mechanical properties of materials

stress

$\sigma=\frac{F}{A_0}$

F: force (N)

$\sigma$: stress ($Pa$)

$A_0$: Area in square meters ($m²$)

strain

$\char"0190 =\frac{l_1-l_0}{l_0}=\frac{\Delta l}{l_0}$

$\char"0190$: strain (unitless)

$l$, $l_0$, $\Delta l$ : length ($m$)

shear stress

$\tau=\frac{F}{A_0}$

$\tau$: shear stress ($Pa$)

$\sigma$: stress ($Pa$)

$A_0$: square meters ($m²$)

shear strain

$\gamma=\tan\theta$

$\gamma$: shear strain (dimensionless)

$\theta$: angle (in $degrees$ or radians: $rad$)

1$MPa$=

$1MPa=10^6Pa=10^6N$ /$m²$

tension

pulling force that causes elongation

compression

pushing force that causes shortening

torsion

twisting around the longitudinal axis

Example : holding one end of a metal stationary while twisting the other end

load cell

measures applied force $F$ in newtons (N)

extensometer

measures elongation $\Delta l$ in $mm$ or $m$

imagine you have a steel rod holding up a very heavy object.

why do mechanical properties matter?

when force is applied, the rod might:

• stretch (elongate)

• shorten (compress)

• change shape (deform)

• break (fracture)

engineers need to understand how materials respond to forces so they can prevent excessive deformation and fracture.

what affects a material's mechanical behavior?

• nature of load: (type of force) whether the material is pulled, pushes, or sheared, and whether the force changes

• duration: how long the load is applied (from seconds to years)

• environment: temperature and surrounding conditions

a) what is ATSM

b) why does it matter

a) ATSM: American society for testing and materials.

it develops and publishes standards for testing materials.

b) to compare results fairly, we need consistent testing procedures.

what do structural engineers do?

1. determine stresses and stress distributions in loaded structures.

2. use experimental testing and mathematical analysis.

3. select materials with suitable combinations of mechanical properties

4 types of loading

tension tests

a tension test measures how a material behaves when it is pulled until it deforms or fractures

the middle section is narrower than it ends so that we can concentrate the deformation in the middle rather than near the grips

the machine gradually increases its tensile load, while measurements are recorded.

parts of the testing machine

instruments:

• load cell: measures the applied force $F$

• extensometer: measures elongation $\Delta l$

characteristics of a tensile test

• the specimen elongates at a constant rate

• the test typically lasts several minutes.

• it is

Area of a circle

$$A=\pi R^2$$

$$R=\frac{d}{2}$$

Comparison test

A comparison test measures how a material behaves when a pushing force is applied

-ve stress or strain

-ve answers ←→ Compression

When are compressive tests used?

• compressive service loads: when the material will experience pushing forces in real/actual uses

• Large plastic deformation: when studying permanent (plastic) deformation, such as metal forming

• Brittle materials: when tensile testing is impractical, such as with ceramics or cast iron

Which test is more common? Why?

Tensile tests are more common

• they are easier to perform and instrument

• For most structural metals, compression tests provide little additional information

What is shear?

Shear occurs when forces are applied parallel to a material’s surface → causing angular distortion

Normal stress vs shear stress

They have identical formulas but…

Normal stress → force acts perpendicular to the cross section

Shear stress → force acts parallel to the surface

Elastic deformation

Temporary deformation (reversible)

When applied force is removed → material returns to it’s original shape and dimensions

Ex: stretching a spring and (applied force) then releasing it (removing applied force → back to its original form)

Hook’s law

$$\sigma=E\varepsilon$$

$\sigma$: normal stress (MPa)

E: Young’s modulus (modulus of elasticity) (MPa)

$\varepsilon$: normal strain (unitless)

Young’s modulus

Modulus of elasticity $E$: measures stiffness

Higher $E$ → stiffer material → less strain for the same stress

Lower $E$ → less stiff material → more strain for the same stress

Stress - strain graph

$$SLOPE=\frac{\Delta\sigma}{\Delta\varepsilon}=E$$

So if the question gives a straight line stress-strain graph and asks for stiffness ($E$) calculate its slope.

Non Linear elastic behavior

Some materials (cast iron, concrete, polymers) exhibit nonlinear elastic behavior

For a point ($\varepsilon$,$\sigma$) → secant modulus → $E_{\sec ant}=\frac{\sigma}{\varepsilon}$

For a point ($\varepsilon$,$\sigma$) → tangent modulus is the slope at that point→ $E_{\tan gent}=\frac{d\sigma}{d\varepsilon}$

Why are some materials stiffer than others?

Because it depends on interatomic bonding forces.

$$E\varpropto\left(\frac{dF}{dr}\right)_{r_0}$$

F: interatomic force (N)

$r$: atomic separation (m)

$r_0$: equilibrium separation (m)

Stronger bonds → higher modulus $E$

Weaker bonds → lower modulus $E$

Temperature and shear modulus

As temp increases:

• atomic vibrations increase

• Effective bond stiffness decreases

• Modulus $E$ decreases

$T$ increases → $E$ decreases

Shear modulus

$\tau=G\gamma$ → $G=\frac{\tau}{\gamma}$

$G$: shear modulus (MPa)

$\tau$: shear stress (MPa)

$\gamma$: shear strain (unitless)

Anelasticity

Anelasticity: delayed elastic response and delayed recovery (unlike elastic deformation which recovers immediately)

• strain can continue increasing after a load is applied

• Recovery takes time after the load is removed

• The material eventually goes back to its original form