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Hooke’s law was naturally generalized to finite strains by Hill in 1978, by introducing the Seth-Hill strain and its conjugate stress. This paper presents the transversely isotropic relations, which are not only a natural extension of Hill’s theory from isotropic materials to transversely isotropic materials, but also the natural generalization of the transversely isotropic Hooke’s law from infinitesimal strains to moderate strains. This generalization introduces a class of transversely isotropic hyperelastic models, which are adopted to investigate the uniaxial stretch and the simple shear problems. Results show that the material responses for different constitutive equations are significantly different; the stiffening or softening behaviors of materials at moderate deformations can be described by the appropriate model with proper material parameters.

In the context of isotropic finite elasticity, Hooke’s law for infinitesimal deformations is usually generalized to moderate deformations by adopting different stress and strain measures [

The researches mentioned above are mainly concerned with the isotropic cases. In many engineering problems, it is necessary to characterize the mechanical behaviors of anisotropic materials undergoing large deformations. In the past decades, there have been several different studies on the large strain responses of anisotropic materials following the pioneer work of Ericksen and Rivlin [

The present paper is not only a natural extension of Hill’s relations from isotropic materials to transversely isotropic materials, but also the natural generalization of transversely isotropic Hooke’s law from infinitesimal strains to moderate strains. The outline of the paper is as follows. Section

For a deformation body, with

The Seth-Hill Lagrangian strain tensors are defined as

The corresponding Eulerian strain tensors are expressed as

When the parameter

Hill [

When the hyperelastic constitutive equations for the Seth-Hill strains and their conjugate stresses are adopted to solve the actual problems, it is necessary to express the Cauchy stress

The left and right Cauchy-Green deformation tensors

In this section, we will give a natural extension of Hill’s class of Hookean compressible materials to transversely isotropic materials at finite strains by the structure tensor method.

For the isotropic linear hyperelastic materials at finite deformations, we take

From (

The equivalent component formulation (

In this subsection we give a natural extension of Hill’s class of compressible materials to transversely isotropic materials by introducing the structure tensor

From (

The present model should be able to be used in small strain cases. Engineering parameters (

When

In the isotropic case, we have

In this section, the proposed transversely isotropic Hill class of compressible materials is used to analyze two kinds of homogeneous deformations, the uniaxial stretch and the simple shear.

Let

Let

When

For isotropic materials, the structure tensor is

Results of the uniaxial stretch are given above for Hill’s class of materials for several numbers

Firstly, we consider the tensile case where

Secondly, we analyze the compression case in which

In order to analyze the differences between the proposed models with the different number

Relations between nondimensional loading

Relations between nondimensional loading

In applications, once engineering constants (

Comparison of the present results with experimental data [

Comparison of the present results with experimental data [

We consider a homogeneous simple shear in the isotropic plane of a transversely isotropic material. The deformation is defined by

Figure

Relations between nondimensional normal stress

Next we will show how

Relations between nondimensional shear stress

Based on the structure tensor method, the present paper generalized the isotropic Hill theory to transversely isotropic media undergoing moderate deformations. The differences of these models are discussed through the analyses of the uniaxial stretch and the simple shear problems. Such generalizations of the structures of the isotropic theory to the transversely isotropic cases might provide an efficient method to model the stiffening or softening behaviors of materials at moderate deformations. The results may be useful for the mechanical analysis of soft tissue materials.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is supported by National Natural Science Foundation of China (nos. 11472249 and 41572360), the Fundamental Research Funds for the Central Universities (no. 292015080), the Beijing Higher Education Young Elite Teacher Project (nos. YETP0645 and YETP0648), and the National Key Technology Support Program of China (no. 2015BAD20B02). Meanwhile, great thanks also go to former researchers for their excellent works, which give great help for the authors’ academic study.