Finite strain theory

Mathematical model for describing material deformation under stress

In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal 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.

The deformation gradient tensor is related to both the reference and current configuration, as seen by the unit vectors and , therefore it is a two point tensor. Two types of deformation gradient tensor may be defined. Due to the assumption of continuity of , has the inverse , where is the spatial deformation gradient tensor. Then, by the implicit function theorem, the Jacobian determinant must be nonsingular, i. e.

The material deformation gradient tensor is a second order tensor that represents the gradient of the mapping function or functional relation , which describes the motion of a continuum. The material deformation gradient tensor characterizes the local deformation at a material point with position vector , 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 , i. e. differentiable function of and time , which implies that cracks and voids do not open or close during the deformation. Thus we have,

Consider a particle or material point with position vector in the undeformed configuration (Figure 2). After a displacement of the body, the new position of the particle indicated by in the new configuration is given by the vector position The coordinate systems for the undeformed and deformed configuration can be superimposed for convenience. Consider now a material point neighboring , with position vector In the deformed configuration this particle has a new position given by the position vector Assuming that the line segments and joining the particles and in both the undeformed and deformed configuration, respectively, to be very small, then we can express them as and Thus from Figure 2 we have

where is the relative displacement vector, which represents the relative displacement of with respect to in the deformed configuration.

The diagonal components of the Lagrangian finite strain tensor are related to the normal strain, e. g.

where is the normal strain or engineering strain in the direction
The off diagonal components of the Lagrangian finite strain tensor are related to shear strain, e. g.

where is the change in the angle between two line elements that were originally perpendicular with directions and , respectively. 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 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. Most such conditions apply to simply connected bodies. 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. Compatibility conditions for two dimensional fields were found by Janet Blume.