1. Some definitions#
Here we recall some definitions of tensors linked to large deformations.
Transformation \(F\) is called the gradient tensor, the tensor that makes it possible to go from the initial configuration \({\mathrm{\Omega }}_{0}\) to the current deformed configuration \(\mathrm{\Omega }(t)\).
\(\mathrm{F}=\frac{\partial \widehat{\mathrm{x}}}{\partial \mathrm{X}}=\mathrm{Id}+{\nabla }_{\mathrm{X}}\mathrm{u}\) with \(\mathrm{x}=\widehat{\mathrm{x}}(\mathrm{X},t)=\mathrm{X}+\mathrm{u}\) eq 1-1
where \(X\) is the position of a point in \({\mathrm{\Omega }}_{0}\), \(x\) the position of this same point after deformation in \(\mathrm{\Omega }(t)\) and \(u\) the displacement.
Different strain tensors can be obtained by eliminating rotation in the local transformation. This can be done in two ways, either by using the polar decomposition theorem, or by directly calculating the length and angle variations (dot product variation).
In a Lagrangian description (that is to say on the initial configuration), we obtain:
Through polar decomposition:
\(\mathrm{F}=\mathrm{R}\mathrm{U}\) eq 1-2
where \(R\) is the rotation tensor (orthogonal) and \(U\) is the pure right deformation tensor (symmetric and positive definite).
By a direct calculation of the deformations:
\(\mathrm{E}=\frac{1}{2}(\mathrm{C}-\mathrm{Id})\) with \(\mathrm{C}={\mathrm{F}}^{T}\mathrm{F}\) eq 1-3
where \(E\) is the Green-Lagrange strain tensor and \(C\) is the right Cauchy-Green tensor.
Tensors \(U\) and \(C\) are linked by the following relationship:
\(\mathrm{C}={\mathrm{U}}^{2}\) eq 1-4