Stress, Strain and The Young Modulus

Solid Material Properties

Define stress.

Force per unit cross-sectional area of a material.

State the equation for stress.

σ = F/A.

What does σ represent?

Stress.

What does F represent in the stress equation?

Force in newtons (N).

What does A represent in the stress equation?

Cross-sectional area in m².

What is the SI unit of stress?

Pascal (Pa), equivalent to N m⁻².

Define tensile stress.

The tensile force acting per unit cross-sectional area of a material.

Define compressive stress.

The compressive force acting per unit cross-sectional area of a material.

Why is stress more useful than simply comparing forces?

Stress accounts for the cross-sectional area, allowing samples of different sizes to be compared fairly.

How does decreasing the cross-sectional area affect stress for the same force?

Stress increases.

How does increasing the cross-sectional area affect stress for the same force?

Stress decreases.

How do you calculate the cross-sectional area of a circular wire?

A = πr².

If the diameter of a wire is given, how do you find its radius?

r = diameter/2.

Define strain.

The extension or compression of a material divided by its original length.

State the equation for strain.

ε = Δx/x.

What does ε represent?

Strain.

What does Δx represent in the strain equation?

The extension or compression.

What does x represent in the strain equation?

The original length of the material.

Does strain have a unit?

No, strain is dimensionless because it is a ratio of two lengths.

Why does strain have no units?

The units of extension and original length cancel.

How can strain be expressed as a percentage?

Percentage strain = strain × 100%.

Why is strain useful when comparing materials?

It accounts for the original length, allowing samples of different lengths to be compared fairly.

How is extension calculated if the original and final lengths are known?

Δx = final length − original length.

What is the difference between extension and strain?

Extension is the actual change in length, whereas strain is the change in length divided by the original length.

What is the Young modulus?

A measure of the stiffness of a material, equal to stress divided by strain during elastic deformation.

State the equation for the Young modulus.

E = σ/ε.

What does E represent?

The Young modulus.

What is the SI unit of the Young modulus?

Pascal (Pa).

Why does the Young modulus have units of Pa?

Strain has no units, so stress divided by strain retains the unit Pa.

What does a high Young modulus indicate?

A stiff material that undergoes relatively little strain for a given stress.

What does a low Young modulus indicate?

A less stiff material that undergoes greater strain for a given stress.

When can E = σ/ε be used to determine the Young modulus?

When the material is undergoing elastic deformation within the proportional region.

What is the relationship between stress and strain during elastic deformation within the limit of proportionality?

Stress is directly proportional to strain.

What is the constant of proportionality between stress and strain?

The Young modulus.

What happens to the stress-strain relationship beyond the limit of proportionality?

Stress is no longer directly proportional to strain.

Why is the Young modulus a property of a material rather than a particular object?

It accounts for the dimensions of the sample, so different samples of the same material have the same Young modulus.

What is the difference between spring constant and Young modulus?

Spring constant describes the stiffness of a particular object or spring, whereas Young modulus describes the stiffness of the material itself.

What quantities must be known to calculate Young modulus directly?

Force, cross-sectional area, extension and original length.

How can Young modulus be written using force, area, original length and extension?

E = Fx/(AΔx).

How is E = Fx/(AΔx) derived?

Substitute σ = F/A and ε = Δx/x into E = σ/ε.

What happens to the extension if the Young modulus is larger for the same stress and original length?

The extension is smaller.

What happens to strain if stress increases while the material remains within the proportional region?

Strain increases proportionally.

What happens to stress if the same force is applied to a thinner wire?

Stress increases because its cross-sectional area is smaller.

What happens to stress if the same force is applied to a thicker wire?

Stress decreases because its cross-sectional area is larger.

Why can a thin wire stretch or break when a thicker bar of the same material does not under the same force?

The thin wire has a smaller cross-sectional area, so it experiences a greater stress.

How do you convert millimetres to metres?

Divide by 1000, or multiply by 10⁻³.

How do you convert millimetres squared to metres squared?

Multiply by 10⁻⁶.

How do you convert centimetres to metres?

Divide by 100, or multiply by 10⁻².

What is 1 Pa equivalent to?

1 N m⁻².

What does MPa mean?

Megapascal, equal to 10⁶ Pa.

What does GPa mean?

Gigapascal, equal to 10⁹ Pa.