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Example text

The (Bubnov-) Galerkin finite element method is characterized by utilizing the same shape functions to approximate V as were used to approximate Ct . 167) where the CA are nsd-vectors of nodal constants. 168) IThe approximation comes about because the surface r" is in general only approximately equal to its approximation r~, and also because the given field CPt is only nodally interpolated by cpf. 157). 167) resulting in satisfaction ofthis condition. Note that the only restriction on the tp h is that they must result in satisfaction of the homogeneous boundary condition on r u; they are otherwise arbitrary.

1. Fixing attention on some time tEll, and considering the current location of the body 0, the following conditions must hold: = pa on 'Pt(O), = CPt on 'Pt(ru), t = t on 'Pt(r a ), V' . 118) where I is the identity mapping (consistent with the assumption that the reference configuration corresponds to the configuration of the body at t = 0). 117), the divergence operator is to be interpreted as being with respect to spatial coordinates a;. The acceleration a is referred to spatial 32 2. 61). pt, but relies upon it for its own definition.

For our discussion here, we consider two such rates especially prevalent in the literature: the Jaumann rate and the Green-Naghdi rate. 92). For example, let us consider the Jaumann rate of stress, which we denote here as J.. Its definition is given as v . u =u-wu+uw. 104) We can verify that this rate of stress is truly objective by direct calculation, by considering the object as it would appear to observer *: '1* U = iT* - w*u* + u*w*. 106) 30 2. 107) which assures us that indeed &is objective.

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