Decay and other estimates for a semi-infinite magnetoelastic cylinder: Saint-Venant''s principle

An analogue of the well known Toupin's version of Saint-Venant's principle is proved for a semi-infinite magnetoelastic cylinder under very mild assumptions on the asymptotic behaviour of the Dirichlet integral of the magnetic field and of the elastic energy. With regard to the elastic fields, we assign on the base either the stress or the displacement vector while we assume that the lateral surface is either traction free or held fixed at zero displacement. We make use of the first Korn inequality and estimate the total energy of the conductor in terms of the data for all the problems considered.     

一种新的计算各向异性材料裂纹尖端应力强度因子杂交元法

利用有限元特征分析法研究了平面各向异性材料裂纹端部的奇性应力指数以及应力场和位移场的角分布函数,以此构造了一个新的裂纹尖端单元。文中利用该单元建立了研究裂纹尖端奇性场的杂交应力模型,并结合Hellinger-Reissner变分原理导出应力杂交元方程,建立了求解平面各向异性材料裂纹尖端问题的杂交元计算模型。与四节点单元相结合,由此提出了一种新的求解应力强度因子的杂交元法。最后给出了在平面应力和平面应变下求解裂纹尖端奇性场的算例。算例表明,本文所述方法不仅精度高,而且适应性强。     

Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.     

Exponential decay estimates for solutions of the von kármán equations on a semi-infinite strip

This paper is concerned with the analysis of Saint-Venant edge effects for nonlinear elastic plates. The model used is based on the von Kármán plate equations: a coupled system of two nonlinear elliptic partial differential equations with the biharmonic operator as the principal part. Energy methods are used to establish a nonlinear integro-differential inequality for a quadratic functional. Arguments based on comparison theorems are then used to establish exponential decay of end effects.     

Stress and fabric in granular material

It has been well recognized that, due to anisotropic packing structure of granular material, the true stress in a specimen is different from the applied stress. However, very few research efforts have been focused on quantifying the relationship between the true stress and applied stress. In this paper, we derive an explicit relationship among applied stress tensor, material-fabric tensor, and force-fabric tensor; and we propose a relationship between the true stress tensor and the applied stress tensor. The validity of this derived relationship is examined by using the discrete element simulation results for granular material under biaxial and triaxial loading conditions.     

Exact solutions of two semi-infinite collinear cracks in piezoelectric strip

Using the complex variable function method and the conformal mapping technique,the fracture problem of two semi-infinite collinear cracks in a piezoelectric strip is studied under the anti-plane shear stress and the in-plane electric load on the partial crack surface.Analytic solutions of the field intensity factors and the mechanical strain energy release rate are derived under the assumption that the surfaces of the crack are electrically impermeable.The results can be reduced to the well-known solutio...     

The effect of a variation in the elastic moduli on Saint-Venant's principle for a half-cylinder

A semi-infinite prismatic cylinder composed of a linear anisotropic classical elastic material is in equilibrium under zero body force and either zero displacement or zero traction on the lateral boundary. The elastic moduli become perturbed. Under suitable conditions on the base load decay estimates are derived for the difference between corresponding quantities in the unperturbed and perturbed bodies. The amplitude in each estimate involves a multiplicative factor that tends to zero as the perturbation tends to zero. The analysis, based upon a first-order differential inequality, introduces apparently new modifications of Korn's inequalities of the first and second kind.     

Stress analysis for an infinite strip weakned by periodic cracks

Stress analysis for an infinite stripcracks were assumed in a horizontal position,weakened by periodic cracks is studied. The and the strip was applied by tension “p“ in y-direction. The boundary value problem can be reduced into a complex mixed one. It is found that the EEVM ( eigenfunction expansion variational method) is efficient to solve the problem. The stress intensity factor at the crack tip and the T-stress were evaluated. From the deformation response under tension the cracked strip can be equivalent to an orthotropic strip without cracks. The elastic properties in the equivalent orthotropic strip were also investigated. Finally, numerical examples and results were given.     

Moving crack for functionally grated material in an infinite length strip under antiplane shear

Considered is a Yoffe crack in an infinite strip of functionally grated material (FGM) subjected to antiplane shear. The shear moduli in two directions of FGM are assumed to be of exponential form. The dynamic stress intensity factor and strain energy density factor at the crack tip are obtained by using integral transforms and dual-integral equations. The numerical results show that the decrease of the strain energy density factor varies with the shear moduli gradient, and the increase of the strain energy density factor varies with the increase of the moving crack speed. The ratio of shear moduli in material vertical orientation has a great influence on the strain energy density factor.     

金属成型材料参数的反求技术

给出了快速准确地获得材料处于弹塑性大变形状态下各向异性弹塑性本构模型参数反求方法。首次提出了筛选试验测试点的活度规则,并以此来指导试验测试点的位置的选择;提出了仿真先验信息的概念,丰富了获取材料参数先验信息的途径;混合采用Levenberg—Marquardt方法和Gauss—Newton法的优化策略,给出了材料参数反求的基本公式和关键算法。数值算例表明,反求参数的初值以及反求区间的确定对于反求结果有着重要影响,为了确保反求过程的顺利进行,必须充分了解材料模型的先验信息．并充分利用筛选试验测试点的活度规则。同时效值算例计算还表明本文方法具有很高的计算精度和计算效率。     

Anti-plane shear crack in a functionally gradient piezoelectric material

The main objective of this paper is to study the singular nature of the crack-tip stress and electric displacement field in a functionally gradient piezoelectric medium having material coefficients with a discontinuous derivative. The problem is considered for the simplest possible loading and geometry, namely, the anti-plane shear stress and electric displacement in-plane of two bonded half spaces in which the crack is parallel to the interface. It is shown that the square-root singularity of the crack-tip stress field and electric displacement field is unaffected by the discontinuity in the derivative of the material coefficients. The problem is solved for the case of a finite crack and extensive results are given for the stress intensity factors, electric displacement intensity factors, and the energy release rate. Project supported by the National Natural Science Foundation of China (No. 10072041), the National Excellent Young Scholar Fund, of China (No. 10125209) and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P. R. C..     

Spatial decay estimates and upper bounds in elasticity for domains with unbounded cross-sections

In this paper we investigate spatial decay estimates and upper bounds for the solutions of elastic problems when the cross-sections of the three dimensional solid are semi-infinite strips. We obtain spatial decay estimates for the solutions of a static problem in the theories of homogeneous and isotropic linear elasticity and linearised elasticity. Energy bounds and some spatial decay estimates are obtained for the solutions of a dynamical problem in the case of anisotropic linear elasticity. For both kinds of problems we use the energy methods.     

Dynamic stress of a circular cavity buried in a semi-infinite functionally graded piezoelectric material subjected to shear waves

The paper presents a theoretical method to investigate the multiple scattering of shear waves and dynamic stress around a circular cavity in a semi-infinite functionally graded piezoelectric material. The analytical solutions of wave fields are expressed by employing wave function expansion method and the expanded mode coefficients are determined by satisfying the boundary conditions of the cavity. Image method is used to satisfy the free boundary condition of the semi-infinite structure. According to the analytical expression of this problem, the numerical solutions of the dynamic stress concentration factor around the cavity are presented. The effects of the piezoelectric property, the buried depth of the cavity, the incident wave number and the nonhomogeneous parameter of materials on the dynamic stress around the cavity are analyzed. Analyses show that the piezoelectric property has great effect on the dynamic stress in the region of intermediate frequency and the effect increases with increasing wave number. When the nonhomogeneous parameter of materials is less than zero, it has less influence on the maximum dynamic stress around the cavity; however, it has greater influence on the distribution of the dynamic stress around the cavity. When the nonhomogeneous parameter of materials is greater than zero, it has greater influence on both the maximum dynamic stress and the distribution of dynamic stress around the cavity, especially in the case that the buried depth is comparatively small.     

Stress concentration factor expression for tension strip with eccentric elliptical hole

Two explicit expressions of the stress concentration factor for a tension finite-width strip with a central elliptical hole and an eccentric elliptical hole, respectively, are formulated by using a semi-analytical and semi-empirical method. Accuracy of the results obtained from these expressions is better, and application scope is wider, than the results of Durelli’s photo-elastic experiment and Isida’s formula. When eccentricity of the elliptical hole is within a certain range, the error is less than 8%. Based on the relation between the stress concentration factor and the stress intensity factor, a stress intensity factor expression for tension strips with a center or an eccentric crack is derived with the obtained stress concentration factor expressions. Compared with the existing formulae and the finite element analysis, this stress intensity factor expression also has sufficient accuracy.     

The anti-plane shear field in an infinite slab of elasto-damaged material with a semi-infinite crack

This paper deals with an infinite slab with a semi-infinite crack, which is subjected to the anti-plane sheark _III field at infinity. The slab is made of an elasto-damaged material. Analytical solution is obtained by use of conformal mapping. The shape of damaged-zone, the dissipative energy, the shear opening displacement on the crack surface and several stress distribution curves are given. The far field condition is checked, The asymptotic behavior near the crack-tip is given. The project supported by National Natural Science Foundation of China     

Dynamic fracture mechanics analysis for composite material with material non-homogeneity in thickness direction

The problem considered here is the response of a non-homogeneous composite material containing some cracks subjected to dynamic loading. It is assumed that the composite material is orthotropic and all the material properties depend only on the coordinatey (along the thickness direction). In the analysis, the elastic region is divided into a number of plies of infinite length. The material properties are taken to be constants for each ply. By utilizing the Laplace transform and Fourier transform technique, the general solutions for plies are derived. The singular integral equations of the entire elastic region are obtained and solved by the virtual displacement principle. Attention is focused on the time-dependent full field solutions of stress intensity factor(SIF) and strain energy release rate. As a numerical illustration, the dynamic stress intensity factor of a substrate/functionally graded film structure with two cracks under suddenly applied forces on cracks face are presented for various material non-homogeneity parameters.     

A Higher Order Asymptotic Analysis for Orthotropic Plates in Stress Edge Conditions

A higher order asymptotic analysis for orthotropic plates is presented with the leading order interior solution reduced to the well-known Kirchhoff plate theory. The boundary-layer solutions are decoupled into the plane strain and torsional deformations of a boundary-layer plane strip, which is analyzed by the Laplace transform method to formulate Saint-Venant’s principle in plate studies. The necessary and sufficient conditions for stress edge-data generating exponentially decaying solutions are established to derive the stress edge conditions for the higher order plate theory. Mathematics Subject Classifications (2000) 74K20, 74B05, 74G10, 74G50.     

Computational assessment of material structure weaknesses in polycrystalline materials

Summary By incorporating local grain orientation, grain geometry and macroscopic elastic properties, a numerical procedure has been developed for computational prediction of mesoscopic stress and strain distributions in simulated polycrystalline material samples. The numerical procedure is developed on the basis of the concept of grain-average fields, Kröner–Kneer model, Waldvogel-Rodin algorithm and a self-adaptive method. Repeated computer tests were performed to investigate mesoscopic stress variation in the samples, and find coherent interrelations of material structure weaknesses (MSWs) with local microstructure of the samples. It was found that the stronger the single crystal elastic anisotropy, the stronger the inhomogeneity of mesoscopic stress distribution. Not only the elastic anisotropy, but also the grain geometry, may produce significant local stress disturbances. It has been found that the defined orientation-geometry factor and correlation parameter are two adequate physical quantities which account for synergetic interactions due to grain-orientation geometry-induced anisotropy. By using the two quantities, MSWs can be well correlated with local microstructure. Computer tests also show that 250–400 conjoining grains are necessary to homogenize the mesoscopic stress distribution in the considered materials.     

Transversely isotropic hyper-elastic material rectangular plate with voids under a uniaxial extension

IntroductionHyper_elasticmaterials ,suchasrubberandpolyurethane ,havemanyexcellentpropertiesandhavebeenusedwidelyinalmostallregionsofevery_daylifeandindustrialmanufacturing .Thevoidformationandgrowthinhyper_elasticmaterialsduetotheinstabilityofmaterialsplayafundamentalroleinthemechanismsofmaterialfractureandfailure.SotheproblemhasgotacertaindevelopmentinthepasttwentyyearsandtherecentreviewisthatofHorgan[1] .Chou_WangandHorgan[2 ] ,RenandCheng[3 ,4] studiedthegrowthofacentervoidinthecylindero…     

Stochastic analysis of a thermoelastic problem in functionally graded plates with uncertain material properties

The statistics (i.e., mean and variance) of temperature and thermal stress are analytically obtained in functionally graded material (FGM) plates with uncertainties in the thermal conductivity and coefficient of linear thermal expansion. These FGM plates are assumed to have arbitrary nonhomogeneous thermal and mechanical properties through the entire thickness of plate and are subjected to deterministic convective heating. The stochastic temperature and thermal stress fields are analysed by assuming the FGM plate is multilayered with distinct, random thermal conductivity and coefficient of linear thermal expansion in each layer. Vodicka’s method, which is a type of integral transform method, and a perturbation method are employed to obtain the analytical solutions for the statistics. The autocorrelation coefficients of each random property and cross-correlation coefficients between different random properties are expressed in exponential function forms as a non-homogeneous Markov random field of discrete space. Numerical calculations are performed for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304), which have the largest dispersion of the random properties at the place where the volume fractions of the two constituent materials are both 0.5. The effects of the spatial change in material composition, thermal boundary condition and correlation coefficients on the standard deviations of the temperature and thermal stress are discussed.