesive zone length is adopted for both tensile and shear fracture modes. Finally, the energy dissipation at the tip of a prescribed cohesive zone is examined using a bilinear cohesive zone model under the uncoupled tension/shear conditions.

Modeling the fatigue crack growth and propagation life of a joint of two elastic materials using interface elements

In this study, a new approach for the simulation of fatigue crack growth in two dissimilar elastic materials has been developed and specifically, the concept has been applied to a butt-welded joint. The phenomena of crack propagation and interface debonding can be regarded as the formation of new surface. Thus, it is possible to model these problems by introducing the mechanism of surface formation. In the proposed method, the formation of new surface is represented by an interface element based on the interface potential energy. The properties of this interface element represent the bonding strength of the material. As the cyclic load continues, the bonding strength decreases between the interacting surfaces and the crack propagates slowly. Based on this concept, an ANSYS code has been written for the simulation of crack propagation. Using this code, the fatigue crack growth rate and fatigue crack propagation life of a 2D Finite Element Method (FEM) model of a butt-welded joint for different stress ratios have been analyzed and presented in this paper. The variation of the crack opening displacement (COD) over crack lengths and crack tip stress and strain over crack tip stress have also been presented. The trend of fatigue crack growth rate is similar to results found in the literature. For validation, the predictions are compared with experimental results and reasonably good comparisons are found. The method is relatively simple compared to other conventional FEM methods and it eliminates the limitation of crack propagation of one element length per cycle and saves computer time significantly.

Near-tip stress fields and intensity factors for an interface crack in metal/piezoelectric bimaterials

The primary goal of this research is to show the fundamental features of an interface crack in metal/piezoelectric bimaterials via Stroh's theory [Phil. Mag. 7 (1958) 625]. Based on the previous works [Phil. Mag. 7 (1958) 625; J. Mech. Phys. Solids 40 (1992) 739; Int. J. Fract. 119 (2003) L41; Singularities and near-tip field intensity factors of piezoelectric interface cracks, J. Mech. Phys. Solids (in press)] and by considering a metal as a special piezoelectric material with extremely small piezoelectricity and extremely large permittivity (conductor), we obtain the two dominant parameters ε and κ for description of interface crack-tip singularity in such bimaterials. Numerical results show that almost all of such bimaterials have the feature that the first parameter ε vanishes whereas the second parameter κ remains non-zero. An interface crack in these bimaterials always possesses the stress singularity r −1/2± κ at the crack tip. From the physical point of view, this implies that an interface crack in such bimaterials shows a feature with non-oscillating singularity, which is far apart from previous results [Prik. Mat. Mekh. 39 (1975) 145; V.Z. Parton, B.A. Kudryavtsev, Electromagnetoelasticity, Gordon and Breach Science Publishers, New York, 1988; J. Mech. Phys. Solids. 51 (2003) 921] and our classical understanding in dissimilar elastic or anisotropic materials [Bull. Seism. Soc. Am. 49 (1959) 119; ASME J. Appl. Mech. 32 (1965) 400]. On the other hand, there is one exceptional metal/piezoelectric bimaterial with non-zero ε and vanishing κ, the oscillating stress singularity r −1/2±i ε at the crack tip is reached in this bimaterial. Consequently, metal/piezoelectric bimaterials are categorized into two classes: one with non-zero κ and vanishing ε could be called as κ-class metal/piezoelectric bimaterials and the other one with non-zero ε and vanishing κ could be termed as ε-class metal/piezoelectric. Analysis of the crack-tip generalized stress field is performed. Of great significance is that: if a purely electric-induced interface crack growth occurs in metal/piezoelectric bimaterials, it is most likely due to the shear mode II fracture rather than the open mode I, and then a purely remote electrical loading enhances the interface crack extension in such bimaterials. Only when an external tensile loading is applied, could the mode I fracture play a dominant role.

The asymptotic stress field for a rigid circular inclusion at the interface of two bonded dissimilar elastic half-space materials

The paper considers three-dimensional interface inclusion problems. The axisymmetric elastostatics problem of a rigid circular inclusion at the interface between two perfectly bonded dissimilar elastic half spaces is analyzed. Based on the representations of displacements and stresses in terms of Love's strain potential and the Hankel transform technique, the mixed boundary value problem associated with a rigid circular inclusion at the interface reduces to a pair of simultaneous integral equations for the stress jumps across the inclusion, which are further transformed to a single singular integral equation. For the case of uniform axial and radial tensions at infinity, the asymptotic stresses near the inclusion front are obtained and they exhibit the oscillatory singularity. Meanwhile, the magnitude of the singularity for the interface inclusion depends on the material constants of the upper and lower half spaces, the dependence of singularity coefficients on material constants for interface inclusion problems, however, is different from that for interface crack problems.

A revisited criterion for crack deflection at an interface in a brittle bimaterial

An asymptotic analysis is carried out to model the mechanism of deflection of a crack at an interface in a brittle bimaterial. Necessary conditions are derived from an energy analysis based on seeking the path which provides the maximum of additional energy released by the fracture process. In the case of a stationary crack impinging perpendicularly on the interface and submitted to progressive loading, the energy criterion depends on the elastic mismatch of the bimaterial constituents and the ratio of the crack extensions in the deflected and the penetrated directions. The criterion reveals high sensitivity to the value of this ratio. Applying this criterion thus requires an arbitrary assumption concerning the value of the crack extension ratio, as was proposed by He and Hutchinson (He, MY, Hutchinson JW. Crack deflection at an interface between dissimilar elastic materials. Int J Solids Structures 1989;25:1053–67). Considering now a crack propagating towards the interface, an improved criterion which does not require any assumption concerning the extension ratio is established by using a quasi-static approximation and assuming that the deflection mechanism occurs under constant loading.

Debonding of the interface as 'crack arrestor'

In this paper, we studied the interface debonding when a crack perpendicularly approaches an interface between two dissimilar elastic materials. An interface toughness law was first defined according to an adhesive model governing the interface fracture. By analysing the interaction between the normally approaching crack and the interface crack and by tacking account of the adhesive forces at ends of the interfacial crack, a model for studying the interface debonding and the debonding stability was established. It is observed that the interface debonding toughness depends strongly on the mixed mode locally produced over the plastic adhesive zone of the interface. Moreover, the interface debonding may be unstable, i.e. the interface debonding length may jump from an initial value to a certain final value under critical remote loading. This jump may be surprisedly important in certain cases. These results agree with the experimental works gathered so far and can be used to explain the mechanism of 'crack arrestor' formed by an interface.

A cohesive zone model for cracks terminating at a bimaterial interface

Linear elastic fracture mechanics (LEFM) does not provide a realistic propagation criterion for a crack tip touching a bimaterial interface. In fact, LEFM predicts that the crack penetrates the interface at either zero or infinite value of the characteristic applied load, depending on the relative stiffness of the bonded materials. This paper presents a cohesive zone model that provides a propagation criterion for such cracks in terms of the parameters that define the relation between the crack opening displacement and the traction acting along the crack surfaces. Extensive numerical results are presented for the case of constant cohesive traction, σo associated with a critical crack tip opening displacement,ηc. A quantitative evaluation of the effective toughening resulting from the presence of the interface is presented, for both small scale and large scale bridging, in terms of the Dundurs parameters (α and β), and ρ2/L, where ρ2 is proportional to the small scale critical cohesive zone length and L is a characteristic length of the crack problem. In particular, universal results for small scale bridging are presented as where kc and δc are, respectively the critical stress intensity factor and critical cohesive zone length, λ is the power of the stress singularity associated with the elastic crack touching the interface, and A and B∗ are universal functions. These equations generalize those derived from the Dugdale model for a homogeneous medium. It is shown through the analysis of a finite length crack that for a relatively wide range of α-β and ρ2/L values, the presence of the interface has a rather insignificant effect on the critical stress, and the elastic singularity associated with a crack terminating at the interface between two dissimilar elastic materials dominates the stress field within an extremely small near-tip region.

Separation of crack extension modes in orthotropic delamination models

In modeling a crack along a distinct interface between dissimilar elastic materials, the ratio of mode I to mode II stress intensity factors or energy release rates is typically not unique, due to oscillatory behavior of near-tip stresses and displacements. Although methods have been developed for comparing mode mixes for isotropic interfacial fracture problems, this behavior currently limits the applicability of interfacial fracture mechanics in predicting delamination in layered materials without isotropic symmetry. The virtual crack closure technique (VCCT) is a method used to extract mode I and mode II energy release rate components from numerical fracture solutions. Energy release rate components extracted from an oscillatory solution using the VCCT are not unique due to their dependence on the virtual crack extension length, Δ. In this work, a method is presented for using the VCCT to extract Δ-independent energy release rate quantities for the case of an interface crack between two in-plane orthotropic materials. The method does not involve altering the analysis to eliminate its oscillatory behavior and it is similar to existing methods for extracting a mode mix from isotropic interfacial fracture models. Knowledge of near-tip fields is used to determine the explicit Δ dependence of energy release rate parameters. Energy release rates are then defined that are separated from the oscillatory dependence on Δ. A modified VCCT using these energy release rate definitions is applied to results from finite element analyses, showing that Δ-independent energy release rate quantities result. The modified technique has potential as a consistent method for extracting a mode mix from numerical solutions. The Δ-independent energy release rate quantities extracted using this technique can also aid numerical modelers, serving as guides for testing the convergence of finite element models. Direct applications of this work include the analysis of planar composite delamination problems, where plies or debonded laminates are modeled as in-plane orthotropic materials.

Crack deflection at an interface between dissimilar elastic materials: Role of residual stresses

A crack intersecting an interface between two dissimilar materials may advance by either penetrating through the interface or deflecting into the interface. The competition between deflection and penetration can be assessed by comparison of two ratios: (i) the ratio of the energy release rates for interface cracking and crack penetration; and (ii) the ratio of interface to material fracture energies. Residual stresses caused by thermal expansion misfit can influence the energy release rates of both the deflected and penetrating crack. This paper analyses the role of residual stresses. The results reveal that expansion misfit can be profoundly important in systems with planar interfaces (such as layered materials, thin film structures, etc.), but generally can be expected to be of little significance in fiber composites. This paper corrects an earlier result for the ratio of the energy release rate for the doubly deflected crack to that for the penetrating crack in the absence of residual stress.

Deflection and penetration of cracks at an interface between two dissimilar materials

The tendency of a crack to deflect or penetrate at an interface between two dissimilar elastic materials in a finite-sized sample is investigated with the boundary element method. The ratio of the energy release rate of a deflecting crack to the maximum energy release rate of a penetrating crack is computed as a function of the Dundurs' elastic mismatch parameters for several double-edged notch specimen geometries and loading conditions. Over the entire range of material parameters, the results displayed only a weak dependence on the loading conditions imposed on the specimen, the aspect ratio of the specimen, and on Dundurs' parameter, β. For moderate differences in the relative stiffnesses of the two materials and when the crack is advancing toward a stiffer material, we find essentially no difference between the singly and doubly deflected crack, and the numerical calculations are in excellent agreement with recent analytical predictions [1]. However, when the crack is advancing into a material of much lower modulus, the numerical calculations for a doubly deflected crack are smaller than the analytical predictions.

Prediction of arbitrary crack growth from the interface between two dissimilar elastic materials

Based on the universal laws of stress distribution around a crack edge, the general analysis of crack start from the interface is given. The solution for the original crack is assumed to be known. In order to classify different possible cases of crack growth beginning, both a slip-region ahead of the opening zone and a core region near the crack edge are introduced. The former provides non-overlapping of the crack surfaces and removes an undesirable oscillating stress field singularity which is produced mathematically at the assumption of a completely opening crack. The latter means that we consider the damage of an elementary volume (geometrical measure of the microstructure) as a discrete fracture operaton. Two asymptotical cases are studied: the slip-region is much smaller or much larger than the cross-section of the core region. Then the stress-strain state on the core region periphery qualitatively is the same as for a completely opening or shear interface crack, respectively. A characteristic crack opening appears instead of a characteristic length of the slip-region for a blunted crack. The angles of crack departure are predicted according to different known criteria of fracture. In passing, parametric, analysis of stresses and energy density angle distributions are given. Plane and penny-shaped cracks are examined as the illustration.

Singular behaviour and asymptotic stress field of a crack terminating at a bimaterial interface

A crack terminating at an interface of two dissimilar elastic materials is investigated. It is found that the asymptotic stress field near the crack tip is in general composed of two parts with each part being characterized by one singularity. The detailed relation of the two singularities with the bimaterial properties is given for some special cases of the crack.

Finite element analysis of a crack approaching an interface between dissimilar elastic materials

Finite element analysis of a crack approaching an interface between dissimilar elastic materials

Treatment of bimaterial interface crack problems using the boundary element method

Quadratic quarter-point crack-tip elements are introduced in the two-dimensional boundary element analysis of problems in which a crack lies along the interface of dissimilar elastic materials. Such problems present some modelling difficulties using conventional procedures because the stresses are oscillatorily singular in the neighbourhood of the crack-tip. Analytical expressions relating the stress intensity factor to the computed displacements of the crack-tip element or to the computed crack-tip nodal tractions are derived and their veracity demonstrated with some examples. Numerical results, compared with exact solutions where possible, are accurate even with relatively coarse mesh designs.

Crack deflection at an interface between dissimilar elastic materials

A crack impinging an interface joining two dissimilar materials may arrest or may advance by either penetrating the interface or deflecting into the interface. The competition between deflection and penetration is examined in this paper when the materials on either side of the interface are elastic and isotropic. The energy release rate for the deflected crack is compared with the maximum energy release rate for a penetrating crack. The results can be used to determine the range of interface toughness relative to bulk material toughness which ensures that cracks will be deflected into the interface.

An application of M-integral to cracks in dissimilar elastic materials

Here W is the strain energy density, n. is the unit normal to curve C, and u. is displacement vector. T. ~s the traction acting on C. It was foun~ by Smelser and Gurtin [5] ~hat the M-integral conservation law holds likewise for dissimilar materials whose interfaces lie along radial lines of the coordinate system chosen. By combining these results Nachman et al. [6,7] gave an expression of energy release rate for an edge crack along the interface between two elastic wedges with different elastic properties subjected to point loads at the apex. The author has independently examined the applicability of M-integral to energy release rate calculation in dissimilar materials and obtained some closed form expressions of energy release rates for other types of cracks.