Two different Young Modulus depending on the force/displacement

[andersonnardin]

Hi all,
My question is:
How can I impose a different Young modulus depending on the force to which a tetrahedron is subject? Suggestions?

More specifically, I imagined something like:

foamSubTopo.addObject('TetrahedronFEMForceField', template='Vec3', name='FEM', method='large',
                       poissonRatio=poissonRatioFoam, youngModulus=youngModulus_1_or_2)

For the case where I have a deformable material that responds differently when under compression vs under extension.
Have you ever seen this?

Thank you!

[hugtalbot]

Hi @andersonnardin

I am unsure what you mean. If you mean having heterogeneous stiffness, this can be done using the youngModulus as a vector field (see example Heterogeneous-TetrahedronFEMForceField.scn).

Else, if you want your material law to depend on the external forces, this makes a coupled system :
$$\rho \dot{v} = \rho \mathbf{b} + (\nabla \cdot \mathbf{\sigma(b)})$$
and this does not exist (at least in SOFA)

For a material that responds differently when under compression vs under extension, I guess there exists an associated material law, right ? Could you share it with us?

Interesting topic @andersonnardin ;)

[andersonnardin]

Hi @hugtalbot ,

Thank you for your response! Based on your suggestions, the first example seems ideal for scenarios where we need to model materials with varying stiffness, allowing for different Young's modulus values throughout the body. This is a helpful feature, but it doesn't quite match my current requirements.

Your second point, involving the Cauchy Momentum Equation, aligns more closely with what I'm looking for. I appreciate your interest and suggestion. Before diving deeper into this approach, I thought of a potential workaround to achieve the desired effect. My idea is to check the stress tensor or forces on each tetrahedron during each iteration. Depending on the sign (positive or negative), I would set an appropriate Young's modulus value.

Here's a more practical outline of the algorithm:

We start with a body defined as:

unit.addObject('MechanicalObject', name='tetras', template='Vec3d')
unit.addObject('TetrahedronFEMForceField', template='Vec3d', name='FEM', method='large', poissonRatio=poissonratiounit, youngModulus=X)

In the controller class, during the animation update, we could implement the following:

def onAnimateBeginEvent(self, eventType):
    # Access the force field
    field = self.node.unit1.tetras.force.value
    
    # Calculate the divergence of the force field
    div_field = calculateDivergence(field)  # Assume calculateDivergence is a defined function

    # Set Young's modulus based on the divergence sign
    if div_field > 0:
        self.node.unit1.FEM.youngModulus[0] = X  # X is the modulus for positive divergence
    else:
        self.node.unit1.FEM.youngModulus[0] = Y  # Y is the modulus for negative divergence

Disclaimer: As you can see, my approach is based on defining the forces (unfortunately, these are calculated at the nodes) and then updating the Young's modulus (this is applied at the tetrahedrons). My doubt is whether it's possible to directly obtain the stress on each tetrahedron, as this would provide more precise information for decision-making. Alternatively, if I continue using the forces at the nodes, I wonder if there's a way to correlate these nodal forces with the respective tetrahedrons they belong to.

I'm eager to hear your thoughts on this approach and any advice you might have on implementing it effectively.

Thanks again for your assistance!

[hugtalbot]

Hi @andersonnardin

Since the VonMises stress can be rendered on both vertices and tetrahedra in the TetrahedronFEMForceField, it should be feasible (modifying the C++) to compute and access this information as well.

See the code here: https://github.com/sofa-framework/sofa/blob/master/Sofa/Component/SolidMechanics/FEM/Elastic/src/sofa/component/solidmechanics/fem/elastic/TetrahedronFEMForceField.inl#L1726-L1745

Cheers