[Study] Pseudo elasticity

[WeiyiVirtonomy]

A study of stiff-soft contact. The stiffer body has stronger compressibility, while the softer one is nearly incompressible.

Aim: study if it is possible to replace Young's modulus of the stiffer body with a pseudo elasticity, which matches the bulk modulus of the softer body.

[Xiangyu-Hu]

It would very helpful if some figures of results are provided.

[WeiyiVirtonomy]

It would very helpful if some figures of results are provided.

I haven't found references to compare yet, so I tried to use SPHinXsys to run a test case with the true and pseudo elasticity first.
Test case: a stiffer tube is moving upward and come into contact with a softer arc.

Properties:
Arc: rho = 1e-3 g/mm^3, K = 3.33 MPa, G = 0.34 MPa (E = 0.99 MPa, nu = 0.45)
Tube: rho = 1e-3 g/mm^3, E = 1322 MPa, nu = 0.3
The pseudo elasticity of the tube is: rho = 1e-3 g/mm^3, E_p = 4.0 MPa, nu = 0.3

With the pseudo elasticity (left), the tube behaves way softer than the real modulus (right):

By using 10 * E_p = 40 MPa = 40 * E_arc, the deformation of the tube is still larger than the real modulus:

Using 60 * E_p = 240 MPa = 240 * E_arc is better:

Using 100 * E_p:

[Xiangyu-Hu]

The difference is not large when 100 times is used.

[Xiangyu-Hu]

Here is the bending modulus is equivalent to Youngs modulus.
https://en.wikipedia.org/wiki/Flexural_modulus
So we can choose as much as possible Youngs modulus with smaller Poisson ratio and have check what happens.

[WeiyiVirtonomy]

Here is the bending modulus is equivalent to Youngs modulus. https://en.wikipedia.org/wiki/Flexural_modulus So we can choose as much as possible Youngs modulus with smaller Poisson ratio and have check what happens.

Do you mean using Young's modulus close to the real value of 1322 MPa but with a smaller Poisson?

[Xiangyu-Hu]

Here is the bending modulus is equivalent to Youngs modulus. https://en.wikipedia.org/wiki/Flexural_modulus So we can choose as much as possible Youngs modulus with smaller Poisson ratio and have check what happens.

Do you mean using Young's modulus close to the real value of 1322 MPa but with a smaller Poisson?

yes.

[WeiyiVirtonomy]

Here is the bending modulus is equivalent to Youngs modulus. https://en.wikipedia.org/wiki/Flexural_modulus So we can choose as much as possible Youngs modulus with smaller Poisson ratio and have check what happens.

Do you mean using Young's modulus close to the real value of 1322 MPa but with a smaller Poisson?

yes.

Case 1: E = E_real = 1322MPa, nu = 0, K = 441 = 123 K_soft
The stiff tube seems to be quite unstable.

Case 2: E = E_real = 1322MPa, nu = 0.1, K = 551 = 165 K_soft
The deformation is very close to the one with real parameters.

Case3: E = 0.1 * E_real = 132MPa, nu = 0.1, K = 55 = 17 K_soft
The tube behaves softer if Young's modulus is scaled.

To conclude, the bending behavior is mainly determined by Young's modulus, so using the same Young's modulus with a smaller Poisson ratio > 0 seems to give the best result. The problem is that the bulk modulus of the stiff body is still > 100 times larger than that of the softer body for these parameters.
When nu = 0.01 the instability still exists, it's better to use nu = 0.1.

[Xiangyu-Hu]

@WeiyiVirtonomy so we can try the case with maximum Youngs modules, small poison ratio and minimum possible bulk modules.

[WeiyiVirtonomy]

@WeiyiVirtonomy so we can try the case with maximum Youngs modules, small poison ratio and minimum possible bulk modules.

Yes