MCT-229: MECHANICS OF MATERIALS

Lecture no 5: Stress and Strain

Misbah ur Rehman misbah@uet.edu.pk

PREVIOUS LECTURE

Stress on Oblique Planes

Design and Factor of Safety

Strain

Stress – Strain curve

Necking

Brittle vs Ductile materials

STRESS-STRAIN DIAGRAM

Plotting the stress σ = 𝑃𝐴 against

the strain ε = δ𝐿 , generates the

stress-strain curve.

Characteristic of the property of

every material, irrespective of the

dimensions of the material.

Tensile Testing is done to check for the stress-strain properties of a material.

Materials can be broadly classified into brittle and ductile on the basis of stress-strain diagram.

DUCTILE MATERIALS

Ductile materials are classified with their ability to yield

at normal temperatures.

• Yield strength

• Ultimate Strength

• Breaking Strength

• Strain Hardening?

• Examples?

BRITTLE MATERIALS

Rupture occurs without prior change in the rate of elongation.

Absence of necking.

DUCTILE MATERIALS

Yield point is not clear for some ductile materials i.e.

aluminum

Yield strength can be defined by using the offset

method.

For example 0.2% offset

DUCTILE MATERIALS : MEASURE

A standard measure of ductility of a material is Percent

Elongation, defined as :

Percent Elongation = 100 𝑳𝑩−𝑳𝟎

𝑳𝟎

Where L0 = initial length of the test material

And LB = final length at the rupture

Another measure is Percent Reduction in Area , defined as:

Percent Reduction in Area = 100 𝑨𝟎−𝑨𝑩

𝑨𝟎

Where A0 = initial cross-section of the test material

And AB = final cross-section at the rupture

COMPRESSIVE LOADING

For ductile materials, stress-stress curve would be

similar, up till the strain hardening.

Necking cannot occur in the compression.

For brittle materials ultimate strength in compression is

much larger.

TRUE STRESS AND TRUE STRAIN

The stress taken so far is

also known as

engineering stress.

True stress is obtained

by dividing the applied

force by the deformed area. σ = P/A.

Similarly, ‘True Strain’ is

obtained by adding

successive values of

strain:

HOOKE’S LAW; MODULUS OF ELASTICITY

Who? Cap’n hook?

HOOKE’S LAW

For the initial straight line portion of the stress-strain

diagram, the stress is directly proportional to strain:

The relation is known as ‘Hooke’s Law’.

The coefficient ‘E’ is known as ‘modulus of

elasticity’ of ‘Young’s modulus’ . Units?

HOOKE’S LAW

The largest value of stress for which the Hooke’s law

can be used is known as the ‘proportional limit’.

Isotropic materials : properties like modulus of

elasticity, stress, strain are independent of the force

applied.

Anisotropic materials: properties like modulus of elasticity, stress and strain are dependent on the direction of the force applied.

Examples of anisotropic materials?

ELASIC vs PLASTIC BEHAVIOR

If the strain disappears when the force applied is

removed, the material is said to be behaving

‘elastically’.

The largest value of stress for which a material

behaves elastically is known as ‘elastic limit’.

If the deformation is permanent, the material is

said to have undergone ‘plastic deformation’.

ELASIC vs PLASTIC BEHAVIOR

FATIGUE AND REPEATED LOADING

If an object is loaded and unloaded many times,

failure occurs even when the loading is within elastic

limits.

This phenomenon is known as ‘fatigue’.

Fatigue must be considered in design of all

structures.

Examples of cyclic loading?

FATIGUE AND REPEATED LOADING

The number of loading

and unloading cycles

required before the

object fails can be

found experimentally.

The maximum stress is

plotted as abscissa and

the number of cycles

as ordinate to obtain a ‘σ-n’ curve:

Reference :

Chapter no 2 : Stress and Strain – Axial Loading

Mechanics of materials by Beer & Johnston

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ANY QUESTIONS?

BONUS QUESTION

SOLUTION