How do I have a continuously varying Young's Modulus using the pde toolbox?

I have to solve a problem where I need to be able to solve a static structural problem with a continuously varying Young's Modulus So I understand that the Young's Modulus, E, input to

    structuralProperties(____,'YoungsModulus',E)

Must be in the form of a double rather than a function label so that leads me to doing this as a generic PDE, requiring me to specify the coefficients m, d, c, a, and f.

Since the structural equation is:

(λ+μ)∇(∇⋅u)+μ∇^2u+f=0

And the generic PDE equation is:

m ∂2/u∂t2+d ∂u/∂t−∇·(c⊗∇u)+au=f

I'm wondering how I should specify these coefficients. ('m' and 'd' should clearly be zero as it is time invariant)

'c' should just be (λ+μ) I believe so not too difficult.

I am having trouble translating the μ∇2u from the structural PDE to the generic PDE though. Would this have to be specified in the 'a' coefficient and if so how would one define the ∇^2 in a function form on the u vector.

I am also wondering how to define the f coefficient, I'm thinking it should either be a zero matrix or an identity matrix but since part of what I'm doing is interpolating the traction forces of the solution I'm wondering if I should be defining this as a function as well.

Any advice is greatly appreciated!

Cheers

Answer 1

[Ravi Kumar]

Check this documentation page showing how to convert structural PDEs into the coefficient form used in PDE Toolbox.

There is a helper function as mentioned in the same documentation page (at the end of section 3D Linear Elasticity Problem), elasticityC3D. This helper function assumes you are solving a 3-D problem, it takes a constant Young's modulus and Poisson's ratio as inputs and provides c-coefficient as output.

However, this information might still not answer your question as to how to specify non-constant Young's modulus. To get such a problem working you have to follow the approach in the documentation page and construct a function that returns a non-constant c-coefficient, such a function could look like elasticityC3D, but would be much more complicated to account for non-constant material properties.

Having more information about your problem would help provide better answer. hat are you trying to solve, is it nonlinear material model? What is Young's modulus dependent on? Is is a function of spatial coordinates and/or displacements?

[bojoh]

Thanks for your reply. For clarification, in my case the Young's Modulus should be a function of spatial coordinates. I'll take an in depth look at your other suggestions now. Thanks again.