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Stress and strain

Stress describes how strongly internal forces act across an area. Strain describes deformation relative to an original dimension. Force and extension alone depend on an object's size; stress and strain make material responses easier to compare across specimens with different cross-sections and lengths.

Normal stress

For a straight bar under a centered axial force, average engineering normal stress is

\[\sigma=\frac{F_\perp}{A_0}\]

Here \(F_\perp\) is the force perpendicular to the cross-section and \(A_0\) is the original area. Stress has SI units of pascals:

\[1\ \mathrm{Pa}=1\ \mathrm{N/m^2}\]

Tensile stress pulls the bar and compressive stress pushes it. A sign convention may take tension as positive and compression as negative, or report positive magnitudes with labels. Either is acceptable when stated consistently.

The formula gives an average over a section. Stress can vary strongly near holes, sharp corners, contacts, cracks, and load application points. Dividing total force by gross area does not reveal those stress concentrations.

Engineering strain

For original gauge length \(L_0\) and current length \(L\), axial engineering strain is

\[\varepsilon=\frac{\Delta L}{L_0}=\frac{L-L_0}{L_0}\]

Strain is dimensionless because it is a ratio of lengths, though it may be written as metres per metre, a percentage, or microstrain. A strain of \(0.001\) is \(0.1\%\) or one thousand microstrain.

Normalizing by original length matters. Two bars that each extend one millimetre do not have the same strain if their original lengths differ.

import numpy as np

original_length, current_length = 1.0, 1.001
area, force = 1e-4, 2e4
strain = (current_length - original_length) / original_length
stress = force / area
print(f"Strain: {strain:.6f}; stress: {stress:.3g} Pa")

Require positive reference length and area. Retain unrounded inputs and declare whether dimensions describe the unloaded reference state.

Linear elasticity and Young's modulus

Many materials approximately obey a linear relation for sufficiently small axial stress and strain:

\[\sigma=E\varepsilon\]

Young's modulus \(E\) is the slope of the linear elastic stress-strain curve and has pressure units because strain is dimensionless. A larger modulus means more stress is required for the same small strain; it does not by itself mean the material has a higher breaking strength.

Combining definitions gives the small-extension relation

\[\Delta L=\frac{FL_0}{EA_0}\]

This assumes a uniform bar, centered axial load, small deformation, and a constant modulus. It is not a general bending, buckling, or fracture formula.

Worked axial example

A wire of length \(2.0\,\mathrm m\), area \(1.0\,\mathrm{mm^2}\), and Young's modulus \(200\,\mathrm{GPa}\) carries a tensile force of \(100\,\mathrm N\). Convert area:

\[1.0\ \mathrm{mm^2}=1.0\times10^{-6}\ \mathrm{m^2}\]

The stress is

\[\sigma=\frac{100}{1.0\times10^{-6}}=1.0\times10^8\ \mathrm{Pa}=100\ \mathrm{MPa}\]

The linear strain and extension are

\[\varepsilon=\frac{100\ \mathrm{MPa}}{200\ \mathrm{GPa}}=5.0\times10^{-4}\] \[\Delta L=\varepsilon L_0=(5.0\times10^{-4})(2.0\ \mathrm m)=1.0\ \mathrm{mm}\]

The calculation is valid only if the material remains in the stated linear elastic range and the idealized load is axial.

Shear stress and strain

Tangential force across area creates average shear stress

\[\tau=\frac{F_\parallel}{A}\]

For a block of height \(h\) whose top shifts sideways by \(\Delta x\), small shear strain is

\[\gamma\approx\frac{\Delta x}{h}\]

In the small linear regime,

\[\tau=G\gamma\]

where \(G\) has pascal units. Shear strain is an angle in radians in the small-angle approximation. Young's modulus and shear modulus describe different loading modes and are not generally interchangeable.

Bulk compression

Nearly uniform pressure changes volume. Volume strain is

\[\frac{\Delta V}{V_0}\]

and the bulk modulus is commonly defined so compression gives a positive modulus:

\[B=-\frac{\Delta p}{\Delta V/V_0}\]

The minus sign reflects that increasing external pressure usually decreases volume. As with the other moduli, this simple ratio describes a small approximately linear response.

Poisson response

Stretching a bar longitudinally usually makes it narrower. Poisson's ratio is

\[\nu=-\frac{\varepsilon_{\mathrm{transverse}}}{\varepsilon_{\mathrm{axial}}}\]

It is dimensionless. For an isotropic linear elastic material, elastic constants are related; for example,

\[E=2G(1+\nu)\]

This relation is not valid for an arbitrary anisotropic material. Crystals, composites, wood, and biological tissues can respond differently with direction and may require a tensor constitutive law.

Elastic, plastic, and failure regions

Elastic deformation is recoverable after unloading, but elastic response is not necessarily perfectly linear. The proportional limit marks departure from a straight stress-strain relation; the elastic limit concerns recovery. Yielding produces significant permanent deformation. Ultimate tensile strength is the maximum engineering stress on a tensile curve, and fracture ends the specimen's load-bearing continuity.

These points are not all identical, and their reported values depend on material, temperature, strain rate, specimen preparation, and measurement convention. A modulus fit is not a yield or safety criterion.

Loading and unloading curves may differ. Their enclosed area represents dissipated energy per reference volume under the chosen stress-strain measures. Fitting both with one line can erase hysteresis.

Engineering and true measures

Engineering stress uses original area and engineering strain uses original length. At large deformation, area and length change substantially. True normal stress uses current area, while logarithmic true strain for uniaxial stretch is

\[\varepsilon_{\mathrm{true}}=\ln\left(\frac{L}{L_0}\right)\]

The two measures agree approximately for small strain but diverge at large strain. State which measures a data table or constitutive law uses before comparing values.

Fitting a modulus

Select a justified linear elastic interval rather than fitting the whole curve. If the model includes an intercept,

\[\sigma=E\varepsilon+sigma_0\]

a nonzero fitted intercept can indicate sensor offset, preload, or an interval that is not truly linear. Forcing the fit through the origin should be a documented physical choice, not an automatic cleanup.

strain_samples = np.array([0.0, 0.0002, 0.0004, 0.0006, 0.0008])
stress_samples = 200e9 * strain_samples
modulus, intercept = np.polyfit(strain_samples, stress_samples, 1)
residuals = stress_samples - (modulus * strain_samples + intercept)
print(f"Fitted modulus: {modulus/1e9:.1f} GPa")
print("Residuals (Pa):", residuals)

Report uncertainty, sample direction, loading rate, and temperature. A high coefficient of determination alone cannot prove physical linearity if the chosen range is too small or measurement errors are structured.

Tensor viewpoint

At a point in a three-dimensional material, stress and small strain are symmetric second- order tensors under ordinary continuum assumptions. Normal and shear components depend on the orientation of the plane on which they are resolved. One scalar stress cannot describe a general multiaxial state.

Linear elasticity then relates stress and strain tensors through material stiffness. An isotropic solid needs two independent elastic constants; an anisotropic solid can require many more. The one-dimensional formulas are controlled reductions of this richer model.

Reliability checks

Stress and strain separate loading intensity from relative deformation. Their simple ratios are useful, but a constitutive relation becomes physically meaningful only after geometry, material direction, reference state, and deformation regime are declared.

A worked Python notebook

Use an illustrative elastic modulus of 200 gigapascals to calculate axial stress. The question for this calculation is precise: how does engineering stress change when we change engineering strain, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

Linear elasticity applies only within the material's small-strain elastic range. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 0.001 dimensionless. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for engineering strain, expressed in dimensionless. The output is engineering stress, expressed in Pa. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 200000000000.0 q\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 0.0005 dimensionless, the expression gives 1e+08 Pa. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("200e9*q", locals={"q": q})
reference_input = 0.0005
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "dimensionless")
print("Predicted engineering stress:", reference_output, "Pa")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 Pa. The midpoint, at 0.0005 dimensionless, gives 1e+08 Pa. The upper endpoint gives 2e+08 Pa. Read these together with the assumptions: they describe different choices of engineering strain within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 0.0005, 0.001])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} dimensionless -> {output_value:.6g} Pa")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows engineering strain; the vertical axis shows engineering stress. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 0.001, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Engineering strain (dimensionless)")
ax.set_ylabel("Engineering stress (Pa)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does engineering stress change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 0.0005 dimensionless is 2e+11 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 200000000000.0\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 0.000505
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "Pa")
print("Full expression:", full_prediction, "Pa")
print("Approximation difference:", full_prediction - linear_prediction, "Pa")

Here the changed input is 0.000505 dimensionless. The tangent estimate is 1.01e+08 Pa, while the full expression gives 1.01e+08 Pa. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 0.001
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of engineering strain inside the stated interval, predict whether engineering stress will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: Linear elasticity applies only within the material's small-strain elastic range. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of Pa: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(0.001) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 0.001, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

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