# How to introduce the CONSTITUTIVE equation into structure mechanics [closed]

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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• 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.