Stress and strain | PhysicsRank

Two wires stretch by different amounts under the same hanging load. One may be longer or thinner; that alone does not make its material softer. Stress and strain let us separate the specimen’s shape from its material response.

We first build the small-deformation model and distinguish stiffness from strength. Then we design a wire-extension inference, use a graph to remove a fixed reading offset and examine how diameter and extension uncertainty affect the reported modulus.

What you will learn to do

Define stress and strain with consistent original dimensions.
Distinguish Young’s modulus from the specimen’s spring constant.
Read linearity, recovery and energy from a stress–strain relation.
Choose axial, shear or bulk response according to the loading.
Infer Y from a force–extension slope with a stated reference.
Propagate geometry and extension uncertainty and assess experimental validity.

Useful before this chapterWork, stored energy and power →Make an experiment speak through a graph →Propagation of uncertainty →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

BEFORE YOU READ

Bring back one idea

Try a sentence or a sketch before opening the comparison. If this is new, use the explanation to get started.

Why is extension alone a poor measure for comparing the elasticity of two different wires?

Compare your reasoning

Extension depends on initial length, area and load as well as material. Strain divides extension by initial length; stress divides force by area, allowing a meaningful comparison within the model.

Revisit the explanation →

01 / Elastic response and measuring Young’s modulus

Describe both the load and the relative change of shape

A solid can deform even when it appears rigid at ordinary viewing scales. Pulling a wire changes its length; pushing sideways on a block changes its shape; pressure applied on all sides changes its volume. Elasticity concerns how the material responds and whether it recovers when the load is removed.

The same force can have very different effects on a thick and a thin wire. Stress accounts for this by dividing force by the area carrying that force. Likewise, a 1 mm extension means a different relative deformation for a 10 cm sample and a 10 m sample. Strain divides the change by the relevant original dimension.

Elastic recovery and a linear stress–strain relation are different properties. A material can recover its original shape after a nonlinear deformation. Hooke’s law describes a linear proportional region; the elastic limit describes whether permanent deformation remains after unloading. Neither should be confused with the load at which the sample finally breaks.

\sigma=\frac FA,\qquad\varepsilon=\frac{\Delta L}{L_0}

For uniform small uniaxial tension, σ is tensile stress and ε is longitudinal strain. These engineering definitions use original area and length.

WORKED EXAMPLE

Separate a force, a stress and a strain

A straight wire of original length 2 m and cross-sectional area 0.50 mm² is stretched by a 25 N axial force. Its extension is 0.50 mm. Find its tensile stress and longitudinal strain.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

STOP & THINK

Which quantity is dimensionless?

Longitudinal strain
Tensile stress
Young’s modulus

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Describe both the load and the relative change of shape
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
\sigma=\frac FA,\qquad\varepsilon=\frac{\Delta L}{L_0}
Meaning: For uniform small uniaxial tension, σ is tensile stress and ε is longitudinal strain. These engineering definitions use original area and length.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

02 / Elastic response and measuring Young’s modulus

Young’s modulus describes the material’s linear response

Within the small-strain linear tensile regime, stress is proportional to longitudinal strain. Their ratio is Young’s modulus Y. A larger Y means a larger stress is required for the same strain: the material is stiffer in this particular deformation mode.

Substituting the definitions gives ΔL = FL/(YA). The extension increases with force and original length and decreases with area and modulus. These dependencies separate material response from sample geometry. Doubling a wire’s length does not halve its Young’s modulus; it doubles its extension under the same force and area.

Young’s modulus is not the breaking stress or a general measure of toughness. A stiff material can be brittle, while another can deform substantially before failure. The modulus can depend on temperature, structure and loading direction, so comparisons assume the specified material state and the relevant linear regime.

Y=\frac{\sigma}{\varepsilon}=\frac{FL}{A\Delta L},\qquad k_{\mathrm{wire}}=\frac{YA}{L}

Y is a material-response parameter under stated conditions. The effective axial spring constant k depends on the specimen’s geometry too.

WORKED EXAMPLE

Change the specimen without changing its material

Two wires have the same material and experience the same axial force. Wire B is twice as long as wire A and has twice its diameter. Both remain in the linear regime. Compare their extensions and effective axial spring constants.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Young’s modulus describes the material’s linear response
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
Y=\frac{\sigma}{\varepsilon}=\frac{FL}{A\Delta L},\qquad k_{\mathrm{wire}}=\frac{YA}{L}
Meaning: Y is a material-response parameter under stated conditions. The effective axial spring constant k depends on the specimen’s geometry too.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

03 / Elastic response and measuring Young’s modulus

Read a stress–strain curve before applying a constant modulus

A stress–strain curve shows how response changes with loading. In the initial straight segment, its slope gives Y. Past the proportional region, a single constant ratio may no longer describe the material. If unloading leaves a permanent strain, plastic deformation has occurred.

The peak stress reached before failure and the slope near the origin answer different questions. A curve can begin steeply but terminate after little strain, or rise more gradually and extend through substantial deformation. Stiffness, elastic range, strength and energy absorption should not be treated as interchangeable words.

The area under a loading stress–strain curve is work supplied per original volume in the small-strain uniaxial description. For reversible linear elasticity, the stored energy density is ½σε = ½Yε². In hysteretic loading, the area enclosed by a load–unload cycle measures dissipated energy density rather than recoverable storage.

u_{\mathrm{elastic}}=\int_0^{\varepsilon}\sigma\,d\varepsilon=\frac12Y\varepsilon^2=\frac{\sigma^2}{2Y}

The final two forms require a reversible linear relation. Energy per volume has the same dimensions as pressure.

WORKED EXAMPLE

Check stored energy using force and extension

A linearly elastic wire has effective axial stiffness 2.0 × 10⁴ N/m and is slowly extended by 1.0 mm from its unloaded state. Find the final force and stored elastic energy.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Read a stress–strain curve before applying a constant modulus
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
u_{\mathrm{elastic}}=\int_0^{\varepsilon}\sigma\,d\varepsilon=\frac12Y\varepsilon^2=\frac{\sigma^2}{2Y}
Meaning: The final two forms require a reversible linear relation. Energy per volume has the same dimensions as pressure.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

04 / Elastic response and measuring Young’s modulus

Match the modulus to the kind of deformation

Shear stress is a tangential force per area, and small shear strain is the lateral displacement divided by the separation of the sheared faces, approximately the shear angle in radians. The shear modulus relates these quantities in a linear shear response.

Bulk modulus describes response to an increase of uniform compressive pressure. Define K = −Δp/(ΔV/V) for a small change. The minus sign makes K positive because an increased pressure normally decreases volume. Compressibility is the reciprocal 1/K.

A fluid at rest cannot sustain a permanent static shear stress in the ordinary fluid model, but it can resist compression and have a finite bulk modulus. Young’s, shear and bulk moduli describe different loading geometries. Do not substitute one for another merely because they share pascal units.

Choose the response that matches the loading
Mode	Stress	Small strain	Modulus
Axial tension	Fnormal/A	ΔL/L	Young’s Y
Shear	Ftangent/A	Δx/L	Shear modulus
Uniform compression	Increase Δp	Fractional volume decrease −ΔV/V	Bulk modulus K

WORKED EXAMPLE

Predict a small volume decrease

A liquid has bulk modulus 2.0 × 10⁹ Pa in the stated pressure range. Its pressure increases by 4.0 × 10⁶ Pa. Estimate the fractional volume change, assuming the small-change modulus is constant.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Match the modulus to the kind of deformation
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

05 / Elastic response and measuring Young’s modulus

Turn a wire-extension experiment into an inference model

To determine Young’s modulus, measure a straight wire’s effective length L, its cross-sectional area A and the change in extension produced by a known change in axial force. In the linear regime, ΔL = (L/YA)F plus any fixed displacement offset in the reading system.

Measure extension relative to a stable reference after the wire is straightened by a small initial load. Add loads gradually, allow motion to settle and keep within a reversible range. Record loading and unloading values rather than assuming every extra load leaves the specimen unchanged. Permanent set or a systematic difference between the two paths calls for investigating the model and apparatus.

For a circular wire, measure diameter at several positions and orientations with a suitable calibrated micrometer. A single reading can miss nonuniformity or an oval cross-section. Convert diameter to area using A = πd²/4. Using diameter as radius produces a factor-of-four area error and therefore a factor-of-four error in the inferred modulus.

A reference wire or independent displacement reference can help reveal support motion and common thermal changes, but it does not eliminate every error automatically. The inference assumes the measured extension belongs to the defined test length, not to bending clamps, slipping grips or motion of the support. State those boundaries before interpreting the slope.

\Delta L=c+CF,\qquad C=\frac{L}{YA},\qquad Y=\frac{L}{AC}

For an extension-versus-force graph, slope C is compliance in m/N. The offset c may represent the chosen reading zero; do not force it to zero without evidence.

STOP & THINK

If a measured circular-wire diameter is mistakenly used as the radius in A = πr², how does the inferred Y = FL/(AΔL) change?

It becomes four times too large
It becomes four times too small
It is unchanged

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Turn a wire-extension experiment into an inference model
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
\Delta L=c+CF,\qquad C=\frac{L}{YA},\qquad Y=\frac{L}{AC}
Meaning: For an extension-versus-force graph, slope C is compliance in m/N. The offset c may represent the chosen reading zero; do not force it to zero without evidence.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Back to chapter contents ↑

06 / Elastic response and measuring Young’s modulus

Use changes across a graph rather than one offset reading

A graph can separate a constant zero offset from the linear response. For ΔL on the vertical axis and F on the horizontal axis, fit the straight-line region and calculate slope from widely separated points on the fitted line. The slope has units m/N and equals L/(YA).

If the axes are reversed, the slope is force per extension, YA/L. Write the slope definition before substituting. A formula remembered for the other axis choice can invert the inferred relation while leaving the algebra superficially plausible.

Use enough observations to inspect linearity and residuals. A visually straight line over a narrow range is limited evidence. Check whether the unloaded or baseline reading returns, whether the support is stable and whether the diameter varies along the loaded length. A numerical fit cannot repair a physically incorrect area or test length.

Original idealised extension readings for a slope calculation
Additional force (N)	Extension-indicator reading (mm)
20	0.24
40	0.44
60	0.64
80	0.84

Find the compliance despite a fixed indicator offset

Inspect a plotted point: Additional force (N) = 0

Indicator: 0.04

The line has a 0.04 mm intercept. Its changes, rather than reading divided by force at one point, determine the wire response. These are idealised teaching data, not a claimed laboratory measurement. Lines connect the plotted samples.

WORKED EXAMPLE

Infer the modulus from offset data

The idealised readings above describe a wire with effective test length 1.6 m and area 0.80 mm². Assume its extension response is linear and all other apparatus effects are negligible. Infer Y from the slope.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Use changes across a graph rather than one offset reading
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

07 / Elastic response and measuring Young’s modulus

Diameter uncertainty enters twice because area is quadratic

For a circular wire using a force/extension measurement, Y = 4FL/(πd²ΔL). A first-order worst-case relative bound therefore adds the relative force, length and extension bounds plus twice the relative diameter bound. The factor two is a sensitivity to d², not an instruction to measure two diameters and add them.

Independent random standard uncertainties can instead be combined in quadrature with the corresponding sensitivity coefficients, when that statistical model is appropriate. A worst-case bound and a standard uncertainty are different reports. State which one is being calculated.

A longer wire can produce a larger extension for the same force and area, improving relative extension resolution, but it can also introduce support stability, temperature or nonuniformity concerns. More readings reduce some random effects; they do not remove a calibration bias in diameter. Because diameter enters squared, its measurement often deserves special attention.

\frac{\Delta Y}{Y}\lesssim\frac{\Delta F}{F}+\frac{\Delta L}{L}+2\frac{\Delta d}{d}+\frac{\Delta(\Delta L)}{\Delta L}

This is a first-order worst-case bound for small input bounds. Here Δ(ΔL) denotes the uncertainty in the measured extension, not another physical extension.

WORKED EXAMPLE

Propagate a stated worst-case measurement budget

A modulus calculation uses relative uncertainty bounds of 1% in force, 0.5% in test length, 1% in diameter and 2% in extension. Estimate the first-order worst-case percentage uncertainty in Y.

MEMBERSThe approach, step-by-step working and check are for members.Unlock →

STOP & THINK

If diameter alone has a small 0.8% uncertainty bound and all other inputs are treated as exact, what first-order percentage uncertainty bound does it contribute to Y?

Answer in %.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: Diameter uncertainty enters twice because area is quadratic
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
\frac{\Delta Y}{Y}\lesssim\frac{\Delta F}{F}+\frac{\Delta L}{L}+2\frac{\Delta d}{d}+\frac{\Delta(\Delta L)}{\Delta L}
Meaning: This is a first-order worst-case bound for small input bounds. Here Δ(ΔL) denotes the uncertainty in the measured extension, not another physical extension.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

08 / Elastic response and measuring Young’s modulus

A credible result includes evidence that the model held

Report the modulus with units, the tested load range, the sample geometry and an uncertainty interpretation. Show which readings support the linear region and whether unloading returned to the baseline. A large number of calculator digits cannot replace that evidence.

If the extension graph curves, investigate rather than simply fitting one line through everything. Possible explanations include leaving the proportional regime, a varying cross-section, grip motion or instrumental response. If loading and unloading differ, the difference can reveal hysteresis or permanent change; it should not be hidden by averaging without explanation.

A useful final check compares the measured compliance with L/(YA), checks that the inferred strain is within the assumed small-deformation range and tests how a changed length or diameter should affect extension. These checks connect the laboratory reading to the material model rather than leaving the experiment as a formula-substitution exercise.

Study with a tutor prompt

Copy this into your AI assistant, then describe where you got stuck. It asks the tutor to coach you one step at a time. Check its reasoning against the lesson.

Topic: Elastic response and measuring Young’s modulus
Focus: A credible result includes evidence that the model held
Learning goals:
- Define stress and strain with consistent original dimensions.
- Distinguish Young’s modulus from the specimen’s spring constant.
- Read linearity, recovery and energy from a stress–strain relation.
- Choose axial, shear or bulk response according to the loading.
- Infer Y from a force–extension slope with a stated reference.
- Propagate geometry and extension uncertainty and assess experimental validity.
Equations in this section (LaTeX):
Build the physical reasoning before choosing an equation.

Start by checking the prerequisite idea I need most. Use a simple example, then change one condition to test my understanding.

Tutoring approach:
Act as a patient physics tutor. Ask what I understand and what I have tried, then wait for my reply. Teach one step at a time; do not give the final answer or a complete solution immediately. Start with a physical picture. Define unfamiliar symbols and operators, explain signs and units, and state the model's assumptions. Give progressively stronger hints only when I need them. Ask me to predict a change before calculating it. When I finish, help me check dimensions and a limiting case, then ask one related transfer question. If information is missing or ambiguous, ask rather than inventing it. Distinguish exact results from approximations and acknowledge uncertainty.

Back to chapter contents ↑

BRING THE IDEAS TOGETHER

What should stay with you

Strain is a relative deformation; stress has pressure units.
Young’s modulus describes linear material response, while YA/L describes a wire’s axial stiffness.
Elastic recovery, linear proportionality and strength are different properties.
Graph-axis choices determine whether slope is stiffness or compliance.
Circular-wire area uses πd²/4, so diameter uncertainty has doubled first-order sensitivity.
A result needs evidence for stable references, suitable geometry and reversible response.

RETRIEVE, THEN EXPLAIN

Try it without the examples

Work these out before checking. If an answer surprises you, identify which assumption or step needs another look. A correct choice is most useful when you can explain why the alternatives fail.

SELF CHECK 1

A sample in its linear tensile regime has stress 40 MPa and strain 0.00020. What is Young’s modulus in GPa?

Answer in GPa.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 2

A wire has L = 2 m, A = 1 mm² and Y = 200 GPa. Under 50 N axial force, what is its linear-model extension in millimetres?

Answer in mm.

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 3

For extension plotted vertically against force horizontally, what is the slope?

YA/L
L/(YA)
Y alone

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 4

A material has a larger Young’s modulus than another. What does this alone establish?

It always withstands a larger breaking stress
It requires greater tensile stress for the same small elastic strain in the compared conditions
It always absorbs more energy before breaking

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

SELF CHECK 5

A force–extension indicator has a constant zero offset. Which analysis can separate it from the linear response?

Use one reading divided by its force without subtraction
Fit a line with an intercept and use its slope
Set every reading to zero before fitting

MEMBERSAnswer checking, feedback and the explanation are for members.Unlock →

APPLY THE CHAPTER

Take the ideas into a problem

Try a complete problem next. Commit to a method before revealing a hint, then compare your reasoning with the worked solution.

Stretching a thin wireIntermediate · Free sample · Hints and guided reasoning Only the heating after contact creates stressAdvanced · Hints and guided reasoning Series rods share force, not stressAdvanced · Hints and guided reasoning Read stiffness from extension dataIntermediate · Hints and guided reasoning A block shifts sidewaysIntermediate · Hints and guided reasoning Energy stored during gradual loadingIntermediate · Hints and guided reasoning

COME BACK TO THE IDEA

What can you explain later?

Try these from memory at your next study session, then again a few days later. Say your reasoning aloud or sketch it before looking. Adjust the gap to what you can explain.

Next study session

For the same material and load within the linear range, how does doubling wire length affect extension?

Compare your reasoning

Extension doubles if cross-sectional area stays fixed. The strain remains the same because stress and Young’s modulus are unchanged.

Revisit the explanation →

A few days later: change the situation

A specimen no longer follows a straight stress–strain line. Does the original Young’s-modulus equation still predict every later extension?

Compare your reasoning

No. The constant linear slope applies over its stated regime. Beyond it, response can be nonlinear, and unloading may reveal permanent deformation. Read the actual curve and loading history.

Revisit the explanation →

Compare the physical reason, the assumptions and the conclusion. Your wording can differ. If one part is missing, revisit that section and try explaining it again after a break.

Back to chapter contents ↑