Nonlinear finite elements/Homework 11/Solutions

Given:

For small strains, the strain tensor is given by

ε = \frac{1}{2} [\nabla 𝐮 + (\nabla 𝐮)^{T}] or ε_{i j} = \frac{1}{2} (u_{i, j} + u_{j, i}) .

In classical (small strain) rate-independent plasticity we start off with an additive decomposition :of the strain tensor

ε = ε^{e} + ε^{p} or ε_{i j} = ε_{i j}^{e} + ε_{i j}^{p} .

Assuming linear elasticity, we have the following elastic stress-strain law

σ = 𝖢 : ε^{e} or σ_{i j} = C_{i j k l} ε_{k l}^{e} .

Let us assume that the

J_{2}

theory applies during plastic deformation of the material. :Hence, the material obeys an associated flow rule

(1) {\dot{ε}}^{p} = \dot{γ} \frac{\partial f (σ, α, T)}{\partial σ}

where

\dot{γ}

is the plastic flow rate,

f

is the yield function,

T

is the temperature, and

α

is an internal variable.

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