FVM for Solids

Hooman · December 3, 2012, 12:21

Can anyone suggest a good paper or book on the use of finite volume methods for solids?

Rami · December 4, 2012, 03:25

Quote:

Originally Posted by Hooman

Can anyone suggest a good paper or book on the use of finite volume methods for solids?

Which equations are you wishing to solve? Heat transfer? Stress analysis? Other?
Usually FEM is more common than FVM for solids, although FVM may certainly be used too.

Engr.RZA · December 4, 2012, 11:10

Hi

I think FVM is preferably right choice to analyze solid structures.

Most of the FVM literature is available in the perspective of computational fluid dynamics. So you should consult CFD books.

1. Computational Fluid Dynamics: Principles & Applications By J. Blazek

2. Computational Fluid Dynamics By Hoffman

Hooman · December 5, 2012, 04:04

I am interested in the governing equations and their discretizations for stress analysis problems. I'm familiar with fvm for fluids, solid equation are somewhat different. In particular I am interested in the stress term ...

Rami · December 5, 2012, 05:09

Hi Hooman,

The equations used in stress analysis for solids are nearly identical to those of the momentum equations for viscous fluid flow, except that for solids

stands for the displacement vector, whereas for fluids - it is the velocity (the stress is related to strain for solid and to strain-rate in fluids).

You should also choose whether you use an Eulerian or Lagrangian formulation. The Lagrangian formulation in Cartesian coordinates is simply $\rho Du_i / Dt = \sigma_{ik,k} + \rho G_i$ (using tensor notation where D/Dt is the material derivative, comma is the covariant derivative - which degenerates to simple spatial derivative in Cartesian coordinates,

is the displacement, $\sigma_{ik}$ is the stress,

is the body force and $\rho$ is the density).

If you treat small displacement and small strain, the various stress and strain measures (Cauchy, Piola-Kirchhoff, Green, Lagrange, etc.) are identical and their relation to each other are practically the same as those for fluids, e.g. for elasticity $\sigma_{ik} = \lambda \epsilon_{mm} \delta_{ik} + 2 \mu \epsilon_{ik}$ , where $\lambda$ and $\mu$ are the Lame modulii and $\epsilon_{ik} = 1/2 ( u_{i,k} + u_{k,i} )$ .

Nevertheless, I still suggest to look into FEM, which has much more literature on solids. It is in many ways similar (and more consistent and general) to FVM. Actually, FVM can be viewed as a special case of FEM, using piecewise-constant shape-functions and some additional minor approximations.

December 3, 2012, 12:21	FVM for Solids	#1
Hooman Senior Member Join Date: Jun 2010 Posts: 111 Rep Power: 15	Can anyone suggest a good paper or book on the use of finite volume methods for solids?

December 5, 2012, 05:09		#5
Rami Senior Member Rami Ben-Zvi Join Date: Mar 2009 Posts: 155 Rep Power: 17	Hi Hooman, The equations used in stress analysis for solids are nearly identical to those of the momentum equations for viscous fluid flow, except that for solids stands for the displacement vector, whereas for fluids - it is the velocity (the stress is related to strain for solid and to strain-rate in fluids). You should also choose whether you use an Eulerian or Lagrangian formulation. The Lagrangian formulation in Cartesian coordinates is simply $\rho Du_i / Dt = \sigma_{ik,k} + \rho G_i$ (using tensor notation where D/Dt is the material derivative, comma is the covariant derivative - which degenerates to simple spatial derivative in Cartesian coordinates, is the displacement, $\sigma_{ik}$ is the stress, is the body force and $\rho$ is the density). If you treat small displacement and small strain, the various stress and strain measures (Cauchy, Piola-Kirchhoff, Green, Lagrange, etc.) are identical and their relation to each other are practically the same as those for fluids, e.g. for elasticity $\sigma_{ik} = \lambda \epsilon_{mm} \delta_{ik} + 2 \mu \epsilon_{ik}$ , where $\lambda$ and $\mu$ are the Lame modulii and $\epsilon_{ik} = 1/2 ( u_{i,k} + u_{k,i} )$ . Nevertheless, I still suggest to look into FEM, which has much more literature on solids. It is in many ways similar (and more consistent and general) to FVM. Actually, FVM can be viewed as a special case of FEM, using piecewise-constant shape-functions and some additional minor approximations. Last edited by Rami; December 5, 2012 at 06:20.

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December 4, 2012, 11:10		#3
Engr.RZA New Member RZA Join Date: Nov 2012 Posts: 25 Rep Power: 13	Hi I think FVM is preferably right choice to analyze solid structures. Most of the FVM literature is available in the perspective of computational fluid dynamics. So you should consult CFD books. 1. Computational Fluid Dynamics: Principles & Applications By J. Blazek 2. Computational Fluid Dynamics By Hoffman

December 5, 2012, 04:04		#4
Hooman Senior Member Join Date: Jun 2010 Posts: 111 Rep Power: 15	I am interested in the governing equations and their discretizations for stress analysis problems. I'm familiar with fvm for fluids, solid equation are somewhat different. In particular I am interested in the stress term ...