# Tangent Moduli

• Robert M. Hackett
Chapter

## Abstract

Clearly, in the case of small strain linear elasticity, the tangent modulus is constant regardless of deformation, i.e., since the stress–strain curve is linear, the stiffness does not change as deformation changes. However, for a hyperelastic model, differentiating the strain-energy function with respect to either the finite strain tensor or one of the two Cauchy-Green deformation tensors yields elastic “constants,” the magnitude of which depend upon the level of deformation. Mathematically, taking the second derivative of the strain-energy function is equivalent to taking the first derivative of the stress–strain curve, yielding the fourth-order tangent stiffness tensor. Hence, for any point on the stress–strain curve the tangent to the curve at that point, i.e., at that amount of deformation, is obtained. Thus, the elastic “constants” obtained by differentiating the strain-energy function twice are referred to as the tangent elastic properties. Depending upon the combination of stress and strain tensors employed, corresponding constitutive models are derived. The correct employment of these model-developing procedures is very critical to solving problems in large deformation nonlinear elasticity. A very important new approach to deriving the fourth-order “first elasticity tensor” is given. A numerical example is presented to augment the developed theory.

## Keywords

Tangent modulus Strain-energy function Stress–strain curve Cauchy-Green deformation tensors Constitutive models Fourth-order tangent stiffness tensor Numerical example

## References

1. Belytschko T, Liu WK, Moran B (2000) Nonlinear finite elements for continua and structures. Wiley, New York
2. Sussman T, Bathe KJ (1987) A finite element formulation for nonlinear incompressible elastic and inelastic analysis. Comput Struct 26:357–409