Finite strain theory
In this case, the undeformed and deformed configurations of the continuum are significantly different, requiring a clear distinction between them.
This is commonly the case with elastomers, plastically deforming materials and other fluids and biological soft tissue.
, i.e., deformation at neighbouring points, by transforming (linear transformation) a material line element emanating from that point from the reference configuration to the current or deformed configuration, assuming continuity in the mapping function
The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience.
in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as
, and assuming continuity on the displacement field, it is possible to use a Taylor series expansion around point
, neglecting higher-order terms, to approximate the components of the relative displacement vector for the neighboring particle
A geometrically consistent definition of such a derivative requires an excursion into differential geometry[2] but we avoid those issues in this article.
The derivative on the right hand side represents a material velocity gradient.
It is common to convert that into a spatial gradient by applying the chain rule for derivatives, i.e.,
If the spatial velocity gradient is constant in time, the above equation can be solved exactly to give
The material time derivative of the inverse of the deformation gradient (keeping the reference configuration fixed) is often required in analyses that involve finite strains.
To transform quantities that are defined with respect to areas in a deformed configuration to those relative to areas in a reference configuration, and vice versa, we use Nanson's relation, expressed as
To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:
Physically, the Cauchy–Green tensor gives us the square of local change in distances due to deformation, i.e.
are stretch ratios for the unit fibers that are initially oriented along the eigenvector directions of the right (reference) stretch tensor (these are not generally aligned with the three axis of the coordinate systems).
The Eulerian finite strain tensor, or Eulerian-Almansi finite strain tensor, referenced to the deformed configuration (i.e. Eulerian description) is defined as
A measure of deformation is the difference between the squares of the differential line element
In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is
In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is
Replacing this equation into the expression for the Lagrangian finite strain tensor we have
B. R. Seth from the Indian Institute of Technology Kharagpur was the first to show that the Green and Almansi strain tensors are special cases of a more general strain measure.
[12] The Seth–Hill family of strain measures (also called Doyle-Ericksen tensors)[13] can be expressed as
is the change in the angle between two line elements that were originally perpendicular with directions
Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the infinitesimal strain tensor The stretch ratio for the differential element
as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have
The problem of compatibility in continuum mechanics involves the determination of allowable single-valued continuous fields on bodies.
These allowable conditions leave the body without unphysical gaps or overlaps after a deformation.
Additional conditions are required for the internal boundaries of multiply connected bodies.
General sufficiency conditions for the left Cauchy–Green deformation tensor in three-dimensions were derived by Amit Acharya.
