# Numerically computing deflection due to thermal expansion

Using linear elasticity formulation, I am attempting to numerically compute the displacement due to thermal expansion. This is done for a 3-D isotropic material.

The governing equations are simply:

$$\nabla \cdot \sigma = 0 \\ \sigma = \sigma_{\text{thermal}} - \sigma_{\text{mechanical}} \\ \sigma_{\text{thermal}} = C'\epsilon_{\text{thermal}} \\ \epsilon_{\text{thermal}} = \begin{bmatrix} \alpha \Delta T \\ \alpha \Delta T \\ \alpha \Delta T \\ 0 \\ 0 \\ 0 \\ \end{bmatrix} \\ C' = \text{4th order stiffness tensor reduced to a 2nd order 6x6 matrix} \\ \sigma_{\text{mechanical}} = C'\epsilon_{\text{mechanical}} \\ \epsilon_{\text{mechanical}} = \begin{bmatrix} \frac{du}{dx} \\ \frac{dv}{dy} \\ \frac{dw}{dz} \\ \frac{dv}{dz} + \frac{dw}{dy} \\ \frac{du}{dz} + \frac{dw}{dx} \\ \frac{du}{dy} + \frac{dv}{dx} \end{bmatrix}$$

From this, we can formulate an expression for the displacements, $$u$$,$$v$$,$$w$$. After writing these equations, I realized that it appears that the stiffness tensor has no bearing on the deflection generated from a thermal load. Mathematically, because $$C'$$ is invertible, we are left with the equation:

$$\nabla \cdot \epsilon_{\text{mechanical}} = \nabla \cdot \epsilon_{\text{thermal}} .$$ I am being a little lazy here with the notation where I previously used $$\epsilon$$ to represent a $$6\times1$$ vector, here it is a $$3\times 3$$ matrix.

I was able to verify this in ANSYS by varying the structural properties of the material and it demonstrated no change in the deflection from the thermal loading.

But physically, why is the deflection due to thermal expansion independent of the structural properties?

• I guess you constrained only the rigid body motions in your model, right? In that case there are no stresses induced by the thermal expansion and thus the structural properties like the Young's modulus don't affect the result. Jun 5 '19 at 8:44
• By rigid body motions, do you mean the 3 displacement components? If so, yes. What else besides rigid body motions could I constraint? Jun 5 '19 at 17:36
• You might want to read about rigid body motions if you are using FE programs like ANSYS. It's important to understand that part of the theory. Jun 6 '19 at 8:15
• @P.G. I know about rigid body motions, which typically consist of the 3 translational and 3 rotational DoFs. ANSYS allows for constraining the translation DoFs by fixing the displacement values, but I do not see it allowing for fixing rotational DoFs. Hence why I ask if you are referring to the 3 displacement components. Jun 6 '19 at 16:56
• Don't mix up rigid body motions and DoFs. Yes, in 3D you have 3 translational and 3 rotational rigid body motions. The rotational rigid body motions can be constrained by constraining translational DoFs. In 3D you need (at least) 6 constraints to constrain the 6 rigid body motions. Jun 7 '19 at 13:24