5. Mechanical Properties of Materials

Mechanics of Materials

Dr. Rami Zakaria

Reference:

1. Mechanics of Materials: R.C. Hibbeler, 9th ed, Pearson

2. Mechanics of Materials: J.M. Gere & B.J. Goodno, 8th ed, Cengage Learning

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Term Definition Notes

Specimen Typical piece of the material under study/ Sample.

Elasticity Being able to recover original shape after deformation

To yield To give up/ to bend

Ultimate Maximum

To Fracture/ Fail To break

Permanent Cannot be recovered / everlasting

Ductile materials … deform (strain) under tensile forces / can be thinned out

Brittle materials … break suddenly / don’t deform before breaking

Stiffness Rigidity / doesn’t bend easily

Resilience Ability to recover to original form or shape/ Energy per unit

volume during the elastic deformation

To creep To act slowly

Fatigue Weariness / weakening of a material because of repeated stresses

Introduction

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We will see how stress and strain are connected together.

In order to perform tension or compression test

on a material, we make a standard shape

specimen (constant circular cross-section) and

measure strain against stress.

We are interested in stress-strain diagram,

which can be obtained by experiment (as the

strength of the material connected to its physical properties).

For tension test, we apply force and measure the

change in length (using optical-extensometer for

example), or we can read the strain directly from

an electrical-resistance strain gauge.

The stress-Strain Diagram

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1- Conventional Stress-Strain Diagram

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P

Let’s take an example, the Conventional Stress-Strain Diagram for steel:

We notice 4 regions depending on the material

behaviour:

Elastic Behavior Curve is a straight line (for most of the region)

until a proportional limit stress . Then the

curve bend and flatten out until the elastic limit,

just before the yield stress (usually difficult

to find as it is very close the proportional

limit) . If the load removed until this point, the

specimen will return back to its original shape.

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Y

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Yielding This is when any increase in stress will cause a permanent deformation (we call it plastic

deformation). At this region the specimen will continue changing length (strain) without

any increase in load

Strain Hardening Yielding is ended, and an increase in load will

make the curve to rise until it reaches a

maximum stress, we call ultimate stress .

Up to this point the length is increasing and the

cross-section area is decreasing in a (kind of)

uniform manner.

u

Necking After the ultimate stress, the cross-section area

will decrease in a local region, making what

looks like a neck. Eventually the graph curve

downward until the specimen breaks at the

fracture stress . f

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2- True Stress-Strain Diagram

Instead of calculating the (engineering) stress and strain using the original area

A0 and length L0 , we can calculate the true stress and true strain at the instant

when the load is measured.

The difference between the two graph become more obvious in the strain-

hardening region, and we can see in the true diagram that stress keeps

increasing (because the area is decreasing at the necking area) .

For most application we are interested in the elastic region, because the

deformation there is not sever and not permanent. The difference between the

two graphs in this region is very small.

Ductile and Brittle Materials

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Ductile Materials:

They have 4 distinct behaviours (elastic, yielding, strain hardening,

and necking) when they are loaded.

This means they strain a lot before they fracture.

Examples: mild steel, brass, and zinc.

Some materials do not have a well-

defined yielding point, so we assume it

(graphically) using an offset method, and

call it a yield strength σYS (for example

aluminium alloy yield strength at 0.2%)

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Brittle Materials

They have very little or no yielding before they fracture.

This means they fracture and fail suddenly.

When they are under tension, they

don’t have a well-defined tensile

fracture stress σtf (it takes place in a

random imperfection or microscopic crack,

and it is relatively small). What we can

calculate is the average of that stress

from many cases.

When they are under compression,

they show a much higher

compressive fracture stress σcf .

Examples: gray cast iron , glass, and concrete.

Notes:

1- Because the fracture point of concrete under tensile forces is small, we almost always

reinforce the concrete structures with steel bars.

2- In most cases materials can shift between the two behaviours (Ductile & Brittle)

depending on the material components (such as the carbon percentage), or the temperature

(high temperature make some materials softer – more ductile), etc…

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Hooke’s Law

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We saw that the relationship between stress and strain is linear within the elastic region.

We can describe this relationship using Hooke’s law

E

This relationship represents (only) the straight line at

the beginning of the σ-ε graph.

E is a constant called the modulus of elasticity, (it

represents the slope of the line), and it has the same

unit as stress (Pa, psi, …). It describes the stiffness of

a material (stiff materials have larger modulus of

elasticity E).

Notice in the example of steel alloys that all grades

have almost the same E=29(103) ksi.

Strain Hardening:

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We saw that when a specimen is loaded in the plastic

region, and then unloaded, the elastic strain is recovered

but the plastic strain remains (permanent set).

In the graphs, notice that the slope OA is similar to the

slope O’A’.

What happened that the new yielding point now (A’) is

higher that the original yielding point (A), also the

elastic region is larger, while the plastic region is

smaller than originally.

13 [2] James M.Gere & Barry J.GooDNO

When a load is applied to a member, the deformations cause

strain energy to be stored in the material. The strain energy

per unit volume or strain energy density is equivalent to the

area under the stress–strain curve.

This area up to the yield point is called the modulus of

resilience, ur.

It represents the total energy per unit volume can be

absorbed without causing permanent damage (useful when

designing shock absorbers).

The entire area under the stress– strain diagram is called the

modulus of toughness, ut.

It represents the total energy per unit volume can be

absorbed before failing (useful when designing members

which could be overloaded accidently).

Strain Energy:

Eu

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plplr

2

2

1

2

1

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(Classroom Exercise) A bar having a length of 5 in. and cross-sectional area of 0.7 in2 is subjected

to an axial force of 8000 lb.

If the bar stretches 0.002 in., determine the modulus of elasticity of the material (the material

has linear-elastic behavior).

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(Classroom Exercise) The stress–strain diagram for polyethylene, is determined from testing a

specimen that has a gauge length of 10 in. a) Determine the Modulus of Elasticity E

b) Determine the modulus of resilience, ur

c) If a load P on the specimen develops a strain of ε=0.024 . Determine the approximate length of the

specimen, when the load is removed. Assume the specimen recovers elastically.

σpl

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Poisson’s Ratio:

rLlatlong

,

Only within the elastic range of a homogenous and isotropic materials

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When materials are in service for long periods of time, considerations of

creep become important. Creep is the time rate of deformation, which occurs

at high stress and/or high temperature. Design requires that the stress in the

material not exceed an allowable stress which is based on the material’s

creep strength.

Fatigue can occur when the material undergoes a large number of cycles of

loading. This effect will cause microscopic cracks to form, leading to a brittle

failure. To prevent fatigue, the stress in the material must not exceed a

specified endurance or fatigue limit.

Failure Due to Creep & Fatigue