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
![{\displaystyle \mathbf {x} ={\boldsymbol {\varphi }}(\mathbf {X} )=\mathbf {x} (\mathbf {X} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/29149491ce2229e2857eda6e0cb44116f67a4c34)
where
is the position of a point in the RVE in the initial
configuration and
is the location of the same point in the
deformed configuration.
The deformation gradient is given by
![{\displaystyle {\boldsymbol {F}}={\frac {\partial \mathbf {x} }{\partial \mathbf {X} }}={\boldsymbol {\nabla }}_{0}~\mathbf {x} ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccf10ea4476520885010c7a2fb6a0f315ed25584)
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
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/69f43d2dbc69998fcb930d0d02b22849a626a87d)
In the above
is the right Cauchy-Green deformation tensor,
is the first Piola-Kirchhoff stress tensor, and
is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related to the Cauchy stress tensor by
![{\displaystyle {\boldsymbol {P}}=\det({\boldsymbol {F}})~{\boldsymbol {F}}^{-1}\cdot {\boldsymbol {\sigma }}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/72186ba233420049da35070d4ba1e45bee758091)
The gradient and divergence operators are defined such that
![{\displaystyle {\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}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff90a30384f2c9bd21eb13b8e3ae5929b072207)
where
is a vector field,
is a second-order tensor field, and
are the components of an orthonormal basis in the reference
configuration.
With respect to the deformed configuration, the governing equations are
![{\displaystyle {\begin{aligned}{\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{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/562d39a3deac75c851dfcb542542558805e3c567)
Here,
is the left Cauchy-Green deformation tensor,
is the
Cauchy stress, and
is the mass density in the deformed configuration.
The gradient and divergence operators are defined such that
![{\displaystyle {\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}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dad10d69729de7bb9b3715fd1e81bea100c1d84b)
It is convenient to express all (unweighted) volume average quantities
for finite deformation in terms of integrals over the reference volume
(
) and the reference surface (
).
Note that the strain measures used for finite deformation contain products
of the deformation gradient. For example,
![{\displaystyle {\boldsymbol {C}}={\boldsymbol {F}}^{T}\cdot {\boldsymbol {F}}~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9e832b8e87728e98f7807154bbeee4ac44ed8d)
A volume average of
may be defined in two ways:
![{\displaystyle \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 ~.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b65770f2f7463f3c2b85df4fa92fdbc8b7bef172)
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
, its rate
, the first
Piola-Kirchhoff stress
, and its rate
.