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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.
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-
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.
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