Hooke’s law and Young’s modulus

The simplest constitutive relation is linear,

\[ \boxed{\sigma = E\varepsilon} \tag{8.6.1}\]

known as Hooke’s law, in which the constant of proportionality \(E\) is the Young’s modulus, or modulus of elasticity. It has the same units as stress, since strain is dimensionless, and it is the stiffness of the material in the same way that a spring constant is the stiffness of a spring.

8.6.1 holds only in the elastic region, the part of the response in which deformation grows linearly with load and vanishes when the load is removed. Not every material has a well defined elastic region, and some have none worth speaking of, so the law is a model rather than a fact about matter. Where it applies, the definition of the modulus follows from the definitions of stress and strain,

\[ E := \frac{\Delta\sigma}{\Delta\varepsilon} = \frac{F/A_0}{\Delta L / L_0} = \frac{F L_0}{A_0 \Delta L} \tag{8.6.2}\]

valid as long as \(\Delta L \ll L_0\), which is the same small-deformation assumption that produced 8.5.1. Typical values are \(E \approx 210\,000~\text{MPa}\) for steel, \(68\,000~\text{MPa}\) for aluminium and \(115\,000~\text{MPa}\) for titanium. Printed polymers sit two orders of magnitude lower, around \(2\,000\) to \(3\,500~\text{MPa}\) for PLA and ABS. These figures vary with composition, temperature, strain rate and, for cast or printed material, with position inside the part.

The tensile test

A real record looks less tidy than Figure 8.6.1. Figure 8.6.2 shows a measured curve for a polymer specimen, plotted against machine extension rather than strain, with the elastic slope and a yield level drawn on by hand.

The tensile test is the experiment that produces \(E\) and everything else on Figure 8.6.1. A specimen of known cross section is gripped at both ends and pulled apart at a controlled rate while the machine records the force \(F\) it applies and the extension \(\Delta L\) of a gauge length \(L_0\). The specimen is waisted so that failure occurs in the middle rather than in the grips, which is why tensile specimens have the dogbone shape.

The machine measures force and displacement, but neither is a material property: both depend on how big the specimen happens to be. Dividing by the original cross-sectional area \(A_0\) and the original gauge length \(L_0\) removes the geometry and leaves the engineering stress and engineering strain

\[ \sigma = \frac{F}{A_0}, \qquad \varepsilon = \frac{\Delta L}{L_0} \tag{8.6.3}\]

which are properties of the material alone. Both use the original geometry, which is why they are called engineering rather than true quantities. As the specimen stretches its cross section shrinks, so the true stress on the actual area is higher than the engineering stress, and the difference becomes large once necking begins. For determining the elastic constants the distinction does not matter, because the strains involved are a fraction of a percent.

Reading the curve

Four quantities are normally extracted from a test record. The Young’s modulus is the slope of the initial straight portion. The yield strength marks the end of elastic behaviour. The ultimate tensile strength \(R_m\) is the highest engineering stress the specimen reaches, and the elongation at break is the strain at which it separates, a measure of ductility.

The yield strength needs care because most materials do not yield abruptly. Mild steel is the exception, showing a distinct upper and lower yield point that can simply be read off. Aluminium, titanium and most polymers curve over gradually, with no point that is obviously the end of the elastic region. The convention adopted by the testing standards is the offset yield strength \(R_{p0.2}\): draw a line parallel to the elastic slope but shifted to the right by a plastic strain of \(0.2\%\), and take the stress where that line crosses the measured curve. The definition is arbitrary, which is exactly why it is standardised: everyone who follows it gets the same number from the same data. The construction and the \(0.2\%\) value are prescribed for metals by ISO 6892-1 [1] and ASTM E8 [2], and for plastics by ISO 527-1 [3], which also fixes the strain window used to fit the modulus.

Example: Processing a test record

The evaluation is four steps, stated here before any code. First convert the two raw columns into stress and strain using the specimen geometry. Second fit a straight line to the elastic part and read its slope as \(E\). Third shift the strain axis so that the fitted line passes through the origin, which removes the machine’s initial slack. Fourth construct the offset line and find where it crosses the measured curve.

We work through those steps on a specimen of rectangular section \(10 \times 4~\text{mm}\), measured over a gauge length of \(50~\text{mm}\).

rng = np.random.default_rng(3)

# Underlying material, unknown to the person evaluating the test
E_true, sigma_y, hardening, sigma_u = 3200.0, 50.0, 9.0, 62.0

sigma_model = np.linspace(0, sigma_u, 900)
eps_model = sigma_model/E_true + 0.002*(sigma_model/sigma_y)**hardening

# What the machine writes to the file: force and extension, with load cell
# noise and an initial toe from the grips settling.
F  = sigma_model*(10.0*4.0) + rng.normal(0, 1.0, sigma_model.shape)
dL = eps_model*50.0 + 0.008*(1 - np.exp(-sigma_model/0.8)) \
     + rng.normal(0, 0.001, sigma_model.shape)

b0, h0, L0 = 10.0, 4.0, 50.0        # specimen width, thickness, gauge length [mm]
A0 = b0 * h0                        # original cross-sectional area [mm^2]

sigma = F / A0                      # engineering stress [MPa]
eps = dL / L0                       # engineering strain [-]

window = (eps >= 0.0005) & (eps <= 0.0025)          # the ISO 527 strain window
E_fit, intercept = np.polyfit(eps[window], sigma[window], 1)

ltx(r"E &=", E_fit, r"~\text{MPa}",
    r"\\ \text{points used} &=", int(window.sum()), aligned=True)

eps_corrected = eps + intercept/E_fit

offset_line = E_fit * (eps_corrected - 0.002)
difference = sigma - offset_line

crossing = np.where(np.diff(np.sign(difference)))[0][-1]      # last sign change
t = -difference[crossing] / (difference[crossing+1] - difference[crossing])

R_p02 = sigma[crossing] + t*(sigma[crossing+1] - sigma[crossing])
R_m = sigma.max()                                             # ultimate strength
eps_break = eps_corrected[-1]                                 # elongation at break

ltx(r"E &=", E_fit, r"~\text{MPa}",
    r"\\ R_{p0.2} &=", R_p02, r"~\text{MPa}",
    r"\\ R_m &=", R_m, r"~\text{MPa}",
    r"\\ \varepsilon_{\text{break}} &=", 100*eps_break, r"~\%", aligned=True)

fig, (ax, axz) = plt.subplots(1, 2, figsize=(11, 4.5))

for a in (ax, axz):
    a.plot(100*eps_corrected, sigma, 'k', linewidth=1.2, label="measurement")
    a.plot(100*eps_corrected, offset_line, 'b--', linewidth=1,
           label=r"$0.2\%$ offset line")
    a.plot(100*eps_corrected, E_fit*eps_corrected, 'g:', linewidth=1.2,
           label="fitted elastic slope")
    a.grid(True)
    a.set_xlabel(r"$\varepsilon$ [%]")
    a.set_ylabel(r"$\sigma$ [MPa]")

ax.plot(100*eps_corrected[np.argmax(sigma)], R_m, 'ro', label=r"$R_m$")
ax.set_ylim(0, 1.15*R_m)
ax.set_title("Full record")
ax.legend(loc='lower right')

axz.plot(100*(eps_corrected[crossing] + t*np.diff(eps_corrected)[crossing]),
         R_p02, 'ro', label=r"$R_{p0.2}$")
axz.set_xlim(0, 2.5)
axz.set_ylim(0, 1.15*R_p02)
axz.set_title("Elastic region and offset construction")
axz.legend(loc='lower right')

plt.tight_layout()
plt.show()

Transversal strain and Poisson’s ratio

Stretching a bar makes it thinner. The contraction transverse to the load is proportional to the strain along it, and the constant of proportionality is Poisson’s ratio

\[ \boxed{\nu = -\frac{\varepsilon_{\text{lat}}}{\varepsilon_{\text{long}}} = -\frac{\varepsilon_y}{\varepsilon_x}} \qquad\Longleftrightarrow\qquad \varepsilon_y = -\nu\,\varepsilon_x \tag{8.6.4}\]

with the minus sign inserted so that \(\nu\) comes out positive for ordinary materials. Measuring it requires a second instrument on the specimen, either a transverse extensometer or a digital image correlation system, which is why \(\nu\) is quoted from tables far more often than it is measured.

Poisson’s ratio is bounded above by \(0.5\) for an isotropic material, which is the value at which the volume stops changing. Rubber sits at \(0.4999\) and is effectively incompressible, aluminium at \(0.32\), steel between \(0.27\) and \(0.33\), and cork at almost exactly zero, which is why a cork can be pushed into a bottle neck without the sides bulging. We show in Chapter 8.8 where the limit of \(0.5\) comes from, and why \(\nu\) is the parameter that ties \(E\) and \(G\) together.

Isotropy and anisotropy

Everything above assumes that the material behaves the same in every direction, which makes it isotropic. A rolled steel plate, a drawn aluminium extrusion, a fibre composite and a 3D printed part are all anisotropic to some degree: they have different stiffness and different strength along different axes.

For printed parts the anisotropy is severe and it is a design variable rather than a nuisance. Material deposited within a layer is continuous, whereas adjacent layers are joined only where they fused, so a specimen loaded across the layers can fail at a fraction of the strength of one loaded along them. A tensile test therefore characterises a material and a print orientation together, and reporting one without the other says little. This is the reason the specimens in the laboratory are printed to a specified orientation, and the reason that comparing results between groups requires the print parameters to be reported alongside the mechanical data.