Hello,
Anyone could explain the physical meaning of the partial differential equation coefficients in the propagation of elastic waves in solid media?
Equations for isotropic solid media in 2D:
Equation 1: ρ (∂ ^ 2 u_x) / (∂t ^ 2) = (2G + λ) (∂ ^ 2 u_x) / (∂x ^ 2) + G (∂ ^ 2 u_x) / (∂y ^ 2) + (G + λ) (∂ ^ 2 u_y) / ∂x∂y
Equation 2: ρ (∂ ^ 2 u_y) / (∂t ^ 2) = G (∂ ^ 2 u_y) / (∂x ^ 2) + (2G + λ) (∂ ^ 2 u_y) / (∂y ^ 2) + (G + λ) (∂ ^ 2 u_x) / ∂x∂y
Recalling that I understand what the physical parameters (G; λ) themselves represent for the specific physical problem, but I do not know what is the physical implication of its use as a multiplier of each partial derivative.
Thanks in advance for the help.
Renato
Compare with the differential equation of a spring:
m d²x/dt²=-kx
where k is the stiffness of the spring.
Here, it is basically the same, but you need to look at the u-derivative on the rhs because a material is only strained if adjacent points are displaced differently, otherwise, it is just a rigid-body-movement.
So the multiplier tells you the "stiffness" of this mode. For example, look at
G (∂ ^ 2 u_y) / (∂x ^ 2)
The derivative tells you that you have a change in the y-displacement when you look into the x-direction. Thus, this is a shear deformation, so the stiffness is given by the shear stiffness. For
(2G + λ) (∂ ^ 2 u_x) / (∂x ^ 2)
this is a uniaxial deformation, so the stiffness is Young's modulus.
Dear Martin Baeker, I really appreciate your help! Your explanation is very clear, but I believe there is something I couldn't get it.
Taking your analogy of the spring as a reference, then you have:
mass * acceleration = force (Newton's second law),
but force = stiffness * displacement
In the elastic wave equations you have:
density (kind of mass) * acceleration = stiffness_type1 * (second order derivative of displacement in the space domain) + other derivatives...
The rhs of the elastic wave equation shouldn't result in a kind of force or stress? Additionally the rhs is formed of 3 parts, including the cross-derivative, which I believe should have, as a multiplier, a kind of combined stiffness.
If what I said is correct, could you give me a tip on how should I think to get the force or stress in the rhs of the equation?
Thanks a lot for your kind attention!
Sorry, I do not understand exactly what you are getting at.
The rhs is formed of 3 parts because in a continuous medium, deformation (strain) is a tensor (see any textbook on mechanics of materials; a book on theoretical mechanics that explains mechanics of continua will also be helpful).
And yes, the rhs is a stress.
If you are worried why you need a 2nd derivative: The first derivative of displacement is the strain. If the strain is homogeneous, there is no driving force for an oscillation.
You may find the following link helpful:
http://www.people.fas.harvard.edu/~djmorin/waves/transverse.pdf
Ok Mr. Baeker, you helped me with this matter.
Thank you!
Regards.
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