1. Mechanical Properties: 1
Eyres: MSC101

2. Chapter Learning Objectives
Identify the key features of the uniaxial stress-strain curve (elastic modulus, yield stress, ultimate tensile strength, necking, etc.)
Construct a stress-strain curve given load and displacement data and calculate the ductility of the sample
Calculate engineering and true stress and strain and explain why two systems of stress and strain exist
Predict how the mechanical properties of a material will change when a material is subjected to different temperatures and strain rates
Calculate the flexural strength and flexural modulus for a material subjected to a bend test
Define the materials property of hardness and explain how it is measured at various length scales
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3. Terminology for Mechanical Properties
Stress: Force acting per unit area over which the force is applied
Strain: Change in dimension per unit length
Elastic strain: Fully recoverable strain resulting from applied stress
Young’s Modulus or modulus of elasticity (E): Slope of linear section of tensile stress-strain curve
Shear modulus (G): Slope of the linear part of the shear stress- strain curve
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4. Definitions and units Stress: S= 𝐹 𝐴 0 Units are: 𝑁 𝑚 2 =𝑃𝑎
Strain: 𝑒= ∆𝑙 𝑙 0 Units are: 𝑚 𝑚 =dimensionless or unitless
Young’s Modulus or modulus of elasticity:
𝐸= ∆𝑆 ∆𝑒 Units are: 𝑃𝑎 1 =𝑃𝑎
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5. Illustration of Stresses and Strains
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6. The Tensile Test: Use of the Stress-Strain Diagram
In a tensile test, a specimen is placed in the testing machine and a force, called the load, is applied slowly
A strain gauge or extensometer is used to measure the amount that the specimen stretches between the gauge marks
Thus, the applied load is known and the change in length Δl is measured with respect to the original length l0
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7. Properties Obtained from the Tensile Test
The critical stress value needed to initiate plastic deformation is defined as the elastic limit of the material
The proportional limit is defined as the as the level of stress above which the relationship between stress and strain is not linear
Since neither limit can be determined precisely, they are defined at an offset strain value (typically 0.2%)
The corresponding stress value is defined as the offset yield strength, also sometimes known as just yield strength
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Elastic vs. Plastic
Elastic strain: Fully recoverable strain resulting from applied stress
Plastic deformation, which is permanent, in a material is known as plastic strain
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9. Tensile Strength and Elastic Properties
The stress obtained at the highest applied force is the tensile strength, or ultimate tensile strength
At some point in a tensile test, specimens undergo necking, which is a large local decrease in cross-sectional area
The relationship between stress and strain is given by Hooke’s Law, which relates stress and strain linearly through Young’s modulus E
Poisson’s ratio ν relates the longitudinal deformation and lateral deformation that occur simultaneously
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10. Tensile Toughness, Ductility, and Temperature Effects
The energy absorbed by a material prior to fracture is known as tensile toughness, the area under the true stress-strain curve
Ductility is the ability of a material to be permanently deformed without breaking when a force is applied
There are two common ways of measuring ductility, percent elongation and percent reduction, and both are described by the equations below
Mechanical properties depend on temperature. Yield strength, tensile strength, and modulus of elasticity decrease at higher temperatures, whereas ductility commonly increases
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Question
Modulus of Elasticity is defined as both 𝐸= 𝑆 𝑒 and 𝐸= ∆𝑆 ∆𝑒 Can you Explain?
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Table 6-1: The results of a tensile test of a cm diameter aluminum alloy test bar, initial length (l0) = 5 cm.
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14. Load (N)
Change L (cm)
Strain
Stress (MPa)
0.0000
4450
0.0025
0.0005
35.52
13350
0.0075
0.0015
106.56
22240
0.0125
177.52
31150
0.0175
0.0035
248.63
33360
0.075
0.0150
266.27
35150
0.2
0.0400
280.56
35580
0.3
0.0600
283.99
35360
0.1
0.0200
282.24
33800
0.5125
0.1025
269.79

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Identify the important values and terms in the problem statement and their connection to the diagram.
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Strain Rates
Strain vs Time graphs When materials are subjected to high strain rates, we refer to the type of loading as impact loading
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18. Viscous material
A viscous material: strain develops over a period of time; material does not return to its original shape after the stress is removed
A viscoelastic material: a response between that of viscous and elastic materials
Viscoelastic materials under constant strain: level of stress decreases over time. This is known as stress relaxation (tennis racket example)
Can you show Strain Recovery on the diagram?

19. True Stress and True Strain
The decrease in engineering stress beyond the tensile strength on an engineering stress-strain curve is related to the definition of engineering stress
We used the original area A0, but this is not precise since that area changes.
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20. True Stress and True Strain
Assuming constant volume, we can convert between definitions using the following equations
𝑒= ∆𝑙 𝑙 0 = 𝑙− 𝑙 0 𝑙 0 = 𝑙 𝑙 0 −1
𝑨𝒍= 𝑨 𝟎 𝒍 𝟎
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21. The Bend Test for Brittle Materials
In many brittle materials, the normal tensile test cannot be performed because of the presence of flaws at the surface
These materials are tested using the bend test
By applying load at three points and causing bending, a tensile force acts on the material opposite the midpoint. Fracture begins at this location
The flexural strength σbend describes a materials strength
The flexural modulus Ebend is the modulus of elasticity in bending
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Hardness of Materials
The hardness test measures the resistance to penetration of the surface of a material by a hard object
Most common are Rockwell test and Brinell Test
Force steel balls (though in some Rockwell test cases, it is a diamond cone) into the surface of a material
Brinell test measures the diameter of the impression while the Rockwell test measures its depth
Both tests produce a hardness number for qualitative comparison
Knoop test, is a microhardness test, microscope for measurement
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Nanoindentation
Microhardness
Nanoindentation
Dimensions of millimeters or microns
Quick, easy, and inexpensive
Only for macroscale samples
Hardness is the only materials property that can be directly measured
Nanometer length scale
Small diamond tip
Measure both hardness and elastic modulus at much smaller length scales
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