Game Development Reference

In-Depth Information

component is positive if it acts in the positive direction of the coordinate axes

and the plane where it acts has an outward normal vector pointing in the positive

coordinate direction.

It is important to note that the Cauchy stress has been obtained considering ob-

jects that experience small deformations. For large deformations, other measures

of stress are required, such as the first and second Piola-Kirchoff stress tensors, the

Biot stress tensor, and the Kirchoff stress tensors that accurately model nonlinear

elasticity. However, as we have mentioned previously, large deformations lead

to less-intuitive equations that are, besides, computationally expensive. Muller

and Gross have shown accurate results for soft body simulation using simply lin-

ear elasticity [Muller and Gross 04]. Their approach is based on a corotational

formulation, which we will describe later in Section 10.3.4.

10.2.3

Linear Elasticity

According to the principle of conservation of linear and angular momentum, equi-

librium requires that the addition of moments with respect to an arbitrary point is

zero, which leads to the fact that the stress tensor is symmetric. Therefore, in-

stead of using nine stress components, we can reduce the stress to only six inde-

pendent stress components. Similarly, the strain assumes orthogonal infinitesimal

displacements, which allow us to reduce its nine components to only six. This

symmetry property is very convenient, since it allows us to reduce memory con-

sumption during the simulation.

The relationship between the strain and the stress is defined by the constitutive

equations of the object, which in the simplest case is defined by Hooke's law,

where such relationship is linear. This is known as linear elasticity and can be

expressed as

σ
=
E
ε,

where
E
is a tensor of 81 experimental elastic constants that are independent of

stress or strain. Since the strain and the stress can be represented using only six

independent components, we can reduce the elasticity matrix to only 36 constants.

We can simplify further the elasticity matrix if we make the following assump-

tions:

1. We consider that the constitutive equations of the object are the same for

any point within the object. This is usually known as the
homogeneous

property
.

2. The elastic properties are independent of temperature changes.