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Topic-Elasticity

Topic-Elasticity. Tensors. Cartesian tensor of order 1. A cartesian tensor of order 1 is an entity that may be represented by components a i

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Topic-Elasticity

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  1. Topic-Elasticity

  2. Tensors

  3. Cartesian tensor of order 1 A cartesian tensor of order 1 is an entity that may be represented by components ai (i=1,2,3) w.r.t a R.C.C.S oxi and the same entity be represented by ai’(i=1,2,3) w.r.t R.C.C.S oxi’ which is obtained by rotating oxi about an angle. Then ai and ai’ obey the law

  4. ai’=lijaj This law is called tensorial law of order 1.or it can be written as ai=lij aj’

  5. Examples of 1st order tensors

  6. Every vector is a tensor of order 1. Pf:-As a=a1e 1+a2e 2+a3 e3 Also a=a1’e1’+a2’e2’+a3’e3’ Therefore aei’=a1e1ei’+a2e2ei’+a3e3ei’ =li1a1+li2a2+l13a3 =lijaj (j=1,2,3)

  7. But aei’=ai’ aei’=a1e1’ei’ +a2e2’ei’+a3e3’ei’ =ai’ So we have aei’=ai’=lijaj This shows that vector is a tensor of order 1.

  8. Velocity is a tensor of order 1. • Gradient of a scalar pt. function is a tensor of order 1.

  9. Tensor of order 2

  10. Let a physical entity be represented by aij w.r.t R.C.C.S oxi and same entity be represented by apq’ w.r.t R.C.C.S oxi’ and if they are related by the law aij’=lipljqapq aij=lpilqjapq’ (I,j,p,q=1,2,3)

  11. Sum of Two Tensors

  12. If aij and bij are two tensors of order two.then their sum is again a tensor of order two,and it is defined as cij=aij+bij (I,j=1,2,3)

  13. Difference of two Tensors

  14. If aij and bij are two tensors of order two then their difference is again atensor of order two.It is defined as: cij=aij-bij (i,j=1,2,3)

  15. Scalar Invariant If an entity be represented by a w.r.t R.C.C.S oxi and by a’ w.r.t another R.C.C.S oxi’ and a’=a then that entity is called scalar invariant and simply scalar.

  16. Zero Tensor • Tensor of order n is called zero tensor if all the components of the tensor w.r.t co-ordinate system are zero. • If all the components of a tensor are zero w.r.t some co-ordinate system oxi then it has zero components w.r.t every co-ordinate system oxi’.

  17. Equality of Tensors Two Tensors A and B are equal if the corresponding components of A and B are equal. Let A=[aij] B=[bij] then A=B if aij=bij (i,j=1,2,3)

  18. Product of Two Tensors

  19. Let ai and bj & ap’and bq’are the components of first order tensors relative to two co-ordinate system oxi and oxi’ Let cij=aibj cpq’=ap’bq’ As ai and bj are tensors of order 1

  20. So ap’=lpiai bq’=lqjbj Multiply these equations: ap’bq’=lpilqjaibj cpq’=lpilqjcij This is tensorial law of order two.

  21. Conclusions • If A is a tensor of order α and B is a tensor of order α,then their sum and difference is again a tensor of order α. • If A is a tensor of order α and B is a tensor of order β then their product is a tensor of order α+β.

  22. Contraction

  23. When we identify two suffixes in a Tensor,then this operetion is called Contraction. We know that if a and b are two tensors of order 1 then their product is a tensor of order 2.and it can be written as (ab)ij=aibj (i,j=1,2,3)

  24. Then we put j=i then this process is called contraction and expressed as (ab)ii=aibi (i=1,2,3) =a1b1+a2b2+a3b3

  25. Definition • Tensor Product:If A is a tensor of order 2 and C is a tensor of order 1,then AC=aijck This is a tensor of order 3 and it is called outer product or tensor product.

  26. Inner Product:If A is a tensor of order 3 and C is a tensor of order 1. then AC=aijcj is a tensor of order 1.and It is called Inner Product i.e I.P

  27. Scalar Product:If A and B are two tensors of order 2.then A.B=aijbij is a tensor of order 0. It is called Scalar Product.

  28. Isotropic Tensor • If we change the frame of refrence or transformation and the values of components of tensor do not change. Then this type of tensor is called Isotropic Tensor. It is denoted by δij.It is also called Scalarinvariant or isotropic tensor.

  29. Alternate Tensor • Alternate Tensor Is defined as 1 if i,j,kare in cyclic order ijk = -1 if i,j,k are in anticyclic order. 0 if any two suffixes are equal.

  30. ijk is a tensor of order 3. 123= 231= 312=1 213= 321= 132=-1 222=0….. i.e. out of 27,6 values are not zero.

  31. TEST-1

  32. State and Prove Law of Transfomation in three dimension. • Prove that Velocity is a tensor of order 1. • State and Prove Contraction theorem. • Define Isotropic Tensor and Prove that it is Scalar Invariant.

  33. TEST-2

  34. Derive Strain quadric of Cauchy. Explain Cubical dilatation with diagram. Derive Saint-Venant’s Equation of Compatibility.Discuss them in every case. Derive Stress-Strain relation for a Homogeneous Isotropic Elastic Solid Medium. OR Derive Stress Strain Relation for an Orthotropic Medium.

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