A Elastically anisotropic sphere under pressure

[camilo]

Does a sphere made of an elastically anisotropic material (eg. a material of cubic symmetry) subject to an hydrostatic pressure retains its spherical shape ?

 

[DrDu]

cubic symmetry is still isotropic, but for lower symmetries (like, e.g., orthorhombic), the pressure will lead to a deformation of the sphere.

 

[camilo]

cubic symmetry is still isotropic, but for lower symmetries (like, e.g., orthorhombic), the pressure will lead to a deformation of the sphere.

How a cubic symmetric is isotropic ?
In the cubic symmetry there are three independent elastic constants, s_11, s_12 and s44. In a cubic crystal structure there are directions along which the material is softer and others along which is is stiffer. For instance, Silver, which has an fcc structure has a Young modulus of 94 GPa along (110), whereas along (100) it is 50 GPa.

 

[DrDu]

How a cubic symmetric is isotropic ?
In the cubic symmetry there are three independent elastic constants, s_11, s_12 and s44. In a cubic crystal structure there are directions along which the material is softer and others along which is is stiffer. For instance, Silver, which has an fcc structure has a Young modulus of 94 GPa along (110), whereas along (100) it is 50 GPa.

Sorry, I had optical properties in mind. As the constitutive stress strain equation involves a fourth order tensor (as opposed to the second order dielectric tensor), a cubic material will not behave isotropically.

 

[Orodruin]

That being said, the symmetries of the fourth order tensor for cubic symmetry are such that the resulting strain tensor must be isotropic if the stress tensor is, which is the case when you subject an object to hydrostatic pressure. As such, the material with cubic symmetry would deform isotropically.

 

[camilo]

That being said, the symmetries of the fourth order tensor for cubic symmetry are such that the resulting strain tensor must be isotropic if the stress tensor is, which is the case when you subject an object to hydrostatic pressure. As such, the material with cubic symmetry would deform isotropically.

So it would retain its spherical shape ?

 

[Orodruin]

If the lattice is cubic, yes. If it has other types of symmetries, not necessarily.

 

[DrDu]

Just did a bit of reading. I think the situation is much more easy in orthotropic materials. The Voigt vector of stresses is proportional to (1,1,1,0,0,0) for isotropic pressure. In orthoscopic materials, the stiffness tensor is 3x3 block diagonal in a certain basis, so that the strain vector will be of the form (a,b,c,0,0,0). I.e. the sphere will be deformed into an ellipsoid with their main axes are the symmetry axes of the material.

 

[Spinnor]

I.e. the sphere will be deformed into an ellipsoid with their main axes are the symmetry axes of the material.

So a sphere made of silver, subject to a hydrostatic pressure, will deform into an ellipsoid?

Thanks.

 

[DrDu]

As Orodruin pointed out already, Silver having a cubic crystal lattice, it won't deform into an ellipsoid under hydrostatic pressure.