Dynamic fracture behavior of functionally graded magnetoelectroelastic solids by BIEM

Views Icon Views Article contents Figures & tables Video Audio Supplementary Data Peer Review Share Icon Share Twitter Facebook Reddit LinkedIn Tools Icon Tools Reprints and Permissions Cite Icon Cite Search Site Citation Y. D. Stoynov; Dynamic behavior of cracked magnetoelectroelastic composites at different boundary conditions. AIP Conf. Proc. 2 October 2012; 1487 (1): 256–263. https://doi.org/10.1063/1.4758966 Download citation file: Ris (Zotero) Reference Manager EasyBib Bookends Mendeley Papers EndNote RefWorks BibTex toolbar search Search Dropdown Menu toolbar search search input Search input auto suggest filter your search All ContentAIP Publishing PortfolioAIP Conference Proceedings Search Advanced Search |Citation Search

The aim of this study is to develop an efficient numerical technique using the non-hypersingular, traction boundary integral equation method (BIEM) for solving wave propagation problems in an anisotropic, viscoelastic plane with cracks. The methodology can be extended from the macro-scale with certain modifications to the nano-scale. Furthermore, the proposed approach can be applied to any type of anisotropic material insofar as the BIEM formulation is based on the fundamental solution of the governing wave equation derived for the case of general anisotropy. The following examples are solved: (i) a straight crack in a viscoelastic orthotropic plane, and (ii) a blunt nano-crack inside a material of the same type. The mathematical modelling effort starts from linear fracture mechanics, and adds the fractional derivative concept for viscoelastic wave propagation, plus the surface elasticity model of M. E. Gurtin and A. I. Murdoch, which leads to nonclassical boundary conditions at the nano-scale. Conditions of plane strain are assumed to hold. Following verification of the numerical scheme through comparison studies, further numerical simulations serve to investigate the dependence of the stress intensity factor (SIF) and of the stress concentration factor (SCF) that develop in a cracked inhomogeneous plane on (i) the degree of anisotropy, (ii) the presence of viscoelasticity, (iii) the size effect with the associated surface elasticity phenomena, and (iv) finally the type of the dynamic disturbance propagating through the bulk material.

This work addresses the scattering of time‐harmonic waves by a finite, blunt nano‐crack in a graded isotropic bulk material with a free surface. The mechanical model is based on classical elastodynamic theory and describes the elastic, isotropic, graded half‐plane with quadratically varying material parameters along the depth coordinate. This model is extended to nano‐mechanics by using non‐classical boundary conditions and localized constitutive equations at the interface between crack and matrix material following the Gurtin‐Murdoch surface elasticity theory. The computational technique employed is based on the non‐hypersingular traction boundary integral equation method (BIEM) using a closed form Green's function for a half‐plane consisting of a functionally graded material (FGM). The dependences of the diffracted and scattered waves and of the local stress concentration fields on key problem parameters such as size and depth of the embedded crack, surface elasticity effects, type and characteristics of the incident wave, and the dynamic interaction phenomenon between the crack and the free‐surface boundary are all examined through parametric studies.

Mathematical modeling of dynamic problems of continuum mechanics in domains of complicated geometry for various types of boundary conditions usually leads to boundary value problems for hyperbolic systems. The method of boundary integral equations is a convenient tool for solving such problems; it allows one to reduce the original differential boundary value problem in a domain to a system of boundary integral equations on the boundary, thus reducing the dimension of the equations, improving the stability of numerical solution methods, and so on. Now this method is widely used for a large class of problems in mechanics and mathematical physics. The construction of fundamental solutions and kernels of boundary integral equations of the system is a key point of the method of boundary integral equations. The fundamental solutions of classical equations of elliptic and hyperbolic types have been studied quite comprehensively [1, 2]. Less is known about fundamental solutions of systems of partial differential equations. Related papers mainly deal with specific equations of continuum mechanics [3–7]. In the present paper, we consider strictly hyperbolic systems of M equations with second-order derivatives in an (N + 1)-dimensional space. For such systems, we construct the Green matrix and analyze its properties. We also study systems invariant under the orthogonal group.

The use of quarter point element in dynamic crack analysis

The asymptotic problem of a conducting crack emanating from the vertex of a wedge in an anisotropic dielectric material under purely electric loading is investigated. The wedge crack in an isotropic material is solved using a conformal mapping technique. The solution of the wedge crack in an anisotropic dielectric material is obtained from that for the transformed isotropic problem after applying a linear transformation method. The electric field intensity factor for the anisotropic wedge crack is obtained in the closed form. The effects of crack and wedge angles as well as anisotropic parameters on the electric field intensity factor are illustrated. The electric field intensity factor is numerically calculated by using the J integral and finite element analysis to validate the exact solution of the electric field intensity factor.

Existence of series solutions for certain nonlinear systems of time fractional partial differential equations

Computation of dynamic stress intensity factors by the time domain boundary integral equation method—II. Examples

Antiplane crack analysis of a functionally graded material by a BIEM

Efficient and accurate stress analysis methods are required to perform routine fracture mechanics analysis of surface and corner cracks in structures. Such analysis capabilities form the basis of fatigue life predictions for growing cracks in gas turbine engine structures. Boundary-integral equation (BIE) methods offer significant advantages to the fracture mechanics analyst in ease of use, accuracy, and modeling efficiency. Techniques for modeling three-dimensional fracture mechanics problems using the boundary-integral equation method are reviewed. Strain energy release rate modeling, together with crack opening displacement data, are found to give accurate and systematic estimates of crack front stress intensity factor distributions. Accuracy of the techniques are evaluated for well-known buried elliptical crack problems. Numerical models are extended to surface and corner cracks.

This paper presents the general solutions of the antiplane electro‐mechanical field for a bimaterial piezoelectric wedge subjected to a pair of concentrated forces and surface charges. Identical wedge angles are assumed here. Using the Mellin transform method, the singularity orders and the generalized stress, strain, electric displacement and electric field intensity factors are derived analytically. The discussion of a semi‐infinite crack at the interface of two bonded piezoelectric half‐planes is also included. Some degenerated cases are investigated to check the accuracy of the obtained solutions.

Stress intensity factors for interacting cracks

Effects of nonhomogeneity on singular electroelastic field near electrodes for a functionally graded piezoelectric material

It is well known that elliptic boundary value problems containing boundary singularities are notoriously difficult to treat numerically, as standard numerical methods lose accuracy in the neighbourhood of the singular point. Consequently, considerable effort has been made in seeking modifications of such numerical methods enabling special treatment to be given to the singularities. In this paper we discuss the use of the Motz technique [4] and the singularity subtraction technique incorporated with the local mesh refinement technique for the numerical solution of singular boundary value problems which are relevant to fracture mechanics. Both techniques for determining the stress intensity and higher order factors (SIFs) are described and successfully applied to the Motz anti-plane crack problem [4]. The problem has been used as a model problem by many applied mathematicians to test different numerical methods for the treatment of elliptic boundary value problems with boundary singularities. Whiteman and Papamichael [8], Rosser and Papamichael [6] used conformal transformation methods to tackle the problem, Symm [7] and Xanthis et al [9] used the boundary integral equation methods, El Misiery and Ortiz [2] used the Tau-lines method [3].

Boundary integral equation crack-tip analysis and applications to elastic media with spatially varying elastic properties