An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.     

Effect of elastomer slight compressibility

This paper is concerned with the constitutive equation for slightly compressible elastic material under finite deformations. We show that material slight compressibility can be effectively taken into account in the case of high hydrostatic pressure or highly confined material. In all other situations the application of the incompressible and nearly incompressible material theories gives practically the same results. Therefore it is of interest to consider the problem in which allowing for material slight compressibility leads to results essentially different from those obtained with help of the incompressible material model. In the present paper this difference has been demonstrated for the problem of high hydrostatic pressure causing an increase of the ‘bulk’ and ‘shear’ material moduli. The behavior of a long hollow cylinder of real material under finite deformations is analyzed. The cylinder is subjected to internal pressure and axial and circular displacements at the outer surface. This problem has been solved analytically using the small parameter method. The solution obtained predicts a decrease of the axial and circular displacements of the outer surface under fixed shear (axial and circular) forces when the internal pressure is applied.     

Interaction of bursts in exponentially graded materials characterized by parametric plots

The nonlinear interaction of tone bursts in functionally graded materials with strongly variable properties is studied resorting to the five constant nonlinear theory of elasticity in the 1D setting. The problem is solved numerically for an exponentially graded material. The influence of the material properties variation on the bursts profiles evolution is traced on the boundaries of the sample. A special case of the bursts interaction by which oscillations evoked by counterpropagating bursts disappear in the homogeneous material is proposed as a reference case for the problem of nondestructive material characterization. The deviation from this special case caused by inhomogeneity in material properties is analyzed by parametric plots. Obtained results may be used by qualitative nondestructive determination of the inhomogeneity in material properties.     

Evolution of the interface in a stratified anisotropic porous material

It is shown that the boundary-value problem describing the evolution of the interface during impregnation of a stratified inhomogeneous anisotropic porous material with a viscous fluid can be reduced to a similar problem for a stratified inhomogeneous isotropic material by nonorthogonal transformation of the coordinates. As a result, the well-known estimates of the problem parameters determining the interface configuration for impregnation of an isotropic material can be extended to the anisotropic case.     

Behavior of brittle anisotropic materials with different orientation of mechanical properties at the edge of piercing

The problem of normal interaction between a compact steel isotropic cylindrical projectile and an orthotropic plate at the edge of piercing in the range of impact velocity from 50 m/s to 400 m/s. The obstacle is made of an organoplastic material with some initial orientation of its mechanical properties or the same material whose properties are obtained by a 90° rotation of the initial material about the axis OY. The destruction of obstacles is studied and the efficiency of their protective properties is comparatively analyzed depending on the orientation of the elastic and strength properties of the anisotropic material. The problem is solved numerically by the finite element method in the three-dimensional statement. The behavior of the projectile material is described by an elastoplastic model, while the response of the obstacle anisotropic material is described in the framework of the elastic-brittle model with different tensile and compressive strengths.     

考虑嵌入移动孔洞的多相材料布局优化

工程结构设计问题中经常需要预先嵌入一个或多个固定形状的孔洞以满足某些功能性或者制造性设计要求.为了有效求解这种带有嵌入可移动孔洞的多相材料连续体结构布局优化问题,通常需要同时优化这些嵌入孔洞的位置和方向及多相材料结构的拓扑构型,以改善结构的整体性能.为此,本文采用参数化的水平集函数描述嵌入孔洞的几何形状,并将定义多相材料结构拓扑的材料密度以及描述嵌入孔洞的位置和方向的几何参数视为所考虑优化问题的设计变量.为了避免由于孔洞移动造成的重新划分网格的繁琐及改善计算效率,使用平滑化的Heaviside函数将所有嵌入孔洞映射为固定网格上的密度场.同时,提出了一种在有限元水平上调用的类SIMP材料插值格式,用于优化问题的材料参数化,进而实现多相材料结构拓扑构型和嵌入孔洞位置和方向的同步优化.这种材料插值格式便于几何变量的解析灵敏度分析,使得当前的优化问题可以用基于梯度的优化算法求解.优化算例证明所提方法可以有效地处理带有多个嵌入孔洞的多相材料结构布局优化问题.      

Generalization of the Prandtl solution to the case of axisymmetric deformation of materials obeying the double shear model

A semianalytic solution of the problem on the compression of an annular layer of a plastic material obeying the double shear model on a cylindrical mandrel is obtained. The approximate statement of boundary conditions, which cannot be satisfied exactly in the framework of the constructed solution, is based on the same assumptions as the statement of the classical plasticity problem of compression of a material layer between rough plates (Prandtl’s problem). It is assumed that the maximum friction law is satisfied on the inner surface of the layer. The solution is singular near this surface. The strain rate intensity factor is calculated, and its dependence on the process and material parameters is shown.     

Analytical Study of Fluid Flow and Heat Transfer during Forced Convection in a Composite Channel Partly Filled with a Brinkman–Forchheimer Porous Medium

In this paper, the problem of fully developed forced convection in a parallel-plate channel partly filled with a homogeneous porous material is considered. The porous material is attached to the walls of the channel, while the center of the channel is occupied by clear fluid. The flow in the porous material is described by a nonlinear Brinkman–Forchheimer-extended Darcy equation. Utilizing the boundary-layer approach, analytical solutions for the flow velocity, the temperature distribution, as well as for the Nusselt number are obtained. Dependence of the Nusselt number on several parameters of the problem is extensively investigated.     

A Eulerian approach to the finite element modelling of neo-Hookean rubber material

A Eulerian approach is applied to the finite element modelling of neo-Hookean rubber material. Two major problems are encountered. The first problem is the construction of an algorithm to calculate stresses in the rubber material from velocities instead of displacements. This problem is solved with an algorithm based on the definition of the velocity gradient. The second problem is the convection of stresses through the finite element mesh. This problem is solved by adapting the so-called Taylor-Galerkin technique. Solutions for both problems are implemented in a finite element program and their validity is shown by test problems. Results of these implementations are compared with results obtained by a standard Lagrangian approach finite element package and good agreement has been found.     

功能梯度板条断裂分析

现存文献关于功能梯度材料断裂问题的研究大都假设材料性质为坐标的指数函数或幂函数，而对其它函数形式较少采用。本文假设功能梯度材料剪切模量为坐标的双曲函数，而泊松比为常量，研究功能梯度板条的混合型裂纹问题。利用Fourier积分变换技术将混合边值问题转化为一对奇异积分方程，通过数值求解奇异积分方程获得含裂纹功能梯度板条分别在剪切和法向载荷作用下的I型和Ⅱ型应力强度因子，并讨论了材料的非均匀性和裂纹相对尺寸对裂纹尖端应力强度因子的影响。