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ha1 - 16.225 Computational Mechanics of Materials Homework...

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16.225 - Computational Mechanics of Materials Homework assignment # 1 Handed out: 9/10/03 Due: 9/24/03 September 22, 2003 1. Verify that the Euler-Lagrange equations corresponding to the Hu- Washizu functional are the field equations of linear elasticity. 2. Consistency test for constitutive models with a potential struc- ture (Computing assignment) Constitutive models that have a poten- tial structure suggest a useful verification test of their implementations. The idea is to make use of the fact that in these models stresses and tangent moduli are obtained by direct differentiation of the strain en- ergy density and compute the derived quantities in two different ways in order to compare them. The first way is directly from their implemen- tation; the second, by numerical differentiation of the function. This is explained in more detail in the algorithm described in the following box: 1
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(a) Generate random strain ± (b) Compute the stresses σ ij ( ± ) and the tangent moduli C ijkl ( ± ) di- rectly from their implementation (c) Recompute these quantities by numerical differentiations, e.g., using central differences: f ( a ) = f ( a + h ) f ( a h ) h 2 f ��� ( η ) + 2 h 6 for some η [ a h, a + h ]. For the case of the stresses
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