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Dr. Wang Xingbo Fall ， 2005PowerPoint Presentation

Dr. Wang Xingbo Fall ， 2005

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Method in Mechanical Engineering

Introduction to Tensors

- Concept of Tensors
- Tensor Algebra
- Tensor Calculus
- Application of Tensors

Method in Mechanical Engineering

Concept of Tensors

A coordinate transformation in an n-dimensional space

Method in Mechanical Engineering

Concept of Tensors

T is a quantity with ns components represented by one of the following three forms

Method in Mechanical Engineering

Concept of Tensors

Components of T are represented by the first one and transformed by

T is called a contravariant tensor of order s.

Method in Mechanical Engineering

Concept of Tensors

Components of T are represented by the second one and transformed by

T is called a covariant tensor of order s.

Method in Mechanical Engineering

Concept of Tensors

Components of T are represented by the third one and transformed by

T is called a mix-variant tensor of order s.

Method in Mechanical Engineering

Concept of Tensors

Tensor product

Let U, V be two vector spaces of dimension m, n,

Tensor product of U and V is an mn–dimensional vector space W denoted by W=UV.

Symbol is used to denote a tensor product

Method in Mechanical Engineering

Concept of Tensors

Symbol is used to denote a tensor product

Method in Mechanical Engineering

Concept of Tensors

Since UV is an mn-dimensional space, it has mn basis vectors.

All pairs (i,j) produce exactly mn pairs of (ui,vj)

It often uses symbol to denote the basis of W=UV.

The elements of W=UV are

Method in Mechanical Engineering

Concept of Tensors

Tensor basis

Covariant tensor basis

are defined by tensor product of covariant vector basis

Method in Mechanical Engineering

Concept of Tensors

Tensor basis

Contravariant tensor basis

are defined by tensor product of contravariant vector basis

Method in Mechanical Engineering

Concept of Tensors

Tensor basis

Mix-variant tensor basis

are defined by tensor product of covariant and contravariant vector basis

Method in Mechanical Engineering

Concept of Tensors

A vector is a first-order tensor

Take s =1, the two forms of components

The transformations

This is what a contravariant vector or a covariant vector is!

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

contravariant tensor T can be represented by

The transformations

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

covariant vector T can be represented by

The transformations

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

Mix covariant vector T can be represented by

The transformations

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

Matrix Form

Method in Mechanical Engineering

Second order tensor

Sample

is a covariant tensor of order 2

Method in Mechanical Engineering

Second order tensor

Quantity to illustrate strain in an elastic material is a covariant tensor of order 2

Let A, B be two points in an elastic body and let .

Method in Mechanical Engineering

Second order tensor

Let us change the Cartesian coordinate transformation O toAssume

Method in Mechanical Engineering

Tensor algebra

Addition and Subtract of Two Tensors

Contraction of Tensors :

Forcing one upper index equal to a lower index

and invoking the summation convention

A special operation on mix-variant tensors

Method in Mechanical Engineering

Geometric Meanings of Cross Product

Contraction of Tensors

Method in Mechanical Engineering

Out Product of Two Tensors

The product of two tensors is a tensor whose order is the sum of the orders of the two tensors, and whose components are products of a component of one tensor with any component of the other tensor.

Method in Mechanical Engineering

Out Product of Two Tensors

- A=AikEik , B=BlmElm,
- Ciklm= AikBlm

Method in Mechanical Engineering

Inner product of Two Tensors

Multiplying two tensors and then contracting the product with respect to indices belonging to different factors

Method in Mechanical Engineering

Quotient Law

Assume are two arbitrary tensors.

IF

Then A is a tensor

Method in Mechanical Engineering

Some Useful and Important Tensors

- Metric tensors

Method in Mechanical Engineering

The Alternating Tensor of Third Order

εjkl= 1, if j, k, l cyclic permutation of 1, 2, 3

εjkl= -1, if j, k, l cyclic permutation of 2, 1, 3

εjkl= 0, otherwise.

Method in Mechanical Engineering

Absolute Derivatives & Differential

be a vector field

in covariant frame-vectors

dA is called absolute differential or

covariant differential of vector field A

Method in Mechanical Engineering

Absolute Derivatives & Differential

Absolute differential of A is composed of two parts

reflects the relationship of the contravariant components changing with spatial position

Method in Mechanical Engineering

Absolute Derivatives & Differential

reflects that of frame-vectors components changing with spatial position

Method in Mechanical Engineering

Absolute Derivatives & Differential

Let

then

absolute derivative represented by contravariant components

Method in Mechanical Engineering

Absolute Derivatives & Differential

We also can derive absolute derivative represented by covariant components

Method in Mechanical Engineering

Absolute Derivatives & Differential

the absolute derivative of a vector field can be represented by either contravariant components or covariant components.

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

are vectors , be linear combination of frame-vectors

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

the first kind of Christoffel symbols

the second kind of Christoffel symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Partial Derivative Operator

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Relationships between and the metric tensor

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

The Christoffel symbols are only symbols but not components of any tensor though they look like the form of tensor-components

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

The derivatives of the contravariant frame-vectors

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Recalling the gradient of a scalar field

where

denote

Let the symbol

absolute differential

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Since it yields

Let ,one can see

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Let us take to be a tensor of order 2 as an example . Suppose

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Let then

We callcovariant derivative of

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Similarly, we have

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Therefore,let then

Method in Mechanical Engineering

Tensors of Order 2

Transpose, Symmetry, Skew-symmetry and Unit

Transpose of a Tensor:DT

Symmetric Tensors

skew-symmetric tensor

Method in Mechanical Engineering

Tensors of Order 2

Inner Product of Two Tensors with Lower Order

A linear transformation

Let D, E be two tensor of order two

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants

then

Let

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants

shows that, the principal value and direction of tensor T can be found by the eigenvalue and vector of the matrix [Tij]

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants

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