Invariants of Strain
Principal Strains
The principal strains are the eigenvalues of the strain tensor (ε), computed from the eigenvalue equation
The three principal strains are sorted so that
This sorting is true also for the 2D cases. The corresponding vectors in the principal directions, vpi, are orthonormal.
•
The internal variables for the principal strains are named solid.ep1, solid.ep2, and solid.ep3.
•
The internal variables for the components of the directions of the first principal strains are named solid.ep1X, solid.ep1Y, and solid.ep1Z. The direction vectors for the other two principal strains are named analogously.
The Green–Lagrange strain tensor is used in a geometrically nonlinear analysis. The orientations of the principal directions will thus be with respect to the material directions. If you plot the principal strains as arrows, you should thus use an undeformed plot.
Principal Stretches
The principal stretches are the eigenvalues of the stretch tensor U, and are also sorted by size:
The internal variables for the principal stretches are named solid.stchp1, solid.stchp2, and solid.stchp3. The elastic principal stretches are named solid.stchelp1, solid.stchelp2, and solid.stchelp3.
The different invariants of the strain tensor form are useful for constitutive modeling and result interpretation. The three fundamental invariants for any tensor are
The invariants of the strain deviator tensor is also useful.
As defined above J2 ≥ 0. I1 represents the relative change in volume for infinitesimal strains and J2 represents the magnitude of shear strain.
In tensor component notation, the invariants can be written as
The volumetric strain is defined as
and the equivalent deviatoric strain as
The internal variable for the volumetric strain is solid.evol. The internal variables for the components of the deviatoric strain tensor in the local coordinate system are solid.eldev11, etc. The internal variable for the equivalent deviatoric strain is solid.edeve.
In terms of the principal strains, the strain invariants can be written as
The principal strains are the roots of the characteristic equation (Cayley–Hamilton theorem)
Strain Rate and Spin
The spatial velocity gradient is defined in components as
where vk(x,t) is the spatial velocity field. It can be shown that L can be computed in terms of the deformation gradient as
where the material time derivative is used.
The velocity gradient can be decomposed into symmetric and skew-symmetric parts
where
is called the rate of strain tensor, and
is called the spin tensor. Both tensors are defined on the spatial frame.
It can be shown that the material time derivative of the Green–Lagrange strain tensor can be related to the rate of strain tensor as
The spin tensor Lw(x, t) accounts for an instantaneous local rigid-body rotation about an axis passing through the point x.
Components of both Ld and Lw are available as results and analysis variables under the Solid Mechanics interface.
•
The components of the rate of strain tensor are named solid.Ldx, solid.Ldxy, and so on.
•
The components of the spin tensor are named solid.Lwx, solid.Lwxy, and so on.