R. Field 10/31/2013 University of Florida

PHY 2053 Page 1

Definition of Strain

V

V

• System Response – Linear Deformation:

• System Response – Volume Deformation:

L

L

Suppose that a wire of length L is stretched by a force F then:

Linear Strain =

Consider an object immersed in a fluid. The fluid will exert a pressure P from all sides causing a volume deformation.

Define variables the cause deformation of a material and the response of the material.

F L L

Volume Strain =

R. Field 10/31/2013 University of Florida

PHY 2053 Page 2

Definition of Stress

PA

F

• Linear Deformation:

• Volume Deformation:

A

F

Suppose that a wire of length L is stretched by a force F then:

Linear Stress =

Consider an object immersed in a fluid. The fluid will exert a pressure P from all sides causing a volume deformation.

Define variables the cause deformation of a material and the response of the material.

F L L

Volume Stress =

Area A

R. Field 10/31/2013 University of Florida

PHY 2053 Page 3

Hooke’s Law Spring• Linear Restoring Force:

xkFspring

Spring Constant kSpring Force

Stress = C × Strain

Generalized Hooke’s Law!

Ideal Spring

LkFext

extF

L

LC

L

L

A

kL

A

Fext

Constant

R. Field 10/31/2013 University of Florida

PHY 2053 Page 4

Stress Proportional to Strain

V

VBP

A

F

• Linear Deformation:

• Volume Deformation:

L

LY

A

F

Suppose that a wire of length L is stretched by a force F then:

Stress = Y × Strain

Consider an object immersed in a fluid. The fluid will exert a pressure P from all sides causing a volume deformation.

F L L

Area A

Stress = -B × Strain

Bulk Modulus Units = Pa

Young’s Modulus Units = Pa

R. Field 10/31/2013 University of Florida

PHY 2053 Page 5

Example Problem: Young’s Modulus

• Linear Deformation:

L

LY

A

F

A wrecking ball with mass M is to be lifted by a crane with a steel cable that has a diameter of 1.5 cm and an unstretched length of 30 m. The Young’s modulus of steel is 2.0 × 1011 Pa. Ignoring the weight of the cable itself, when the ball is lifted and held at rest the cable stretches by 1.66 cm, what is the mass M of the wrecking ball?

Mg

L

L

22 )2/(DrA MgF

L

LAYMgF

kgmsm

mmNm

gL

LYD

gL

LAYM

000,2)30)(/8.9(

0166.0)/102()0075.0(

)2/(

22112

2

R. Field 10/31/2013 University of Florida

PHY 2053 Page 6

Example Problem: Bulk Modulus

V

VBP

A

F

• Volume Deformation:The atmospheric pressure at the surface of a clear lake is 101 kPA. By what percentage does the density of lake water increase at a depth of 1.0 km below the surface of the lake? The bulk modulus for water is 2.2×109 Pa?

B

VPV dd

gdPPd 0

BP

VM

BVPV

M

VV

M

V

M

/1

/

)/( 00000

BP

VM

BVPV

M

VV

M

V

M

ddddd /1

/

)/(

Pamsmmkggd 623 108.9)000,1)(/8.9)(/000,1(

00445.01091.9102.2

108.9

)(1

/1

/169

6

0

00

0

0

gdPB

gd

PB

PP

BP

BP

d

d

d

d

0.445%