1. Continuum Mechanics (MTH487)
Lecture 3
Instructor
Dr. Junaid Anjum

2. Recap Addition of vectors Multiplication of a vector by a scalar
Dot (scalar) product of two vectors
Cross product of two vectors
Triple scalar product (box product)
Triple cross product

3. Quiz…
Show that
(i) (ii)

4. Aims and Objectives Tensor Algebra Tensor product (dyad)
Notation for dyad
Dyadic Algebra
Vector-Dyad product
Dyad-Dyad product
Vector-Tensor product
Tensor-Tensor product
Tensor Rank (Order)
Contraction
Symmetric and Antisymmetric Tensor

5. Tensor Algebra ….
Tensor product

6. Tensor Algebra …. Tensor product Notation for dyad product
which is called a Tensor product. The tensor product of vectors a & b is defined by how maps all vectors u

7. Tensor Algebra ….
Notation for dyad product

8. Dyadic Algebra ….
A dyad can be multiplied by a vector giving a vector-dyad product

9. Dyadic Algebra ….
A dyad can be multiplied by a vector giving a vector-dyad product

10. Tensor Algebra ….
A dyad can be multiplied by a vector giving a vector-dyad product

11. Tensor Algebra ….
A dyad can be multiplied by a vector giving a vector-dyad product

12. Tensor Algebra ….
Dyads can be multiplied with each other to yield another dyad

13. Tensor Algebra ….
Vectors can be multiplied by a Tensor to give a vector
Two tensors can be multiplied resulting in a tensor

14. Rank (Order) of a Tensor
Tensor rank (order) is the number of free indices.
scalar : no free index
vector : free index
2nd order tensor : 2 free indices
Scalar (0th order tensor) 3
Vector (1st order tensor)
3 components
Dyad (2nd order tensor)
9 components
Dyadic (2nd order tensor)
Triadic (3rd order tensor)
27 components
Tetradic (4th order tensor)
81 components

15. Contraction
The process of identifying (that is setting equal to one another) any two indices of a tensor term.
An Inner tensor product is formed from an outer tensor product by one or more contractions.
Outer products
Contraction(s)
Inner products
Note that the rank of a given tensor is reduced by 2 for each contraction

16. Symmetric and Antisymmetric tensors
Symmetric tensors in i & j
Antisymmetric in i & j
Antisymmetric in all indices

17. Aims and Objectives Tensor Algebra Tensor product (dyad)
Notation for dyad
Dyadic Algebra
Vector-Dyad product
Dyad-Dyad product
Vector-Tensor product
Tensor-Tensor product
Tensor Rank (Order)
Contraction
Symmetric and Antisymmetric Tensor

18. Quiz… Write expressions for: Vector-Dyad products Dyad-Dyad products
Vector-Tensor product
Tensor-Tensor product

19. Solution… Write expressions for: Vector-Dyad products
Dyad-Dyad products
Vector-Tensor product
Tensor-Tensor product

20. Quiz… Double dot products of dyads are defined by
Expand these products and compare the component form

21. Quiz…
Show that the inner product of of a symmetric tensor and an antisymmetric tensor is zero.

22. Quiz…
By direct expansion of the expression determine the components of vector in terms of the components of the tensor
What if the tensor is symmetric?