On Saint-Venant's principle for a linear poroelastic material in plane strain

In this paper we consider the state of plane strain in an elastic material with voids occupying a rectangular strip. Such a strip is maintained in equilibrium under self-equilibrated traction and equilibrated force applied on one of the edges, while the other three edges are traction-free and subjected to zero volumetric fraction or zero equilibrated force. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The both cases of homogeneous and inhomogeneous poroelastic materials are considered. The results of the present paper prove how the spatial-decay rate varies with the constitutive profile.     

End effects for a generalized biharmonic equation with applications to functionally graded materials

In this paper we consider a generalized biharmonic equation modeling two-dimensional inhomogeneous elastic state in the curvilinear rectangle a<r<b, 0<θ<α, where (r,θ) denote plane polar coordinates. Such an arch-like region is maintained in equilibrium under self-equilibrated traction applied on one of the edges, while the other three edges are traction free. Our aim is to derive some explicit spatial estimates describing how some appropriate measures concerning the specific Airy stress function evolve with respect to the distance to the loaded edge. Two types of smoothly varying inhomogeneity are considered: (i) the elastic moduli vary smoothly with the polar distance, (ii) they vary smoothly with the polar angle. Such types of smoothly varying inhomogeneous elastic materials provide a model for technological important functionally graded materials. The results of the present paper prove how the spatial decay rate varies with the constitutive profile.     

On Saint‐Venant's principle in a poroelastic arch‐like region

In this paper we consider the state of plane strain in an elastic material with voids occupying a curvilinear strip as an arch‐like region described by R: a<r<b, 0<θ<ω, where r and θ are polar coordinates and a, b, and ω (<2π) are prescribed positive constants. Such a curvilinear strip is maintained in equilibrium under self‐equilibrated traction and equilibrated force applied on one of the edges, whereas the other three edges are traction free and subjected to zero volumetric fraction or zero equilibrated force. In fact, we study the case when one right or curved edge is loaded. Our aim is to derive some explicit spatial estimates describing how some appropriate measures of a specific Airy stress function and volume fraction evolve with respect to the distance to the loaded edge. The results of the present paper prove how the spatial decay rate varies with the constitutive profile. For the problem corresponding to a loaded right edge, we are able to establish an exponential decay estimate with respect to the angle θ. Whereas for the problem corresponding to a loaded curved edge, we establish an algebraical spatial decay with respect to the polar distance r, provided the angle ω is lower than the critical value $\pi\sqrt{2}$. The intended applications of these results concern various branches of medicine as for example the bone implants. Copyright © 2010 John Wiley & Sons, Ltd.     

一类热弹性板的空间衰减估计

研究了二维空间中半无限带形区域上一类含有双调和算子的热弹性系统板解的空间性质.首先构造一个能量表达式,然后利用微分不等式技术,推导出该能量表达式是可由它本身的一阶导数控制的微分不等式,最后得到解的空间衰减估计.该结果可看成是Saint-Venant原则在双曲抛物耦合双调和方程组上的应用.     

Convexity considerations and spatial behavior for the harmonic vibrations in thermoelastic plates

In this paper we study the spatial behavior of the steady-state solutions for the approach of thin thermoelastic plates developed by Lagnese and Lions [J.E. Lagnese, J.-L. Lions, Modelling, Analysis and Control of Thin Plates, Collection RMA, vol. 6, Masson, Paris, 1988]. The model leads to a coupled complex system of partial differential equations, one of fourth order in terms of the amplitude of the vertical deflection and the other of second-order in terms of the amplitude of temperature field. Coupling in an appropriate way the two equations in an integral identity we are able to identify some cross-sectional line integral measures associated with the amplitudes of the vertical deflection and temperature vibrations, provided that the exciting frequency is less than a certain critical frequency. Furthermore, we are able to establish a second-order differential inequality whose integration furnishes a Saint-Venant type decay estimate for a bounded strip and an alternative of Phragmén-Lindelöf type for a semi-infinite strip. The critical frequency is individuated by means of the use of some Wirtinger and Knowles inequalities.     

Stability,Steady‐State Bifurcations,and Turing Patterns in a Predator–Prey Model with Herd Behavior and Prey‐taxis

In this paper, we consider a predator–prey model with herd behavior and prey‐taxis subject to the homogeneous Neumann boundary condition. First, by analyzing the characteristic equation, the local stability of the positive equilibrium is discussed. Then, choosing prey‐tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of nonconstant solutions bifurcating from the positive equilibrium by an abstract bifurcation theory, and find the stable bifurcating solutions near the bifurcation point under suitable conditions. We have shown that prey‐taxis can destabilize the uniform equilibrium and yields the occurrence of spatial patterns. Furthermore, some numerical simulations to illustrate the theoretical analysis are also carried out, Turing patterns such as spots pattern, spots–strip pattern, strip pattern, stable nonconstant steady‐state solutions, and spatially inhomogeneous periodic solutions are obtained, which also expand our theoretical results.     

Influence of Initial Stresses on Stress Intensity Factors at Crack Tips in a Composite Strip

The influence of initial tension or compression along cracks on the stress intensity factor (SIF) at crack tips under the action of additional normal forces on crack edges is studied for infinite bodies. A strip made of a composite material is considered. The strip ends are simply supported, and the strip contains a crack whose edges are parallel to its face planes. The strip is first stretched or compressed along crack edges, and then additional uniformly distributed normal forces are applied to the crack edges. The influence of the initial tension (compression) on the SIF caused by the additional normal forces is studied. The corresponding boundary-value problems are modelled with the use of the three-dimensional linearized theory of elasticity. All the investigations are carried out numerically by employing the finite-element method. The values of SIF are calculated by the energy release method.     

The symmetry principle in continuum mechanics

For problems of the mechanics of an anisotropic inhomogeneous continuum, theorems are given concerning the uninterrupted symmetrical and antisymmetrical analytical continuation of the solution through the plane part of the boundary surface of the medium. Theorems are given for two types of mechanics problem; in the first of these both symmetrical and antisymmetrical continuations of the solution are allowed, while in the second only symmetrical continuation of the solution is allowed. Problems of the first type include problems which are reduced to linear thermoelastic dynamic differential equations of motion of an inhomogeneous anisotropic medium possessing a plane of elastic symmetry, to linear thermoelastic dynamic differential equations of motion of an inhomogeneous Cosserat medium, to non-linear differential equations describing the static elastoplastic stress state of a plate, etc. The second type includes problems which are reduced to non-linear differential equations describing geometrically non-linear strains of shells, to Navier–Stokes equations, etc. These theorems extend the principle of mirror reflection (the Riemann–Schwartz principle of symmetry) to linear and non-linear equations of continuum mechanics. The uninterrupted continuation of the solutions is used to solve problems of the equilibrium state of bodies of complex shape.     

The equations of thermoelasticity with time-dependent coefficients

We consider an inhomogeneous thermoelastic system with second sound in one space dimension where the coefficients are space- and time-dependent. For Dirichlet-Neumann type boundary conditions the global existence of smooth solutions is proved by using the theory of Kato. Then the asymptotic behavior of the solutions is discussed.     

A BEM for transient thermoelastic analysis of a functionally graded layer on a homogeneous substrate under thermal shock

Transient thermoelastic analysis of isotropic and linear thermoelastic bimaterials, which are constituted by a functionally graded (FG) layer attached to a homogeneous substrate, subjected to thermal shock is presented in this paper. For this purpose, a boundary element method for transient linear coupled thermoelasticity is developed. The material properties of the FG layer are assumed to be continuous functions of the spatial coordinates. The boundary-domain integral equations are derived by using the fundamental solutions of linear coupled thermoelasticity for the corresponding isotropic, homogeneous and linear thermoelastic solids in the Laplace-transformed domain. For the numerical solution, a collocation method with piecewise quadratic approximation is implemented. Numerical results for the dynamic stress intensity factors are presented and discussed. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Exponential stability of a network of elastic and thermoelastic materials

In this paper, we study the stability of a network of some elastic and thermoelastic edges. We establish the exponential decay rate of the whole system. Copyright © 2012 John Wiley & Sons, Ltd.     

论修正的Castigliano定理

本文给出了修正的Castigliano定理,与经典的Castigliano定理相比较,修正的Castigliano定理应用方便而且广泛。对经典Castigliano定理的修正是在两个方面,第一个修正是以外载荷与影响函数乘积的表达式代替经典Castigliano定理中余能密度对集中力偏导数的表达式,这一修正为我们在复杂载荷作用下的计算带来极大的简便;第二个修正是在经典Castigliano定理中引入了非齐次边界位移与影响函数乘积的表达式,这一修正为求解复杂边界条件的问题提供了理论基础。我们还说明了如何应用修正的Castigliano定理求解表面结构力学问题的方法。最后,作为修正的Castigliano定理的应用算例,我们求解了两邻边固定另两邻边自由的矩形板的挠曲面方程。     

变刚度矩形薄板的弯曲与稳定计算

本文研究两对边简支,其它两边任意支承,板的刚度沿简支边方向按任意规律变化的矩形板.采用文[1]提出的有限板条元素法求解,其特点与习用的有限元法或有限单条法不同.不是先建立单元或单条的刚度矩阵,然后拼装总刚阵再求解,而是确立各个板条元素的变位和内力的传播关系.实例计算表明,该法是一个简便有效的方法.     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.     

Parameter identification for rubber materials with artificial higher dimensional data

In general, laboratory test render only a limited number of experimental data. Consequently, the prediction of material behaviour becomes a difficult task and, moreover, a statistical analysis with a statistically based approach is almost impossible. As a remedy to increase the number of data, artificial data are generated by stochastic simulation. As a consequence an arbitrary number of data is available and the process of parameter identification can be analysed statistically. Here, the special challenge is the consideration of spatial and inhomogeneous problems. In this work artificial data are generated for a elastomer strip with hole under tension. The inhomogeneous stress/strain fields are optically measured with an Aramis/GOM system and have to be fitted to a stochastic model in order to generate artificial data. B-Splines are applied to fit the geometry of the test specimen and the measured data in space as well as in time. Parameter identifications applied and the resulting material parameters are statistically analysed. In the example, a statistical analysis of an Ogden model is performed. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Thermal effects in orthotropic porous elastic beams

This paper is concerned with the linear theory of anisotropic porous elastic bodies. The extension and bending of orthotropic porous elastic cylinders subjected to a plane temperature field is investigated. The work is motivated by the recent interest in the using of the orthotropic porous elastic solid as model for bones and various engineering materials. First, the thermoelastic deformation of inhomogeneous beams whose constitutive coefficients are independent of the axial coordinate is studied. Then, the extension and bending effects in orthotropic cylinders reinforced by longitudinal rods are investigated. The three-dimensional problem is reduced to the study of two-dimensional problems. The method is used to solve the problem of an orthotropic porous circular cylinder with a special kind of inhomogeneity.      

Quasi-three-dimensional theory for solution of dynamic thermoelastic problem of laminated composite plates

An uncoupled dynamic thermoelastic problem for laminated composite plates has been considered. The hypotheses used take into account the nonlinear distribution of temperature and displacements over the thickness of a laminated plate. On the basis of these hypotheses a quasi-three-dimensional (layerwise) theory is constructed that makes it possible to investigate the internal thermal and stress-strain states, as well as the edge effects of the boundary layer type for laminated plates. Systems of the heat conduction and motion equations are derived using the variational method. The order of the equations depends on the number of layers and terms in expansions of temperature and displacements of each layer. An analytical solution of the dynamic thermoelastic problem is presented for a cross-ply laminated rectangular plate with simply supported edges. The reliability of the results is confirmed by a comparison with the known exact solutions. The results based on the proposed theory can be used for verifying various two-dimensional plate theories when solving the dynamic thermoelastic problems for laminated composite plates.     

A strip yield model solution for an internally cracked piezoelectric strip

An analysis of the crack closure and fatigue crack growth rate have been carried out for an infinitely long poled piezoelectric ceramic strip weakened by a straight hair line internal crack. The ceramic under consideration is assumed to be mechanically more brittle. The crack faces are perpendicular to the poled direction of the strip. The crack faces open in Mode-I deformation on account of in-plane tension applied to the edges of the strip together with either an in-plane electric displacement prescribed on edges of the strip or a uniform constant electric field prescribed on its edges. As a result, a yield zone is formed ahead of each tip of the crack. The yield zones developed are then arrested by applying a normal, cohesive, linearly varying yield point-stress to their rims. For each case, the Fourier transform method is used to find a solution. The resulting integral equations are solved numerically. Expressions are derived for the crack opening displacement and the crack growth rate. The variations in these quantities are plotted in relation to the affecting parameters, viz., the strip thickness, the yield zone length, the electric displacement, and material constants. A case study is presented graphically for PZT-4, PZT-5H, and BaTiO₃ ceramics. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 44, No. 5, pp. 647–664, September–October, 2008.     

关于非局部场论的几个新观点及其在断裂力学中的应用(Ⅱ)——重论非线性非局部热弹性本构方程

按照理性力学的本构公理系统严格推导了非线性非局部热弹性体的本构方程,修正、补充和完善了以前的工作,所得的结果表明,非局部热弹性体的本构响应是与其物质空间的曲率及其高阶梯度相关联的,且存在满足零平均条件的反对称应力;文中给出了反对称应力与局部化残余的表达式,导出了热传递方向与温度降方向一致的结论,简要论述了局部化残余以及非局部能量守恒定律的客观性。