A typical stress-strain curve of a nonlinear material model is:
For the particular case of stress history as related to proportional loading, where components of stress tensor vary monotonically in constant ratio to each other, the strains can be expressed in terms of the final state of stress in the following form:
Ds is the secant material matrix, Es is the secant modulus, ν is the Poisson's ratio
To incorporate this model, the Poisson's ratio and a material stress-strain curve should be defined.
The total strain vector ε is used to compute the effective strain ε(bar) to get the secant modulus from the user-defined material (stress-strain) curve. For the three dimensional case:
The stress-strain curve from the third (compressive) to the first (tensile) quadrants are applicable to this model for two and three dimensional elements with some modifications. A method of interpolation is used to get the secant and tangent material moduli. Defining a ratio R which is a function of the volumetric strain Φ, effective strain, and the Poisson's ratio, R has the following expression:
It is noted that R = 1 represents the uniaxial tensile case and R = -1 is for the compressive case. These two cases are set to be the upper and lower bound such that when R exceeds these two values, the program will push it back to the limit. The nonlinear elastic material model can be used with solid and shell meshes.