Hello all,
I am looking to implement a plane-stress isoparametric user-element routine that I intend to use with non-linear material behaviour. For linear elastic cases, the out-of-plane strain (epsilon_33) can be obtained in closed form using Hooke's law.
epsilon_ij = ( ((1+nu)*sigma_ij) / E ) - (nu*tr(sigma)*delta_ij / E), where tr(.) is the trace and delta denotes the kronecker operator.
In the above, as sigma_13 and sigma_23 are zero, we can clearly show that epsilon_13 and epsilon_23 are zero and can therefore be neglected.
My question is, can we also adopt the same idea for large strain case with non-linear but isotropic material behaviour?
i.e., can we also approximate F_13, F_23, F_31 and F_32 (components of def_grad) as unity assuming stress measure to be 1st Piola-Kirchoff for example in a hyperelastic material model?
or
the fact that epsilon_13 and epsilon_23 are zero for Hooke's law is just a result of linear elastic assumption?
I hope my question is clear and of course I can provide more information if needed.
Thank you for your time
Shree
Hello Shree,
Because for large strain case with non-linear but isotropic material behaviour the Cauchy stress tensor and the strain tensor are coaxial, your idea can be adopted.
The answer is slightly more difficult. Plane stress does -- for large strains -- not really exist. Think of a plate with an inhomogeneous deformation. Due to the Poisson behavior there will be a tapering in some region. This, however, is connected to the shear components you talked about. Of course, if one discusses the problem only on the level of equations, any assumption to fix components of the deformation gradient or stress components will lead to some resulting shear strains and shear stresses.
If one assumes plane stress, which might be justified for moderate strains, this leads to a (implicit) system of non-linear equations in the case of elasticity and to a system of differential-algebraic equations in the case of rate-independent plasticity, viscoplasticity or viscoelasticity (see "Krämer, S., Rothe, S., Hartmann, S.: Homogeneous stress-strain states computed by 3D-stress algorithms of FE-codes. Application to material parameter identification, Engineering with Computers 31(1), 2015, 141-159", where we discussed such problems within the context of parameter identification for elastic and inelastic materials). The question is "what are moderate strains" since there does not exist a standard telling us when the plane stress case is violated.
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