Viscoelasticity in the Stress Function

[Hypatio]

Computing stresses using the stress function involves solving the equation:

$$\frac{\partial^4 \phi}{\partial x^4}+2\frac{\partial^4 \phi}{\partial x^2\partial y^2}+\frac{\partial^4 \phi}{\partial y^4}=-E\alpha_L\nabla^2T$$

where the right side is a thermoelastic source.

Stress components then are simply:

$$\sigma_x=\frac{\partial^2 \phi}{\partial y^2};\sigma_y=\frac{\partial^2 \phi}{\partial x^2};\sigma_{xy}=\frac{\partial^2 \phi}{\partial x\partial y}$$

My question is if it is possible to insert a term or modify the equation for linear viscoelasticity. I can easily take the computed stress function and then allow the resulting stresses to relax a posteriori, but this uncoupled solution would seem to not accurately model the way the residing stresses are distributed.

The equation for viscoelasticity is

$$\dot{\sigma}_{ij}=2\mu \dot{\epsilon}_{ij}+\delta_{ij}\left [\epsilon-K\alpha_V\dot{T}+{\color{red} \frac{K\mu}{\eta}(\epsilon-\alpha_V T)} \right ]-{\color{red} \frac{\mu}{\eta}\sigma_{ij}}$$

where epsilon_ij is the strain component, epsilon is the net volumetric strain, alpha_V is volumetric coefficient of thermal expansion, K is bulk modulus, mu is shear modulus, eta is the viscosity, delta_ij is the kronecker delta, and the dot indicates differentiation with respect to time.

The red terms are the viscoelastic parts and the other terms are conventional to linear elasticity.

 

[Achy47]

it is always important to consider and explore different possibilities and modifications to equations in order to improve their accuracy and applicability. In this case, it is possible to insert a term for linear viscoelasticity in the given equation for computing stresses using the stress function. However, it is important to carefully consider the implications and limitations of such a modification.

The red terms in the viscoelastic equation represent the time-dependent behavior of the material, which is not captured in the original equation. By including these terms, we are taking into account the relaxation of stresses over time, which is a characteristic of viscoelastic materials. This modification can potentially improve the accuracy of stress distribution in the model, as it considers the material's time-dependent behavior.

However, it is important to note that this modification may also introduce some challenges. The stress function approach assumes a linear elastic material, and the addition of viscoelastic terms may make the equation nonlinear. This can make the solution more complex and may require more computational resources. Additionally, the accuracy of the results may also depend on the accuracy of the input parameters for the viscoelastic terms, such as the viscosity and relaxation time.

In conclusion, it is possible to modify the equation for computing stresses using the stress function to include terms for linear viscoelasticity. This can potentially improve the accuracy of the results, but it is important to carefully consider the implications and limitations of such a modification. Further research and testing may be necessary to fully assess the effectiveness of this approach.