Nonlinear finite elements/Homework 11/Solutions


 * Problem 1: Small Strain Elastic-Plastic Behavior
 *  Given:
 * For small strains, the strain tensor is given by

\boldsymbol{\varepsilon} = \frac{1}{2}\left[\boldsymbol{\nabla} \mathbf{u} + (\boldsymbol{\nabla} \mathbf{u})^T\right] \qquad\text{or}\qquad \varepsilon_{ij} = \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

\boldsymbol{\varepsilon} = \boldsymbol{\varepsilon}^e + \boldsymbol{\varepsilon}^p \qquad\text{or}\qquad \varepsilon_{ij} = \varepsilon_{ij}^e + \varepsilon_{ij}^p ~. $$
 * Assuming linear elasticity, we have the following elastic stress-strain law

\boldsymbol{\sigma} = \boldsymbol{\mathsf{C}} : \boldsymbol{\varepsilon}^e \qquad\text{or}\qquad \sigma_{ij} = C_{ijkl}\varepsilon_{kl}^e ~. $$
 * Let us assume that the $$J_2$$ theory applies during plastic deformation of the material. :Hence, the material obeys an associated flow rule
 * $$\text{(1)} \qquad

\dot{\boldsymbol{\varepsilon}}^p = \dot{\gamma}\frac{\partial f(\boldsymbol{\sigma},\alpha,T)}{\partial \boldsymbol{\sigma}} $$
 * where $$\dot{\gamma}$$ is the plastic flow rate, $$f$$ is the yield function, $$T$$ is the temperature, and $$\alpha$$ is an internal variable.


 * Problem 1: Part 1: Evolution rule for plastic flow
 * Problem 1: Part 2: Energy equation
 * Problem 1: Part 3: Rate form constitutive relation
 * Problem 1: Part 4: Consistency condition
 * Problem 1: Part 5: Consistency condition - 1
 * Problem 1: Part 6: Continuum elastic-plastic tangent modulus
 * Problem 1: Part 7: Flow rule
 * Problem 1: Part 8: Consistency condition - 2
 * Problem 1: Part 9: Elastic-plastic tangent modulus
 * Problem 1: Part 10: Discrete evolution equations
 * Problem 1: Part 11: Discrete Kuhn-Tucker
 * Problem 1: Part 12: Trial elastic stress - 1
 * Problem 1: Part 13: Trial elastic stress - 2
 * Problem 1: Part 14: Return mapping
 * Problem 1: Part 15: Plastic flow parameter
 * Problem 1: Part 16: Newton iterations
 * Problem 2: Billet Upset Forging
 * Problem 3: Taylor Impact Tests