Question

4. Consider a viscoplastic stress versus strain rate relationship of the form ơi,-CijklDk1 where where λ and μ are constants. (a) Express σ as a function of λ and μ. (b) S (3A+ 2a) Dkk and σ;,-2μ0% where σ0 and Dij are the deviatoric how that ơi : parts of the Cauchy stress and rate of deformation tensors, respectively.

Answer 1

• Initially I am listing out some properties of Kronecker delta (δij) and Contraction of Tensors (summation over a common index) .The possible contractions of a 4^th order tensor given by
  AijBkt
  are:

Where the ‘.’ is represents matrix multiplication.

Terms are rearranged until summed index is adjacent, when this happens, we can write it as a product of matrices.

• Definition of Kronecker delta is
  0 リ
  
• Now the Kronecker Delta is a second order tensor which can thought of as equivalent to the identity matrix

Therefore, just as we have

• The property is contraction of Kronecker Delta given by

Solution for part (a)

The work is shown below, some of the properties discussed above are used,

(20 hawr

2 ik Drj we

So we obtain

say this is equation (7)

Answer for part (b)

• Now further contracting the index's in equation (7), that is we set i=j , then we obtain:

  

Note that D_ii=D_kk

• Now  
  Dkk
  is the Volumetric strain when only Deviatoric part is only required the Volumetric part becomes zero therefore setting volumetric strain to zero in equation (7) we obtain :

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