the constraints . It is then easy to show by exactly the same arguments like those used at the end of the previous Section that equation yields $$\mathbf{P}\cdot \nabla =0,
\quad
-w_{\beta ^{\mathfrak{a}}}+\nabla \cdot w_{\nabla \beta ^{\mathfrak{a}}}=\frac{\partial D}{\partial \dot{\beta }^{\mathfrak{a}}}, \quad \nabla \cdot w_{\nabla l^{\mathfrak{a}}} =\frac{\partial D}{\partial \dot{l}^{\mathfrak{a}}},$$ which are subjected to the boundary conditions and . The constitutive equations remain exactly the same as -.

Small strain theory
===================

Let us simplify the above theory for small strains. In this case, instead of the placement $\mathbf{y}(\mathbf{x})$ we regards the displacement $\mathbf{u}(\mathbf{x})$ that is related to the former by $$\mathbf{u}(\mathbf{x})=\mathbf{y}(\mathbf{x})-\mathbf{x}$$ as the unknown function. Thus, the total compatible deformation is $$\mathbf{F}=\frac{\partial \mathbf{y}}{\partial \mathbf{x}} =\mathbf{I}+\mathbf{u}\nabla .$$ We assume that the displacement gradient $\mathbf{u}\nabla $ (called distortion) is small compared with $\mathbf{I}$. Concerning the plastic deformation given by we also assume that the plastic slips $\beta ^{\mathfrak{a}}$ are much smaller than 1. Using the the multiplicative resolution to express $\mathbf{F}^e$ through $\mathbf{F}$ and $\mathbf{F}^{p-1}$ and neglecting the small nonlinear terms in it, we get $$\mathbf{F}^e=\mathbf{F}\cdot \mathbf{F}^{p-1}=\mathbf{I}+{\boldsymbol{\beta}}^e$$ where $${\boldsymbol{\beta}}^e =\mathbf{u}\nabla -\sum_{{\mathfrak{a}}=1}^n \beta ^{\mathfrak{a}}\mathbf{s}^{\mathfrak{a}}\otimes \mathbf{m}^{\mathfrak{a}}.$$ The last equation can be interpreted as the additive resolution of the total distortion into the plastic and elastic 