For a right stretch tensor , define . A standard normalized choice has , , and strictly increasing; requiring gives a nonsingular stretch-to-strain parameterization. This strain vanishes exactly for a rigid rotation, is independent of superposed spatial rotations, and agrees to first order with the infinitesimal symmetric displacement gradient. Indeed gives , and Taylor expansion of proves the last statement.
The Biot strain tensor uses , so . Its principal values are extension ratios minus one. It is a spectral strain measure, since , and . Its reference-volume work conjugate is the symmetric Biot stress.
For , differentiation of gives . Its reference-power conjugate is . In a convected coordinate basis , these are the covariant components of the Kirchhoff stress tensor. The second Piola-Kirchhoff stress tensor instead gives its contravariant components, since .
With the right Cauchy-Green deformation tensor , the Green-Lagrange strain tensor measures the change in the material metric. Since , its work-conjugate stress and strain partner is the second Piola-Kirchhoff stress tensor . The formula follows by cycling factors in , not by identifying nominal and symmetric material stress.
In plane strain, the deformation map has , and . The out-of-plane stretch is one and out-of-plane shear vanishes. This kinematic condition need not make the out-of-plane stress vanish.
Simple shear displaces one coordinate direction in proportion to a transverse coordinate while preserving volume. Its deformation gradient has a single off-diagonal component . Simple shear flow describes the corresponding velocity field rather than the finite deformation.
The principal stretches are the eigenvalues of a stretch tensor, or equivalently the singular values of the deformation gradient. They measure local length ratios along the principal reference directions.
The Jacobian matrix of a deformation map carries an infinitesimal reference line element to its current image. Its determinant is the local volume ratio. An orientation-preserving deformation has , and incompressibility imposes .
For an orientation-preserving deformation gradient , oriented area transforms by its cofactor matrix: . Taking the dot product with proves this from the determinant identity for the scalar triple product. Applying it to tangent vectors of a surface proves Nanson's formula. Equating current and reference tractions then gives the material-first nominal stress tensor for symmetric Cauchy stress tensor .
The factor can encode an internal relaxed configuration, while determines the instantaneous elastic energy. Differentiating at fixed or fixed produces the corresponding nominal stress tensor and thermodynamic force conjugate to an internal variable. This is a kinematic decomposition, not an assertion that each factor separately derives from a global deformation map.
For an orientation-preserving invertible deformation gradient, the polar decomposition of an invertible real matrix separates a rotation from the right stretch tensor or the left stretch tensor . The rotation is shared by the two factorizations, with .
The right Cauchy-Green tensor measures stretch on the reference configuration. Its eigenvalues are the same squared principal stretches as those of the left Cauchy-Green deformation tensor, but its eigenvectors are reference directions.
The left Cauchy-Green tensor measures stretch on the current configuration. It is a positive-definite symmetric matrix for an invertible deformation gradient, with eigenvalues equal to the squares of the principal stretches.
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Finite strain theory is a framework used in the field of continuum mechanics to describe the behavior of materials undergoing large deformations. Unlike small strain theory, which assumes that deformations are infinitesimally small and uses linear approximations, finite strain theory accounts for significant changes in shape and size of materials. Key aspects of finite strain theory include: 1. **Large Deformations**: It is specifically designed to handle situations where the deformations are not minor and where geometric nonlinearity cannot be ignored.