Continuum Elasticity

A deformable object is a body whose shape may change dynamically under the action of a force field. The way its shape changes depends on a set of parametersK that represents itsmechanical properties.

Theundeformed shape, orrest shape, is the initial configuration in which the object is when no external force is applied.

An equilibrium configuration is a state where the energy due to deformation is at a local minimum. In particular, rest shape can be classified as an equilibrium configuration where that energy term is zero.

The rest shape is described by a continuous connected subset of IR³ referred as material coordinates orM, while points belonging to such subset are usually calledmaterial points.

The parametersK characterize how the object changes its shape when external events per-turbs its equilibrium configuration. Intuitively, such parameters determine the mechanical behavior of a deformable object, for example they differentiate a soft from a stiff object or fluid from a solid object.

Under the action of applied forces, the object deforms, according to K, to reach a new equilibrium position. In particular, each material point x originally located at its rest position m(x) ∈ IR³, moves to a new coordinates w(x) ∈ IR³, which are called deformed coordinates. We can express deformed coordinates as the sum of material coordinatesm(x) and a displacement vectoru(x):

w(x) =m(x) +u(x) (2.1) The displacement field u(x),∀x ∈ M encode the entire body’s deformation. It is impor-tant to notice that not every possible displacement field produces a deformation; a rigid transformation for example, such as a rotation or uniform displacement, does not produce any deformation.

The ”amount of deformation” is expressed in terms of spatial variations of the displace-ment fields, so that the contribute provided by rigid transformations is nullified.

The Strain tensor ε express the ”amount of deformation” in terms of gradient of the displacement field ∇u, that is theJacobian of the displacement field:

∇u=

A popular choice consist of evaluating strain trough theGreen strain tensor _g, or its linear approximation, the Cauchy’s strain tensor _c:

_g = 1

2(∇u+∇u^T +∇u^T∇u) (2.3)

_c= 1

2(∇u+∇u^T) (2.4)

Cauchy’s stress principle asserts that when a force acts on a continuum body, then internal reactions (coded as force vectors) rise between the material points. In order to simulate the dynamic of a deforming object, it is important to quantify such internal forces.

The Stress measures the amount of force applied per area-unit. It is a measure of the intensity of the total internal forces acting within M across imaginary internal surfaces.

The state of stress at a point in M is defined by the nine components of the Cauchy stress tensor, σ ∈ IR^3×3, that, for isotropic purely elastic materials, is linearly related to strain by the Hook’s law:

σ=E (2.5)

The coefficients E ∈ IR^3×3 depend on intrinsic material characteristics K, and determine the stiffness of simulated object. If we assume that the material is isotropic, then E is univocally determined by two independent values, Young’s modulus and Poisson’s ratio.

Young’s modulus E is the ratio of stress to strain on the loading plane along the loading

direction, while Poisson’s express the ratio of lateral strain and axial strain. The Strain Energy Density U(x) defines the amount of energy stored on material points:

U(x) = 1

2(x)·σ(x) (2.6)

consequently, the total elastic energy U is obtained by integrating U(x) over the entire domain:

U = Z

M

U(x) (2.7)

Finally, the Elastic Force, F(x), acting on a material point, is the negative gradient of elastic strain density with respect to material point’s displacement.

In linear elasticity this relation can be expressed as:

F(x) = −∇_uU(x) (2.8)

2.1.2 Discretization

The total potential energy Π of a deformable object is described by the following equation:

Π =U +W (2.9)

Where W is the load due to external forces (gravity or contact constraints for example).

The potential energy reaches a local minimum (defining an equilibrium configuration of the deformable object), when the derivative of Π with respect to the material points dis-placements functions is zero.

The minimization process leads to the resolution of the following differential equation, commonly referred as Equilibrium Equation, which describes the dynamics of a material point x:

ρ·w(x, t) =¨ ∇ ·σ+f_ext (2.10) where ρ is the density of the material, fext represents an externally applied force The divergence operator turns the 3 by 3 stress tensor back into a 3 vector:

∇ ·σ=

The first term of equation 2.10 represents the internal force acting on material point x. It is defined by multiplying the second derivative of it’s world position, which represents the acceleration, by its local density. The second term of equation is the sum of internal forces (which are described in terms of stress tensor) and the external applied forces.

While is possible to solve the PDE expressed by Equation 2.10 directly for very simple cases which provides an analytic description of the domain M (such as a sphere or a bar), it not possible to do so for the cases where the shape is more complex. For the majority of the real objects we have no analytic description of the domain (the integral described by Equation 2.7 is analytically insoluble), then in order simulate their elastic behavior we need to approximate somehow their domain.

Following these considerations, it becomes essential to discretize the continuum-mechanic based model, (described in Section 2.1.1), in a way that the PDE of Equation 2.10 is locally soluble on each discrete sample.

There are two main classes of methods related to the discretization of deformable objects domain, mesh-based and mesh-less. The following paragraphs give an overview of the major advantages and drawback of both classes, along with a detailed description of the discrete models that are the most popular in computer animation.

The derivation of the forces, due to deformation, is clearly not sufficient to animate a deforming body. Animating its dynamics requires the knowledge of time-dependent world coordinates of material points w(x, t).

To make the whole simulation suitable for computer animation, time must be discretized by sampling at fixed interval δt, usually called time steps.

Then the ”time-step dependent” sequence of world coordinates:

w(x, t₀), w(x, t₁)..., w(x, t(n−1)), w(x, t(n−1)) (2.12) can be used to generate the frame sequence of the scene.

Next section introduces the different numerical methods to express time-dependent world coordinates during the simulation, focusing on advantages and drawbacks of each of them.