Material (stress-strain) relations

We have described stress, the forces in a body, and strain, a measure of geometric distortion of a body. What remains of linear elasticity is to describe the relationship between the two, which is a function of the material properties of a body.

In the one-dimensional case, Hooke's law states that stress is linearly dependent on strain: $\sigma = E \varepsilon$ , where the constant $E$ is the modulus of elasticity or Young's modulus. Even in the one-dimensional case, this is a major simplification. Figure XXX shows a measured stress-strain curve reproduced from [Gould, 1994]. (TODO) The term elasticity means that if a material is strained and then released, it returns along the same path in the stress-strain diagram. The path taken is not necessarily a straight line, but if it is, the term linear elasticity applies. It is remarkable that many materials do exhibit linear elasticity for some range of strains. However, all materials have a limit to the amount of strain they can react to linearly. As strain is increased, the proportional limit marks the start of non-linear behaviour where the slope decreases, finally flattening to zero at the plastic limit. Once the plastic limit is reached, further strain does not increase the stresses inside the material at all, and the material deforms permanently. If the material is released after the strain has exceeded the plastic limit, it will return along a different path in the stress-strain diagram than the initial elastic path, and its final stress-free state will have some non-zero strain (deformation).

Feynman [Feynman et al., 1989] explains that plastic deformation is a result of layers of molecules in the material slipping past each other and out of their initial formation. As long as this slippage does not occur, the inter-molecular forces induce the elastic behaviour of the material as a whole, but once slippage has ocurred, the material is permanently deformed. Once the plastic deformation has occurred, then, the initial undeformed configuration has changed and the stress-strain diagram should realistically be given a new origin. Modelling plasticity as a change in the undeformed state is a reasonable approximiation for computation in particular.

The proceeding discussion is for the one-dimensional case. Similar things can be expected for a three-dimensional solid when examined along a single axis, but in general the situation is unsurprisingly more complicated. For the rest of our discussion we will be assuming that materials are linearly elastic. This is the most common assumption in computer graphics.

It is assumed that the stresses are governed by a strain energy density $W$ that is quadratic in the strains:

$$\sigma_{ij} = \ensuremath{\frac{\partial W}{\partial \varepsilon_{ij}}} = E_{ijkl}\varepsilon_{kl}$$ (2.67)

Clearly $E_{ijkl}$ is a fourth-order tensor that mixes the components of $\varepsilon$ to get a component of $\sigma$ . $E$ starts with $3^4=81$ components, but this is rapidly reduced by the following points and assumptions:

1. Both the stress and strain are symmetric, and we want to be able to invert the stress-strain relationship (get strains from stresses), hence $E$ is symmetric. (Alternatively, $W$ is continuous, hence $E$ is symmetric.) This material is the most general considered, called anisotropic and $E$ has 36 coefficients.
2. The material behaviour has one plane of symmetry. Having a plane of symmetry means that a stress applied normal or parallel to the plane induces displacements only in directions normal and/or parallel to the plane. This is called monoclinic and $E$ has 13 coefficients.
3. The material behaviour has two planes of symmetry. This is called orthotropic and $E$ has 9 coefficients.
4. The material has only three directions of independence. This is called cubic and $E$ has three coefficients. The vast majority of engineering materials fall into this class.
5. The material has no directional dependence at all. This is called isotropic and $E$ has only two coefficients. This is a commonly-assumed model of materials.

Without derivation, the components of $E$ for an isotropic material take the form

$$\varepsilon_{ij} = 2\mu\varepsilon_{ij} + \lambda \delta_{ij} \varepsilon_{kk},$$ (2.68)

where $\mu$ and $\lambda$ are called the Lamé constants.