Stress intensities at interface corners in anisotropic bimaterials

Asymptotic stress and displacement fields near the tip of a sharp anisotropic bimaterial interface corner are computed using a combination of the Stroh formalism and the Williams eigenfunction expansion method. From the asymptotic fields, the path independent H-integral is developed and implemented to calculate anisotropic interface corner stress intensities. The calculation procedure is demonstrated for two glass–silicon interface corner configurations that commonly arise in practice in the microsensor industry. In each case, the bimaterial interface corner experiences mixed mode I and II loading, and accurate estimates of both stress intensities are obtained.     

Calculation of elastic T-stresses near interface crack tip under in-plane and anti-plane loading

The relations between the J-based mutual integral and the T-stress are derived for in-plane and anti-plane problems, respectively. The mutual integral is shown to be evaluated without solving the boundary value problems. The T-stress, a useful parameter for crack stability, is thus obtained easily by a suitable application of the concept of the conservation integral. As an example to show this, the fundamental interface crack problems for an infinite dissimilar solid and two infinite strips are presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture analysis of anisotropic materials using enriched crack tip elements

Enriched finite element methodology, which employs special crack tip elements, is extended for cracks in anisotropic materials. Enrichment formulation is described briefly and three validation examples using single crystal, directionally solidified, and orthotropic material properties are presented to demonstrate the accuracy and effectiveness of the methodology. In addition to validation examples, the effect of material anisotropy on stress intensity factors is investigated using the common compact tension specimen and the results are compared to the ASTM solution for isotropic materials. It is shown that the effect of anisotropy on the computed stress intensity factors can be significant, depending on the degree of anisotropy, material orientation, and a/W ratio in the compact tension specimen geometry.     

T-stress solutions for two-dimensional crack problems in anisotropic elasticity using the boundary element method

The importance of a two‐parameter approach in the fracture mechanics analysis of many cracked components is increasingly being recognized in engineering industry. In addition to the stress intensity factor, the T stress is the second parameter considered in fracture assessments. In this paper, the path‐independent mutual M‐integral method to evaluate the T stress is extended to treat plane, generally anisotropic cracked bodies. It is implemented into the boundary element method for two‐dimensional elasticity. Examples are presented to demonstrate the veracity of the formulations developed and its applicability. The numerical solutions obtained show that material anisotropy can have a significant effect on the T stress for a given cracked geometry.     

Crack tip stress intensity factors for a crack emanating from a sharp notch

Accurate calibrations are provided for the crack tip stress intensity factor for a crack of finite length emanating from the symmetric tip of a sharp notch, of arbitrary angle, in terms of the generalised stress intensity quantifying remote loading of the notch. The solution is applied to example problems and shown to be accurate for cases where the crack is much shorter then the notch depth.     

Evaluating crack tip stress field in a thin glass plate under thermal load

The stress field around a propagating crack tip in a quenched thin glass plate is discussed through experimental and theoretical analyses. Instantaneous phase-stepping photoelasticity using a CCD camera equipped with a pixelated micro-retarder array is used for measuring the crack tip stress field. From the successive phase maps of principal direction, the position and the velocity of the crack tip are evaluated. On the other hand, the fracture parameters, that is, the stress intensity factors and the T-stress are determined from the phase maps of the retardation. Experimental results obtained for a straight crack show good agreement with those obtained by theory of elasticity. The results also indicate that the direction of the crack propagation arising in the quenching process is not determined by the direction of the maximum principal stress. Furthermore, the results show that the T-stress criterion is inappropriate to evaluate the crack path instability in a quenched thin glass plate.     

Singular intensity factors at bimaterial anisotropic interfaces

The singular intensity factors at bimaterial anisotropic interfaces in bonded joints with composite adherends are found by using a hybrid method based on numerical and elasticity solutions. The method is applicable to the solution of problems having complex geometry, loading and boundary conditions, which is the case in typical composite structures. Results are given in terms of the singular intensity factor, which is a generalization of the stress intensity factor commonly used with cracks. Both closed and open wedges, which are found, respectively, in bonded joints with or without adhesive fillets, are considered. Equivalent singular intensity factors in modes I, II and III are defined, and the results indicate that the mode III factor, which arises due to out-of-plane coupling, is negligible in all cases studied. Moreover, use of the Erdogan–Sih failure criterion indicates that the direction of crack propagation in lap joints with fillets remains constant beyond a very small region near the point of singularity, while for joints without fillets crack initiation always occurs in a direction parallel to the adhesive–adherend interface.     

A constraint based parameter for quantifying the crack tip stress fields in welded joints

By means of finite element analyses of plane strain crack tip stress fields from homogeneous and heterogeneous modified boundary layer formulations, as well as homogeneous and mismatched full field solutions, a new constraint parameter β_m has been established for overmatched welded joints, allowing the material mismatching effect on the crack tip stress fields to be quantified. In the case of complete specimens, both geometry and material mismatching affect the crack tip stress fields, and a total constraint parameter β_T can be defined. This approach allows to quantify the stress fields directly from the values of the remote applied load.     

Computing stress singularities in transversely isotropic multimaterial corners by means of explicit expressions of the orthonormalized Stroh-eigenvectors

Composite materials reinforced by unidirectional long fibers behave macroscopically as homogeneous transversely isotropic linear elastic materials. A general, accurate and computationally efficient procedure for the evaluation of singularity exponents and singular functions characterizing singular stress fields in multimaterial corners involving this kind of material is presented in this paper. To take full advantage of the sextic Stroh formalism of anisotropic elasticity applied to this particular problem, the complete set of explicit expressions of the eigenvalues and eigenvectors of the real 6 × 6 fundamental elasticity matrix N has been deduced for all the non-degenerate and degenerate (repeated roots of the sextic Stroh equation) cases. These expressions will also facilitate further applications of the Stroh formalism to these materials. Several numerical examples of singularity analysis of multimaterial corners appearing in adhesively bonded joints and damaged cross-ply laminates of composite materials are presented.     

A wedge crack in an anisotropic material under antiplane shear

A crack emanating from the apex of an infinite wedge in an anisotropic material under antiplane shear is investigated. An isotropic wedge crack subjected to concentrated forces is first solved by using the conformal mapping technique. The solution of an anisotropic wedge crack is obtained from that of the transformed isotropic wedge crack based on a linear transformation method. Expressions for the stress intensity factor for the anisotropic wedge crack with both concentrated and distributed loads are derived. The stress intensity factors are numerically calculated for generally orthotropic wedge cracks with various crack and wedge angles as well as anisotropic parameters.     

Fast, accurate, and stable algorithm for the stress field around a zig-zag-shaped crack

A new scheme decreases memory usage and execution time for the evaluation of certain weighted Cauchy type singular integrals. The scheme is incorporated into an algorithm based on an integral equation of Fredholm second kind. The algorithm computes mode I and II stress intensity factors of cracks in infinite elastic domains. Problems with analytical solutions are solved with relative errors less than 10⁻¹⁵. Earlier investigators' results for kinked cracks with one to four corners are improved. Cracks with a large number of corners can also be studied. In an example the stress intensity factors of a crack with 100 corners are calculated with a relative error of less than 10⁻⁶.     

Stress intensification due to an edge crack in an anisotropic elastic solid

Integral transform techniques are used to determine the stress intensity factors of a crack at the edge of an anisotropic elastic half space under generalized plane strain conditions. Numerical results are given for a carbon fibre reinforced epoxy in uniaxial tension.     

An analytical expression for the J2‐integral of an interfacial crack in orthotropic bimaterials

This paper presents a new analytical expression relating the J₂‐integral and stress intensity factors (SIF) in an in‐plane traction‐free crack between two orthotropic elastic solids using the complex function method. The singular oscillatory near tip field of a bimaterial interfacial crack is usually characterized by a pair of SIFs. In linear elastic interfacial fracture mechanics, the majority of numerical and experimental methods rely on the analytical equations relating J_k‐integrals and SIFs. Although an analytical equation relating J₁‐integral or strain energy release rate and SIFs is available, a similar relation for J₂‐integral in debonded anisotropic solids is non‐existent. Using this new analytical expression, in conjunction with the values of J_k, the SIFs can be computed without the need for an auxiliary relation. An example with known analytical solutions for SIFs is presented to show the variation of the J₂‐integral near the crack tip of a bimaterial orthotropic plate. Different bimaterial combinations are considered, and the effect of material mismatch on J_k is demonstrated.     

Kinked cracks in anisotropic elastic materials

The problem of a kinked crack is analysed for the most general case of elastic anisotropy. The kinked crack is modelled by means of continuous distributions of dislocations which are assumed to be singular both at the crack tips and at the kink vertex. The resulting system of singular integral equations is solved numerically using Chebyshev polynomials and the reciprocal theorem. The stress intensity factors for modes I, II and III and the generalised stress intensity factor at the vertex are obtained directly from the dislocation densities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Analysis of a mode III crack problem in a functionally graded coating-substrate system with finite thickness

This paper presents an analysis of the static problem of model III crack of a functionally graded coating-substrate system with an internal crack perpendicular to the interface under antiplane shear loading when the coating layer and substrate have finite thickness. After the Fourier transform method is employed, the expressions of the displacement components can be obtained. Integral transforms are employed to reduce the problem to a singular integral equation that can be solved numerically. The influences of the nonhomogeneity constant, relative crack length and thickness ratio are quantitatively studied.     

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.     

Dynamic stress field for torsional impact of a penny-shaped crack in a transversely isotropic functional graded strip

The torsional impact response of a penny-shaped crack in a transversely isotropic strip is considered. The shear moduli are assumed to be functionally graded such that the mathematics is tractable. Laplace and Hankel transforms are used to reduce the problem to solving a Fredholm integral equation. The crack tip stress field is obtained by considering the asymptotic behavior of Bessel function. Investigated are the effects of material nonhomogeneity and orthotropy and strip’s highness on the dynamic stress intensity factor. The peak of the dynamic stress intensity factor can be suppressed by increasing the shear moduli’s gradient and/or increasing the shear modulus in a direction perpendicular to the crack surface. The dynamic behavior varies little with the increasing of the strip’s highness.     

The M-integral for calculating intensity factors of an impermeable crack in a piezoelectric material

In this study, a conservative integral is derived for calculating the intensity factors associated with piezoelectric material for an impermeable crack. This is an extension of the M-integral or interaction energy integral for mode separation in mechanical problems. In addition, the method of displacement extrapolation is extended for this application as a check on results obtained with the conservative integral. Poling is assumed parallel, perpendicular and at an arbitrary angle with respect to the crack plane, as well as parallel to the crack front. In the latter case, a three-dimensional treatment is required for the conservative integral which is beyond the scope of this investigation. The asymptotic fields are obtained; these include stress, electric, displacement and electric flux density fields which are used as auxiliary solutions for the M-integral.Several benchmark problems are examined to demonstrate the accuracy of the methods. Numerical difficulties encountered resulting from multiplication of large and small numbers were solved by normalizing the variables. Since an analytical solution exists, a finite length crack in an infinite body is also considered. Finally, a four point bend specimen subjected to both an applied load and an electric field is presented for a crack parallel, perpendicular and at an angle to the poling direction. It is seen that neglecting the piezoelectric effect in calculating stress intensity factors may lead to errors.     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

含不完美界面的磁电复合材料倾斜裂纹问题

研究了含非完美界面的双层压电/压磁复合材料中压电相存在一个倾斜于界面的Ⅲ型裂纹问题。采用弹簧型耦合界面模型模拟非完美界面,运用Fourier积分变换法将裂纹面条件转化为奇异积分方程,并使用Lobatto-Chebyshev方法数值求解了裂纹尖端应力强度因子（SIF）。详细地研究了裂纹尖端SIF与界面参数、压电/压磁材料参数和材料的层厚、裂纹的倾斜角、裂纹与界面的距离等几何参数的关系。结果表明：力学不完美性可以独立地增大SIF,而磁学、电学不完美性只有与力学不完美性耦合时才会减小SIF;力学-电学、力学-磁学不完美性的耦合会减小SIF,而磁学-电学不完美性的耦合不会影响SIF;磁场作用下,增大压磁层弹性模量会减小SIF,而增大压电层压电系数,减小压电层弹性模量和介电常数,均会减小SIF;界面不完美性会影响SIF随裂纹倾斜角度或裂纹与界面之间距离的变化规律;在一定范围内增加压电层或压磁层厚度可以减小SIF。