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Novozhilov's equilibrium equations
I’ve been solving finite deformation problems (small strain but large rotations) using equations I found in “Foundations of Nonlinear Theory of Elasticity” by V. V. Novozhilov. The attraction of Novozhilov’s book is that he didn’t use tensors, a subject I’ve never mastered. I was able to use his equations in an FEA solver (FlexPDE) and get results that look reasonable. As I’ve moved along with my research, I’ve realized that I need to learn tensors. As a first step, I’m attempting to relate what I learned from Novozhilov to what I’m finding in modern tensor-based developments of finite deformation theory – for example in “Applied Mechanics of Solids” by A. F. Bower.
There is no problem with strain. Novozhilov’s equations for finite strain are exactly the same as the Green-Lagrange strain when you expand the tensor. But, stress is another matter.
Novozhilov develops equations of equilibrium that can be applied to the reference coordinates (undeformed coordinates). The stresses in these equations seem to bear no connection to either of the Piola-Kirchoff stresses.
Novozhilov’s 2D x-direction equilibrium equation with no body forces is,
There is, of course, a similar equation for the y-direction.
The A ratios are the ratios of the deformed to undeformed areas. Variables u and v are the x and y displacements. Ex and Ey are the relative elongations. The epsilon symbols are the Green-Lagrange strains. The asterisked sigma symbols are the ratios of the forces in a deformed volume element relative to the undeformed areas of the element. The sigma symbols without the asterisk are the Cauchy stresses. My problems involve only large rotations but small strains. So the equilibrium equation can be simplified by taking the A ratios as unity and the relative elongations as equal to the strain components. In that case the equilibrium equation becomes,
I had assumed that I should be able to start with one or the other of the PK stresses in the equation of equilibrium (the equations in Bower’s book, for example), expand the tensors and arrive at Novozhilov’s first equation. I don’t get anything like that. For example, if I use the PK 1stress I get this for the 2D x-derivative of the x-direction equilibrium equation.
The PK 2 stress is even worse.
Am I just making a mathematical mistake or do I have a conceptual misunderstanding? Surely, Novozhilov isn’t wrong. I’ve been using his simplified equation and producing plausible results. I’ve run problems that can be solved with beam theory and gotten good agreement.
Re: Novozhilov's equilibrium equations
You've probably got the math wrong.
If you start with F_{ik} = u_i,k + \delta_{ik} and note that equilibrium in material form is given by (S_{jk} F_{ik}), j = 0 you get equations of the form
\partial/\partial X_j [S_j1 (u_i.1 + delta_{i1}) + S_j2 (u_i,2 + delta_i2) + S_{j3} (u_i,3 + delta_i3)]
In 2D, these simplify to one equation of the form (for i=1)
\partial/\partial x_1 (S_11 (u_1,1 + 1) + S_{12} u_1,2) +
\partial/\partial x_2 (S_21 u_1,1 + S_{22} (u_1,2 + 1) ) = 0
and a similar one for i=2.
-- Biswajit
Thank you very much for
Thank you very much for responding.
I think my problem must be in understanding the meaning of the PK2 stress, because, if I interpret S in your equilibrium equation as PK2, it agrees with the references I've been using. As I understand it, the PK2 stress is the current force relative to the reference area. I've assumed that in order to apply the constitutive equations for small strains and large rotations (Hook's law), that I needed to express it in terms of the Cauchy stress and that it is the Cauchy stress (current force relative to the current area) that is represented by sigma in Novozhilov's equilbrium equations. In other words,
It’s when I expand this that everything goes south.
More on Novozhilov's equilibrium equations
The S in my equations is the 2nd PK stress. The derivatives are with respect to material coordinates. I can't see an easy way of going from those equations to similar equations in terms of spatial derivatives of the Cauchy stress a la Novozhilov. Several other ad-hoc assumptions may be needed, e.g., spatial and material derivatives are identical, the deformation gradient is identity, etc.
One way of dealing with small strains and large rotations is to use hypoelasticity (which is not really elasticity but works reasonably well for some problems). If you want to avoid that and still use Hooke's law, I'd suggest the Green-St Venant form of the elastic constitutive equation. Solve everything in material coordinates and then transform back to spatial coordinates. The elastic properties in spatial coordinates are not the same as those in material coordinates. But for small strains you can assume that they are the same. Ideally you should transform the constitutive equation from material coordinates to spatial cordinates and see what the changed expressions for the elastic moduli turn out to be.
-- Biswajit
The Piola Kirchhoff Stresses
I was taught that the PK1 stress is the current force relative to the reference area and the PK2 stress is the reference force relative to the reference area. I've also seen a different equation for the PK2 stress.
sigPK1=det F sigC F-T
sigPK2=det F F-1 sigC F-T
Perhaps the PK2 stress is also reachable by applying Hooke's Law to the Green-Lagrange strain.
http://imechanica.org/node/6857
The deformation gradient can be easily derived from the diagram.
It finally makes sense
I'm a little further down the road with tensors and can now understand. You were right, Biswajit. The quantities inside the brackets of Novozhilov's equilibrium equations equal the product of the deformation gradient and the PK2 stress. For small strains and large rotations, the components of the PK2 stress tensor coincide with the components of the Cauchy tensor expressed in the local set of orthogonal coordinates (approximately orthogonal because of the small strain) that results from the rotation (not small) caused by the deformation. Multiplying this stress by the deformation gradient pulls the stress area back to the inititial coordinates so the problem can be solved in terms of the initial coordinates.
Geez, this stuff is hard to get your head around.
Thanks, Jerry Brown