mechanical behavior of materials 01 pdf

1Mechanical Behavior of Materials

Introduction and Course Syllabus

Ittipon Diewwanit, Sc.D.Department of Metallurgical Engineering,

Chulalongkorn University

Copyright 2013 by Ittipon

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2Textbooks and References

Textbooks

Mechanical Metallurgy, George E. Dieter, SI Metric edition, McGraw-Hill, 1988.

References

Mechanical Metallurgy: Principle and Applications, Marc A. Meyers and Krishan K. Kumar, Prentice Hall, 1984

Metal Forming, W.F. Hosford and R.M. Caddell, 4th ed., Cambridge University Press 2011

Introduction to Dislocations, Derek Hull and David Bacon, Pergamon Press, 1984.

The Mechanics of Crystals and Textured Polycrystals, William F. Hosford, Oxford University Press, 1993.

Fracture and Fatigue Control in Structures: Applications of Fracture Mechanics, Butterworth-Heinemann, 1999


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3Grading Policy

Midsemester Examination 50%

Final Examination 50%

Assignments: weekly handouts of problem sets with solution following.

There will be no grading for assignments.


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4Course Contents

Introduction

Theory of Elasticity

Introduction to Plastic Deformation and Elementary Theory of Plasticity

Theory of Dislocation

Plastic Deformation of Crystalline Materials

Mid-semester Examination

Characterization of Mechanical Behavior

Deformation of Polymeric Materials

Introduction to Fracture Mechanics

Fatigue of Metals

Deformation at Elevated Temperature

Applications to Material Processing and Material Selection


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5Mechanical Behavior

The response of materials to mechanical loads.

Half of the subject deals with the relationship between force and

deformation (displacement) of materials.

The other half deals with internal structure and their influence on material

properties especially mechanical ones.


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6Conceptual Mathematical Space

Force Displacement

Stress Strain

Mechanical Properties

of Materials

measurable

conceptual

1st Rank Tensor: Vector

2nd Rank Tensor

3rd Rank Tensor


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7

Simple Cantilever Beam

Force in y direction = -9,800 N

No displacement at this end

2 by 2 cm beam made of steel with Youngs modulus of 200 GPa


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Normal Stress in the z Direction


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Linear Strain in the x Direction


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Displacement in the y Direction

11

Definition of Stress

Stress may be described as a mathematical quantity indicating the severity

of mechanical load at a certain location of material.

As defined in continuum mechanics, stress is considered as a second rank

tensor having 9 components.

It may be loosely defined as force divided by area.

The unit of stress is N m-2 or Pascal (abbreviated Pa) in SI system.


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12

Type of Stresses

Stress may be classified to two types based on geometry of applied force with relative to the surface of interest.

If the force is acting normal to the area (surface) of interest, the stress is said to be normal stress.

If the force is acting parallel to the area (surface) of interest, the stress is said to be shear stress.

normal component of stress

shear component of stress


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13Copyright 2013 by Ittipon

DiewwanitFrom M. F. Ashby, Engineering Materials Vol 1

14

From M. F. Ashby, Engineering Materials Vol 1


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15

Type of Stresses (Cont.)

Normal stress may be classified further to two types. Normal stress

creating tension is termed tensile stress and is assigned an algebraic

positive sign.

Normal stress creating compression is termed compressive stress and is

assigned an algebraic negative sign.


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DiewwanitFrom M. F. Ashby, Engineering Materials Vol 1


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17

Mechanical Behavior of Materials

Definition of Stress

Ittipon Diewwanit

Department of Metallurgical Engineering,


Elongation: Response of material when subjected to axial tension. Tensile

stress along the direction of tension force is defined as F/A.


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Shearing: Response of material when subjected to shear force

Shear stress on the surface is defined as F/A


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20

21

Definition of State of Stress

Stress is a second rank tensor, reference axes must be well defined.

State of stress is a mathematical function dependant on one variable:

position vector.

x, x1, 1

y, x2, 2

z, x3, 3

Static force F

Cantilever beam

with a fixed end


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22

State of Stress in 3-Dimension

x, x1, 1

y, x2, 2

z, x3, 3

zzxz

yz

xxzy

zx

xy

yy

yx


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Matrix Representation of Stress

With relative to the reference axes, stress may be written using matrix symbol:

At static equilibrium condition, there are only six independent components according to the relationship:

zzzyzx

yzyyyx

xzxyxx

ij

zyyzzxxzyxxy and ; ;


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24

Static Equilibrium for Rotational

y

x

xy xy

yx

yx

This results in the symmetry of stress tensor matrix: jiij


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25

Components of Stress Tensor

Diagonal components in the matrix represent the three normal stress

according to the orthogonal reference coordinate system.

The rest are shear components.

Due to the symmetry of the matrix under static equilibrium, there are only

six independent components of stress tensor.


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26

Axes Transformation for 2-D Stress

In 2-D we deal with only 4 components of stress tensor ( )

This condition occurs in many real engineering applications such as thin

wall vessels and other sheet metal components.

Transformation of orthogonal reference coordinate results in the change of

stress components.

Analytically, we can do this by using eq. 2-5 to 2-7 (in Dieters) but we can

also use graphical method called Mohrs circle of stress.

yxxyyyxx ,,,


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2cos2sin2

2sin2cos22

2sin2cos22

xy

xxyy

xyyx

xy

yyxxyyxx

yy

xy

yyxxyyxx

xx

xx

y y


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Principal Stresses occur at a special rotation angle. At this angle of rotation,

the shear components vanishes (eq. 2-8 in Dieters).

yyxx

xy

22tan

2/1

2

2

1max22

xy

yyxxyyxx

2/1

2

2

2min22

xy

yyxxyyxx


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A

B

A

B

2 C

C

D

D

note the direction of rotation

normal

stress

shear

stress

30

Principal Stresses and Maximum Shear Stress

Principal Stresses

Maximum Shear Stress

2/1

2

2

1max22

xy

yyxxyyxx

2/1

2

2

2min22

xy

yyxxyyxx

2/1

2

2

max2

xy

yyxx


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A

B

A

B

2 C

C

D

D

max

12

F

E

E

FCopyright 2013 by Ittipon

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32


The root of cubic equation [eq.2-14] yields the three values of principal

stresses in 3-D.

The first invariant of stress tensor, I1

0)2(

)()(

222

22223

xyzzxzyyyzxxxzyzxyzzyyxx

xzyzxyzzxxzzyyyyxxzzyyxx

ii

i

iizzyyxx I

3

1

1)(


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Transformation (Summary)

The maximum and minimum values of normal stress on three principal

orthogonal planes occur when shear stress on the three planes are zero.

Shear stresses alone occur at angles which are halfway between the three

principal planes.

The value of the maximum shear stress is

2

31max


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Principal Stresses and Maximum Shear Stresses in 3D

From Mechanics of Sheet Metal Forming, Z. Marciniak, J.L. Duncan, S.J. Hu, 2nd ed., Butterworth-Heinemann 2002


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The second invariant of stress tensor, I2

The third invariant of stress tensor, I3

The value of I3 is equal to the determinant of the stress tensor matrix.

2

222 )( Ixzyzxyzzxxzzyyyyxx

3

222 )2( Ixyzzxzyyyzxxxzyzxyzzyyxx


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Mechanical Behavior of Materials

Definition of Strain

Ittipon Diewwanit, Sc.D.Department of Metallurgical Engineering,



Diewwanit


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Deformation of Materials

Deformation behavior of a material may be loosely defined as the response

of the material under applied stresses.

Applied stresses may be external or internal.

Deformation is represented by a measurable vector quantity defined as

displacement.

Strain is a higher rank quantity defined in a differential form based on the

displacement.


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Deformation of Materials

Deformation behavior of materials may be divided into two types: elastic and plastic.

Elastic deformation is temporary. Material will resume its original shape and dimensions after removing the applied stresses. The deformation when the material is under applied stresses is termed recoverable elastic deformation.

Plastic deformation is permanent. If the amount of applied stresses exceeds a certain limit (known as elastic limit), material cannot resume its original shape and dimensions after removing the applied stresses. The remaining, permanent deformation is termed plastic deformation.


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39

Engineering Linear Strain

Engineering linear strain (e):

00

0

l

l

l

lle

0l

l

ncontractio 0

extension 0

l

l

y

x


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Engineering Shear Strain

Engineering shear strain ():

a

h

tanh

a

y

x


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Definition of Strain at a Point

A

BA

B

y

x

x dx

Au

x

u

AB

ABBAe

dxx

udxudxuBA

xx

AB

dxx

uuu AB

Au is the displacement vector of point A


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Position and Displacement Vector

A

A

x

z

y

),,(

ofnt vector displaceme

wvu

A

),,(

ofctor positon ve

zyx

A

),,(

ofctor positon ve

zyx

A


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Translational Motion

A

A

x

z

y

B

B

Displacement vectors at any point within the body are equal.


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44

Pure Rotational Motion

A

A

x

z

y

B

B

No strain but displacement vector varies as a function of position.


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Displacement Vector and Displacement Tensor

Displacement vector is a function of position,

In vector format where

In matrix format

),,(

),,(

),,(

zyxww

zyxvv

zyxuu

jiji

zzzyzx

yzyyyx

xzxyxx

xeu

zeyexew

zeyexev

zeyexeu

or

x

we

x

ve

x

ue zzyyxx

, ,

z

ue

x

ve

y

ue xzyzxy

, ,

zzzyzx

yzyyyx

xzxyxx

ij

eee

eee

eee

e


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Displacement, Strain, and Rotation Tensors

Displacement tensor, , can be decomposed into two parts. One is a

symmetric tensor and the other is skew-symmetric tensor.

We define the symmetric tensor as strain tensor

ije

)(2

1)(

2

1jiijjiijij eeeee

)(2

1jiijij ee


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Strain Tensor

z

w

y

w

z

v

x

w

z

u

y

w

z

v

y

v

x

v

y

u

x

w

z

u

x

v

y

u

x

u

ij

2

1

2

1

2

1

2

1

2

1

2

1

jiij


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Rotation Tensor

02

1

2

1

2

10

2

1

2

1

2

10

y

w

z

v

x

w

z

u

y

w

z

v

x

v

y

u

x

w

z

u

x

v

y

u

ij

jiij


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Definition of Strain Tensor

Strain tensor is defined as,

In full matrix form,

ijijij e

z

w

y

w

z

v

x

w

z

u

y

w

z

v

y

v

x

v

y

u

x

w

z

u

x

v

y

u

x

u

ij

2

1

2

1

2

1

2

1

2

1

2

1

total displacement

rigid-body rotation

jiij


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Displacement, Strain, and Rotation Tensors

We define the skew-symmetric tensor as rotation tensor

The two tensors play important roles in the analysis of deformation and

motion of bodies.

)(2

1jiijij ee

ijijije

Copyright 2013 by Ittipon Diewwanit 51

General Equation of Motion and Deformation

Combining the decomposition of displacement tensor

with the vector equation for displacement vector

We have a general vector equation describing the motion and deformation

of a body as

ijijije

jiji xeu

jijjiji xxu

Copyright 2013 by Ittipon Diewwanit 52

Stress and Strain Relationship in Shear

Pure rotation without shear

yxxy ee -

x

y

z

ue

x

ve

y

ue xzyzxy

, ,


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x

y

z

ue

x

ve

y

ue xzyzxy

, ,

Simple shear with rotationSimple shear

yxxy ee

0

yx

xy

e

e


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Considering the definition of engineering shear strain which is based on

simple shear, it is obvious that our definition of shear component in the

strain tensor is related to the engineering shear strain by

whereas

ijij 2

z

u

x

w

z

v

y

w

x

v

y

uzxyzxy

and , ,

mechanical behavior of materials 01 pdf

Documents

ittipon diewwanitcopyright

ittipon diewwanit2textbooks

unit of stress

mechanical ones

mechanical loads

severity of mechanical

area surface

material processing