Question

Show that displacement gradient is splitted into strain tensor and rotation tensor?

Answer 1

The rate of rotation is given by : 

b · R · = ⋅ B W b 

now sir , 

Note that coincides with the spin tensor if b is aligned along one of the  

principal directions of the rate of deformation tensor. We now focus on the

corotational frame which transforms with respect to the undeformed

configuration as . A family of the corotational Cauchy stress tensor, denoted

as , can be defined : 

sigma ^r = r^t ×sigma ×r

For hypoelastic materials the constitutive equation is given by 

sigma dash = L / d 

where  L elasticity 4th order tensor and d is the rate of deformation now sir , 

premultiplying and postmultiplying the result with r^t and r yields 

sigma ^r = r^t r ×L/d

and sir we assume that d^r denote the set of corotational rate of deformation tensors defined as 

d^r = r^t ×d ×r 

which means sir , 

It can be seen that the form of the constitutive equations in the corotational frame is identical to that in small deformation theory. Note that for isotropic materials the constitutive,

properties are rotation independent and thus L^r = L 

hence prove that displacement gradient is splitted into strain tensor and rotation tensor

sir if you have any doubt ask your doubts in comment section 

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