Numerical integration on the sphere and its effect on the material symmetry of constitutive equations-A comparative study

作者：Ehret A E^*; Itskov M; Schmid H

来源：International Journal for Numerical Methods in Engineering, 2010, 81(2): 189-206.

DOI：10.1002/nme.2688

• solids
• constitutive equations
• numerical integration
• microplane models
• microsphere models
• anisotropy

In the present paper, we consider a class of constitutive models based on numerical integration on the unit sphere. The directional behaviour of the quadrature schemes and its effect on the symmetry properties of these constitutive models are Studied by subjecting the set of integration points on the sphere to arbitrary rigid rotations. We investigate a number of recently proposed integration schemes in application to a full network model of rubber elasticity and to an exponential model for soft tissues. In order to assess and compare these schemes, statistical methods are presented and applied. The analysis discloses a number of integration schemes that offer a good compromise between the numerical error and the number of integration points. However, as a general result it turns Out that numerical integration is prone to introduce strong anisotropy into originally isotropic constitutive equations, in particular, for highly non-linear integrand functions. The consequences for application of the investigated class of constitutive models in finite element calculations are highlighted in a benchmark-like numerical example.

• 出版日期2010-1-8