While the former way is instructive and provides useful geometric insight, the latter way is mathematically shorter and has the additional advantage of automatically ensuring energy conservation, i . e ., guaranteeing that the second-order work of the stress increment tensor on the strain increment tensor be correct ( work conjugacy requirement ).
2.
The incremental response of this nonlinear material is assumed for simplicity linear, so that it can be expressed as a relation between a stress increment " \ dot \ sigma " and a strain increment " \ dot \ varepsilon ", through a fourth-order constitutive tensor " \ mathbb { C } " as
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