# Nonlinear finite elements/Homework 11/Solutions

< Nonlinear finite elements | Homework 11(Redirected from Nonlinear finite elements/Homework11Solutions)

- Problem 1: Small Strain Elastic-Plastic Behavior

**Given:**- For small strains, the strain tensor is given by
- In classical (small strain) rate-independent plasticity we start off with an additive decomposition :of the strain tensor
- Assuming linear elasticity, we have the following elastic stress-strain law
- Let us assume that the theory applies during plastic deformation of the material. :Hence, the material obeys an associated flow rule
- where is the plastic flow rate, is the yield function, is the temperature, and 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