Time-harmonic and unsteady elasticity equations
Elastodynamics
In time-harmonic domain, we are solving the following equation :
![](equation/eqelastic_py1.png)
where C is the elasticity tensor appearing in Hooke's law, u is the displacement and ε(u) is the strain tensor. This tensor can be isotropic, and Lame coefficients (or Young's modulus/Poisson's ratio) are provided in the data file through field Materiau Dielec :
# you can enter Lame coefficients : # MateriauDielec = ref ISOTROPE rho lambda mu MateriauDielec = 1 ISOTROPE 1.2 2.4 0.8 # you can specify Young's modulus and Poisson's ratio # MateriauDielec = ref YOUNG_POISSON rho E nu MateriauDielec = 2 ISOTROPE 2700 6.9e10 0.34 # you can directly enter elasticity tensor C for orthotropic material # in 2-D : MateriauDielec = ref ORTHOTROPE rho c0000 c0011 c1111 c0101 # in 3-D : MateriauDielec = ref ORTHOTROPE rho c0000 c0011 c0022 c1111 c1122 c2222 c0101 c0202 c1212 MateriauDielec = 3 ORTHOTROPE 1.2 2.4 4.5 1.3 3.7 # in 2-D : MateriauDielec = ref ANISOTROPE rho c0000 c0011 c0001 c1111 c1101 c0101 # in 3-D : MateriauDielec = ref ANISOTROPE rho c0000 c0011 c0022 c0001 c0002 c0012 \ # c1111 c1122 c1101 c1102 c1112 c2222 \ # c2201 c2202 c2212 c0101 c0102 c0112 \ # c0202 c0212 c1212 MateriauDielec = 4 ANISOTROPE 1.2 2.4 4.5 1.3 3.7 0.2 0.5
Time-harmonic elastic equation is specified in class HarmonicElasticEquation. You can also solve static elastic equation with class ElasticEquation, which can be written as :
![](equation/eqelastic_py2.png)
This last equation is solved in real numbers.
Reissner-Mindlin
In the case of thin plates, a classical model is the so-called Reissner-Mindlin model, which can be written as :
![](equation/eqelastic_py3.png)
where the tensor C is defined with Young's modulus E and Poisson's ratio ν by the relationship
![](equation/eqelastic_py4.png)
E and ν as entered as for classical elastic equations (with MateriauDielec field). You can also solve static Reissner-Mindlin equations :
![](equation/eqelastic_py5.png)
and time-harmonic Reissner-Mindlin equations
![](equation/eqelastic_py6.png)
Fluid-structure interaction
We are considering the following set of equations
![](equation/eqelastic_py7.png)
where u and σ are the displacements and the stresses and exist only in the solid, p and v the primitive of the pressure and displacement of the fluid and exist only in the fluid. The transmissions conditions between the fluid and the solid are the following ones
![](equation/eqelastic_py8.png)
This problem is solved with explicit time-schemes only (leap frog or Runge-Kutta-like schemes).
Vibroacoustics
We also propose a coupling between Reissner-Mindlin equations (on a plate) and acoustics equation (in the volume), in order to avoid the meshing of a thin 3-D domain (the plate). The system of equations can be written as :
![](equation/eqelastic_py9.png)
where ρf is the density of the fluid, c the velocity of sound waves in the fluid, p the primitive of the pressure, v the displacement of the fluid, θ and u displacements of the plate. We assume here that the normale is outward from Ω- to Ω+, and we denote the jump as :
![](equation/eqelastic_py10.png)
For this problem, one can only use hexahedral elements with first-order absorbing conditions