Extension and torsion of incompressible non-linearly elastic solid circular cylinders
Extension and torsion of incompressible non-linearly elastic solid circular cylinders
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不可压缩非线性弹性实心圆柱体的伸长和扭转
DOI：
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Jeremiah G. Murphy
中科院分区：
文献类型：
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作者：
C. Horgan;Jeremiah G. Murphy
在Rivlin开创性的理论研究之后，关于非线性弹性的文献中已经广泛地研究了由不可压缩的各向同性超弹性材料组成的实心圆柱体的轴向延伸上叠加的扭转问题。Rivlin的经典结果是以依赖于柯西-格林变形张量的主不变量的一般应变能密度给出的。在这里，我们重新制定这些结果的一般应变能密度取决于替代不变量，即拉伸张量的不变量。这样的方法是由Rivlin在这本杂志上的一篇论文中对纯扭转的特殊情况进行的。正如在那里所指出的，这是一个相对简单的过程，并且是从拉伸张量的不变量的第一性原理发展理论的可行替代方案。作为一个说明性的例子，我们考虑著名的Varga模型的不可压缩材料。对于这种材料模型，它表明，拉伸圆柱总是倾向于进一步伸长扭转。还简要地考虑了纯扭转的特殊情况。对于Varga模型，保持纯扭转所需的合成轴向力是压缩力。在没有这种力的情况下，杆在扭转时倾向于伸长，反映了著名的坡印廷效应。
The problem of torsion superimposed on axial extension of a solid circular cylinder composed of an incompressible isotropic hyperelastic material has been extensively investigated in the literature on non-linear elasticity following the pioneering theoretical investigations of Rivlin. The classical results of Rivlin were given in terms of a general strain—energy density that depends on the principal invariants of the Cauchy—Green deformation tensor. Here we reformulate these results in terms of a general strain—energy density that depends on alternative invariants, namely the invariants of the stretch tensor. Such an approach was carried out in a paper in this journal by Rivlin for the special case of pure torsion. As was remarked there, this is a relatively straightforward procedure and is a viable alternative to development of the theory from first principles in terms of the invariants of the stretch tensor. As an illustrative example, we consider the well-known Varga model for incompressible materials. For this material model, it is shown that the stretched circular cylinder always tends to further elongate on twisting. The special case of pure torsion is also briefly considered. The resultant axial force necessary to maintain pure torsion is compressive for the Varga model. In the absence of such a force, the bar tends to elongate on twisting, reflecting the celebrated Poynting effect.