Suppose I have stress and strain relationship and it can be simplified to the equation $\sigma=Function(\epsilon)$. This is the CONSTITUTIVE equation of the material I want to do research.

Assume that I need to do the finite element analysis and it has dynamic(nonlinear) deformation, I have the model and I have the CONSTITUTIVE equation which I mentioned the 1st paragraph.
How can we put those things into a structural equation to do the simulation and what structural equation I need to use?

Can we put this CONSTITUTIVE equation into the following equation $\nabla\cdot \sigma =Force $? Then we can get $\epsilon$, we get displacement and $\sigma$ explicitly.

Also, I think the above is the linear equation, what about nonlinear equation?

Thank you very much.

closed as unclear what you're asking by OpticalResonator, Wasabi Dec 7 at 0:56

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  • 1
    I'm not sure what you are doing which is "new" here. For a general material you already have $\sigma_{ij} = C_{ijkl}\epsilon_{kl}$ in tensor notation where because of the symmetry of $\sigma$ and $\epsilon$, the 81 terms of $C$ contain 21 independent parameters in general, all of which can be functions of $\epsilon$ if you like. (In general, they will be functions of other things as well). You can solve the nonlinear equations by any method that works - numerically, this will require an iterative method. – alephzero Dec 6 at 9:44

The balance of linear momentum is $$ \nabla \cdot \boldsymbol{\sigma} + \rho \mathbf{b} = \rho \mathbf{a} $$ where $\boldsymbol{\sigma}$ is the Cauchy stress, $\rho$ is the mass density, $\mathbf{b}$ is the body force, and $\mathbf{a}$ is the acceleration.

Given a nonlinear stress-strain relation $$ \boldsymbol{\sigma} = g(\boldsymbol{F}) $$ where $\boldsymbol{F}$ is the deformation gradient, we can write the linear momentum equation as $$ \nabla \cdot [g(\boldsymbol{F})] + \rho \mathbf{b} = \rho \mathbf{a} $$ Note that various strain definitions can be derived from $\boldsymbol{F}$ (see, for example, https://en.wikipedia.org/wiki/Finite_strain_theory#Seth%E2%80%93Hill_family_of_generalized_strain_tensors).

The deformation gradient tensor ($\boldsymbol{F}$) is related to the displacement vector ($\mathbf{u}$) as explained in great detail in https://en.wikipedia.org/wiki/Finite_strain_theory#Material_coordinates_(Lagrangian_description). For some problems (but not all) we can write $$ \nabla \cdot [g(\nabla \mathbf{u} + \boldsymbol{I})] + \rho \mathbf{b} = \rho \ddot{\mathbf{u}} $$ where $\boldsymbol{I}$ is the second-order identity tensor. This is clearly a highly nonlinear problem that can be solved for the displacement $\mathbf{u}$ given the appropriate boundary conditions.

So, the answer to your question is that the balance of linear momentum holds whether the problem is linear or not.

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