test stress and tangent by strain energy function for large strain models

[koehlerson]

@lijas I got this invariant formulation of the strain energy function, but I'm getting for the tangent a result which doubles all entries. Could it be that your AD of the stress needs a factor of 2? Otherwise, I need to double check the strain energy formula again

and maybe we should declare one ElasticState which is shared by all hyperelastic materials, if you agree, I can include it in this PR

We can also think about using the analytic forms in material_response and use the AD ones only for testing

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Merging #34 (ee5859a) into main (2a4ef29) will increase coverage by 0.10%. The diff coverage is 100.00%.

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src/FiniteStrain/stvenant.jl	100.00% <100.00%> (ø)

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[lijas]

Good to add proper testing :P I dont know where the factor 2 is coming from... C = 4 d^2PhidC^2 as you have written, and S = 2dPhidC. I will check the _SPK function

1. I kind of like that all materials have their own state, even though it adds some extra code
2. Sounds good to AD only for testing.

[koehlerson]

do you have some pdf with the analytic derivatives of Yeoh and/or NeoHook? Too lazy to do them on my own :D

[lijas]

No PDF unfortunately.

In Nonlinear continuum mechanics for finite elment analysis by Javier Bonet, I found derivations of the tangent and stress for the NeoHook material:

function check_tangents_AD(material::NeoHook,loading::Vector)
    state = initial_material_state(material)
    for C in loading

        invC = inv(C)
        J = sqrt(det(C))

        S = material.μ*(one(SymmetricTensor{2,3}) - inv(C)) + material.λ*log(J)*inv(C)
        T = material.λ*(invC⊗invC) + 2*(material.μ-material.λ*log(J))*otimesu(invC,invC)
        T = symmetric(T)

        ∂²Ψ∂C², ∂Ψ∂C, _ =  hessian(C -> MaterialModels.ψ(material, C), C, :all)
        S_AD = 2*∂Ψ∂C
        T_AD = 4*∂²Ψ∂C²

        @show S_AD ≈ S
        @show T_AD ≈ T
    end
end

I had to take symmetrize the analytical tangent for some reason. I remember my proffessor mentioning this at one time, but i never understood why.

I also found analytical derivations for Yeoh material in an old matlab code (voigt notiation):

S2 = 2*( mu/2*I + ...
    2*c2*(Ic-3)*I + ...
    3*c3*(Ic-3)^2*I + ...
    -mu*dlnJdc + ...
    lambda * log(J) * dlnJdc );

dS2_dE = 4 * (...
         2*c2*(I*I') + ...
         6*c3*(Ic-3)*(I*I') + ...
         -mu*d2lnJdcdc + ...
         lambda*(log(J)*d2lnJdcdc + (dlnJdc*dlnJdc')) );

with

invC=m_2_v9( inv(v9_2_m(C) ) );
detC=det(v9_2_m(C)); 
J=sqrt(detC);

dlnJdc = 1/2*invC; %Transpose C mayby.
d2lnJdcdc = 1/2*inv((f9_open_u_9(-C,C)));

%2nd order identity tensor
I=[1 1 1 0 0 0 0 0 0]';
Ic = trace(v9_2_m(C));

No derivation for StVenant material though :P

[lijas]

I (or someone else) can add these for analytical derivations for the meterial_response instead of the using AD as we are doing now

[koehlerson]

hm it's true that the symmetric statement is needed, otherwise the test fails, also for Yeoh model

[lijas]

Can this be merged? Tests are passing, but I guess we are unsure why we need Symmetric for Yeoh and StVenand material...

[koehlerson]

Other than the symmetric issues this should be good to go. Im not sure where or how to investigate into this, if you ever find it out please let me know :)