1. Elastic & Plastic behavior of Materials
Lecture 2

2. Which material to choose?
An engineer has a vast range of materials at his disposal:
metals and alloys, polymers, glasses and ceramics, composites
How does he go about selecting the material, or combination of materials, which best suit his purpose?
by selecting properties
Mistakes can cause disasters.
SS John P. Gaines split in two (1943)

3. Bulk Mechanical Properties
Modulus
Yield strength, tensile strength, and ductility
Hardness
Impact strength and fracture toughness
Fatigue strength and thermal fatigue resistance
Creep

4. What happens to material when it is loaded with a mechanical force?
Stress and Strain
Tension
Compression
Shear
Torsion
Elastic deformation
Plastic Deformation
Yield Strength
Tensile Strength
Ductility
Toughness
Hardness

5. The concept of stress and strain
The mechanical behaviour of material under applied force may be ascertained by a simple stress – strain diagram or, load – deformation diagram
Stress - Force or load per unit area of cross- section over which the force or load is acting
Strain - Change in dimension (elongation) per unit length
Stress and strain are considered positive for tensile loads, negative for compressive loads
One of the most commonly performed mechanical stress-strain test is known as the tensile test.

6. Stress-Strain Behavior
Engineering Stress:  = F / Ao
F: is applied load
A0:Cross-sectional area before loading
Engineering Strain:  = l / lo ( 100 %)
l: change in length
lo: original length.
Elastic deformation
Reversible (For small strains):
Stress removed  material returns to original size
Plastic deformation
Irreversible:
Stress removed  material does not return to original dimensions.

7. Stress-Strain Test: Tensile Test
The machine
Two categories of machines are available:
Screw-driven: allows selection and control of the strain rate (de/dt)
Hydraulically driven: allows selection and control of the loading rate (ds/dt)
specimen
Universal Testing Machine
(UTM)
The sample
“505 bar” — Nickname for the ASTM standard specimen most commonly used in tensile testing; a cylindrical specimen, 0.505" dia. along 2" gauge length (i.e., the length of the straight section between threaded ends). This diameter gives a convenient 0.20 in2 cross-sectional area.

8. Tensile Testing Apparatus

9. F - d characteristics are dependent on the size of specimen
The material’s response to the applied tensile or compressive load is a change in length.
During tension test, instantaneous applied load/force (F) and elongation or deformation (d) data are recorded, and the output of test is given as F – d chart
 To minimise these geometric factors, load and elongation parameters are normalised to the respective parameters of engineering stress and engineering strain.
F - d characteristics are dependent on the size of specimen
For example, for a doubled cross-sectional area, to generate the same elongation, the load must be doubled.

10. Stress-Strain Diagram
ultimate
tensile strength
necking 3
Slope=E
Strain
Hardening
yield
strength
Fracture 5 2
Elastic region
slope =Young’s (elastic) modulus
yield strength
Plastic region
ultimate tensile strength
strain hardening
fracture
Plastic
Region
Stress (F/A)
Elastic
Region 4 1
Strain ( ) (DL/Lo)

11. Properties obtained from the tensile test
Elastic limit
Young’s Modulus
Yield Strength
Tensile strength
Necking
Hooke’s law
Modulus of resilience
Toughness
Ductility

12. Important Mechanical Properties from Tensile Test
Young's Modulus: This is the slope of the linear portion of the stress-strain curve, it is usually specific to each material; a constant, known value.
Yield Strength: This is the value of stress at the yield point, calculated by plotting young's modulus at a specified percent of offset (usually offset = 0.2%).
Ultimate Tensile Strength: This is the highest value of stress on the stress-strain curve.
Percent Elongation/ Ductility: This is the change in gauge length divided by the original gauge length.

13. Basic Definitions Tensile Strength
This is the ability of a material to withstand tensile loads without rupture when the material is in tension.
Compressive Strength
This is the ability of a material to withstand Compressive (squeezing) loads without being crushed when the material is in compression .
Shear Strength
This is the ability of a material to withstand offset or traverse loads without rupture occurring .
Elasticity
This is the ability of a material to deform under load and return to its original size and shape when the load is removed. The property is required for springs
Plasticity
This is the property of a material to deform permanently under the application of a load. This is the exact opposite to elasticity.

14. Basic Definitions Ductility
This is the ability of a material to stretch under the application of tensile load and retain the deformed shape on the removal of the load. A ductile material combines the properties of plasticity and tensile strength. All materials which are formed by drawing are required to be ductile.
Malleability
This is the property of a material to deform permanently under the application of a compressive load. A material which is forged to its final shape is required to be malleable.
Fatigue Strength
This is the property of a material to withstand continuously varying and alternating loads .
Hardness
This is the property of a material to withstand indentation and surface abrasion by another hard object. It is an indication of the wear resistance of a material .e.g Diamonds are very hard.

15. Elastic Deformation: Atomic Perspective
Initial state F
Small load applied
Load removed
return to initial
bond stretch d F d
Linear elastic
Elastic means reversible!!
Non-linear elastic
This happens when strains are small
(except for the case of plastic materials)

16. s = E e Elastic deformation
Initially, stress and strain are directly proportional to each other atoms can be thought of as masses connected to each other through a network of springs
 In tensile test, if the deformation is elastic, the stress-strain relationship follows the Hooke’s law:
s = E e
E is known as the Young’s modulus, the modulus of elasticity, or simply the modulus
 E has the same unit as those of stress, MPa or psi, although GPa (109 Pa) is commonly used.

17. Young’s modulus of elasticity
 E is a measure of :
 bond strength (on the atomic level)
 intrinsic stiffness of material
Very stiff materials : Ceramics, steels, W ….
Medium stiff materials : Cu, Al, ….
Low stiff materials : Plastics, ….
 For single phase (or, nearly single phase) materials, E is insensitive to :
 degree of plastic deformation
 microstructure (i.e., grain size, inclusion)
 Hooke’s law applied for only a small value of e (typically < ~ %)
 ceramic materials follow Hooke’s law up to fracture
 E is decreased with increasing T
 In the elastic region, E does not vary with the applied stress, i.e., E ≠ E(s)

18. Example: Design of a Suspension Rod
An aluminum rod is to withstand an applied force of 45,000 pounds. To assure a sufficient safety, the maximum allowable stress on the rod is limited to 25,000 psi. The rod must be at least 150 in. long but must deform elastically no more than 0.25 in. when the force is applied. Design an appropriate rod.
SOLUTION :
Using the definition of engineering stress, the required cross-sectional area of the rod F s
A0 = = (45000 lbs) / (25000 psi) = 1.8 in2
p d2 4
A0 = = 1.8 in2 or d = in

19. Example: Design of a Suspension Rod
However, the minimum length or rod is specified as 150 in. To produce a longer rod, we might make the cross-sectional area of the rod larger.
e = = = in/in Dl l0
0.25 in
150 in
The minimum strain allowed for the 150 in rod is
A0 = = = in2 F s
45000 lbs
16670 psi
Now, using the Hook’s law
s = E e = (10x106 psi) ( in/in) = psi
Then, the area required to withstand this stress
Thus, in order to satisfy both the maximum stress and the minimum elongation requirements, cross-sectional area of the rod must be at least 2.7 in2 , or a minimum diameter of 1.85 in.