Constitutive Response and Material Testing

Prerequisite knowledge: force equilibrium, SI dimensional analysis, algebra and first-year calculus.

Learning objectives: state a model and its validity range, calculate physically meaningful engineering quantities, reconcile equations with a free-body diagram or deformation compatibility, and identify at least two failure modes excluded by the idealization.

Unit context and engineering interpretation

Materials data require careful interpretation. Elastic modulus is a slope fitted over a strain interval, not simply the largest reported stress divided by elongation at fracture. Offset yield strength is defined using a prescribed offset strain and a measured elastic slope. Ultimate tensile engineering stress is the maximum test force divided by original specimen area, and it can differ greatly from true stress in the necked region. Percent elongation measured after fracture depends on gage length and specimen geometry, so it should not be compared blindly across standards.

A Poisson ratio describes how an initially uniaxially stressed isotropic sample contracts or expands transversely. Under a fully constrained plane-strain condition, the same elastic modulus and Poisson ratio produce a different relationship between applied stress and deformation. This is why a concrete pier confined by surrounding soil, an elastomer bonded to rigid plates, and an unconstrained tensile coupon cannot be modeled by one scalar spring stiffness without checking the state of stress. Anisotropic composites demand directional elastic constants instead of one E and nu.

Plastic strain is path dependent. In an ideal bilinear material the first loading branch is linear, and subsequent loading follows a reduced tangent stiffness. Unloading commonly follows a steeper elastic slope. The crossing of the unloaded line with the zero-stress axis illustrates residual strain. This simplified graph is valuable for seeing how permanent deformation develops, but it cannot predict damage, low-cycle fatigue, reverse yielding or rate-dependent creep. Appropriate model complexity follows the engineering question.

Lesson 3.1 — Tensile testing and engineering curves

Physical mechanism and governing idea

A tensile test measures force and extensometer displacement over a defined gage length. Engineering stress divides load by original area; engineering strain divides gage extension by original length. Elastic slope, offset proof strength, ultimate strength and fracture elongation answer different questions. A testing machine reports a curve only after calibration and careful specimen preparation.

Governing relationship

sigma_eng = F/A0; epsilon_eng = delta/L0

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

A 20 kN test force acting on a 100 mm² original section means 200 MPa engineering stress.

What the simplified formula misses

Engineering stress differs from true stress when the area changes significantly. Post-necking curve interpretation needs more than a nominal conversion.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 3.1 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 3.2 — Poisson ratio and multiaxial elastic response

Physical mechanism and governing idea

An axially stretched isotropic linear elastic specimen generally contracts laterally. The Poisson ratio relates transverse strain to axial strain under uniaxial stress. In three-dimensional elasticity, strain depends on all three normal stresses. Material properties must be assessed for the appropriate direction in composites and other anisotropic solids.

Governing relationship

epsilon_trans = -nu epsilon_axial; epsilon_x = (sigma_x - nu(sigma_y + sigma_z))/E

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

An axial strain of 900 microstrain and nu=0.30 predicts a transverse strain of -270 microstrain in uniaxial linear elasticity.

What the simplified formula misses

The Poisson formula alone is not sufficient for a plane-strain constraint or a layered material that responds differently in different directions.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 3.2 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 3.3 — Yield, plasticity and unloading

Physical mechanism and governing idea

When load increases beyond initial linearity, permanent strain can develop. A bilinear tangent model idealizes strain hardening by assigning a second lower slope after the yield point. Unloading from plastic strain often occurs approximately along the elastic modulus, leaving a residual strain. That is why a bar can return to zero load without returning to its original length.

Governing relationship

Elastic: sigma=E epsilon; ideal bilinear after yield: sigma=fy+Et(epsilon-fy/E)

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

For E=200 GPa and fy=250 MPa, the elastic yield strain is 0.00125.

What the simplified formula misses

A teaching bilinear model is not a calibrated constitutive law. Cyclic loading, Bauschinger effects and localization are omitted.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 3.3 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Lesson 3.4 — Temperature, strain rate and property scatter

Physical mechanism and governing idea

A strength value comes from a test population under specified conditions. Temperature, loading rate, specimen orientation, defects, processing route and manufacturing history may change measured strength and ductility. Designers need characteristic or specified properties appropriate to their code and product form, not the single highest laboratory result.

Governing relationship

Use a documented property distribution and the applicable design standard

Reading the equation: Start from a properly oriented free body or a measured constitutive response, identify the force or displacement controlled externally, and decide which geometric quantity belongs in the denominator. Check units before substituting numbers, and compare the sign of the result with the physical action.

Worked interpretive example

If two specimens yield at 242 and 258 MPa, reporting the average as a guaranteed minimum would be an unsupported inference.

What the simplified formula misses

The workbook simulations use illustrative materials. They are not material certification data or replacements for manufacturer design allowables.

Engineering reasoning exercise

1. Explain in your own words which observable physical quantity the formula in Lesson 3.4 is estimating. Specify where it would be measured.
2. Change exactly one of the underlying dimensions or loading parameters, predict the direction of change before calculating, and say which other variables are held constant.
3. Invent a connection, defect or boundary condition under which the approximation in this lesson becomes unreliable. Suggest the simplest more faithful model.
4. State a dimensional consistency check and one limiting case that should hold if the stated equations are applied correctly.

Worked numerical casebook

The following 18 worked cases are intended for dimensional fluency and model interpretation, not as a substitute for unique field measurements. Cases change materials, loads and geometry and deliberately restate their validity conditions.

When reproducing a result, document a complete free body or section cut, record the input units, and calculate at least one quantity without looking at the worked result.

Case 3.01: Elastic-plastic material idealization for a maintenance bracket

An illustrative bilinear test has initial E=200 GPa, yield strength 240 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.001200. At total engineering strain 0.0008, the bilinear loading stress is 160.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.01: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.02: Elastic-plastic material idealization for a temporary support

An illustrative bilinear test has initial E=200 GPa, yield strength 355 MPa and tangent Et=18 GPa. The elastic limit strain is fy/E=0.001775. At total engineering strain 0.0026, the bilinear loading stress is 369.85 MPa. Unloading along the original elastic slope leaves an estimated 750.7 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.02: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.03: Elastic-plastic material idealization for a pipe rack

An illustrative bilinear test has initial E=210 GPa, yield strength 240 MPa and tangent Et=12 GPa. The elastic limit strain is fy/E=0.001143. At total engineering strain 0.0012, the bilinear loading stress is 240.69 MPa. Unloading along the original elastic slope leaves an estimated 53.9 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.03: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.04: Elastic-plastic material idealization for a solar panel rail

An illustrative bilinear test has initial E=210 GPa, yield strength 240 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.001143. At total engineering strain 0.0008, the bilinear loading stress is 168.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.04: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.05: Elastic-plastic material idealization for a transit shelter

An illustrative bilinear test has initial E=69 GPa, yield strength 400 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.005797. At total engineering strain 0.0018, the bilinear loading stress is 124.20 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.05: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.06: Elastic-plastic material idealization for a lightweight crane hook

An illustrative bilinear test has initial E=210 GPa, yield strength 355 MPa and tangent Et=2 GPa. The elastic limit strain is fy/E=0.001690. At total engineering strain 0.0004, the bilinear loading stress is 84.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.06: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.07: Elastic-plastic material idealization for a water-tank frame

An illustrative bilinear test has initial E=200 GPa, yield strength 320 MPa and tangent Et=2 GPa. The elastic limit strain is fy/E=0.001600. At total engineering strain 0.0008, the bilinear loading stress is 160.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.07: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.08: Elastic-plastic material idealization for a portal-frame member

An illustrative bilinear test has initial E=105 GPa, yield strength 355 MPa and tangent Et=2 GPa. The elastic limit strain is fy/E=0.003381. At total engineering strain 0.0012, the bilinear loading stress is 126.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.08: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.09: Elastic-plastic material idealization for a cable termination

An illustrative bilinear test has initial E=210 GPa, yield strength 250 MPa and tangent Et=12 GPa. The elastic limit strain is fy/E=0.001190. At total engineering strain 0.0004, the bilinear loading stress is 84.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.09: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.10: Elastic-plastic material idealization for a maintenance walkway

An illustrative bilinear test has initial E=105 GPa, yield strength 250 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.002381. At total engineering strain 0.0018, the bilinear loading stress is 189.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.10: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.11: Elastic-plastic material idealization for a girder

An illustrative bilinear test has initial E=69 GPa, yield strength 160 MPa and tangent Et=18 GPa. The elastic limit strain is fy/E=0.002319. At total engineering strain 0.0008, the bilinear loading stress is 55.20 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.11: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.12: Elastic-plastic material idealization for a roof tie

An illustrative bilinear test has initial E=210 GPa, yield strength 160 MPa and tangent Et=12 GPa. The elastic limit strain is fy/E=0.000762. At total engineering strain 0.0008, the bilinear loading stress is 160.46 MPa. Unloading along the original elastic slope leaves an estimated 35.9 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.12: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.13: Elastic-plastic material idealization for a machine anchor

An illustrative bilinear test has initial E=200 GPa, yield strength 320 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.001600. At total engineering strain 0.0008, the bilinear loading stress is 160.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.13: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.14: Elastic-plastic material idealization for a laboratory frame

An illustrative bilinear test has initial E=200 GPa, yield strength 160 MPa and tangent Et=18 GPa. The elastic limit strain is fy/E=0.000800. At total engineering strain 0.0012, the bilinear loading stress is 167.20 MPa. Unloading along the original elastic slope leaves an estimated 364.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.14: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.15: Elastic-plastic material idealization for a small footbridge

An illustrative bilinear test has initial E=69 GPa, yield strength 240 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.003478. At total engineering strain 0.0012, the bilinear loading stress is 82.80 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.15: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.16: Elastic-plastic material idealization for a service gantry

An illustrative bilinear test has initial E=210 GPa, yield strength 250 MPa and tangent Et=8 GPa. The elastic limit strain is fy/E=0.001190. At total engineering strain 0.0004, the bilinear loading stress is 84.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.16: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.17: Elastic-plastic material idealization for a construction prop

An illustrative bilinear test has initial E=210 GPa, yield strength 320 MPa and tangent Et=12 GPa. The elastic limit strain is fy/E=0.001524. At total engineering strain 0.0018, the bilinear loading stress is 323.31 MPa. Unloading along the original elastic slope leaves an estimated 260.4 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.17: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Case 3.18: Elastic-plastic material idealization for a testing rig

An illustrative bilinear test has initial E=200 GPa, yield strength 160 MPa and tangent Et=18 GPa. The elastic limit strain is fy/E=0.000800. At total engineering strain 0.0008, the bilinear loading stress is 160.00 MPa. Unloading along the original elastic slope leaves an estimated 0.0 microstrain of permanent strain. The result is a model output, not certification of an actual material heat or temper.

Extension task 3.18: Replace one assumed boundary condition with a compliant or imperfect connection. Which input would you need to measure, and which claim above would cease to be justified?

Engineering case study: Material substitution for a small lifting bracket

A project team considers replacing a steel component with aluminum to reduce mass. The two candidate materials may have different yield strengths, moduli, density and fatigue behavior. Equal dimensions do not imply equal stiffness or equal safety. If applied load is prescribed, elastic extension scales inversely with E; if assembly displacement is prescribed, reactions can increase with stiffness.

Deliverable: Create a decision matrix separating stiffness, strength, deformation limit, corrosion exposure, joining, cost and standard-compliant inspection. Do not assign a single winner until required limits and the actual load control method are known.

Case-study discussion

Start by differentiating data from assumptions: measured loads and geometry should be separated from guessed restraints and material properties. Build one simple calculation whose result can be checked independently. List at least three mechanisms by which the idealization could fail, and prioritize them by whether they would change the engineering conclusion. Translate the numerical demand into a practical question about safety, serviceability, testing or maintenance. Keep a clear audit trail so a reviewer can follow the unit conversions, load path and model limitations.

Calculation verification

• Obtain E from the local elastic slope, not from ultimate strength divided by total elongation.
• On a bilinear model, unload from two distinct peak strains and compare residual strain predictions.
• Check positivity, zero-load behavior, symmetry, and asymptotic scaling whenever applicable.
• Do not use a graph or slider response as the only evidence that an equation is correct.

Structured unit assessment

Answer the following without access to the provided worked solutions. Each question needs a diagram, assumptions and a clear interpretation.

1. Use a load or material value from Case 3.02, change one geometric dimension by 20%, and calculate the resulting response with explicit units.
2. Explain how the result of Lesson 3.1 would change if one apparently rigid support had finite stiffness.
3. Choose a nonuniform or damaged member for which the idealization in Lesson 3.2 becomes inaccurate; describe a better analytical or experimental representation.
4. Derive a useful limiting case from the relationship in Lesson 3.3 and compare with a limiting setting in any associated simulation.
5. A reviewer reports a one-thousandfold discrepancy. Show one plausible N–kN, m–mm or N·m–N·mm conversion error that could create it for a calculation in this unit.
6. For the case study in Unit 3, distinguish two possible failure modes and explain why demonstrating adequate performance against one does not guarantee the other.
7. Create an experimental plan to validate one of the numerical predictions in Unit 3; name the sensor, its calibration requirement and an expected source of uncertainty.
8. State two ways that manufacturing tolerance, support flexibility or temperature could alter a result computed from one equation in Unit 3.

Associated interactive laboratories

Laboratory 02: Material Stress–Strain & Unloading. A simplified uniaxial bilinear material has elastic modulus E, yield strength fy and post-yield tangent Et. Its ideal unloading slope is E. The plotted curve is not a universal material law.

Investigation: Change fy without changing E. Compare the stress at a prescribed total strain; then investigate residual strain as a function of post-yield tangent modulus.

Open the [working HTML model](../labs/lab-02-material.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Laboratory 11: Strain-Gauge Bridge Measurement. A longitudinal strain gauge follows axial strain in an elastic prismatic member. A quarter bridge produces a small voltage approximately proportional to gauge factor, excitation voltage and strain for small resistance changes.

Investigation: Double the excitation voltage without changing the bar load. Predict why bridge voltage doubles but physical strain does not. Compare signal amplitude with a 10-microvolt noise floor.

Open the [working HTML model](../labs/lab-11-gauge.html), record inputs and CSV data, and compare numerical outputs with an independent calculation. Identify the limiting assumptions before discussing how a real physical experiment would differ.

Unit research project

Prepare a technical notebook on Constitutive response and material testing for a civil engineering structure of your choice. Include a drawing with geometric dimensions, a carefully numbered input table and a free-body diagram. Develop two load or restraint scenarios, compute the most relevant stress or deformation variables, cross-check them with an independent limiting case, and identify all omissions that would prevent formal design approval. Where an associated laboratory exists, export a CSV series and include an annotated plot. Write a short conclusion that distinguishes what your simplified model genuinely predicts from what would need a professional code-based check.

Project marking rubric