## 3.2. Equations of State

The material model developed in [1] provides a mechanism in an orthotropic material to calculate:

- The contributions to pressure from the isotropic and deviatoric strain components

- The contributions to the deviatoric stress from the deviatoric strains

Further, this methodology gives rise to the possibility for incorporating nonlinear effects (such as shock effects) that can be attributed to the volumetric straining in the material. To use this model ‘Ortho’ is selected as the equation state for the material and either ‘Polynomial’ or ‘Shock’ for the volumetric response option.

The incremental linear elastic constitutive relations for an orthotropic material can be expressed, in contracted notation, as:

| (3–10)

In order to include nonlinear shock effects in the above linear relations, it is first desirable to separate the volumetric (thermodynamic) response of the material from its ability to carry shear loads (strength). To this end, it is convenient to split the strain increments into their average, , and deviatoric, , components.

| (3–11)

Now, defining the average direct strain increment, , as a third of the trace of the strain tensor,

| (3–12)

and assuming, for small strain increments, the volumetric strain increment is defined as

| (3–13)

The total strain increments can be expressed in terms of the volumetric and deviatoric strain increments resulting in the following orthotropic constitutive relation.

| (3–14)

If the above relations are expanded and the deviatoric and volumetric terms grouped, the following expressions for the direct stress increments results.

| (3–15)

To find the equivalent pressure increment, we first define the pressure as a third of the trace of the stress increment tensor;

| (3–16)

Substituting Equation 3–15 into Equation 3–16 results in an expression for the pressure increment of the form

| (3–17)

from which the contributions to the pressure from volumetric and deviatoric components of strain can clearly be identified.

For an isotropic material, the stiffness matrix coefficients can be represented in terms of the material bulk modulus, K, and shear modulus, G. Thus,

| (3–18)

Substituting Equation 3–18 into Equation 3–16 gives

| (3–19)

and given

| (3–20)

Equation 3–19 reduces to

| (3–21)

which is immediately recognizable as the standard relationship between pressure and volumetric strain (Hooke’s law) at low compressions.

The first term of Equation 3–17 can therefore be used to define the volumetric (thermodynamic) response of an orthotropic material in which the effective bulk modulus of the material K’ is

| (3–22)

For the inclusion of nonlinear shock effects, the contribution to pressure from volumetric strain is modified to include nonlinear terms. The final incremental pressure calculation becomes

| (3–23)

where the pressure contribution ΔPEOS from volumetric strains can include the nonlinear shock (thermodynamic) effects and energy dependence as in a conventional equation of state.

A form of equation of state that is used extensively for isotropic solid continua is known as the Mie-Grüneisen form:

| (3–24)

Where the Grüneisen gamma is defined as

| (3–25)

The functions pr(v) and er(v) are assumed to be known functions of v on some reference curve.

Two Mie-Grüneisen forms of equation of state are available for coupling with an orthotropic response in the AMMHIS model and are now described.

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