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Mathematical & Mechanical Method in Mechanical Engineering. Dr. Wang Xingbo Fall , 2005. Mathematical & Mechanical Method in Mechanical Engineering. Introduction to Tensors . Concept of Tensors Tensor Algebra Tensor Calculus Application of Tensors. Mathematical & Mechanical

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Mathematical & Mechanical

Method in Mechanical Engineering

Dr. Wang Xingbo

Fall,2005


Mathematical & Mechanical

Method in Mechanical Engineering

Introduction to Tensors

  • Concept of Tensors

  • Tensor Algebra

  • Tensor Calculus

  • Application of Tensors


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

A coordinate transformation in an n-dimensional space


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

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


Mathematical & Mechanical

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.


Mathematical & Mechanical

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.


Mathematical & Mechanical

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.


Mathematical & Mechanical

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=UV.

Symbol  is used to denote a tensor product


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Symbol  is used to denote a tensor product


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Since UV 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=UV.

The elements of W=UV are


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Tensor basis

Covariant tensor basis

are defined by tensor product of covariant vector basis


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Tensor basis

Contravariant tensor basis

are defined by tensor product of contravariant vector basis


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Tensor basis

Mix-variant tensor basis

are defined by tensor product of covariant and contravariant vector basis


Mathematical & Mechanical

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!


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

contravariant tensor T can be represented by

The transformations


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

covariant vector T can be represented by

The transformations


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

Mix covariant vector T can be represented by

The transformations


Mathematical & Mechanical

Method in Mechanical Engineering

Concept of Tensors

Second-order tensor

Matrix Form


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor

Sample

is a covariant tensor of order 2


Mathematical & Mechanical

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 .


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor

After deformation


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor

Let us change the Cartesian coordinate transformation O toAssume


Mathematical & Mechanical

Method in Mechanical Engineering

Second order tensor


Mathematical & Mechanical

Method in Mechanical Engineering

Strain tensor


Mathematical & Mechanical

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


Mathematical & mechanical

Method in Mechanical Engineering

Geometric Meanings of Cross Product

Contraction of Tensors


Mathematical & mechanical

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.


Mathematical & mechanical

Method in Mechanical Engineering

Out Product of Two Tensors

  • A=AikEik , B=BlmElm,

  • Ciklm= AikBlm


Mathematical & mechanical

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


Mathematical & mechanical

Method in Mechanical Engineering

Quotient Law

Assume are two arbitrary tensors.

IF

Then A is a tensor


Mathematical & mechanical

Method in Mechanical Engineering

Some Useful and Important Tensors

  • Metric tensors


Mathematical & mechanical

Method in Mechanical Engineering

Metric tensors

In 3-dimensional space


Mathematical & mechanical

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.


Mathematical & mechanical

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


Mathematical & mechanical

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


Mathematical & mechanical

Method in Mechanical Engineering

Absolute Derivatives & Differential

reflects that of frame-vectors components changing with spatial position


Mathematical & mechanical

Method in Mechanical Engineering

Absolute Derivatives & Differential

Let

then

absolute derivative represented by contravariant components


Mathematical & mechanical

Method in Mechanical Engineering

Absolute Derivatives & Differential

We also can derive absolute derivative represented by covariant components


Mathematical & Mechanical

Method in Mechanical Engineering

Absolute Derivatives & Differential

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


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

are vectors , be linear combination of frame-vectors


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

the first kind of Christoffel symbols

the second kind of Christoffel symbols


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Partial Derivative Operator


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Relationships between and the metric tensor


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

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


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

The derivatives of the contravariant frame-vectors


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

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


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Since it yields

Let ,one can see


Mathematical & Mechanical

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


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Let then

We callcovariant derivative of 


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Similarly, we have


Mathematical & Mechanical

Method in Mechanical Engineering

Derivatives of Frame-vectors and Christoffel Symbols

Therefore,let then


Mathematical & Mechanical

Method in Mechanical Engineering

Tensors of Order 2

Three Representations


Mathematical & Mechanical

Method in Mechanical Engineering

Tensors of Order 2

Transpose, Symmetry, Skew-symmetry and Unit

Transpose of a Tensor:DT

Symmetric Tensors

skew-symmetric tensor


Mathematical & Mechanical

Method in Mechanical Engineering

Circulation

Unit


Mathematical & Mechanical

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


Mathematical & Mechanical

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants

then

Let


Mathematical & Mechanical

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants


Mathematical & Mechanical

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]


Mathematical & Mechanical

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants


Mathematical & Mechanical

Method in Mechanical Engineering

Tensors of Order 2

Principal Value, Direction and Invariants


Mathematical & Mechanical

Method in Mechanical Engineering

Class is Over!

See you next time!


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