A Concise Proof of the Representation Theorem for Fourth-Order Isotropic Tensors

We present a new proof of the representation theorem for fourth-order isotropic tensors that does not assume the tensor to have major or minor symmetries at the outset.     

Linear Isotropic Relations in Finite Hyperelasticity: Some General Results

The form of the classical stress–strain relations of linear elasticity are considered here within the context of nonlinear elasticity. For both Cauchy and symmetric Piola-Kirchhoff stresses, conditions are obtained for the associated strain fields so that they are independent of the material constants and compatible with existence of a strain–energy function. These conditions can be integrated in both cases to obtain the most general strain field that satisfies these conditions and the corresponding strain–energy function is obtained. In both cases, a natural choice of form of solution is suggested by the special case of the compatibility conditions being satisfied identically. It will be shown that some strain–energy functions previously introduced in the literature are special cases of the results obtained here. Some recent linear stress–strain relations, proposed in the context of Cauchy elasticity, are examined to see if they are compatible with hyperelasticity.      

冻土的有限变形本构关系的实验研究

根据有限变形理论,给出了冻土三轴蠕变实验数据处理时所需Green应变和Kirchhoff应力的计算公式,并根据冻土三轴蠕变实验结果给出冻土的有限变形本构关系及其蠕变参数。通过对冻土的实验数据对比分析可知:在相同试验条件下,应变较小时,小应变ε1 和Green应变Ez的数值几乎相等,随着应变的增加,两种表征方式的计算结果差别越来越大,可见对于冻土如果不考虑试件变形前后长度的变化,计算得出的应变偏离实际变形情况较大,因此冻土材料的本构关系采用有限变形表征更确切。根据冻土的有限变形本构关系计算得到的冻结壁内侧最大径向位移更接近实测结果,因此冻结壁设计计算依据冻土的有限变形本构关系更为合理。对今后冻结工程的设计具有参考价值。     

Representation Theorems for Hemitropic and Tranversely Isotropic Tensor Functions

A representation theorem for transversely isotropic tensor-valued functions of a symmetric tensor variable is proved. The theorem holds in any finite dimension. The proof is based on the decomposition of a symmetric tensor of dimension N into a scalar, a vector, and a symmetric tensor of dimension N-1, and on the fact that the transverse isotropy of the original function is equivalent to the hemitropy of three functions, one scalar-valued, one vector-valued, and one tensor-valued, of the last two terms in the decomposition. Representation theorems for the three functions are obtained as generalizations of two theorems of W. Noll on isotropic functions. The proofs make use of an appropriate algebraic structure based on alternating forms. The three-dimensional case, as well as those of linear and of hyperelastic functions, are treated as special cases. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite Inhomogeneous Radial Expansion (Contraction) and Longitudinal Shearing of an Isotropic Compressible Elastic Circular Cylindrical Shell

We have first obtained that the equations of equilibrium governing the finite radial expansion (contraction) and longitudinal shearing of a circular cylindrical shell become uncoupled for a class of harmonic materials (a class of isotropic homogeneous compressible elastic materials). Next it has been assumed that the dilatation is uniform. Following this the exact solutions of the uncoupled equations of equilibrium have been obtained for a simple harmonic material which is reduced to the Neo-Hookean material for the incompressible case. The deformation is nonhomogeneous in nature. The stresses have been obtained. This revised version was published online in August 2006 with corrections to the Cover Date.     

Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids

Based on the Theory of Porous Media (TPM), a formulation of a fluid-saturated porous solid is presented where both constituents, the solid and the fluid, are assumed to be materially incompressible. Therefore, the so-called point of compaction exists. This deformation state is reached when all pores are closed and any further volume compression is impossible due to the incompressibility constraint of the solid skeleton material. To describe this effect, a new finite elasticity law is developed on the basis of a hyperelastic strain energy function, thus governing the constraint of material incompressibility for the solid material. Furthermore, a power function to describe deformation dependent permeability effects is introduced.After the spatial discretization of the governing field equations within the finite element method, a differential algebraic system in time arises due to the incompressibility constraint of both constituents. For the efficient numerical treatment of the strongly coupled nonlinear solid-fluid problem, a consistent linearization of the weak forms of the governing equations with respect to the unknowns must be used.