SCIENCE · INVESTIGATE AND EXPLAIN

Separate material modulus from specimen geometry and validity

A material can deform even when its overall translational motion is in equilibrium. For uniform axial tension, stress is force divided by cross-sectional area and strain is extension divided by original length. Stress has pressure units; strain is dimensionless. The Young modulus E is stress/strain within a stated linear regime, not the same quantity as a particular specimen’s spring stiffness.

Learning outcome: Compare areas1,2,4 mm² at 20 N and compare20 with 40 N at 1 mm². Retain SI area, stress, conditional strain and whether the linear model is valid. Explain why a bounded elastic relation does not establish breaking strength.

Starting recap

Declare the axis, angular units, recipient and valid contact or material branch. Distinguish supplied shape factors from independently measured inertia.

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At F=20 N,A=1 mm², the predicted strain is exactly0.001, the stated boundary, and extension 0.0005 m. This is a permitted model endpoint, not a measured material yield point.

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Predict

One square millimetre equals:

Your prediction and reason

Experiment

Compare areas1,2,4 mm² at 20 N and compare20 with 40 N at 1 mm². Retain SI area, stress, conditional strain and whether the linear model is valid. Explain why a bounded elastic relation does not establish breaking strength.

Separate material strain from specimen stiffness

Worked teaching model · synthetic values · Original hypothetical linear material E=2e10 Pa, length .5 m, uniform axial tension. Linear response is stipulated only for strain at most .001. Requested states beyond this limit are labeled unsupported; no strength, fracture or safe-load claim. Areas converted from mm² to m².

Tensile force: 20 N Cross-sectional area: 2 mm²

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Supported small-strain elastic law

Unlike graph quantities rescale independently; read units and values, not screen angles. 1: linear range only (solid). Joined curves show the supplied model, not additional observations.

Equivalent numerical coordinates for Supported small-strain elastic law

linear range only

strain (ratio)	tensile stress (Pa)
0	0
0.000025	500000
0.00005	1000000
0.000075	1500000
0.0001	2000000
0.000125	2500000
0.00015	3000000
0.000175	3500000
0.0002	4000000
0.000225	4500000
0.00025	5000000
0.000275	5500000
0.0003	6000000
0.000325	6500000
0.00035	7000000
0.000375	7500000
0.0004	8000000
0.000425	8500000
0.00045	9000000
0.000475	9500000
0.0005	10000000
0.000525	10500000
0.00055	11000000
0.000575	11500000
0.0006	12000000
0.000625	12500000
0.00065	13000000
0.000675	13500000
0.0007	14000000
0.000725	14500000
0.00075	15000000
0.000775	15500000
0.0008	16000000
0.000825	16500000
0.00085	17000000
0.000875	17500000
0.0009	18000000
0.000925	18500000
0.00095	19000000
0.000975	19500000
0.001	20000000

At most 82 representative computed coordinates including endpoints; the plotted curve can use a finer grid. A table of a revealed generator is not independent measurement evidence.

Calculated quantities under the declared measurement model
Quantity	Value	Unit
Cross-sectional area in SI	0.000002	m²
Applied tensile stress	10000000	Pa
Conditional linear strain	0.0005	ratio
Stipulated linear range valid	1	flag
Conditional axial stiffness	80000	N/m
Predicted extension	0.00025	m
Elastic stored energy	0.0025	J

Stress and strain describe material response; axial stiffness also depends on specimen geometry.

Stress=F/A, strain=DeltaL/L and E=stress/strain, giving DeltaL=FL/(EA) and k=EA/L only within the declared linear range. One mm² is 10⁻⁶ m², not 10⁻³. A smaller area increases stress and compliance at the same force. The .001 limit is a supplied model boundary, not a measured yield or breaking strength.

Predict before inspecting the values. Save contrasting records with assumptions and one alternative explanation. This virtual display does not certify practical measurement technique.

Stress and strain describe material response; axial stiffness also depends on specimen geometry.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

Choose the axis, reference or material model

Build the algebraic relationship

The original hypothetical material has E=2×10^10 Pa and specimen length 0.5 m. The area control is in mm², requiring conversion by 10⁻⁶ to m². The linear prediction is ΔL=FL/(EA) and axial stiffness k=EA/L. Doubling area doubles stiffness and halves extension at the same force; doubling length does the reverse without changing the material modulus.

Check the original numerical packet

The supplied linear model is asserted only through strain 0.001. At F=20 N and A=2 mm², stress is 10^7 Pa, strain 0.0005 and extension 0.00025 m. At F=40 N and A=1 mm², extrapolated linear strain would be0.002, outside that regime. The workbench labels the state unsupported and does not report an actual extension, yield or fracture outcome.

Test the boundary and independent evidence

Small-strain shear uses shear stress and sideways displacement divided by thickness, with a distinct shear modulus. Compression of volume uses pressure change and fractional volume decrease with bulk modulus. These are different deformations, not opportunities to substitute Young modulus into every formula. For an original supplied shear modulus4×10^9 Pa and stress2×10^6 Pa, shear strain is 0.0005; for bulk modulus2×10^9 Pa and pressure increase10^6 Pa, fractional volume change is−0.0005. Predict the signs and ratios before checking. No strength or safe working load is inferred from any of these elastic ratios.

Worked original case

At F=20 N,A=1 mm², the predicted strain is exactly0.001, the stated boundary, and extension 0.0005 m. This is a permitted model endpoint, not a measured material yield point.

Bounded investigation

Changed-case numerical practice

Try the changed case before feedback

Within a linear regime, a specimen has E=10^10 Pa,L=1 m,A=2 mm² and tension10 N. Find extension in millimetres.

Numerical result in the requested units

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and limits

Original content checked 30 October 2026 against OpenStax College Physics 2e. All records are supplied or generated teaching evidence. No rotating equipment, applied loads, material tests or physical assembly is requested. An admissible virtual model state does not certify real equipment safety, supervised practical skill or independent mastery.

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Public practice checks a specific claim. It does not establish independent mastery.

Why do a material modulus and a specimen spring constant have different units and geometry dependence?

Your explanation and limitations

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