Scattering of love waves by an interface crack between a piezoelectric layer and an elastic substrate

The scattering of Love waves by an interface crack between a piezoelectric layer and an elastic substrate is investigated by using the integral transform and singular integral equation techniques. The dynamic stress intensity factors of the left and the right crack tips are determined. It is found from numerical calculation that the dynamic response of the system depends significantly on the crack configuration, the material combination and the propagating direction of the incident wave. It is expected that specifying an appropriate material combination may retard the growth of the crack for a certain crack configuration. Project supported by the National Natural Science Foundation of China (No. 19891180), the Fundamental Research Foundation of Tsinghua University (JZ 2000.007) and the Fund of the Education Ministry of China.     

Effective elastic moduli of inhomogeneous solids by embedded cell model

An embedded cell model is presented to obtain the effective elastic moduli for three-dimensional two-phase composites which is an exact analytic formula without any simplified approximation and can be expressed in an explicit form. For the different cells such as spherical inclusions and cracks surrounded by sphere and oblate ellipsoidal matrix, the effective elastic moduli are evaluated and the results are compared with those from various micromechanics models. These results show that the present model is direct, simple and efficient to deal with three-dimensional two-phase composites. The project supported by the National Natural Science Foundation of China (No. 19704100) and the National Natural Science Foundation of Chinese Academy of Sciences (No. KJ951-1-201)     

Dynamic growth of martensitic plates in an elastic material

The growth of martensitic plates under conditions of anti-plane shear is considered for a particular isotropic hyperelastic material. An asymptotic solution is presented for the displacement field near the tip of a plate growing at an arbitrary velocity up to the shear wave speed of the austenite. An energy balance shows that the rate of energy dissipation is essentially the same as for the quasi-static motion of a normal equilibrium shock. Numerical solutions illustrate how the martensitic plates develop in an initial boundary value problem.This work was supported by the National Science Foundation through grant MSM-8658107 and through a grant of supercomputer resources at the John von Neumann Center.     

The scattering of harmonic elastic anti-plane shear waves by two collinear cracks in the piezoelectric plate by using the non-local theory

In this paper, the dynamic interaction between two collinear cracks in a piezoelectric material plate under anti-plane shear waves is investigated by using the non-local theory for impermeable crack surface conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. These equations are solved using the Schmidt method. This method is more reasonable and more appropriate. Unlike the classical elasticity solution, it is found that no stress and electric displacement singularity is present at the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress at the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The project supported by the Natural Science Foundation of Heilongjiang Province and the National Natural Science Foundation of China(10172030, 50232030)     

Scattering of elastic waves in an elastic matrix containing an inclusion with interfaces

Based on the theory of elastic dynamics, the scattering of elastic waves and dynamic stress concentration in fiber-reinforced composite with interfaces are studied. Analytical expressions of elastic waves in different medium areas are presented and an analytic method of solving this problem is established. The mode coefficients are determined by means of the continuous conditions of displacement and stress on the boundary of the interfaces. The influence of material properties and structural size on the dynamic stress concentration factors near the interfaces is analyzed. It indicates that they have a great influence on the dynamic properties of fiber-reinforced composite. As examples, numerical results of dynamic stress concentration factors near the interfaces are presented and discussed. This paper provides reliable theoretical evidence for the study of dynamic properties in fiber-reinforced composite. Project supported by the National Natural Science Foundation of China (No. 19972018).     

The OPCM strain gauges for strain and stress measurement of orthotropic material structures

The orthotropic mechanical sensor of piezoelectric composite material made from piezoelectric ceramic and resin materials and their sensing mechanism are presented. The sensing equations of the adhered-and embedded-type sensing units are deduced, which are used to detect the stresses in orthotropic material structures. The surface strain of the orthogonal plate is measured under the action of the planar stress field, and the error is analyzed. Supported by the National Natural Science Foundation (No. 59635640), the Science Foundation of Jiangsu Education Committe (99KJD130001) and the Science Foundation of Jiangsu Province (BK99116).     

Interaction between a screw dislocation and an elastic elliptical inhomogeneity with interfacial cracks

The elastic interaction between a screw dislocation and an elliptical inhomogeneity with interfacial cracks is studied. The screw dislocation may be located outside or inside the inhomogeneity. An efficient complex variable method for the complex multiply connected region is developed, and the general solutions to the problem are derived. As illustrative examples, solutions in explicit series form for complex potentials are presented in the case of one or two interfacial cracks. Image forces on the dislocation are calculated by using the Peach-Koehler formula. The influence of crack geometries and material properties on the image forces is evaluated and discussed. It is shown that the interfacial crack has a significant effect on the equilibrium position of the dislocation near an elliptical-arc interface. The main results indicate, when the length of the crack goes up to a critical value, the presence of the interfacial crack can change the interaction mechanism between a screw dislocation and an elliptical inclusion. The present solutions can include a number of previously known results as special cases.The project supported by the National Natural Science Foundation of China(10272009 and 10472030) and the Natural Science Foundation of Hunan Province(02JJY2014)     

Dynamical equations of various types for finite deformable polar elastic media

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Notes on the nonlinearly elastic Boussinessq problem

The nonlinearly elastic Boussinesq problem is to find the deformation produced in a homogeneous, isotropic, elastic half space by a point force normal to the undeformed boundary, using the exact equations of elasticity for an incompressible or compressible material. First we derive the governing equations from the Principle of Stationary Potential Energy and then we examine some of the implications of the conservation laws of elastostatics when applied to the entire half space, assuming that the well-known linear Boussinesq solution is valid at large distances from the point load. Next, we hypothesize asymptotic forms for the solutions near the point load and, finally, we seek solutions for two specific materials: an incompressible, generalized neo-Hookean (power-law) material introduced by Knowles and a compressible Blatz-Ko material. We find that the former, if sufficiently stiffer than the conventional neo-Hookean material, can support a finite deflection under the point load, but that the latter cannot.This research was supported by the U.S. Army Research Office under Grant DAAL 03-91-G-0022 and by the National Science Foundation under Grant MSS-9102155.     

Boundary element method for arbitrary elastic thin shell

In this paper, a boundary element scheme for arbitrary elastic thin shells is elaborated, Based on BEM of 3D linear elasticity and Kirchhoff's hypothesis, boundary integral equations for shells are deduced. As a result, only Kelvin's solution is used, the difficulty in finding a fundamental solution of arbitrary shells is successfully avoided. The project supported by National Natural Science Foundation of China     

Near-tip fields for a crack growing along an elastic perfectly plastic interface

The stress and deformation fields near the tip of an anti-plane crack growing quasi-statically along an interface of elastic perfectly plastic materials are given in this paper. A family of solutions for the growing crack fields is found covering all admissible crack line shear stress ratios. The project supported by the National Natural Science Foundation of China     

DYNAMIC STRESS FIELD AROUND THE MODE Ⅲ CRACK TIP IN AN ORTHOTROPIC FUNCTIONALLY GRADED MATERIAL

The problem of a Griffith crack in an unbounded orthotropic functionally graded material subjected to antipole shear impact was studied. The shear moduli in two directions of the functionally graded material were assumed to vary proportionately as definite gradient. By using integral transforms and dual integral equations, the local dynamic stress field was obtained. The results of dynamic stress intensity factor show that increasing shear moduli’s gradient of FGM or increasing the shear modulus in direction perpendicular to crack surface can restrain the magnitude of dynamic stress intensity factor.     

Anti-plane crack moving along the interface of dissimilar piezoelectric materials

The problem of an anti-plane Griffith crack moving along the interface of dissimilar piezoelectric materials is solved by using the integral transform technique. It is shown from the result that the intensity factors of anti-plane stress and electric displacement around the crack tip are dependent on the speed of the Griffith crack as well as the material coefficients. When the two piezoelectric materials are identical, the present result will be reduced to the result for the problem of an anti-plane moving Griffith crack in homogeneous piezoelectric materials. Supported by the National Natural Science Foundation and the National Post-doctoral Science Foundation of China.     

Cavitation in hookean elastic membranes

An exact solution to cavitation is found in tension of a class of Cauchy elastic membranes. The constitutive relationship of materials is based on Hookean elastic law and finite logarithmic strain measure. A variable transformation is used in solving the two-point boundary-value problem of nonlinear ordinary differential equation. A simple formula to calculate the critical stretch for cavitation is derived. As the numerical results, the bifurcation curves describing void nucleation and suddenly rapidly growth of the cavity are obtained. The boundary layers of displacements and stresses near the cavity wall are observed. The cata-strophic transition from homogeneous to cavitated deformation and the jumping of stress distribution are discussed. The result of the energy comparison shows the cavitated deformation has lower energy than the homogeneous one, thus the state of cavitated deformation is relatively stable. All investigations illustrate that cavitation reflects a local behavior of materials. Project supported by the National Natural Science Foundation of China (No. 19802012) the Scientific Research Foundation for Returned Overseas Chinese Scholars, and the Scientific Research Foundation for Key Teachers in Chinese Universities.     

Pure bending problem for a vibrating elastic plate with non-uniform temperature

This work was performed in the University of Delaware Center for Composite Materials under the auspices of the University/Industry Research Program Application of Composite Materials to Industrial Products. The author(s) acknowledge the support of the University of Delaware/Rutgers University National Engineering Research Center for Composites Manufacturing Science and Engineering. The Center is a National Science Foundation cross-disciplinary research program.     

Surface cracking of a piezoelectric strip bonded to an elastic substrate (Mode I crack problem)

Summary A piezoelectric material layer bonded to an elastic substrate is investigated. The piezoelectric layer contains an edge crack that is perpendicular to the surface of medium. The poling axis of the piezoelectric layer is parallel to its edge. The elastic layer can be an ideal insulator or an ideal conductor. The Fourier transform technique is used to reduce the problem to a solution of singular integral equations. Both impermeable crack and permeable crack assumptions are considered. Stress and electric displacement intensity factors are investigated for different dimensions of the medium. A double-edge cracked symmetric piezoelectric laminate under symmetric electro-mechanical load is also investigated. BLW would like to thank the National Science Foundation of China (#10102004) and the City University of Hong Kong for the support of this work (DAG #7100219). YGS also thanks the Multidiscipline Scientific Research Foundation Project (HIT. MD 2001. 39) of the Harbin Institute of Technology and the SRF for ROCS, SEM.     

The approximate solution of penny-shaped cracks periodically distributed in infinite elastic body

The penny-shaped cracks periodically distributed in infinite elastic body are studied. The problem is approximately simplified to that of a single crack embedded in finite length cylinder and the stress intensity factor is obtained by solving a Fredholm integral equation. Numerical results are given and the effects of crack interaction on the stress intensity factor are discussed. The project suppoted by National Natural Science Foundation of China     

Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases. The project supported by the National Natural Science Foundation of China (10372083, 90405016), 973 Program (2006CB601205) and the Aeronautical Science Foundation (04B53080). The English text was polished by Keren Wang.     

MULTIPLE SCATTERING AND DYNAMIC STRESS ANALYSIS OF ELASTIC WAVES IN A FIBER—REINFORCED COMPOSITE WITH INTERFACES

Based on the theory of elastic dynamics, multiple scattering of elastic waves and dynamic stress concentrations in fiber-reinforced composite are studied. The analytical expressions of elastic waves in different regions are presented. The mode coefficients of elastic waves are determined in accordance with the continuous conditions of displacement and stress on the boundary of the multi-interfaces. By using the addition theorem of Hankel functions, the formula of scattered wave fields in different local coordinates are transformed into those in one local coordinate to determine the unknown coefficients and dynamic stress concentration factors (DSCFs). The influences of the distance between two inclusions, material properties and structural size on the DSCFs near the interfaces are analyzed. As examples, the numerical results of DSCFs near the interfaces for two kinds of fiber-reinforced composites are presented and discussed. The project supported by the National Natural Science Foundation of China (19972018)     

The behavior of a crack in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading

Summary In this paper, the behavior of a crack in functionally graded piezoelectric/piezomagnetic materials subjected to an anti-plane shear loading is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using a Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. These equations are solved using the Schmidt method. The relations among the electric displacement, the magnetic flux and the stress field near the crack tips are obtained. Numerical examples are provided to show the effect of the functionally graded parameter on the stress intensity factors of the crack.The authors are grateful for financial support from the Natural Science Foundation of Hei Long Jiang Province (A0301), the National Natural Science Foundation of China (50232030, 10172030), the Natural Science Foundation with Excellent Young Investigators of Hei Long Jiang Province(JC04-08) and the National Science Foundation with Excellent Young Investigators (10325208).