Micromechanics of composites/Finite deformations

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Finite deformations[edit | edit source]

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

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

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

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

The gradient and divergence operators are defined such that

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

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

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,

A volume average of may be defined in two ways:

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 .