关于受内约束超弹性物质的响应函数

On the Response Functions of Hyperelastic Materials with Internal Constraints

提出了一种直接根据正规化条件正规化受内约束超弹性物质的响应函数的一般方法。形式上就是贮能函数φ的Dtφ代替其Gateaux导数。如果使用共轭的应力和应变量就相当于用贮能函数的协变导数代替Gateaux导数。其本质是这两种情况的正规化条件不同。使用正规化响应函数的优点之一是,如果其线性化也使用协变导数,则得到的受内约束超弹性物质的线性化响应函数其线性化反作用应力对线性化应变将不做功,从而使线性化的受内约束超弹性物质也满足关于反作用应力不做变形功率的假设。

In this paper an approach is proposed so that the normalized response functions of hyperelastic materials with internal constraints can be computed. Mathematically the method is that the Gateaux derivative Dφ of stored energy φ is replaced with Dtφ. If the work -conjugate stress and strain measures are used, Dtφ is in fact the covariant derivative Dφ of φ. The reason is that in these two different cases the normalization conditions are different. One advantage of using normalized response functions is, if the covariant derivatives are applied when the linearization of response functions are made, the linearized hyperelastic materials with internal constraints will satisfy the assumption that the reaction stresses are workless.