Stress and strain

Stress measures force spread over an area. Strain measures relative deformation, which makes it comparable across objects with different original sizes. They are not interchangeable labels: stress has pressure units and strain is dimensionless; a constitutive relation connects them only under specified geometry, loading, material, and deformation regime.

Think like a programmer

Store force sign convention, loaded cross-sectional area, original and current length, material/loading direction, temperature/rate assumptions, and measurement uncertainty. Derive engineering stress and strain in pure functions with explicit units. Keep force-extension data separate from a fitted modulus; fitting a straight line is a modeling choice over a selected elastic interval, not a property of every point in a plot.

Model checklist

Inputs: Axial force and sign convention, loaded area, reference/current lengths, material/loading direction, force-extension samples, elastic-fit interval, temperature/rate policy, and units.
State: Stress, strain, optional modulus fit/residual, loading/unloading path, and validity flags.
Rule: Calculate stress as force divided by area and strain as extension divided by original length; compare measurements to a declared constitutive relation only in its stated regime.
Output: Signed stress/strain data, elastic slope/modulus estimate, fit residual, geometry metadata, and yield/nonlinearity warning.
Check: Doubling area halves stress at fixed force; scaling both original length and extension equally preserves strain; zero extension gives zero engineering strain; linear elastic fixture gives stress equal to modulus times strain; loading/unloading differences are recorded rather than fitted as one reversible line.

\[\sigma=\frac{F}{A}\qquad\varepsilon=\frac{\Delta L}{L_0}\]

Spring force and stored energy

Change stiffness and extension. The spring force points back toward equilibrium, while the stored energy stays positive.

Stiffness: 40 N/mExtension: 0.35 m

Stiffness 40 N/m; extension 0.35 m; restoring force -14.00 N; elastic energy 2.45 J.

For a small, axial, linear-elastic deformation, Hooke-style material response is sigma equals E epsilon, where E is Young's modulus. It has pressure units, so a strain multiplied by E returns stress. This unit check catches use of length change as though it were normalized strain. It does not say a material remains linear to fracture or that the same E applies in every direction.

\[\sigma=\frac{F}{A},\qquad\varepsilon=\frac{L-L_0}{L_0},\qquad\sigma=E\varepsilon\;\text{(small, linear axial regime)}\]

Choose an engineering convention and preserve it. Tension/compression signs, current versus original area, and engineering versus true strain can differ in larger deformations. If reporting a fitted modulus, record sample range, loading rate, temperature, uncertainty, and residual. A single slope from a visibly nonlinear curve is a reduction whose scope needs a name.

Try this experiment

Prediction: At fixed force, a thicker bar has lower stress, while scaling original and changed length together leaves engineering strain unchanged.

Vary force, area, original length, and extension one at a time. Check units and the area/length scaling fixtures. Create a small linear data set, fit its slope, and identify the stress/strain interval used. Add a loading/unloading difference and explain why one reversible elastic fit no longer tells the complete energy story.

Where this model breaks

These scalar definitions describe a simple one-dimensional loading case. Real materials can be anisotropic, plastic, viscoelastic, damaged, temperature/rate dependent, or subject to shear, bending, buckling, and changing area. Large deformation needs a stated finite-strain/current-geometry convention; a linear fit is not a safety limit or fracture prediction.

Summary

Use stress for load density and strain for normalized shape change. Preserve geometry, sign, units, regime, and path metadata; treat a modulus as a fitted/limited constitutive parameter rather than a universal display label.

Glossary

Stress: force per area.
Strain: fractional change in length.
Cross-sectional area: area resisting an axial load.
Young's modulus: small-strain axial stiffness parameter with pressure units.
Constitutive relation: model relating stress, strain, and sometimes history/rate.

Self-check

What are the units of stress?
Why is strain dimensionless?
How does area affect stress?
Which data defines the valid interval of a fitted modulus?

Sources

OpenStax, University Physics Volume 1, elasticity.
L. D. Landau and E. M. Lifshitz, Theory of Elasticity.

Model contract

Treat the lesson as a small function before treating it as a fact to memorize. Give every value a unit, keep only the state needed for the next step, and make the output easy to inspect.

\[\text{observable output} = f(\text{inputs},\,\text{state})\]

Inputs: Quantities you set or measure, with units and useful bounds.
State: Values the program must retain to reproduce the next result.
Rule: The relationship or update that turns inputs and state into a result.
Check: A known limit, unit check, invariant, or measured result that can expose a bad model.

Implement the idea as a model

For Stress and Strain, write down the quantities you can control, the values your program must retain, and the result a reader could inspect. In Equilibrium and Elasticity, the useful program is not the drawing: it is the smallest explicit model that makes a prediction you can test.

Guided experiment

Prediction: changing one declared input while holding the others fixed should change only the outputs that the model connects to that input. Choose one input, predict the direction of change, then check a limiting case such as zero, a symmetric arrangement, or a familiar low-speed or small-change approximation.

Where this model breaks

This lesson is a teaching model, not a complete simulator. Before using it outside the stated question, check which interactions, scales, uncertainties, boundary conditions, and measurement limits it leaves out.

Summary

Treat Stress and Strain as a contract: named inputs and units enter a rule, the rule produces an observable result, and a known limit or invariant checks whether the implementation deserves trust.

Glossary

Input: a measured value or chosen parameter supplied to a model.
State: the smallest set of values needed to continue or reproduce a model.
Validation: comparing an output with a known result, limit, invariant, or measurement.

Self-check

Which values are inputs, and which values must remain state?
What observable result would tell you the model is behaving as expected?
Which assumption would you test first before applying the model to a real system?

Model review: turn Stress and Strain into a test

Use force per area and normalized deformation to model simple material response.

Name the inputs and units that the equilibrium and elasticity model needs.
Separate the state you must keep from values you can calculate when needed.
Write one rule that maps the current state and inputs to an observable result.
Choose a limiting case, unit check, invariant, or known result before trusting an output.
State one assumption you would change before using this simplified model for a real decision.

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