A material is said to be elastic if it deforms under external forces (stress), but returns to its original shape when the stress is removed. For small deformations, the stress is roughly proportional to the strain in many solids. The constant of proportionality between stress and strain is given by the Young’s modulus (E) which is a measure of stiffness. This linear relationship between stress and strain is called Hooke’s law and is the basis for the theory of linear elasticity.

The general relationship (generalized Hooke’s law) between multi-axial stress and strain is described by using 2nd order tensors for stress and strain and the 4th order elasticity tensor for their relation.

In solid mechanics, the Young’s modulus E can be experimentally determined from the slope of a stress-strain curve created during tensile tests conducted on a test specimen of the material. Most metals and ceramics are isotropic. i.e., their mechanical properties are the same in all directions.

If e.g., metals are treated in a special way for example by deep drawing, they become anisotropic, so that the Young’s modulus depends on the direction from which the force is applied. Some materials, which are composites of two or more constituents, such as wood or reinforced concrete, are strongly anisotropic materials and display widely different mechanical properties when load is applied in different directions. For example, they exhibit a higher Young’s modulus (stiffness) when loaded parallel to the fibers. This is true under the assumption that the fibers have a higher Young’s modulus than the surrounding matrix material.

Based on the input parameters, GeoDict solves six load cases or experiments (three compressions in x, y and z, and three shearing experiments) which are sequentially computed by the ElastoDict solver to predict the entries of the 6x6 effective elasticity tensor. Each of these six simulations is done by assigning load case-specific displacements on the boundaries of the structure and calculating the corresponding stresses. By averaging the stresses over the structure, the Hooke’s Law in the general anisotropic case is obtained,

where c is a symmetric fourth order tensor, called also elasticity tensor. Usually, the elasticity tensor and the stresses and strains are written in Voigt notation (see Wikipedia). This way, the elasticity tensor is reduced to a 6x6 matrix, and the stresses and strains are written as 6x1 vectors. This leads to a more compact and readable notation.

Because of these symmetry properties, it is convenient to use the so-called “reduced suffix notation” (e.g., Chadwick et. al.) with the following index assignment:

Then, taking into account also the symmetry of the elasticity tensor,

The stress tensor can be written as in Voigt notation:

The strain tensor can be written as in Voigt notation:

With the Voigt notation, the symmetric fourth order elasticity tensor can be reduced to a symmetric matrix with the following entries (coefficients):

Therefore, the generalized Hooke’s law can be written as a matrix-vector product:

with stress tensor, elasticity or stiffness tensor, and deformation tensor.

This elasticity or stiffness tensor describes the most general stress-strain relations for a linear elastic anisotropic solid (e.g., triclinic solid which has no material symmetry, Nayfeh, A.H., 1995). According to Nayfeh, the constitutive relations for the different symmetry classes can be listed as follows:

The 6x6 matrix for the homogeneous material “nearest” to the inhomogeneous original material is shown in the *.gdr result file after the solver has finished the stiffness estimation computations.

When using ElastoDict with linear laws for the constituent materials, it is important to keep in mind that the elasticity solver is working in the range of linear elasticity. To compute large deformations, plastic yield, damage effects, possible crack initiation, crack growth, and other nonlinear effects one has to use nonlinear material laws, which describe the effects under consideration, for the constituent materials. For such cases, the Hooke’s law does not hold.

Nonlinear materials, e.g., for plastic deformation and damage laws are described in section Theoretical Background: Material Models in ElastoDict. Additionally, nonlinear material laws can be defined by an Abaqus UMAT. With the GeoDict installation several examples for UMATs are available.