## 3.1. Orthotropic Elastic Model

For conventional materials the constitutive relations, or strength model, defines the deviatoric response of the material as a function of any combination of deviatoric strain, deviatoric strain rate, temperature, pressure, and damage.

| (3–1)

For many other materials, including composites, the macroscopic properties are not identical in all directions. In general, the behavior of such materials is represented through a set of orthotropic constitutive relations. Constitutive relations for this type of material are conventionally based on a total stress formulation, as opposed to dividing the total stress into hydrostatic and deviatoric components. Thus, the incremental stress-strain relations can be expressed as

| (3–2)

where

[C] = stiffness matrix

= strain rate tensor

= time step

The linear elastic constitutive relations for a general anisotropic (triclinic) material can be expressed in relation to a Cartesian co-ordinate system, in contracted notation, as

| (3–3)

In which there are 21 independent elastic constants, Cij . If there is one plane of material symmetry, the above stress strain relations reduce to

| (3–4)

where the plane of symmetry is x33 = 0. Such a material is termed monoclinic and there are 13 independent elastic constants. This is the basic form assumed for textiles and composites which generally have a plain of symmetry in the through thickness direction while the symmetry in the plane of the material is dependant on the lay-up and weave of the fibers.

If a second plane of symmetry exists in the plane of the fiber-composite then symmetry will also exist in a third mutually orthogonal plane. Such a material is said to be orthotropic and the constitutive relations are of the form:

| (3–5)

The inverse of the above stiffness matrix for a three-dimensional orthotropic configuration, the compliance matrix, is

| (3–6)

where,

Eii are the Young’s Moduli in the principal material directions,

Gij are the shear moduli,

are the Poisson’s ratios, where is defined as the transverse strain in the j-direction when stressed in the i-direction, that is:

; note that

Note: In Autodyn 2D the directions 23 and 31 do not apply, so:

G23 and G31 are not required

= = 0

σ23= σ31= 0

The above, and similar, constitutive relations for orthotropic materials are commonly applied to fiber reinforced composites in which the fibers are set in a solid matrix material. Inherent in the model is the assumption of a linear volumetric elastic response of the material. This may not be representative of actual material response under the high pressures experienced during a hypervelocity impact event.

There are, of course, restrictions on the elastic constants that can be used for an orthotropic material and these restrictions are more complex than those for isotropic materials. These restrictions result from the fact that the sum of work done by all stress components must be positive in order to avoid the creation of energy. The first condition states that the elastic constants are positive:

E11 ,E22 ,E33 ,G12 ,G23 ,G31 > 0 | (3–7)

Second, the determinant of the stiffness matrix should be positive:

| (3–8)

Finally the requirement for positive stiffnesses leads to:

, , | (3–9)

Whenever the elastic constants in an Autodyn model are defined or redefined the three conditions above are tested and the user is informed if any of them are violated.

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