Department of Mechanical and Manufacturing Engineering, University of Calgary, 2500 University Drive NW, Calgary, Alberta, Canada.
J R Soc Interface. 2010 Jun 6;7(47):955-66. doi: 10.1098/rsif.2009.0502. Epub 2010 Jan 6.
The elastic strain energy potential for nonlinear fibre-reinforced materials is customarily obtained by superposition of the potentials of the matrix and of each family of fibres. Composites with statistically oriented fibres, such as biological tissues, can be seen as being reinforced by a continuous infinity of fibre families, the orientation of which can be represented by means of a probability density function defined on the unit sphere (i.e. the solid angle). In this case, the superposition procedure gives rise to an integral form of the elastic potential such that the deformation features in the integral, which therefore cannot be calculated a priori. As a consequence, an analytical use of this potential is impossible. In this paper, we implemented this integral form of the elastic potential into a numerical procedure that evaluates the potential, the stress and the elasticity tensor at each deformation step. The numerical integration over the unit sphere is performed by means of the method of spherical designs, in which the result of the integral is approximated by a suitable sum over a discrete subset of the unit sphere. As an example of application, we modelled the collagen fibre distribution in articular cartilage, and used it in simulating displacement-controlled tests: the unconfined compression of a cylindrical sample and the contact problem in the hip joint.
非线性纤维增强材料的弹性应变能势通常通过基质和各向同性纤维的势的叠加来获得。具有统计取向纤维的复合材料，如生物组织，可以被视为由连续的无穷多纤维家族增强，其取向可以通过定义在单位球面上（即立体角）的概率密度函数来表示。在这种情况下，叠加过程导致弹性势的积分形式，使得积分中的变形特征因此无法预先计算。因此，这种势的解析使用是不可能的。在本文中，我们将弹性势能的这种积分形式实现到一个数值程序中，该程序可以在每个变形步骤中评估势能、应力和弹性张量。通过球型设计方法在单位球上进行数值积分，其中积分的结果通过单位球上的离散子集的合适和来近似。作为应用实例，我们对关节软骨中的胶原纤维分布进行了建模，并在模拟位移控制测试中使用了它：圆柱形样品的无约束压缩和髋关节的接触问题。
