Solid Mechanics Theory

In the following, the theory for the Solid Mechanics interface is described. To a large extent, this theory covers other structural mechanics physics interfaces, such as Shell and Beam, which are included with the Structural Mechanics Module. For these other interfaces, only the details which are specific to the interface are described in its additional documentation.

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Theory for the Shell and Plate Interfaces

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Theory for the Membrane Interface

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Theory for the Beam Interface

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Theory for the Beam Cross Section Interface

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Theory for the Truss Interface

Tensor Notation

Some of the theory is developed using tensor notation. In most cases, explicit index notation is avoided. This means that the order of a tensor usually must be understood from the context. As an example, Hooke’s law for linear elasticity is usually written like

Here, the stress tensor σ and the strain tensor ε are second-order tensors, while the constitutive tensor C is a fourth-order tensor. The ‘:’ symbol means a contraction over two indices. In a notation where the indices are shown, the same equation would read

where the Einstein summation convention has been used as a shorthand for

In a few cases, nonorthonormal coordinate systems must be considered. It is then necessary to keep track of the covariance and contravariance properties of tensors. In such a case, Hooke’s law is written

The stress and constitutive tensors have contravariant components, while the strain tensor has covariant components.