I feel obliged to response to this because I think there are something conceptually important. If you find this annoying, hit the delete key now! My response is in [  ]

Dazhi

From: Tectonics & structural geology discussion list [mailto:[log in to unmask]] On Behalf Of Robert J. Twiss
Sent: Wednesday, March 30, 2011 4:22 PM
To: [log in to unmask]
Subject: Re: a plea and a new question?

A clarification of the diagram in my last post.  I neglected to show the deformed shape of the outer box in the right diagram.  In this revision, the black shows the undeformed shapes; red shows the deformed shapes.  Here's the description from the last post:

I have revised my diagram slightly to show the shear stresses on the squares.  And I have drawn a material square on the "Pure shearing" diagram with sides parallel and perpendicular to the "boundary" so that it encloses the square I had previously drawn.  The stress state is exactly the same as before (see the principal stresses on the boundary), but I have shown the shear stress on the boundary and represented the state of stress on the squares by the shear stresses on the sides of each square.  As before, the normal stresses on horizontal and vertical planes are zero.  This diagram emphasizes how the state of stress in the two situations for simple shearing and for pure shearing are the same.  

If you just look at the shear stresses on the squares, and forget for a minute the boundary conditions on the whole body, can you say whether the deformation will be represented by the square on the left or the one on the right?  

[Yes, the whole set of equations will select the left not the right one. What you show here is a graphic representation of the instantaneous stress and strain rate relation. This is only the constitutive equation alone. There are compatibility etc. for the deformation to obey – my main point to John Waldron’s question.]

According to the mechanical equations, both represent possible solutions, and it is only when you specify boundary conditions can you determine which of the two will result.  

[No, the right one is not a possible solution. See more of my comments below the diagram]

I suggest that with stress boundary conditions, the deformation would be represented by the progressive pure shear in the diagram on the right.  It is only when you impose the displacement boundary condition for progressive simple shearing (boundary conditions that I have not represented in this diagram, since I only show stress boundary conditions) that the deformation would be represented by the square on the left.

The symmetry principle simply states that there must be a symmetry relationship between causes and effects in the mechanical process, and it specifies how causes and effects must be related symmetrically.  For the inverse problem of inferring the mechanical situation from the deformed material, the symmetry principle tells you that the pure shearing must have been produced by the orthorhombic stress boundary conditions illustrated, and that the symmetry of the simple shearing is too low to be consistent with these orthorhombic boundary conditions.  The simple shearing deformation is only consistent with the monoclinic velocity boundary conditions, such as would be imposed by a rigid plate on one boundary moving parallel to itself relative to a rigid plate on the opposite boundary. 

[I cannot see how the pure shearing can continue while maintaining compatibility with the neighbors. Is this still the thought experiment case but with different BC? If so, the bottom boundary is kept fixed and traction (shear and normal stress is applied to the top boundary) to achieve the same thing as the velocity boundary condition. In this case the pure shearing is not a possible choice because it cannot maintain compatibility.]