Hypoelasticity

The hypoelastic material model:

can only be used in implicit analyses;
is intended to be used when the load path is monotonic; and
must be defined by user subroutine UHYPEL if temperature dependence is to be included.

Hypoelasticity

In a hypoelastic material the rate of change of stress is defined as a tangent modulus matrix multiplying the rate of change of the elastic strain:

d σ = D^{e l} : d ε^{e l}

where

d σ

is the rate of change of the stress (the “true,” Cauchy, stress in finite-strain problems),

D^{e l}

is the tangent elasticity matrix, and

d ε^{e l}

is the rate of change of the elastic strain (the log strain in finite-strain problems).

The entries in $D^{e l}$ are provided by giving Young's modulus, $E$ , and Poisson's ratio, $ν$ , as functions of strain invariants. The strain invariants $(I_{1}, I_{2}, I_{3})$ are defined as

I_{1} = t r a c e (ε^{e l})

I_{2} = \frac{1}{2} (ε^{e l} : ε^{e l} - I_{1}^{2})

I_{3} = \det (ε^{e l}) .

For plane stress and uniaxial stress states out-of-plane strain components are not computed. For the purpose of defining the above invariants, it is assumed that $I_{1} = 0$ ; that is, the material is assumed to be incompressible. For example, in a uniaxial stress case (such as a truss element) this assumption implies that

I_{1} = 0

I_{2} = \frac{3}{4} {(ε_{11}^{e l})}^{2}

I_{3} = \frac{1}{4} {(ε_{11}^{e l})}^{3}

You can define the material coefficients directly.

Table 1. Type=Material Coefficients
Input Data	Description
Young's Modulus	$E$ .
Poisson's Ratio	$ν$ .
First Strain Invariant	$I_{1}$ .
Second Strain Invariant	$I_{2}$ .
Third Strain Invariant	$I_{3}$ .

Alternatively, you can define the behavior with a user subroutine (see UHYPEL ). No data lines are used with this option when the Type=User option is specified.