# Pure shear decomposition of the deformation gradient for finite strain measures

Hai,

I am wondering about deformation gradient for pure shear decompositions. As i saw much literature on simple shear, I couldn't able to track one on pure shear.

Please some one in this forum provide me with literaure and fine details.

Regards

Venkat

### Several recent papers in J. Elasticity

I am not an expert in this area, but there are a couple of papers discussing pure shear in J. Elasticity:

Belik, P., Fosdick, R.: The state of pure shear. J. Elast. 52, 91–98 (1998)
Boulanger, P., Hayes, M.: On pure shear. J. Elast. 77, 83–89 (2004)
Ting, T.C.T.: Further study on pure shear. J. Elast. 83, 95–104 (2006)

### Pro. Zheng has done some works in this field.

You can see some papers from him as follow. I also have seen these papers and they give me a new viewpoint. Maybe they will help you.

Decomposition of large incompressible deformations
He, Q.-C. (Laboratoire de Mecanique, Universite de Marne-la-Vallee); Zheng, Q.-S. Source: Journal of Elasticity, v 85, n 3, December, 2006, p 175-187

### Pure shear is not "pure" in finite deformation

In finite deformation, people only discuss simple shear because pure shear is not "pure" anymore. Consider a square under "pure" shear, it will change its shape to a diamond. It is considered as "pure shear" in infinitesimal theory because the length of the four edges of the square do not change. But in finite deformation, it is not "pure" shear anymore. Although we can keep the length of the four edges of the square unchanged, the finite shear angle will change the area of the square. If the material is incompressible, the thickness of the square will change. For compressible material, the thickness will also change unless you put extra constraint to prevent it.

Similar to Prof. Zheng's paper in 2006, a more general decomposition scheme is developed simultaneously for incompressible transversely isotropic material:

Z.Y. Guo, X.Q. Peng and B. Moran (2006). A composites-based hyperelastic constitutive model for soft tissue with application to the human annulus fibrosus, Journal of the Mechanics and Physics of Solids, 54(9), 1952-1971.

It can be used to model soft tissue with a proper fiber-matrix shear interaction. The theoretical advantage of this decomposition is also discussed:

Z.Y. Guo, X.Q. Peng and B. Moran (2007). Mechanical response of neo-Hookean fiber reinforced incompressible nonlinearly elastic solids, International Journal of Solids and Structures, 44(6), 1949-1969.

We are now working on the compressible material and a similar decomposition is also possible.

Similar models and/or frameworks can be found in papers by Criscione, deBottun (see references in the papers above).

Zaoyang

### Very Helpful - Thanks !

I found this discussion very helpful, especially the simple shear decomp. paper. I have been trying to find a more basic decomposition (than SVD) of U in F=RU. Thanks !  John.