THE CONVERGENT CONDITION AND UNITED FORMULA OF STEP REDUCTION METHOD

The step reduction method was first suggested by Prof. Yeh Kai-yuan. This method has more advantages than other numerical methods. By this method, the analytic expression of solution can he obtained for solving nonuniform elastic mechanics. At the same time. its ealculuting time is very short and convergent speed very fast. In this paper, the convergent condition and nited formula of step reduction method are given by mathemutical method. It is proved that the solution of displacement and stress resultants obtained by this method can eonverge to exact solution uniformly, when the convergent condition is sutisfied. By united formula, the analytic solution solution can be expressed as matrix form, and therefore the former complicated expression can be avoMed. Two numerical examples are given at the end of this paper which indicate that. by the theory in this paper, a right model can be obtained for step reduction method.     

THE BOUNDARY EXPANDING-CONTRACTING METHOD

The boundary element method(BEM)which a used to solve the elastic problems has more advantages than other numerical methods.Especially,it can resolve rapidly varying internal stress and strain fields more accurately.However,it is of ten fails in the region near the boundary because of the singularity of the solutions.Though we can increase the boundary meshes more and more,the solutions of stress on the boundary can‘t be given directly;which has obstructed the applications of the BEM to some extent.In this paper we proposed the boudary expanding-contracting principle and the boundary expanding-contracting method(BECM)based on the principle.With this method,not only the solutions in the region near or on the boundary can be obtained directly,but the iterative processes can also be used conveniently to improve the accuracy of the solutions.     

BOUNDARY VALUE PROBLEMS OF TWO-DIMENSIONAL ANISOTROPIC BODY WITH A PARABOLIC BOUNDARY

Based upon Stroh formalism we derive a novel and convenient scheme for determiningthe elastic fields of a two-dimensional anisotropic body with a parabolic boundary subject to two kindsof boundary conditions, which are free surface and rigid surface, respectively. The correspondingGreen's functions are found by using the conformal mapping method. When the parabolic curve de-generates into a half-infinite crack or rigid inclusion, the singular stress fields near the tip of defectsare obtained. In particular, those Green's functions for a concentrated moment M_0 applied at a pointon the parabolic curve are also studied. It is easily found that arbitrary parabolic boundary value prob-lems can be solved by using these Green's functions and associate integrals.     

Crack buckling in soft gels under compression

Recent interest in designing soft gels with high fracture toughness has called for simple and robust methods to test fracture behavior. The conventional method of applying tension to a gel sample suffers from a difficulty of sample gripping. In this paper, we study a possible fracture mechanism of soft gels under uni-axial compression. We show that the surfaces of a pre-existing crack, oriented parallel to the loading axis, can buckle at a critical compressive stress. This buckling instability can open the crack surfaces and create highly concentrated stress fields near the crack tip, which can lead to crack growth. We show that the onset of crack buckling can be deduced by a dimensional argument com- bined with an analysis to determine the critical compression needed to induce surface instabilities of an elastic half space. The critical compression for buckling was verified for a neo-Hookean material model using finite element simulations.     

Indentation of a compressible soft electroactive half-space: Some theoretical aspects

We theoretically study the indentation response of a compressible soft electroactive material by a rigid punch. The half-space material is assumed to be initially subjected to a finite deformation and an electric biasing field. By adopting the linearized theory for incremental fields, which is established on the basis of a general nonlinear theory for electroelasticity, the appropriate equations governing the perturbed infinitesimal elastic and electric fields are derived particularly when the material is subjected to a uniform equibiaxial stretch and a uniform electric displacement. A general solution to the governing equations is presented, which is concisely expressed in terms of four quasi-harmonic functions. By adopting the potential theory method, exact contact solutions for three common perfectly conducting rigid indenters of flat-ended circular, conical and spherical geometries can be derived, and some explicit relations that are of practical importance are outlined.     

Scattering of SH-wave by cracks originating at an elliptic hole and dynamic stress intensity factor

The method of complex function and the method of Green‘s function are used to investigate the problem of SH-wave scattering by radial cracks of any limited length along the radius originating at the boundary of an elliptical hole, and the solution of dynamicstress intensity factor at the crack tip was given. A Green‘s function was constructed for the problem, which is a basic solution of displacement field for an elastic half space containing a half elliptical gap impacted by anti-plane harmonic linear source force at any point of its horizontal boundary. With division of a crack technique, a series of integral equations can be established on the conditions of continuity and the solution of dynamic stress intensity factor can be obtained. The influence of an elliptical hole on the dynamic stress intensity factor at the crack tip was discussed.     

DIFFRACTION OF ELASTIC WAVES IN THE PLANE MULTIPLY-CONNECTED REGION AND DYNAMIC STRESS CONCENTRATION

This paper deals with the problem of diffraction of elastic waves in the plane multiply-connected regions by the theory of complex functions.The complete function series whichapproach the solution of the problem and general expressions for boundary conditions aregiven.Then the problem is reduced to the solution to infinite series of algebraic equationsand the solution can be directly obtained by using electronic computer.In particular,for thecase of weak interaction,an asymptotic method is presented here,by which the problem of pwaves diffracted by a circular cavities is discussed in detail.Based on the solution of thediffracted wave field the general formulas for calculating stress concentrationfactor for a cavity of arbitrary shape in multiply-connected region are given.     

THE NEW CRITERIA OF ELASTIC AND FATIGUE FAILURE IN THE COMPONENT OF COMPLEX STRESS STATES

In this paper, a total criterion on elastic and fatigue failure in complex stress, that is, octahedral stress strength theory on dynamic and static states on the basis of studying modern and classic strength theories. At the same time, an analysis of an independent and fairly comprehensive theoretical system is set up. It gives generalized failure factor by 36 materials and computative theory of the 11 states of complex stresses on a point, and derives the operator equation on generalized allowable strength and a computative method for a total equation can be applied to dynamic and static states. It is illustrated that the method has a good exactness through computation of eight examples of engineering. Therefore, the author suggests applying it to engineering widely.     

General solution for large deflection problems of nonhomogeneous circular plates on elastic foundation

In this paper, a new method is presented based on [1]. It can be used to solve the arbitrary nonlinear system of differential equations with variable coefficients. By this method, the general solution for large deformation of nonhomogeneous circular plates resting on an elastic foundation is derived. The convergence of the solution is proved. Finally, it is only necessary to solve a set of nonlinear algebraic equations with three unknowns. The solution obtained by the present method has large convergence range and the computation is simpler and more rapid than other numerical methods.Numerical examples given at the end of this paper indicate that satisfactory results of stress resullants and displacements can be obtained by the present method. The correctness of the theory in this paper is, confirmed.     

SURFACE EFFECTS ON ELASTIC FIELDS AROUND SURFACE DEFECTS

<正>There are always severe stress concentrations around surface defects like grooves or bugles,which might induce the failure of solid materials and structures.In the present paper,we consider the elastic fields around nanosized bugles and grooves on solid surfaces.The influence of surface elasticity on the elastic deformation is addressed through a finite element method.It is found that when the size of defects shrinks to nanometer,the stress fields around such defects will be affected significantly by surface effects.     

The null field approach to two-dimensional elastostatics

The vector basis functions, necessary for solving two-dimensional inclusion problems in an elastic solid under time independent conditions by means of the null field approach (T-matrix method), are obtained as a zero frequency limit of the corresponding basis functions commonly used in elastodynamics. The expansion of the fields appearing in the surface integral representation of the static displacement can thus be achieved, leading to the T-matrix equations of 2d-elastostatics. We specialize the problem to the simple boundary condition case of a single cavity and develop the analytical expressions as much as possible before numerical implementation. A numerical test for the ellipse and some examples for the superellipse, with applied static pressure or shear stress at infinity, are given.     

A novel hybrid finite element analysis of inplane singular elastic field around inclusion corners in elastic media

This paper deals with the inplane singular elastic field problems of inclusion corners in elastic media by an ad hoc hybrid-stress finite element method. A one-dimensional finite element method-based eigenanalysis is first applied to determine the order of singularity and the angular dependence of the stress and displacement field, which reflects elastic behavior around an inclusion corner. These numerical eigensolutions are subsequently used to develop a super element that simulates the elastic behavior around the inclusion corner. The super element is finally incorporated with standard four-node hybrid-stress elements to constitute an ad hoc hybrid-stress finite element method for the analysis of local singular stress fields arising from inclusion corners. The singular stress field is expressed by generalized stress intensity factors defined at the inclusion corner. The ad hoc finite element method is used to investigate the problem of a single rectangular or diamond inclusion in isotropic materials under longitudinal tension. Comparison with available numerical results shows the present method is an efficient mesh reducer and yields accurate stress distribution in the near-field region. As applications, the present ad hoc finite element method is extended to discuss the inplane singular elastic field problems of a single rectangular or diamond inclusion in anisotropic materials and of two interacting rectangular inclusions in isotropic materials. In the numerical analysis, the generalized stress intensity factors at the inclusion corner are systematically calculated for various material type, stiffness ratio, shape and spacing position of one or two inclusions in a plate subjected to tension and shear loadings.     

Stochastic functional expansion in elasticity of heterogeneous solids

The Volterra-Wiener functional expansion is employed to the analysis of statistic properties for random heterogeneous solids. For simplicity, the technique is displayed on an elastic suspension of spheres. The basis function in the expansion is chosen as that corresponding to the so-called “perfect disorder” of spheres (PDS), recently introduced by the authors. An infinite hierarchy of equations for the kernels in the expansion is derived whose truncating after the nth equation is shown to yield results for the averaged statistical characteristics which are valid to order cⁿ_f, where c_f is the volume fraction of the spheres. The kernels for the first and the second approximations, n = 1, 2, are found and related to the displacement fields in an infinite elastic body containing, respectively, one and two spherical inhomogeneities. Within the frame of the so-called singular approximation the overall tensor of elastic moduli for a suspension of perfectly disordered spheres is shown to coincide to the order c²_f with a formula, earlier obtained by means of the method of the effective field.     

Stress concentration around a small spherical inhomogeneous inclusion on the axis of a circular cylinder in torsion

Stresses are calculated for a small elastic inclusion bonded (in a way to ensure continuous displacements and tractions across the surface of the inclusion) to a large circular cylindrical shaft. The inclusion considered is inhomogeneous and of spherical shape. The shear modulus of the inclusion is assumed to be μ₀ exp(−1/2β² r ²). Some special cases are also discussed.     

A Crack with an Electric Displacement Saturation Zone in an Electrostrictive Material

A crack with an electric displacement saturation zone in an electrostrictive material under purely electric loading is analyzed. A strip saturation model is here employed to investigate the effect of the electrical polarization saturation on electric fields and elastic fields. A closed form solution of electric fields and elastic fields for the crack with the strip saturation zone is obtained by using the complex function theory. It is found that the K _I -dominant region is very small compared to the strip saturation zone. The generalized Dugdale zone model is also employed in order to investigate the effect of the saturation zone shape on the stress intensity factor. Using the body force analogy, the stress intensity factor for the asymptotic problem of a crack with an elliptical saturation zone is evaluated numerically.     

In Plane Displacement Formulation for Finite Cracked Plates Under Mode I Using Grid Method and Finite Element Analysis

A grid method is used to experimentally determine the in-plane displacement fields around a crack tip in a Single-Edge-Notch (SEN) tensile polyurethane specimen. Horizontal displacement u _x-exp and vertical displacement u _y-exp are expressed as functions of circular coordinates centred on the crack tip. These are compared with the approximate solutions of linear elastic fracture mechanics with a view to studying the applicability to polymers. The results show that this solution is not in agreement with the experiments at the focused on the vicinity of a crack tip. Taking this into account, an FEA program is developed with CAST3M for the purpose of comparing the experimental displacements and the numerical data. New formulations of displacements u _x and u _y are then developed. These formulations are derived from the principle of superposition and based on Arakawa’s formulation. With the displacement gradients obtained from the FEA and the new formulations, the determination of J-integrals is found to be in very good agreement with those derived from numerical calculation. Consequently, the proposed formulations can give displacement fields compatible with the J-integral calculation for the region near the crack tip. An application based on an experimental test is proposed to evaluate the performances of the proposed formulations.     

The influence of elastic waves on dynamic stress intensity factors (two-dimensional problems)

Summary In this paper, an indirect boundary integral equation method for the solution of dynamic crack problems is presented. The Laplace transform method is used to derive the fundamental solutions for the opening mode (mode I) and the sliding mode (mode II) displacement discontinuity. Accurate dynamic stress intensity factorsK _N (t) (N=I,II) resulting from different time-dependent loads on the crack surface are obtained. The specific influences of the various elastic waves on the stress intensity factors can be clearly seen from the results.On leave Central-South University of Technology Changsha, P.R. China     

A moving dislocation in a magneto–electro-elastic solid

This work considers the generalized plane problem of a moving dislocation in an anisotropic elastic medium with piezoelectric, piezomagnetic and magnetoelectric effects. The closed-form expressions for the elastic, electric and magnetic fields are obtained using the extended Stroh formalism for steady-state motion. The radial components, E_rand H_r, of the electric and magnetic fields as well as the hoop components, D_θ and B_θ, of electric displacement and magnetic flux density are found to be independent of θ in a polar coordinate system. This interesting phenomenon is proven to be is a consequence of the electric and magnetic fields, electric displacement and magnetic flux density that exhibit the singularity r⁻¹ near the dislocation core. As an illustrative example, the more explicit results for a moving dislocation in a transversely isotropic magneto–electro-elastic medium are provided and the behavior of the coupled fields is analyzed in detail.     

Elastic fields and effective moduli of particulate nanocomposites with the Gurtin–Murdoch model of interfaces

A complete solution has been obtained for periodic particulate nanocomposite with the unit cell containing a finite number of spherical particles with the Gurtin–Murdoch interfaces. For this purpose, the multipole expansion approach by Kushch et al. [Kushch, V.I., Mogilevskaya, S.G., Stolarski, H.K., Crouch, S.L., 2011. Elastic interaction of spherical nanoinhomogeneities with Gurtin–Murdoch type interfaces. J. Mech. Phys. Solids 59, 1702–1716] has been further developed and implemented in an efficient numerical algorithm. The method provides accurate evaluation of local fields and effective stiffness tensor with the interaction effects fully taken into account. The displacement vector within the matrix domain is found as a superposition of the vector periodic solutions of Lamé equation. By using local expansion of the total displacement and stress fields in terms of vector spherical harmonics associated with each particle, the interface conditions are fulfilled precisely. Analytical averaging of the local strain and stress fields in matrix domain yields an exact, closed form formula (in terms of expansion coefficients) for the effective elastic stiffness tensor of nanocomposite. Numerical results demonstrate that elastic stiffness and, especially, brittle strength of nanoheterogeneous materials can be substantially improved by an appropriate surface modification.