3. Reminders on major deformations#
3.1. Cinematics#
Let us consider a solid subject to great deformations. Let \({\Omega }_{0}\) be the domain occupied by the solid before deformation and \(\Omega (t)\) the domain occupied at the time \(t\) by the deformed solid.
Figure 3.1-a: Representation of the initial and deformed configuration
In the initial configuration \({\Omega }_{0}\), the position of any particle in the solid is designated by \(X\) (Lagrangian description). After deformation, the position at time \(t\) of the particle that occupied position \(X\) before deformation is given by the variable \(x\) (Eulerian description).
The global motion of the solid is defined, with \(u\) the displacement, by:
\(x=\stackrel{ˆ}{x}(X,t)=X+u\)
To define the metric change in the vicinity of a point, we introduce the gradient tensor of the \(F\) transformation:
\(F=\frac{\partial \stackrel{ˆ}{x}}{\partial X}=\text{Id}+{\nabla }_{X}u\)
The transformations of the volume element and the density equal:
\(d\Omega =\text{Jd}{\Omega }_{o}\) with \(J=\text{det}F=\frac{{\rho }_{o}}{\rho }\)
where \({\rho }_{o}\) and \(\rho\) are the density in the initial and current configurations respectively.
Different strain tensors can be obtained by eliminating rotation in the local transformation. For example, by directly calculating the length and angle variations (dot product variation), we get:
\(E=\frac{1}{2}(C-\text{Id})\) with \(C={F}^{T}F\)
\(A=\frac{1}{2}(\text{Id}-{b}^{-1})\) with \(b={\text{FF}}^{T}\)
\(E\) and \(A\) are respectively the Green-Lagrange and Euler-Almansi deformation tensors and \(C\) and \(b\) are the right and left Cauchy-Green tensors respectively.
In Lagrangian description, deformation will be described by tensors \(C\) or \(E\) because they are quantities defined on \({\Omega }_{0}\), and in Eulerian description by tensors \(b\) or \(A\) (defined on \(\Omega\)).
Note:
Let a solid undergoing two successive transformations, for example the first transformation causes the solid to pass from the initial configuration \({O}_{0}\) to a configuration \({O}_{1}\) (gradient tensor \({F}_{1/0}\) and displacement vector \({u}_{1/0}\) ), then the second transformation of the configuration ), then the second transformation of the configuration \({O}_{1}\) to \({O}_{2}\) (gradient tensor \({F}_{2/1}\) ), then the second transformation of the configuration to (gradient tensor and displacement vector \({u}_{2/1}\) ) .
The transition from configuration \({O}_{0}\) to \({O}_{2}\) is given by the gradient tensor \({F}_{2/0}\) (displacement \({u}_{2/0}={u}_{2/1}+{u}_{1/0}\) ) such that:
\({F}_{2/0}={F}_{2/1}{F}_{1/0}\)
We then get, for example, for the Green-Lagrange strain tensor \(E\)
\({E}_{2/0}={F}_{1/0}^{T}{E}_{2/1}{F}_{1/0}+{E}_{1/0}\)
where \({E}_{2/0}\) , \({E}_{1/0}\) and \({E}_{2/1}\) are the Green-LaGrange deformations of the configurations \({\Omega }_{2}\) with respect to \({O}_{0}\) associated with \({F}_{2/0}\) , \({O}_{1}\) compared to \({O}_{0}\) associated with associated with associated with \({F}_{1/0}\) and \({O}_{2}\) with respect to \({\Omega }_{1}\) associated with \({F}_{2/1}\) , respectively.
This is one of the difficulties encountered when writing a law of behavior in large deformations because you can no longer write a formula similar to that written in small deformations, i.e. \({\varepsilon }_{2/0}={\varepsilon }_{2/1}+{\varepsilon }_{1/0}\) where \(\varepsilon\) is the linearized total deformation tensor.
To find \({\varepsilon }_{2/0}={\varepsilon }_{2/1}+{\varepsilon }_{1/0}\) in small deformations based on the expression of \({E}_{2/0}\) , you must overlook all the 2nd order terms of \(\nabla {u}_{2/0}\) , \(\nabla {u}_{1/0}\) and \(\nabla {u}_{2/1}\). In this case, we have \({E}_{2/0}\simeq {\varepsilon }_{2/0}\) , \({\mathrm{E}}_{1\mathrm{/}0}\mathrm{\simeq }{\varepsilon }_{1\mathrm{/}0}\) and \({F}_{1/0}^{T}{E}_{2/1}{F}_{1/0}\simeq {\varepsilon }_{2/1}\) .
3.2. Constraints#
For the model described here, the stress tensor used is the Eulerian Kirchhoff \(\tau\) tensor defined by:
\(J\sigma =\tau\)
where \(\sigma\) is the Cauchy Eulerian tensor. The \(\tau\) tensor therefore results from a « scaling » by the variation in volume of the Cauchy tensor \(\sigma\); this is not the case with other stress tensors used (first and second Piola-Kirchhoff tensors).
In Eulerian description, the equilibrium equations are given by:
\(\begin{array}{}{\text{div}}_{x}\sigma +\rho f=0\text{sur}\Omega \\ \sigma \text{.}n={t}^{d}\text{sur}\partial {\Omega }^{f}\end{array}\)
where \(f\) is the force density applied to the domain \(\Omega\), \(n\) the external normal at the border \(\partial {\Omega }^{f}\), and \(\partial {\Omega }^{f}\) the part of the border of the domain \(\Omega\) where the surface forces \({t}^{d}\) are applied.
3.3. Objectivity#
When we write a law of behavior in large deformations, we must check that this law is objective, that is to say invariant by any change in the spatial frame of reference of the form:
\({x}^{\text{*}}=c(t)+Q(t)x\)
where \(Q\) is an orthogonal tensor that reflects the rotation of the frame of reference and \(c\) is an orthogonal tensor that reflects the translation.
More specifically, if a tensile test is carried out in direction \({e}_{1}\), for example, followed by a rotation of 90° around \({e}_{3}\), which is equivalent to carrying out a tensile test according to \({e}_{2}\), then the danger with a non-objective law of behavior is not to find a uniaxial stress tensor in direction \({e}_{2}\) (which is in particular the case with the kinematics PETIT_REAC).