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### Transcript of Chapter 2 Basics of Materials and suemasu/files/ of Materials 2 2.1 Stress 2.1.1 Definition of...

• Mechanics of Materials 1

Chapter 2

Basics of Materials

and Mechanics

13:14

• Mechanics of Materials 2

2.1 Stress

2.1.1 Definition of stress

- P

S

SS

P- P

- P

P

P

P2

PP1

V

V1

V2

hypothetical internal

surface S Body is in

equilibrium

Sum of the internal

forces on S

=Applied force P

Stress Vector

Quantity to express the

intensity of internal

force

SS

Pp

0lim (1)

13:14

• Mechanics of Materials 3

Unit of Stress Pa(=N/m2), kgf/m2, psi (=lb/inch2)etc Stress depends not only the internal force (vector) but

also the direction of the surface considered.

dS

dPp

dS

dPp

2

1

sin

cos

Normal componentNormal Stress

Inplane componentShear Stress

13:14

• Mechanics of Materials 4

Positive stresses are defined as shown above. In the other sides

of the cube where the outward normals are opposite to the

coordinate axes, the directions of positive stresses are opposite.

Definition of Stresses in the Cartician

Coordinate System

x

y

z

P

B

A

O

C dx

dy

dz xxy

xz y

yx

yz

z

zx

zy

13:14

• Mechanics of Materials 5

Stress in an arbitrary surface

Equilibrium of the force for infinitesimal tetrahedron

Area of ABCdS Outward normal of ABC : =(l,m,n)

l, m, n are directional cosines

OA=dx, OB=dy, OC=dz

OBC=dydz/2=dSl,

OCA=dzdx/2=dSm,

OAB=dxdy/2=dSn

n

x

y

z

x

y

z

O A

B

C pdS

-pxldS

-pymdS

-pzndS

13:14

• Mechanics of Materials 6

zzyzxz

xzyyxy

xzxyxx

ZYX

,,

,,

,,

,,

p

p

p

p

Stress vectors

Equilibrium of the force when no body force exists.

0 ndSmdSldSdS zyx pppp

ndSmdSldSdS

Z

Y

X

z

zy

zx

yz

y

yx

xz

xy

x

x

y

z

OA

B

C pdS

-pxldS

-pymdS

-pzndS

13:14

• Mechanics of Materials 7

n

m

l

Z

Y

X

zyzxz

zyyxy

zxyxx

Then

Stress vector of an arbitrary surface

Nine components are necessary to express the stress state.

Note) From the equilibrium of an infinitesimal cube in moment,

zy=yz , xz=zx, yx=xy

Six components are practically independent.

yxxyxzzxzyyzzyx ,,,,,,,,pp

13:14

• Mechanics of Materials 8

2.1.2 Transformation of Coordinate system

Let (x, y, z) and (x', y', z') be two Cartesian coordinate systems

with a common origin. The cosines of the angles between the six

coordinate axes are represented in the following tabular form:

x y z

x a11 a12 a13

y a21 a22 a23

z a31 a32 a33

x

y

z

O

p

i

j

i

x y

z k

k

j

Normal to the

x axis

13:14

• Mechanics of Materials 9

333231

232221

131211

332313

322212

312111

'''''

'''''

'''''

aaa

aaa

aaa

aaa

aaa

aaa

zyzxz

zyyxy

zxyxx

zzyzx

yzyyx

xzxyx

The stresses referring to (x', y', z') coordinate

system can be derived from the stresses

referring to (x, y, z) coordinate system .

13:14

• Mechanics of Materials 10

2.1.3 Principal stress

The values of stresses at a point depend on the plane where the

stresses are considered. The direction of the plane where the normal

stress is extremum is called principal axis and the normal stress is

called principal stress. The principal directions and principal stresses

are obtained through the following eigenvalue analysis.

0

zyzxz

zyyxy

zxyxx

222

3

2222

1

xyzzxyyzxzyx

xyzxyzyxxzzy

zyx

J

J

J

The coefficients of the equation are called stress invarient.

13:14

• Mechanics of Materials 11

2.1.4 Maximum Shear Stress

Shear Stress becomes maximum at the surface whose unit

normal is 0,2/1,2/1,, nml

212

1

212

1

The normal stress at the surface of maximum shear stress is

where the coordinate system accords to the principal directions. 1 is

maximum principal stress and 2 is minimum principal stress

13:14

• Mechanics of Materials 12

2.1.5 Equilibrium Equation

0

0

0

Zzyx

Yzyx

Xzyx

zyzxz

zyyxy

zxyxx

x y

z

P

B

A

O

C dx

dy

dz

dxx

xx

dxx

xyxy

dxxxz

xz

dyy

yy

dyy

yxyx

dyy

yzyz

dzz

zz

dz

zzx

zx

dz

z

zyzy

From the equilibrium of the force acting on the

small rectangular parallelepiped

13:14

• Mechanics of Materials 13

A

B C

D

C

A B

dz

D

dy

dx

x

y

z

2.2 Strain 2.2.1 Definition of strain

Intensity of Deformation

Normal strain Shear strain

x

xxx

i

ir

AB

ABBA

''

xy

2

13:14

• Mechanics of Materials 14

Normal strain represents change of length.

When the length is increased, it is called

tensile strain. When the length is decreased,

it is called compressive strain. Normal strain

is usually positive if tensile. Strain being

defined as length/length is non-dimensional

value.

Shear strain represents change of shape. It is

defined as reduction of angle between two

coordinate axes which were normal each

other.

13:14

• Mechanics of Materials 15

Relations between the strain components and

displacements (u,v,w)

x

w

z

w

x

v

z

v

x

u

z

u

x

w

z

u

z

w

y

w

z

v

y

v

z

u

y

u

z

v

y

w

y

w

x

w

y

v

x

v

y

u

x

u

y

u

x

v

z

w

z

v

z

u

z

w

y

w

y

v

y

u

y

v

x

w

x

v

x

u

x

u

zx

yz

xy

z

y

x

222

222

222

2

1

2

1

2

1

13:14

• Mechanics of Materials 16

When u and v is infinitesimal,

x

w

z

u

z

v

y

w

y

u

x

v

z

w

y

v

x

u

zxyzxy

zyx

,,

,,

13:14

• Mechanics of Materials 17

2.2.2 Motion of elastic body

Displacement of P

dz

dy

dx

dz

dy

dx

w

v

u

zzyzzx

yzyxy

zxxyx

zxy

xz

yz

drp

2/2/

2/2/

2/2/

0

0

0

0

0

0

0 uuuu

y

u

x

v

x

w

z

u

z

v

y

wzyx

2

1,

2

1,

2

1

where

, : deformation

: rotation

O

/2

P

O'

P'/2

13:14

• Mechanics of Materials 18

2.2.3 Transformation of strain component under change of coordinate system

Two rectangular Cartesian

coordinate systems x, y, z

and x, y, z

333231

232221

131211

21

21

21

21

21

21

332313

322212

312111

'''21

''21

''21

'''21

''21

''21

'

aaa

aaa

aaa

aaa

aaa

aaa

zyzxz

zyyxy

zxyxx

zzyzx

yzyyx

xzxyx

Table of directional cosines

x y z

x a11 a12 a13

y a21 a22 a23

z a31 a32 a33

13:14

• Mechanics of Materials 19

zyxzyx

zyxyxz

zyxxzy

zxzx

yzyz

xyxy

xyzxyzz

xyzxyzy

xyzxyzx

zxxz

yzzy

xyyx

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2

2.2.4 Compatibility Equation

13:14

• Mechanics of Materials 20

2.3 Generalized Hookes Law and elastic constants

Hookes Law refers that normal strain is directly

proportional to the relating normal stress. However, in

general stress state, the whole strain components are

dependent on whole stress state as

xy

zx

yz

z

y

x

xy

zx

yz

z

y

x

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

13:14

• Mechanics of Materials 21

For three dime