Evaluation of stress intensity factors for bi-material interface cracks using displacement jump methods

The aim of the present work is to investigate the numerical modeling of interfacial cracks that may appear at the interface between two isotropic elastic materials. The extended finite element method is employed to analyze brittle and bi-material interfacial fatigue crack growth by computing the mixed mode stress intensity factors(SIF). Three different approaches are introduced to compute the SIFs. In the first one, mixed mode SIF is deduced from the computation of the contour integral as per the classical J-integral method,whereas a displacement method is used to evaluate the SIF by using either one or two displacement jumps located along the crack path in the second and third approaches. The displacement jump method is rather classical for mono-materials,but has to our knowledge not been used up to now for a bimaterial. Hence, use of displacement jump for characterizing bi-material cracks constitutes the main contribution of the present study. Several benchmark tests including parametric studies are performed to show the effectiveness of these computational methodologies for SIF considering static and fatigue problems of bi-material structures. It is found that results based on the displacement jump methods are in a very good agreement with those of exact solutions, such as for the J-integral method, but with a larger domain of applicability and a better numerical efficiency(less time consuming and less spurious boundary effect).

DYNAMIC BEHAVIOR OF TWO UNEQUAL PARALLEL PERMEABLE INTERFACE CRACKS IN A PIEZOELECTRIC LAYER BONDED TO TWO HALF PIEZOELECTRIC MATERIALS PLANES

The dynamic behavior of two unequal parallel permeable interface cracks in a piezoelectric layer bonded to two half-piezoelectric material planes subjected to harmonic anti-plane shear waves is investigated. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations in which the unknown variables were the jumps of the displacements across the crack surfaces. Numerical results are presented graphically to show the effects of the geometric parameters, the frequency of the incident wave on the dynamic stress intensity factors and the electric displacement intensity factors. Especially, the present problem can be returned to static problem of two parallel permeable interface cracks. Compared with the solutions of impermeable crack surface condition, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller.

SIF-based fracture criterion for interface cracks

The complex stress intensity factor K governing the stress field of an interface crack tip may be split into two parts, i.e.,■ and s^（-iε）, so that K = ■ s^（-iε）, s is a characteristic length and ε is the oscillatory index. ■ has the same dimension as the classical stress intensity factor and characterizes the interface crack tip field. That means a criterion for interface cracks may be formulated directly with■, as Irwin（ASME J. Appl. Mech. 24：361–364, 1957） did in 1957 for the classical fracture mechanics. Then, for an interface crack,it is demonstrated that the quasi Mode I and Mode II tip fields can be defined and distinguished from the coupled mode tip fields. Built upon SIF-based fracture criteria for quasi Mode I and Mode II, the stress intensity factor（SIF）-based fracture criterion for mixed mode interface cracks is proposed and validated against existing experimental results.

WEIGHT FUNCTIONS FOR INTERFACE CRACKS IN DISSIMILAR ANISOTROPIC MATERIALS

Bueckner's work conjugate integral customarily adopted for linear elastic materials is established for an interface crack in dissimilar anisotropic materials.The difficulties in separating Stroh's six complex arguments involved in the integral for the dissimilar materials are overcome and thert the explicit function representations of the integral are given and studied in detail.It is found that the pseudo-orthogonal properties of the eigenfunction expansion form(EEF)for a crack presented previously in isotropic elastic cases,in isotopic bimaterial cases,and in orthotropic cases are also valid in the present dissimilar arbitrary anisotropic cases.The relation between Bueckner's work conjugate integral and the J-integral in these cases is obtained by introducing a complementary stress- displacement state.Finally,some useful path-independent integrals and weight functions are proposed for calculating the crack tip parameters such as the stress intensity factors.

Steady-state growing crack-tip field characteristics of mode Ⅲ elastic-viscoplastic/rigid interface cracks

The existence of viscosity effect at the interface of double dissimilar materials has an important impact on the distribution of the interface crack-tip field and the properties variety of the interface itself. The singularity and viscosity are considered in the crack-tip. The elastic-viscoplastic governing equations of double dissimilar materials at the interface crack-tip field are established. The displacement potential function and boundary condition of interface crack-tip are introduced. The numerical analysis of elastic-viscoplastic/rigid interface for mode Ⅲis worked out. The stress-strain fields are obtained at the crack-tip and the variation rules of solutions are discussed according to each parameter. The numerical results show that the viscosity effect is a main factor of the interface propagating in the crack-tip field, and the interface crack-tip is a viscoplastic field governed by the viscosity coefficient, Mach number （Ma）, and singularity exponent.

Dynamic behavior of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips

The dynamic interaction of two collinear interface cracks between two dissimilar functionally graded piezoelectric/piezomagnetic material strips subjected to the anti-plane shear harmonic stress waves was investigated. By using the Fourier transform, the problem can be solved with the help of a pair of triple integral equations in which the unknown variable is jump of displacement across the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided to show the effect of the functionally graded parameter, the circular frequency of the incident waves and the thickness of the strip upon stress, electric displacement and magnetic flux intensity factors of cracks.

ELASTIC SH WAVE PROPAGATION IN PERIODIC LAYERED COMPOSITES WITH A PERIODIC ARRAY OF INTERFACE CRACKS

The interaction of anti-plane elastic SH waves with a periodic array of interface cracks in a multi-layered periodic medium is analyzed in this paper. A perfect periodic structure without interface cracks is first studied and the transmission displacement coefficient is obtained based on the transfer matrix method in conjunction with the Bloch-Floquet theorem. This is then generalized to a single and periodic distribution of cracks at the center interface and the result is compared with that of perfect periodic cases without interface cracks. The dependence of the transmission displacement coefficient on the frequency of the incident wave, the influences of material combination, crack configuration and incident angle are discussed in detail. Compared with the corresponding perfect periodic structure without interface cracks, a new phenomenon is found in the periodic layered system with a single and periodic array of interface cracks.

The asymptotic field of mode Ⅰ quasi-static crack growth on the interface between a rigid and a pressure-sensitive material

A mechanical model of the quasi-static interface of a mode I crack between a rigid and a pressure-sensitive viscoelastic material was established to investigate the mechanical characteristic of ship-building engineering hi-materials. In the stable growth stage, stress and strain have the same singularity, ie （σ, ε） ∝ r^-1/（n-1）. The variable-separable asymptotic solutions of stress and strain at the crack tip were obtained by adopting Airy＇s stress function and the numerical results of stress and strain in the crack-tip field were obtained by the shooting method. The results showed that the near-tip fields are mainly governed by the power-hardening exponent n and the Poisson ratio v of the pressure-sensitive material. The fracture criterion of mode I quasi-static crack growth in pressure-sensitive materials, according to the asymptotic analyses of the crack-tip field, can be viewed from the perspective of strain.

TRANSIENT RESPONSE OF COPLANAR INTERFACIAL CRACKS BETWEEN TWO DISSIMILAR PIEZOELECTRIC STRIPS UNDER ANTI-PLANE MECHANICAL AND IN-PLANE ELECTRICAL IMPACTS

The dynamic response of multiple coplanar interface cracks between two dissimilar piezoelectric strips subjected to mechanical and electrical impacts is investigated.Solutions to two kinds of electric boundary conditions on crack surfaces,i.e.electric impermeable and electric permeable,are obtained.Laplace and Fourier transforms and dislocation density functions are employed to reduce the mixed boundary value problem to Cauchy singular integral equations, which can be solved numerically.The effects of electrical load,geometry criterion of piezoelectric strips,relative location of cracks and material properties on the dynamic energy release rate are examined.

SCATTERING OF SH-WAVES BY AN INTERACTING INTERFACE LINEAR CRACK AND A CIRCULAR CAVITY NEAR BIMATERIAL INTERFACE

An analytical method is developed for scattering of SH-waves and dynamic stressconcentration by an interacting interface crack and a circular cavity near bimaterial interface.Asuitable Green’s function is contructed,which is the fundamental solution of the displacement fieldfor an elastic half space with a circular cavity impacted by an out-plane harmonic line source loadingat the horizontal surface.First,the bimaterial media is divided into two parts along the horizontalinterface,one is an elastic half space with a circular cavity and the other is a complete half space.Then the problem is solved according to the procedure of combination and by the Green’s functionmethod.The horizontal surfaces of the two half spaces are loaded with undetermined anti-plane forcesin order to satisfy continuity conditions at the linking section,or with some forces to recover cracks bymeans of crack-division technique.A series of Fredholm integral equations of first kind for determiningthe unknown forces can be set up through continuity conditions as expressed in terms of the Green’sfunction.Moreover,some expressions are given in this paper,such as dynamic stress intensity factor(DSIF)at the tip of the interface crack and dynamic stress concentration factor(DSCF)around thecircular cavity edge.Numerical examples are provided to show the influences of the wave numbers,the geometrical location of the interface crack and the circular cavity,and parameter combinations ofdifferent media upon DSIF and DSCF.

Influence of interface crack on dynamic characteristics of CRTSⅢslab ballastless track on bridge

The damage of the self-compacting concrete in CRTSⅢslab ballastless track on bridge will lead to a partial void of the track slab,which will affect the comfort and safety of the train and the durability of the track slab and bridge structure.In order to study the impact of the interface crack on the dynamic response of CRTSⅢballastless track system on bridge,based on the principle of multi-body dynamics theory and ANSYS+SIMPACK co-simulation,the spatial model of vehicle-track-bridge integration considering the longitudinal stiffness of supports,the track structure and interlayer contact characteristics were established.The dynamic characteristics of the system under different conditions of the width,length and position of the interface crack were analysed,and the limited values of the length and width of the cracks at the track slab edge were proposed.The results show that when the self-compacting concrete does not completely void along the transverse direction of the track slab,the crack has little effect on the dynamic characteristics of the vehicle-track-bridge system.However,when the self-compacting concrete is completely hollowed out along the transverse direction of the track slab,the dynamic amplitudes of the system increase.When the crack length is 1.6 m,the wheel load reduction rate reaches 0.769,which exceeds the limit value and threatens the safety of train operation.The vertical acceleration of the track slab increases by 250.1%,which affects the service life of the track system under the train speed of 200 km/h.

Exact solution of plane isolated crack normal to a bimaterial interface of infinite extent

This paper presents an exact solution for the transverse interface crack in the plane strain case. The crack is perpendicular to the interface and in one material. The exact complex stress functions are first obtained with some unknown constants. The satisfactions of all boundary conditions are then checked, the condition at infinity is considered and the unknown constants are determined. Further study may focus on the case with different shear moduli and the influence of the large deformation.

STRESS SINGULARITY ANALYSIS AT CRACK TIP ON BI-MATERIAL INTERFACES BASED ON HAMILTONIAN PRINCIPLE

In this paper, the stress singularity analysis at the crack tip on elastic bi-material interfaces is considered. The governing equations of plane elasticity In sectorial domain are derived to be in Hamiltonian form via variable substitution and variational principle. The methods of separation of variables and conjugate symplectic eigen-function expansion are developed to solve the equations in sectorial domain. The general formulae for the solution of stress singularities at the crack tip on bi-material interfaces are put forward, and a new solution technique for fracture problems is presented.

Effects of electric/magnetic impact on the transient fracture of interface crack in piezoelectric-piezomagnetic sandwich structure: anti-plane case

Due to the incompatibility of the interlaminar deformations,the interface debonding or cracking usually happens in a layered magnetoelectric(ME)structure under an applied load.In this paper,the transient responses of the anti-plane interface cracks in piezoelectric(PE)-piezomagnetic(PM)sandwich structures are studied by the standard methods of the integral transform and singular integral equation.Discussion on the numerical examples indicates that the PE-PM-PE structure under electric impact is more likely to fracture than the PM-PE-PM structure under a magnetic impact.The dynamic stress intensity factors(DSIFs)are more sensitive to the variation of the active layer thickness.The effects of the material constants on the DSIFs are dependent on the roles played by PE and PM media during the deformation process.

Analysis of a permeable interface crack in elastic dielectric/piezoelectric bimaterials

A permeable interface crack between elastic dielectric material and piezoelectric material is studied based on the extended Stroh＇s formalism. Motivated by strong engineering demands to design new composite materials, the authors perform numerical analysis of interface crack tip singularities and the crack tip energy release rates for 35 types of dissimilar bimaterials, respectively, which are constructed by five kinds of elastic dielectric materials： Epoxy, Polymer, Al2O3, SiC, and Si3N4 and seven kinds of practical piezoelectric ceramics： PZT-4, BaTiO3, PZT-5H, PZT-6B, PZT-TA, P-7, and PZT-PIC 151, respectively. The elastic dielectric material with much smaller permittivity than commercial piezoelectric ceramics is treated as a special transversely isotropic piezoelectric material with extremely small piezoelectricity. The present investigation shows that the structure of the singular field near the permeable interface crack tip consists of three singularities： r^-1/2±iε and r^-1/2, which is quite different from that in the impermeable interface crack. It can be concluded that different far field loading cases have significant influence on the near-tip fracture behaviors of the permeable interface crack. Based on the present theoretical treatment and numerical analysis, the electric field induced crack growth is well explained, which provides a better understanding of the failure mechanism induced from interface crack growth in elastic dielectric/piezoelectric bimaterials.

Stress field near interface crack tip of double dissimilar orthotropic composite materials

In this paper, double dissimilar orthotropic composite materials interfacial crack is studied by constructing new stress functions and employing the method of composite material complex. When the characteristic equations＇ discriminants △1 〉 0 and △2 〉0, the theoretical formula of the stress field and the displacement field near the mode I interface crack tip are derived, indicating that there is no oscillation and interembedding between the interfaces of the crack.

Acoustic emission assessment of interface cracking in thermal barrier coatings

In this paper,acoustic emission（AE） and digital image correlation methods were applied to monitor interface cracking in thermal barrier coatings under compression.The interface failure process can be identifie via its AE features,including buckling,delamination incubation and spallation.According to the Fourier transformation of AE signals,there arefourdifferentfailuremodes：surfaceverticalcracks,opening and sliding interface cracks,and substrate deformation.The characteristic frequency of AE signals from surface vertical cracks is 0.21 MHz,whilst that of the two types of interface cracks are 0.43 and 0.29 MHz,respectively.The energy released of the two types of interface cracks are 0.43 and 0.29 MHz,respectively.Based on the energy released from cracking and the AE signals,a relationship is established between the interface crack length and AE parameters,which is in good agreement with experimental results.

THE SCATTERING OF GENERAL SH PLANE WAVE BY INTERFACE CRACK BETWEEN TWO DISSIMILAR VISCOELASTIC BODIES

The scattering of general SH plane wave by an interface crack between two dissimilar viscoelastic bodies is studied and the dynamic stress intensity factor at the crack-tip is computed. The scattering problem can be decomposed into two problems: one is the reflection and refraction problem of general SH plane waves at perfect interface (with no crack); another is the scattering problem due to the existence of crack. For the first problem, the viscoelastic wave equation, displacement and stress continuity conditions across the interface are used to obtain the shear stress distribution at the interface. For the second problem, the integral transformation method is used to reduce the scattering problem into dual integral equations. Then, the dual integral equations are transformed into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Finally, the singular integral equation is solved by Kurtz's piecewise continuous function method. As a consequence, the crack opening displacement and dynamic stress intensity factor are obtained. At the end of the paper, a numerical example is given. The effects of incident angle, incident frequency and viscoelastic material parameters are analyzed. It is found that there is a frequency region for viscoelastic material within which the viscoelastic effects cannot be ignored.

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 con- sideration 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 dif- ferential 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 εt 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 be- tween the bi-material stress intensity factor and the ratio of bi-material Young＇s modulus are given in the numerical analysis.

The dynamic stress intensity factor analysis of adhesively bonded material interface crack with damage under shear loading

This paper studies the dynamic stress intensity factor （DSIF） at the interface in an adhesive joint under shear loading. Material damage is considered. By introducing the dislocation density function and using the integral transform, the problem is reduced to algebraic equations and can be solved with the collocation dots method in the Laplace domain. Time response of DSIF is calculated with the inverse Laplace integral transform. The results show that the mode Ⅱ DSIF increases with the shear relaxation parameter, shear module and Poisson ratio, while decreases with the swell relaxation parameter. Damage shielding only occurs at the initial stage of crack propagation. The singular index of crack tip is -0.5 and independent on the material parameters, damage conditions of materials, and time. The oscillatory index is controlled by viscoelastic material parameters.