Bimaterial Interface Cracks Research Articles

Transient analysis of stationary interface cracks in orthotropic bi-materials using oscillatory crack tip enrichments

The problem of an interface crack between two orthotropic layers under dynamic loading is analyzed. Special crack tip enrichment functions are incorporated into the standard finite element shape functions to exactly reproduce oscillatory stress and displacement fields near the tip of the interface crack. Moreover, kinematics of displacement and its gradients across the crack face and material interface are also modeled by partition of unity enrichments. Special attention is given to extraction of stress intensity factors by utilizing a proper form of the interaction integral for orthotropic bi-materials. Advantages of this method of extracting stress intensity factors over the conventional displacement extrapolation technique are discussed. Several bi-material configurations with both vanishing and non-vanishing oscillatory indexes are solved using the interaction integral and the results are compared with the available data in the literature. Effects of employing oscillatory crack tip enrichments and validation of the path-independent J-integral are also discussed.

Computational modeling of strong and weak discontinuities using the extended finite element method

ABSTRACTIn this article, the extended finite element method is employed to solve problems, including weak and strong discontinuities. To this end, a level set framework is used to represent the discontinuities location, and the Heaviside and Branch function are included in the standard finite element method. The case of two arbitrary curved cracks is solved numerically and stress intensity factor (SIF) values at the crack tips are calculated based on the evaluation of the crack tip opening displacement. Afterwards, J-integral methodology is adopted to evaluate the SIFs for isotropic and anisotropic bi-material interface crack problems. Numerical results are verified with those presented in the literature.

Computation of the Jk-integrals for bimaterial interface cracks using boundary element crack shape sensitivities

This paper presents a new algorithm for the efficient evaluation of Jk-integrals for cracks between bonded homogeneous and isotropic materials using the boundary element crack shape sensitivities (BECSS). The flexibility of this novel method allows for analysis of both curved and straight interface cracks. In contrast to the available algorithms, the present method does not require stress analysis at a series of internal points around the crack or employment of an auxiliary equation. For an interface crack, the J1-integral is the strain energy release rate (SERR) or the derivative of the total potential energy with respect to the crack length extension. Although the J2-integral shows an oscillatory type behaviour and is non-existent at the crack tip, it can also be evaluated by direct differentiation of the structural response. It is well-known that a bimaterial interface crack induces both opening and shearing behaviour even for a single mode loading. Here it is shown that the computed Jk can be used to decouple and estimate the stress intensity factors (SIFs). Here, three example problems are analysed and their Jk values are presented which are in excellent agreement with the corresponding analytical results. Each case includes the contribution to J2 by the jump of displacement derivatives across the interface and the strain energy density discontinuity on the crack surfaces and interface region.

Computation of three-dimensional fracture parameters at interface cracks and notches by the scaled boundary finite element method

This paper presents the computations of fracture parameters including stress intensity factors and T-stress of three-dimensional cracks and notches by the scaled boundary finite element method. The singular stress field along the crack front is approximated by a singularity at a point through a semi-analytical solution. The solution is expressed as a matrix power function which allows direct extraction of the fracture parameters based on their definitions. No singular element or asymptotic solution is required for the extraction process. The numerical examples presented which include bimaterial interface cracks and V-notches illustrate the accuracy and versatility of the proposed approach.

Method for calculating G, GI, and GII to simulate crack growth in 2D, multiple-material structures

A finite element approach for calculating the energy release rate (G) and individual modes (GI and GII) when simulating crack growth through a two-dimensional composite material is presented. A new mesh-substitution scheme is used with basic fracture mechanics concepts to automatically calculate G, GI, and GII for potential growth along material interfaces and at different kink angles. The method is general and can be used when investigating interface cracks, substrate cracks (i.e., cracks contained in a material region), and substrate cracks with tips touching one or more interfaces. For bimaterial interface cracks, a coupled energy release rate (GI–II) is determined, and complex stress intensity factor components are calculated. A complete crack propagation framework, which includes the developed methods, is presented and then used to simulate crack growth through a composite material.

Numerical modelling of plane strain plasticity induced crack closure effects for bimaterial interfacial cracks

The effects of plane strain plasticity induced crack closure on fatigue cracks located at the interface of dissimilar steel materials are presented using finite element modelling. Based on the study, it has been observed that bimaterial cracks produced unsymmetrical residual plastic strains and crack profiles in the crack wakes. It is seen that Young’s modulus and yield stress mismatch have profound effects on the development of unsymmetrical residual plastic strain and crack profiles, whereas the effect of Poisson’s ratio is insignificant. However, it has been found that for the material properties considered, low value of crack closure levels have been identified.

Evaluation of mode III interface cracks in magnetoelectroelastic bimaterials by symplectic expansion

The mode III interface crack between two dissimilar magnetoelectroelastic materials is studied by means of symplectic expansion. Four crack surface electromagnetic assumptions, that is, electrically and magnetically permeable, electrically permeable and magnetically impermeable, electrically impermeable and magnetically permeable, and electrically and magnetically impermeable, are taken into account. In the symplectic space, stress, electric displacement, and magnetic induction are found to be the dual variables of the displacement, electric potential, and magnetic potential. The Hamiltonian governing equations that allow the use of the method of separation of variables are formulated by employing the variational principle of mixed energy. Expressions for the mechanical, electric, and magnetic fields are derived explicitly in the series of symplectic eigensolutions with coefficients to be determined by the boundary conditions, and so are the fracture parameters, such as field intensity factors and energy release rates for dissimilar bimaterials. Numerical results are provided to verify the validity of the present method and reveal the influences of the material constants and boundary conditions.

Analysis of stress intensity factor in orthotropic bi-material mixed interface crack

Adopting the complex function approach, the paper studies the stress intensity factor in orthotropic bi-material interface cracks under mixed loads. With consideration of the boundary conditions, a new stress function is introduced to transform the problem of bi-material interface crack into a boundary value problem of partial differential equations. Two sets of non-homogeneous linear equations with 16 unknowns are constructed. By solving the equations, the expressions for the real bi-material elastic constant ɛt and the real stress singularity exponents λt are obtained with the bi-material engineering parameters satisfying certain conditions. By the uniqueness theorem of limit, undetermined coefficients are determined, and thus the bi-material stress intensity factor in mixed cracks is obtained. The bi-material stress intensity factor characterizes features of mixed cracks. When orthotropic bi-materials are of the same material, the degenerate solution to the stress intensity factor in mixed bi-material interface cracks is in complete agreement with the present classic conclusion. The relationship between the bi-material stress intensity factor and the ratio of bi-material shear modulus and the relationship between the bi-material stress intensity factor and the ratio of bi-material Young’s modulus are given in the numerical analysis.

Analysis of composite material interface crack face contact and friction effects using a new node-pairs contact algorithm

A new node-pairs contact algorithm is proposed to deal with a composite material or bi-material interface crack face contact and friction problem (e.g., resistant coating and thermal barrier coatings) subjected to complicated load conditions. To decrease the calculation scale and calculation errors, the local Lagrange multipliers are solved only on a pair of contact nodes using the Jacobi iteration method, and the constraint modification of the tangential multipliers are required. After the calculation of the present node-pairs Lagrange multiplier, it is turned to next contact node-pairs until all node-pairs have finished. Compared with an ordinary contact algorithm, the new local node-pairs contact algorithm is allowed a more precise element on the contact face without the stiffness matrix singularity. The stress intensity factors (SIFs) and the contact region of an infinite plate central crack are calculated and show good agreement with those in the literature. The contact zone near the crack tip as well as its influence on singularity of stress fields are studied. Furthermore, the frictional contacts are also considered and found to have a significant influence on the SIFs. The normalized mode-II stress intensity factors K̂II for the friction coefficient decrease by 16% when f changes from 1 to 0.

2D dynamic analysis of cracks and interface cracks in piezoelectric composites using the SBFEM

The dynamic stress and electric displacement intensity factors of impermeable cracks in homogeneous piezoelectric materials and interface cracks in piezoelectric bimaterials are evaluated by extending the scaled boundary finite element method (SBFEM). In this method, a piezoelectric plate is divided into polygons. Each polygon is treated as a scaled boundary finite element subdomain. Only the boundaries of the subdomains need to be discretized with line elements. The dynamic properties of a subdomain are represented by the high order stiffness and mass matrices obtained from a continued fraction solution, which is able to represent the high frequency response with only 3–4 terms per wavelength. The semi-analytical solutions model singular stress and electric displacement fields in the vicinity of crack tips accurately and efficiently. The dynamic stress and electric displacement intensity factors are evaluated directly from the scaled boundary finite element solutions. No asymptotic solution, local mesh refinement or other special treatments around a crack tip are required. Numerical examples are presented to verify the proposed technique with the analytical solutions and the results from the literature. The present results highlight the accuracy, simplicity and efficiency of the proposed technique.

Stress Intensity Factor for Interface Cracks in Bimaterials Using Complex Variable Meshless Manifold Method

This paper describes the application of the complex variable meshless manifold method (CVMMM) to stress intensity factor analyses of structures containing interface cracks between dissimilar materials. A discontinuous function and the near-tip asymptotic displacement functions are added to the CVMMM approximation using the framework of complex variable moving least-squares (CVMLS) approximation. This enables the domain to be modeled by CVMMM without explicitly meshing the crack surfaces. The enriched crack-tip functions are chosen as those that span the asymptotic displacement fields for an interfacial crack. The complex stress intensity factors for bimaterial interfacial cracks were numerically evaluated using the method. Good agreement between the numerical results and the reference solutions for benchmark interfacial crack problems is realized.

Simulation of Bi-metallic Interfacial Cracks Using Element Free Galerkin Method

Abstract This paper presents the implementation of element free Galerkin method for the stress analysis of structures having cracks at the interface of two dissimilar materials. The material discontinuity at the interface has been modeled using a jump function with a jump parameter that governs its strength. The jump function enriches the approximation by the addition of special shape function that contains discontinuities in the derivative. The trial and test functions of the weak form are constructed using moving least-square interpolants in each material domain. An intrinsic enrichment criterion with enriched basis has been used to model the crack tip stress fields. The mixed mode (complex) stress intensity factors for bi-material interface cracks are numerically evaluated using the modified domain form of interaction integral. The numerical results are obtained for edge and centre cracks lying at the bi-material interface, and are found to be in good agreement with the reference solutions for the interfacial crack problems.

Fatigue crack growth simulations of homogeneous and bi-material interfacial cracks using element free Galerkin method

In this paper, quasi-static fatigue crack growth simulations of homogeneous and bi-material interfacial cracks have been performed using element free Galerkin method (EFGM) under mechanical as well as thermo-elastic load. The thermo-elastic fracture problem is decoupled into thermal and elastic problems. The temperature distribution obtained by solving heat conduction equation is used as input in the elastic problem to get the displacement and stress fields. Discontinuities in the temperature and displacement fields are captured by extrinsic partition of unity enrichment technique. The values of stress intensity factors have been extracted from the EFGM solution by domain based interaction integral approach. The standard Paris fatigue crack growth law has been implemented for the life estimation of various model problems. The results obtained by EFGM under mechanical and thermo-elastic loads were compared with those obtained by FEM using remeshing approach.

Numerical Computation of Complex Stress Intensity Factors of Interface Cracks in Bi-Materials Based on Photoelastic Theory

This paper presents a new numerical method for obtaining the complex stress intensity factor with an interface crack in bi-materials using photoelastic isochromatic fringe numbers N. The theoretical solution of stress field at the crack tip was deduced from Muskhelishvilis stress function and an undetermined term σ0 which is a function of material properties was added to this theoretical solution. A partial differential iterative equation with fast convergence was formed by applying the photoelastic theory. The complex stress intensity factor K=K1+iK2 and σ0 were obtained by Newton-Raphson iteration method and K domain was discussed. The simulant photoelastic isochromatic fringe pattern could be generated through image processing and numerical calculation according to K and σ0. The simulant isochromatic fringe pattern accords with experimental photoelastic isochromatic fringe pattern, so it is practicable for this numerical method of obtaining the complex stress intensity factor.

Numerical Computation of Complex Stress Intensity Factors of Interface Cracks in Bi-Materials Based on Photoelastic Theory

This paper presents a new numerical method for obtaining the complex stress intensity factor with an interface crack in bi-materials using photoelastic isochromatic fringe numbers N. The theoretical solution of stress field at the crack tip was deduced from Muskhelishvilis stress function and an undetermined term σ0 which is a function of material properties was added to this theoretical solution. A partial differential iterative equation with fast convergence was formed by applying the photoelastic theory. The complex stress intensity factor K=K1+iK2 and σ0 were obtained by Newton-Raphson iteration method and K domain was discussed. The simulant photoelastic isochromatic fringe pattern could be generated through image processing and numerical calculation according to K and σ0. The simulant isochromatic fringe pattern accords with experimental photoelastic isochromatic fringe pattern, so it is practicable for this numerical method of obtaining the complex stress intensity factor.

Stress intensity factors and T-stress for an edge interface crack by symplectic expansion

An analytical method is presented for finding the complex stress intensity factors (SIFs) and T-stress at an edge bi-material interface crack. A Hamiltonian system is first established by introducing dual (conjugate) variables of displacements and stresses whose solutions are expanded in terms of the symplectic series. With the aid of the adjoint symplectic orthogonality, coefficients of the series are determined by the boundary conditions along the crack faces and along the external geometry. Analytical solutions of SIFs and T-stress are obtained simultaneously. Numerical examples including the complex mixed boundary conditions are given. Factors influencing the SIFs are discussed.

Determination of dynamic intensity factors and time-domain BEM for interfacial cracks in anisotropic piezoelectric materials

In this paper, a universal matrix-form displacement extrapolation formula and the explicit form are derived to determine the dynamic intensity factors of an interface crack in general anisotropic piezoelectric bi-materials. It is very feasible for numerical methods, especially for boundary element method (BEM) which can directly present the unknown boundary values. Using the explicit extrapolation formula, the transient response of interfacial cracks in piezoelectric bi-materials under electro-mechanical impacts is studied by time-domain BEM together with a sub-domain technique. The present time-domain BEM uses a quadrature formula for the temporal discretization to approximate the convolution integrals and a collocation method for the spatial discretization. Quadratic quarter-point elements are implemented at the tips of the interface cracks. General cases for the poling axis of each piezoelectric component unparallel to the Cartesian coordinate axes are studied. Numerical examples are presented and discussed the effects of the poling direction, material combination and the load combination on dynamic intensity factors.

Fatigue crack growth simulations of bi-material interfacial cracks under thermo-elastic loading by extended finite element method

In this paper, fatigue crack growth simulations of bi-material interfacial cracks have been performed using extended finite element method (XFEM) under thermo-elastic loading. The material discontinuity (interface) has been modelled by a signed distance function whereas a strong discontinuity (crack) has been modelled by two functions i.e. Heaviside and asymptotic crack tip enrichment functions. The values of stress intensity factors are extracted from the XFEM solution by domain based interaction integral approach. Standard Paris fatigue crack growth law is used for the life estimation of various model problems. The results obtained by XFEM for an interfacial edge and centre cracks are compared with those obtained by finite element method based on a remeshing approach.

Modeling bimaterial interface cracks using the numerical manifold method

Abstract The numerical manifold method (NMM) is explored for simulations of bimaterial interface cracks. Two special types of physical covers with customized cover functions are introduced to describe the weak discontinuity across the material interface, the partially cracked elements as well as the interface crack tip singularity. Three typical bimaterial crack problems are simulated. The mixed-mode stress intensity factors are evaluated by the virtue of the domain form of the interaction integral and compared with the available reference solutions. Good agreements have demonstrated the validity and accuracy of the developed program.

A Singular Finite Element on the Mixed-Mode Bimaterial Interfacial Cracks

A singular finite element is presented to study the mixed-mode Dugdale-model-based bimaterial interfacial cracks. Firstly, the bimaterial interfacial crack problem is led into the symplectic space, and the symplectic dual equation is obtained and solved analytically. The cohesive stresses of the Dugdale model are treated as special solutions. Subsequently, the analytical solution is employed to develop a novel singular finite element, which depicts accurately the characteristic of displacements and singular stress fields near the crack tip. Finally, combining the singular finite element and conventional finite element method, the length of plastic zone, crack tip opening, and/or sliding displacement can be solved by iteration. Numerical examples are given to illustrate the validity of the present method.