Micromechanics of composites/Finite deformations

Finite deformations
If a RVE undergoes finite deformations (i.e., large strains and large rotations), then we have to make a distinction between the initial and deformed configuration. Let us assume that the deformation can be described by a map

\mathbf{x} = \boldsymbol{\varphi}(\mathbf{X}) = \mathbf{x}(\mathbf{X}) $$ where $$\mathbf{X}$$ is the position of a point in the RVE in the initial configuration and $$\mathbf{x}$$ is the location of the same point in the deformed configuration.

The deformation gradient is given by

\boldsymbol{F} = \frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \boldsymbol{\nabla}_0~ \mathbf{x} ~. $$

If we assume that the RVE is small enough, we can neglect inertial and body forces. Then the equations that govern the motion of the RVE can be written with respect to the reference configuration as

{ \begin{align} \boldsymbol{C} & = \boldsymbol{F}^T\cdot\boldsymbol{F} & & \qquad \text{Strain-deformation Relations} \\ \boldsymbol{P} & = \hat{\boldsymbol{P}}(\boldsymbol{C}) & & \qquad \text{Stress-Strain Relations} \\ \rho~\det(\boldsymbol{F}) & = \rho_0 & &\qquad \text{Balance of Mass} \\ \boldsymbol{\nabla}_0 \bullet \boldsymbol{P}^T & = 0& & \qquad \text{Balance of Linear Momentum} \\ \boldsymbol{F}\cdot\boldsymbol{P} & = \boldsymbol{P}^T\cdot\boldsymbol{F}^T& & \qquad \text{Balance of Angular Momentum} \\ \rho_0~\dot{e} & = \boldsymbol{P}^T:\dot{\boldsymbol{F}} - \boldsymbol{\nabla}_0 \bullet \mathbf{q} + \rho_0~s & & \qquad\text{Balance of Energy.} \\ \end{align} } $$ In the above $$\boldsymbol{C}$$ is the right Cauchy-Green deformation tensor, $$\boldsymbol{P}$$ is the first Piola-Kirchhoff stress tensor, and $$rho_0$$ is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by

\boldsymbol{P} = \det(\boldsymbol{F})~\boldsymbol{F}^{-1}\cdot\boldsymbol{\sigma} ~. $$ The gradient and divergence operators are defined such that

\boldsymbol{\nabla}_0~ \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial X_j}\boldsymbol{E}_i\otimes\boldsymbol{E}_j = v_{i,j}\boldsymbol{E}_i\otimes\boldsymbol{E}_j ~; \boldsymbol{\nabla}_0 \bullet \mathbf{v} =\sum_{i=1}^3 \frac{\partial v_i}{\partial X_i} = v_{i,i} ~; \boldsymbol{\nabla}_0 \bullet \boldsymbol{S} = \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial X_j}~\boldsymbol{E}_i = S_{ij,j}~\boldsymbol{E}_i $$ where $$\mathbf{v}$$ is a vector field, $$BS$$ is a second-order tensor field, and $$\boldsymbol{E}_i$$ are the components of an orthonormal basis in the reference configuration.

With respect to the deformed configuration, the governing equations are

{ \begin{align} \boldsymbol{b} & = \boldsymbol{F}\cdot\boldsymbol{F}^T & & \qquad \text{Strain-deformation Relations} \\ \boldsymbol{\sigma} & = \hat{\boldsymbol{\sigma}}(\boldsymbol{b}) & & \qquad \text{Stress-Strain Relations} \\ \rho~\det(\boldsymbol{F}) & = \rho_0 & &\qquad \text{Balance of Mass} \\ \boldsymbol{\nabla} \bullet \boldsymbol{\sigma} & = 0& & \qquad \text{Balance of Linear Momentum} \\ \boldsymbol{\sigma} & = \boldsymbol{\sigma}^T& & \qquad \text{Balance of Angular Momentum} \\ \rho~\dot{e} & = \boldsymbol{\sigma}:(\boldsymbol{\nabla}\mathbf{v}) - \boldsymbol{\nabla} \bullet \mathbf{q} + \rho~s & & \qquad\text{Balance of Energy.} \\ \end{align} } $$ Here, $$\boldsymbol{b}$$ is the left Cauchy-Green deformation tensor, $$\boldsymbol{\sigma}$$ is the Cauchy stress, and $$\rho$$ is the mass density in the deformed configuration. The gradient and divergence operators are defined such that

\boldsymbol{\nabla} \mathbf{v} = \sum_{i,j = 1}^3 \frac{\partial v_i}{\partial x_j}\mathbf{e}_i\otimes\mathbf{e}_j = v_{i,j}\mathbf{e}_i\otimes\mathbf{e}_j ~; \boldsymbol{\nabla} \bullet \mathbf{v} =\sum_{i=1}^3 \frac{\partial v_i}{\partial x_i} = v_{i,i} ~; \boldsymbol{\nabla} \bullet \boldsymbol{S} = \sum_{i,j=1}^3 \frac{\partial S_{ij}}{\partial x_j}~\mathbf{e}_i = S_{ij,j}~\mathbf{e}_i ~. $$

It is convenient to express all (unweighted) volume average quantities for finite deformation in terms of integrals over the reference volume ($$\Omega_0$$) and the reference surface ($$\partial{\Omega}_0$$).

Note that the strain measures used for finite deformation contain products of the deformation gradient. For example,

\boldsymbol{C} = \boldsymbol{F}^T\cdot\boldsymbol{F} ~. $$ A volume average of $$\boldsymbol{C}$$ may be defined in two ways:

\langle\boldsymbol{C}\rangle := \cfrac{1}{V_0} \int_{\Omega_0} \boldsymbol{C}~\text{dV} = \cfrac{1}{V_0} \int_{\Omega_0} \boldsymbol{F}^T\cdot\boldsymbol{F}~\text{dV} \qquad\text{or}\qquad \overline{\boldsymbol{C}} := \left(\cfrac{1}{V_0} \int_{\Omega_0} \boldsymbol{F}^T~\text{dV}\right)\cdot \left(\cfrac{1}{V_0} \int_{\Omega_0} \boldsymbol{F}^T~\text{dV}\right) = \langle \boldsymbol{F}\rangle^T\cdot\langle \boldsymbol{F}\rangle ~. $$ The choice of the definition of a macroscopic average quantity is based on physical considerations. Ideally, { such quantities are chosen such that their unweighted volume averages are completely defined by the surface data.}Unweighted average quantities that satisfy these requirements are the deformation gradient $$\boldsymbol{F}$$, its rate $$\dot{\boldsymbol{F}}$$, the first Piola-Kirchhoff stress $$\boldsymbol{P}$$, and its rate $$\dot{\boldsymbol{P}}$$.