3.11 Recitation #11

November 25, 2003 If there’s anything you’d like covered, please let me know.

Also—please let me know good times to hold a review for the final. I was thinking perhaps Thursday or Friday after classes end. Let me know if this would be appropriate. By email is fine.

Today:

Review vocabulary for mechanical properties of Materials.

Go over stress-strain relationships, plasticity

Example Problems, if time.

Stress-strain diagrams The relationship between loads and deflection/stress-strain in a structure of a member can be obtained from experimental load-deflection/stress-strain curves

1

L

P

P

dL/2

dL/2

P/2 P/2

δ P

P

P

P

T

T

Tension test

Bending test

Compression test

Torsion test

Shear Test

The most common tests are tension test for ductile materials (steel) & compression test for brittle materials (concrete)

2

Tension Test

A

B'

C

C'

The upperstress levelat which the

materialbehaves

elastically

NeckingMaterial canresist more load

increase

YieldingElastic

Material will deformpermanently and will n to its

orginal shape upon unloading. Thedeformation that occurs is called

σ

fσ ′

uσ

fσ

yσ

PLσ: The upper stresslimit that strain

varies linearly withstress. Material

follows

:Stress at which aslight increase in

stress will result inappreciably increas

in strain withoutincrease in stress

Necking

Stress-strain using originalarea to calculate

True Stress-strain usingactual area to calculate

terialwill return to its orginal

shape if material is loadedand unloaded within this

range

ε

:10 - 40 timeselastic strain

yε

Elastic Limit:

Strain Hardening:

Plastic Behaviour: NOT retur

plastic deformation

Proportional Limit

Hooke's Law

Yield Stress

Ultimate stress

Failure stress

Elastic Behaviour: Ma

Yield strain

Stress-strain diagram for ductile materials

Hooke’s Law: Eσ ε=

E is the modulus of elasticity steelE = 200 GPa

concreteE = 29 GPa (21 – 29 GPa)

3

Ductile Materials

Materials that can be subjected to large strains before rupture

Have high percent elongation

− ×

of

o

L LPercent elongation = 100

L

Have high percent reduction in area

o f

o

A APercent reduction in area = 100

A−

×

Have capacity to absorb energy

If structure made of ductile materials is overloaded, it will present large deformation before failing

Some ductile materials do not exhibit a well-defined yield point, we will use offset method to define a yield strength

Some ductile materials do not have linear relationship between stress and strain, we call them nonlinear materials

σ

yσ

ε

0.002 or 0.2% offset

σ

ε

( )fσ ε=

Elastic-plastic Materials

Stress-strain for structural steel will consist of elastic and perfectly plastic region. We call this kind of material elastoplastic material

Analysis of structures on the basis of elastoplastic diagram is called elastoplastic analysis or plastic analysis

4

Bilinear stress-strain diagram having different slopes is sometimes used to approximate the general nonlinear diagrams. This will include the strain hardening.

σ

ε

yσ

yε

σ

ε

Linearly elastic

Perfectly plastic

Linearly elastic

Strain hardening

Nonlinear

Brittle Materials

Materials that do not exhibit yielding before failure

Some materials will show both ductile and brittle behaviours, e.g. steel with high carbon content will demonstrate brittle behaviours while steel with low carbon content will be ductile or steel subjects to low temperature will be brittle while those in the high temperature environment will be ductile

Creep

Deformation which increases with time under constant load (examples: rubber band; concrete bridge deck: sagging between supports due to self weight therefore the deck is constructed with an upward camber)

P

δ

δ

t

oδ

ot

In several situations, creep will associate with high temperature

If creep becomes important, creep strength will be used in design

5

Relaxation

Loss of stress with time under constant strain

Another manifestation of creep

Prestressedwire

t

Creepstrength

σoσ

Cyclic loading and fatigue

Fracture after many cycles of loading

If material is loaded into the plastic region, upon unloading elastic strain will be recovered but plastic strain remains

Loading

Unloading

Permanentset

Elasticrecovery

Elasticregion

σ

ε

6

Strain energy

Energy stored internally throughout the volume of a material which is deformed by an external load

F

x

oF

ox

Work

F

k

x

Consider a linear spring having stiffness k

If we apply a force , the spring will stretch F x . The relationship between and F x is

F kx=

If we apply a force from zero to and the spring stretches to the amount of

oF

ox , the work done is the average force magnitude times the displacement, i.e.

12 o oW F =

x

From the conservation of energy, this work done must be equivalent to the internal work or strain energy stored within the spring when it is deformed

7

dxdy

dz

σ

σ

If an infinitesimal element of elastic material is subjected to a normal stress σ , then tensile force on the element will be

dF dxdyσ=

The change in its length is dzε

The work done, which equals to the strain energy stored in the element, is

( )(12

dU dxdy dzσ ε= ) or 12

dU dVσε=

V

The total strain energy stored in a material will be

V

U dσε= ∫

The strain energy per unit volume or the strain energy density is

12

dUudV

σε= =

If the material is linear elastic ( Eσ ε= , Hooke’s Law holds), the strain energy density will be

21 12 2

uE Eσ σσ = =

σ

ε

PLσ

PLε

ru

If the stress σ reaches the proportional limit, the strain energy density is called the modulus of resilience u r

12r PLu PLσ ε=

8

σ

ε

tu

The total strain energy density which stored in the material just before it fails is called the modulus of toughness tu

Two Example Problems:

9

Answer to 1.

10

Answer to 2.

11

More on crazing and shear deformations and zones next time. (Phenomena in amorphous polymers, as discussed yesterday)

12