Scattering of inhomogeneous wave by viscoelastic interface crack

Summary The reflection, refraction and scattering of inhomogeneous plane waves of SH type by an interface crack between two dissimilar viscoelastic bodies are investigated. The singular integral equation method is used to reduce the scattering problem into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Then, the singular integral equation is solved numerically by Kurtz's piecewise continous function method. The crack opening displacement and dynamic stress intensity factor characterizing the scattered near-field are estimated for various incident angles, frequencies and relaxation times. The differences on crack opening displacement and stress intensity factor between elastic and viscoelastic interface crack are contrasted. And the effects of incident angle, incident frequency and relaxation time of the viscoelastic material are analyzed and explained by the features of phase lag and energy dissipation of the viscoelastic wave.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

Evaluation of crack tip stress and strain fields under nonproportional loading in a viscoelastic material

The crack tip strain and stress fields in a viscoelastic material under nonproportional loading conditions are evaluated. In order to evaluate the strain field, the crack tip displacement field is measured by using the digital image correlation (DIC) technique. This displacement field is then approximated by using the theoretically obtained crack tip displacement field in viscoelastic materials. The result shows that the approximation method can smoothly reconstruct the experimentally obtained displacement field. From the approximated displacement field, the crack tip strain field can be precisely obtained by using the differential form of the theoretical displacement. On the other hand, the crack tip stress field is analyzed by using the stress function. This suggests that the strain and stress fields can be independently evaluated. In addition, different time dependencies between stress and strain fields near the crack tip are observed. Based on this experiment, we can discuss the several criteria for the crack propagation directions in viscoelastic materials.     

Efficient techniques for predicting viscoelastic behavior of sublaminates

Even for linear elastic behavior, stress analysis of thick laminated composites can be very computation intensive if every lamina is modeled discretely. In such cases, modeling of individual lamina is impractical and the homogenization method for sublaminates becomes essential. In the current work, 3D homogenization formulas for an elastic sublaminate, which were derived by the authors in previous work, were utilized to determine the 3D effective properties for a viscoelastic sublaminate. The properties were determined by three methods that exploited the 3D elastic homogenization formulas: (i) quasi-elastic method, (ii) correspondence principle, and (iii) direct time integration of the incremental viscoelastic equations. The finite element method with discrete modeling of the plies was used to obtain reference solutions. The effective viscoelastic properties obtained using the three methods based on the elastic homogenization formulas were in very good agreement with the reference solution. Among these methods, the quasi-elastic method was found to be both accurate and the simplest method in determining the effective properties. The methods were also used to predict the stress response of a sublaminate to different strain histories. The direct time integration method using the 3D elastic homogenization formulas performs accurately and efficiently for this problem.     

Calculation of stress intensity factors based on mode I crack opening integral parameters

We present stress intensity factor assessment using nodal displacements of the crack surfaces determined by the finite element method for cracked bodies. The equation is solved by expanding the crack opening displacement in the Chebyshev function, where crack front asymptotic behavior corresponds to the regulations of the linear elastic fracture mechanics. Results of the stress intensity factor calculations are obtained for test problems with analytical solution. Crack opening displacements are defined with the help of the 3D SPACE software package designed to model mixed variational formulation of the finite element method for displacements and strains of the thermoelastic boundary value problems. Translated from Problemy Prochnosti, No. 6, pp. 122–127, November–December, 2008.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

Dynamic analysis of interfacial crack problems in anisotropic bi-materials by a time-domain BEM

A time-domain boundary element method (BEM) together with the sub-domain technique is applied to study dynamic interfacial crack problems in two-dimensional (2D), piecewise homogeneous, anisotropic and linear elastic bi-materials. The bi-material system is divided into two homogeneous sub-domains along the interface and the traditional displacement boundary integral equations (BIEs) are applied on the boundary of each sub-domain. 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. A displacement extrapolation technique is used to determine the complex dynamic stress intensity factors (SIFs). Numerical examples for computing the complex dynamic SIFs are presented and discussed to demonstrate the accuracy and the efficiency of the present time-domain BEM.     

The time-dependent interaction between inclusions and a cracked matrix subjected to antiplane shear

Summary. The time-dependent interaction between multiple circular inclusions and a cracked matrix in the antiplane viscoelastic problem is discussed in this paper. The fundamental elastic solution is obtained as a rapidly convergent series in terms of complex potentials via successive iterations of Möbius transformation in order to satisfy continuity conditions on multiple interfaces. Based on the correspondence principle, the Laplace transformed viscoelastic solution is then directly determined from the corresponding elastic one. In association with the singular integral technique, the time-dependent mode-III stress intensity factor of the crack tip can be solved numerically in a straightforward manner. Finally, some typical examples of an arbitrary crack lying in a matrix with various material properties under various loading types are also discussed. The results show that, depending on the relative locations and material properties of inclusions, the evolution of the stress intensity factor (SIF) may increase or decrease with time.     

Asymptotic stress field in a cracked orthotropic strip

Christensen's theory of viscoelastic fracture allows the crack propagation velocity to be determined in terms of dissipation whose calculation requires the knowledge of the stress field in the vicinity of the crack tip: the simplest configuration leading to a constant velocity is that of a straight semi-infinite crack contained in an infinitely long strip whose clamped edges are displaced normal to the crack; although experimental data pertaining to this problem have been obtained for a number of materials, no analytical solution is available. When the material is highly anisotropic, an asymptotic solution involving a small parameter related to the ratio of shear modulus to the larger Young's modulus can be attempted. As the corresponding perturbation problem is singular, a matched asymptotic expansion has to be used: it is the sum of outer and inner approximations; both of these are solutions to simple boundary-value problems which can be solved in closed form. The so-constructed asymptotic solution is shown to agree with finite element results, even when the small parameter is as large as 0.2.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Thermo-mechanical behavior of a viscoelastic FGMs coating containing an interface crack

In this paper the plane thermo-mechanical behavior of a crack in a viscoelastic functionally graded materials (FGMs) coating with arbitrary material properties bonded to a homogeneous substrate is studied. In order to avoid the complex forms that describe the viscoelastic properties of FGMs, a multi-layered model for the FGMs coating is developed. The compliance and thermal conductivity in the multi-layered model linearly vary in each layer. In this mixed boundary value problem, the system is reduced to singular integral equations and solved numerically with the Lobatto-Chebyshev collocation technique. Using the correspondence principle and Laplace transform, the problem of an interface crack between a homogeneous substrate and a viscoelastic FGMs is solved. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the multi-layered model. The numerical results confirm that the fracture toughness of materials can be greatly improved by the graded variation of material parameters. It is also confirmed that the specific variation of material parameters greatly influences the fracture behavior of viscoelastic FGMs coating.     

Numerical analysis of the influence of crack surface roughness on the crack path

The influence of the 3D frictional crack surface interaction on the fracture mechanical parameters as well as on the crack path is numerically investigated. For the solution of the boundary value problem the 3D dual boundary element method in terms of the discontinuous formulation is utilized. This method is especially suited for contact problems because it directly deals with the discontinuities at the crack surfaces. The contact problem is solved by the application of the penalty method. Coulomb’s frictional law is utilized for the consideration of the dissipative nature of friction. For discrete steps within one load cycle the stress intensity factors are determined by an extrapolation procedure from the stress field. Based on the analysis of a load cycle, the cyclic stress intensity factors are obtained. For the simulation of crack propagation an implicit time integration scheme of a crack propagation law implemented in terms of a predictor-corrector scheme is applied. The influence of the crack surface roughness on the crack path is shown by numerical examples.     

A time domain boundary element method for modeling the quasi-static viscoelastic behavior of asphalt pavements

A time domain boundary element method (BEM) is presented to model the quasi-static linear viscoelastic behavior of asphalt pavements. In the viscoelastic analysis, the fundamental solution is derived in terms of elemental displacement discontinuities (DDs) and a boundary integral equation is formulated in the time domain. The unknown DDs are assumed to vary quadratically in the spatial domain and to vary linearly in the time domain. The equation is then solved incrementally through the whole time history using an explicit time-marching approach. All the spatial and temporal integrations can be performed analytically, which guarantees the accuracy of the method and the stability of the numerical procedure. Several viscoelastic models such as Boltzmann, Burgers, and power-law models are considered to characterize the time-dependent behavior of linear viscoelastic materials. The numerical method is applied to study the load-induced stress redistribution and its effects on the cracking performance of asphalt pavements. Some benchmark problems are solved to verify the accuracy and efficiency of the approach. Numerical experiments are also carried out to demonstrate application of the method in pavement engineering.     

2D SH modelling of ultrasonic testing for cracks near a non-planar surface

A model of 2D SH ultrasonic nondestructive testing for interior strip-like cracks near a non-planar back surface in a thick-walled elastic solid is presented. The model employs a Green's function to reformulate the 2D antiplane wave scattering problem as two coupled boundary integral equations (BIE): a displacement BIE for the back surface displacement and a hypersingular traction BIE for the crack opening displacement (COD). The integral equations are solved by performing a boundary element discretization of the back surface and expanding the COD in a series of Chebyshev functions which incorporate the correct behaviour at the crack edges. The transmitting ultrasonic probe is modelled by prescribing the traction underneath it, enabling the consequent calculation of the incident field. An electromechanical reciprocity relation is used to model the action of the receiving probe. A few numerical examples which illustrate the influence of the non-planar back surface are given.     

Fracture of a surface layer bonded to a half space

The plane problem of a cracked elastic surface layer bonded to an elastic half space is considered. The surface layer is assumed to contain a transverse crack whose surface is subjected to uniform compression. The problem is formulated in terms of a singular integral equation, the derivative of the crack surface displacement being the density function. By using appropriate quadrature formulas, the integral equation reduces to a system of linear algebraic equations. This system is solved; the stress intensity factors and the crack surface displacement for various crack geometries, namely for internal crack, edge crack, crack touching the interface, and completely broken layer cases, are obtained.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

Smeared-tip superposition method for cohesive fracture with rate effect and creep

A recent formulation of the smeared-tip superposition method presented by Baant [1], which itself was a generalization and modification of an integral equation formulation with an asymptotic series solution derived by Planas and Elices [2], is further improved, generalized and adapted to an efficient finite difference solution scheme. A crack with bridging stresses is modeled as a superposition of infinitely many LEFM cracks with continuously distributed (smeared) tips having infinitely small intensity factors. Knowledge of the stress intensity factor as a function of the location of the crack tip along the crack path is all that is needed to obtain the load-displacement relation. The solution is reduced to a singular integral equation for a function describing the components of applied load associated with crack tips at various locations. The integral equation is complemented by an arbitrary relation between the bridging stress and the crack opening displacement, which can be rate-independent or rate-dependent. Furthermore, using the creep operator method, the equation is extended to aging linearly viscoelastic behavior in the bulk of the specimen. The previously presented finite difference solution is improved and generalized in a form that leads to a system of nonlinear algebraic equations, which can be solved by an optimization method. Application of the smeared-tip method to the analysis of recent measurements of the size effect in three-point-bend fracture specimens of different sizes is presented and a crack opening law that yields the main qualitative characteristics of the test results, particularly an increase of brittleness with a decreasing loading rate, is presented.     

A fast multipole boundary element method for 2D viscoelastic problems

A fast multipole formulation for 2D linear viscoelastic problems is presented in this paper by incorporating the elastic–viscoelastic correspondence principle. Systems of multipole expansion equations are formed and solved analytically in Laplace transform domain. Three commonly used viscoelastic models are introduced to characterize the time-dependent behavior of the materials. Since the transformed multipole formulations are identical to those for the 2D elastic problems, it is quite easy to implement the 2D viscoelastic fast multipole boundary element method. Besides, all the integrals are evaluated analytically, leading to highly accurate results and fast convergence of the numerical scheme. Several numerical examples, including planar viscoelastic composites with single inclusion or randomly distributed multi-inclusions, as well as the problem of a crack in a pressured viscoelastic plane, are presented. The results are verified by comparison with the developed analytical solutions to illustrate the accuracy and efficiency of the approach.     

On the inclined crack problem in an orthotropic medium under biaxial loading

The problem of an inclined crack in an orthotropic medium under biaxial loading is analyzed. A suitable coordinate transformation is introduced and two decoupled systems of the Cauchy–Riemann type are obtained in terms of complex potentials. The crack problem is solved by using the method of analytic continuation and closed form expressions of the near tip stress and displacement fields are derived. The influence of load biaxiality on the stress intensity factors, as well as on the local stress components is studied and graphically represented. Moreover, the action of material orthotropy on various quantities describing the crack characteristic is pointed out.     

The transient response of two cracks at the interface of a layered half space

In this paper, the transient response of two interface cracks, in a layered half space subjected to antiplane stress fields, is analytically studied. The problem is formulated in terms of a coupled set of integral equations, which are then solved by expanding the unknown crack opening displacements in a complete set of Chebyshev's polynomials. The method is coded in FORTRAN program and numerical results for a sample problem are presented. The results show that the response of one crack is significantly influenced by the presence of a bigger neighboring crack.