It is not linear

A metal has a straight portion. Its slope comes from the stiffness of the atomic bonding, it is very nearly the same on loading and on unloading, and a line fitted anywhere inside that portion gives the same answer. None of it holds for a printed PLA specimen, whose slope falls continuously from the first measurement onwards, as Figure 8.6.5 shows, so no region exists in which a line is uniquely determined, and no slope exists that a second laboratory would recover from a different but equally reasonable choice.

What we report as \(E\) is therefore a secant: the average stiffness between two strains we picked. Quoted without those two strains it means nothing, and this is not a caveat to be tucked into a footnote. It is what the number is.

The stiffness does not come back

Hooke’s law has one parameter and it quietly assumes that parameter stays put. Load a printed specimen well into the curve, then let it go, and it does not retrace the line it came up on. It comes back down a shallower one.

The usual way to carry that is a scalar damage variable \(D\) between zero and one, with the stress written

\[ \sigma = (1 - D)\,E\,\varepsilon \]

so that \(D = 0\) is pristine material at its original stiffness and \(D \to 1\) is material that carries nothing at all. What a test measures while unloading is therefore not \(E\) but the degraded stiffness \((1-D)E\). Writing \(E_1\) for the stiffness the specimen started with and \(E_2\) for the one it comes back on, \(E_2 < E_1\), and the gap between them is the damage the specimen accumulated on the way up.

This has a direct consequence for everything below. A specimen pulled straight to fracture cannot show \(D\) at all. It yields one stiffness, measured near the origin on undamaged material, and by the time the specimen is anywhere near its ultimate stress that number has stopped describing it. Seeing damage requires unloading on purpose, once or several times, and reading the slope on each return.

We have tried and cannot yet resolve it. On a specimen taken up and down seven times, the unloading slopes scatter between roughly 2100 and 2700 MPa and, if anything, rise with the load reached, which is the wrong direction for damage. That is a statement about our measurement rather than about the polymer, and it is the sort of thing a better extensometer would settle.

The point for now is smaller and firm: a single number called the modulus describes the material at one instant of its history, near the beginning. Do not mistake the elastic region for the whole story, and do not mistake \(E\) for a property the specimen carries unchanged to failure.

It is not isotropic

A material that behaves the same in every direction is isotropic, and that is what Hooke’s law with a single \(E\) describes. 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.

Isotropy is not the same thing as homogeneity. A homogeneous material has the same properties at every point, an isotropic one the same properties in every direction, and a rolled plate is homogeneous but anisotropic. A printed part is neither, since its properties also change from the solid skin to the interior.

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.

A specimen printed standing, with its axis along the build direction, is loaded across every layer at once, and what it measures is layer adhesion, the strength of the bond between layers. Partial infill raises a related problem. A specimen printed with \(20\,\%\) infill is a solid skin around a sparse lattice, and the force divided by its outer area gives an apparent stress: a property of that printed structure and not of the material. The laboratory specimens are printed at \(100\,\%\) infill so that the outer area is the area that carries the load.

A standard written for a different process

ISO 527 is the tensile standard for plastics and it is a good one. ISO 527-2 fixes the test conditions for moulding and extrusion plastics, with specimens moulded to shape or machined from moulded plate [1]. There is the mismatch, and it is worth stating precisely because the imprecise version is easy to refute: the material class is right, the manufacturing route is not ours. A moulded bar is dense, close enough to isotropic, and carries a skin the process gives it. A printed bar is a bonded assembly of extruded roads with a deliberate internal architecture and a skin whose thickness is a slicer setting.

That is a reason to be careful with the result. It is not a reason to abandon the standard, and the argument for keeping it has nothing to do with whether it is physically true.

A testing standard exists so that two laboratories get the same number from the same material. Because ISO 527 is the established way to test plastics, essentially every published modulus you might compare your printed part against was produced with it. Measure some other way and your number may well be closer to the truth, and it will also be incomparable with every datasheet, every supplier figure and every other student group. A value that is right and alone is worth less than a value that is approximate and shared.

The standard is also much more than a strain window. It prescribes how the specimen is gripped and aligned, how much prestress may be present before the measurement starts, what class of extensometer is required, and what speed the test runs at. Those clauses are not decoration: they are what makes the window work. Criticising the window while ignoring the rest is not a criticism of the standard, it is a description of a test that did not follow it, and the end of this chapter applies that test to our own procedure before it applies it to anything else.

A standard does exist for our process. ISO/ASTM 52903-2 covers material extrusion of plastics and includes guidance on building and testing specimens so that direction dependent properties are captured [2]. It prescribes no fit window, so it does not settle the question this chapter is about, but it is where to start on specimen design and on what to report alongside a number.

Why compute a modulus at all

If the material is neither linear nor isotropic, fitting a linear isotropic constant to it invites the objection that the result describes nothing. The objection is sound and the answer is short: no part can be sized without a stiffness. Deflection under load, buckling, natural frequency and every finite element model require one, and a number carrying its conditions is worth more than no number at all.

There is a second answer, and for a student it is the more useful one. Fitting a linear model to a material that does not obey it is not a search for a hidden true value. It is an exercise in taking awkward data, a prescribed procedure and a model known to be wrong, and producing from them a number you are willing to defend. That is most of engineering.

What the objection does rule out is treating the result as a constant of the material in the way \(210\,000~\text{MPa}\) is a constant for steel. For a printed part the stiffness is a property of the material, the print orientation, the print settings and the strain range over which it was measured. All four belong in the report, and a value quoted without them cannot be checked, compared or reused.

Step 1: The data

The machine writes a block of # comment lines carrying the specimen area, the gauge length and the load at which the strain measurement was zeroed, then one header row, then the data. Because it has already divided force by the original area and marker separation by its original value, the two columns we need are 8.6.3 ready made.

The area in that header is the one typed into the machine before the test. It should be the measured area, width times thickness of the gauge section taken with a caliper, and not the nominal area of the CAD model. Printed specimens do not hold their drawn dimensions, and a section a few percent off moves every stress, and \(E\) with it, by the same few percent. This record was run on the nominal \(80~\text{mm}^2\), so for a measured area \(A\) its stresses scale by \(80~\text{mm}^2/A\).

Two columns carry strain and only one of them belongs to the material. DIC_Cauchy is the separation of two markers on the specimen, tracked by a camera, relative to their separation at zero load. Motor_Strain is crosshead travel divided by gauge length, and the crosshead also moves the frame, the fixture, the grips and the shoulders of the specimen, so it overstates the strain of the gauge section by a factor of three or more. Use DIC_Cauchy.

The camera also drops frames. When the tracker cannot see exactly two markers it writes a blob count other than two, and when a frame is lost it writes a strain of exactly zero. Both appear in a raw plot as a stripe of points stacked on the \(\varepsilon = 0\) axis, and both have to go before anything is fitted.

path = 'testData/UTM_Test_20260830_122259.csv'

n_header = sum(1 for line in open(path) if line.startswith('#'))
data = np.genfromtxt(path, delimiter=',', skip_header=n_header, names=True)

tracked = (data['DIC_Blobs'] == 2) & (data['DIC_Cauchy'] != 0.0)
sigma = data['Stress_MPa'][tracked]        # engineering stress [MPa]
eps = data['DIC_Cauchy'][tracked]          # engineering strain [-]

ltx(r"\text{rows in the file} &=", len(tracked),
    r"\\ \text{rows with both markers tracked} &=", int(tracked.sum()), aligned=True)

Step 2: Fit the modulus

The modulus is the slope of the elastic part, and the only real decision in the whole evaluation is which points belong to it. Too few and the slope is noise. Too many and the line is dragged down as the curve bends over.

A printed polymer makes this harder than the idealised picture suggests, because it has no straight portion worth the name: the slope falls continuously from the first measurement onwards. There is no window that is right in the sense of capturing a genuinely linear region, only windows that are defensible and reproducible.

We use the one the standard prescribes. ISO 527-1 fits between \(0.05\%\) and \(0.25\%\) strain [1], and we follow it for the reason given above: it is what everyone else uses, so the number can be compared with somebody else’s. How well that window survives a printed specimen, and an alternative we have been testing, are taken up at the end of the chapter.

Within the window we fit the line

\[ \sigma = E\varepsilon + b, \qquad \min_{E,\,b}\sum_{0.0005\,\le\,\varepsilon_i\,\le\,0.0025}\left(\sigma_i - E\varepsilon_i - b\right)^2 \]

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

i_u = np.argmax(sigma)                          # the ultimate point, needed further down
R_m, eps_u = sigma[i_u], eps[i_u]

ltx(r"E &=", E_fit, r"~\text{MPa}",
    r"\\ \text{points used} &=", int(window.sum()),
    r"\\ R_m &=", R_m, r"~\text{MPa}",
    r"\\ \varepsilon_u &=", 100*eps_u, r"~\%", aligned=True)

Step 3: Remove the toe

The fitted line does not pass through the origin. Its intercept is the signature of the toe region, the initial slack while the grips settle and the specimen straightens, which adds extension without adding load. Anything that happened before the strain measurement was zeroed ends up there too: this file records that the camera was zeroed after a preload of \(49~\text{N}\), which on \(80~\text{mm}^2\) is \(0.61~\text{MPa}\) the specimen already carried at the moment it was told its strain was zero. That is about a third of the \(1.64~\text{MPa}\) intercept, and comfortably inside the prestress the standard allows, which is \(E_t/2000\) or \(1.7~\text{MPa}\) here.

The toe belongs to the fixture and to the procedure, not to the material, so we slide the strain axis until the fitted line does pass through the origin. Since \(E\varepsilon + b = E(\varepsilon + b/E)\), the corrected strain is

\[ \varepsilon_c = \varepsilon + \frac{b}{E} \]

eps_corrected = eps + intercept/E_fit

Step 4: The offset construction

The offset line is the elastic line moved to the right by a plastic strain of \(0.002\),

\[ \sigma_\text{off} = E(\varepsilon_c - 0.002) \]

and \(R_{p0.2}\) is the measured stress where the difference \(d_i = \sigma_i - \sigma_{\text{off},i}\) changes sign. Between the two points \(i\) and \(i+1\) that bracket the sign change we interpolate linearly,

\[ t = \frac{-d_i}{d_{i+1} - d_i}, \qquad R_{p0.2} = \sigma_i + t(\sigma_{i+1} - \sigma_i) \]

A record that rises to a peak and then softens crosses the offset line more than once, so we take the first crossing beyond the fit window and not the last.

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

crossings = np.where(np.diff(np.sign(difference)))[0]            # every sign change
crossing = crossings[eps_corrected[crossings] > 0.0025][0]        # first beyond the window
t = -difference[crossing] / (difference[crossing+1] - difference[crossing])

R_p02 = sigma[crossing] + t*(sigma[crossing+1] - sigma[crossing])
eps_last = eps_corrected[-1]              # last strain read before the markers were lost

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{last}} &=", 100*eps_last, r"~\%", aligned=True)

Reading the result

The evaluation gives a modulus of \(3462~\text{MPa}\), an offset yield strength of \(35.4~\text{MPa}\) and an ultimate strength of \(40.9~\text{MPa}\).

Two of those deserve comment. The yield strength sits at \(87\%\) of the ultimate, which looks wrong and is not: a curve that bends over this gradually is not met by a line offset by \(0.2\%\) of plastic strain until the material is close to its peak. And the last strain we can report is \(2.9\%\), which is not the elongation at break. The markers separate beyond what the tracker will follow at the moment the specimen fails, so an optical extensometer measures everything except the quantity that needs the specimen to be in one piece.

Figure 8.6.4 shows the record with both constructions drawn on it.

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

# Left: the whole record with both results marked
ax.plot(100*eps_corrected, sigma, 'k', lw=1.2, label='measurement')
ax.plot(100*eps_corrected, offset_line, 'b--', lw=1, label=r'$0.2\%$ offset line')
ax.plot(100*eps_corrected, E_fit*eps_corrected, 'g:', lw=1.4, label='fitted elastic slope')
ax.plot(100*eps_corrected[i_u], R_m, 'ro')
ax.annotate(rf'$R_m$ = {R_m:.1f} MPa', (100*eps_corrected[i_u], R_m),
            xytext=(6, -12), textcoords='offset points', fontsize=9)
ax.plot(100*(eps_corrected[crossing] + t*np.diff(eps_corrected)[crossing]), R_p02, 'bo')
ax.annotate(rf'$R_{{p0.2}}$ = {R_p02:.1f} MPa',
            (100*eps_corrected[crossing], R_p02),
            xytext=(-8, 12), textcoords='offset points', fontsize=9, ha='right', color='b')
ax.set_xlim(0, 100*eps_corrected.max()*1.05)
ax.set_ylim(0, 1.18*R_m)
ax.set_title('Full record', fontsize=11)
ax.legend(loc='lower right', fontsize=9)

# Right: the elastic region, showing which points were actually fitted
axz.plot(100*eps_corrected, sigma, 'k', lw=1.2, label='measurement')
axz.plot(100*eps_corrected[window], sigma[window], 'o', ms=3, color='#C98A2B',
         label=r'fitted points, ISO 527 window')
axz.plot(100*eps_corrected, E_fit*eps_corrected, 'g:', lw=1.4, label='fitted elastic slope')
axz.set_xlim(0, 0.45)
axz.set_ylim(0, 16)
axz.set_title('The elastic region and the fit window', fontsize=11)
axz.legend(loc='lower right', fontsize=9)

for a in (ax, axz):
    a.grid(True)
    a.set_xlabel(r'$\varepsilon$ [%]')
    a.set_ylabel(r'$\sigma$ [MPa]')

plt.tight_layout()
plt.show()

The zoomed view on the right is where the judgment lies. The offset line runs parallel to the fitted slope, so any error in the fit tilts it, and the crossing then slides along a shallow part of the curve where a small vertical error means a large horizontal one. Two students who fit different windows will report different moduli and, through them, different yield strengths. Reporting the window, the strain source and the specimen geometry alongside the numbers is what makes two sets of results comparable at all.

Before criticising the standard: what we did not do

ISO 527-1 prescribes much more than the strain window, and our own records do not satisfy all of it. Two clauses matter here and both are measurable from the files.

Prestress. Clause 9.4 requires the specimen to carry a small positive stress before the measurement starts, enough to pull out slack and seat the grips, and caps it at \(\sigma_0 \le E_t/2000\), a prestrain of \(0.05\%\) in the standard’s own words. For the record evaluated above that cap is \(1.7~\text{MPa}\) and the file records \(0.61~\text{MPa}\), so this one complies. Other records in the same folder do not, and the student sets we have seen run up to twice the limit. Too little prestress leaves a toe; too much starts the measurement inside the curve. Both move the modulus.

Test speed. Clause 9.6 requires a strain rate as close as possible to \(1\%\) of the gauge length per minute for modulus measurement. The record above was pulled at \(6~\text{mm/min}\) on an \(80~\text{mm}\) gauge, which is \(7.5\%\) per minute, seven times too fast. Polymers are strain rate sensitive, so this inflates both the modulus and the strength relative to any catalogue value. The standard anticipates the tedium and allows a two speed test: run at \(1\%\) per minute to \(\varepsilon = 0.25\%\), change speed before \(\varepsilon = 0.3\%\), and continue to fracture at whatever rate you like.

None of this excuses the standard’s window on a printed specimen. It does mean that what follows is evidence about our test, and that the first thing to fix is our procedure rather than the standard.

Where the straight part actually is

Figure 8.6.5 takes the record evaluated above and fits a short line at every point along it, so that the local slope can be read directly.

half = 0.0004                                   # half width of the sliding fit, in strain
centres = np.arange(0.0006, 0.0100, 0.0002)
slopes = []
for c in centres:
    near = (eps >= c - half) & (eps <= c + half)
    slopes.append(np.polyfit(eps[near], sigma[near], 1)[0] if near.sum() >= 8 else np.nan)

fig, ax = plt.subplots(figsize=(7, 4.2))
ax.plot(100*centres, slopes, 'o-', ms=3, lw=1.2, color='#236B8E')
ax.axvspan(0.05, 0.25, color='#C25B4E', alpha=.14, label='ISO 527 window')
ax.axvspan(100*0.10*eps_u, 100*0.50*eps_u, color='#7A9E3F', alpha=.18,
           label=r'$0.10\,\varepsilon_u$ to $0.50\,\varepsilon_u$')
ax.axhline(E_fit, color='k', ls='--', lw=0.9, label=f'fitted $E$ = {E_fit:.0f} MPa')
ax.set_xlabel(r'strain $\varepsilon$ (%)')
ax.set_ylabel(r'local slope $\mathrm{d}\sigma/\mathrm{d}\varepsilon$ (MPa)')
ax.set_xlim(0, 1.0)
ax.legend(fontsize=9)
ax.grid(alpha=.25)
plt.tight_layout()
plt.show()

A material does not become \(40\%\) more compliant and then stop. The steep part at the left is the instrument: the optical extensometer resolves strain in steps of about \(0.006\%\), and the ISO window spans only three pixels of marker separation, so the line is being fitted through a staircase. On this record the standard’s window sits inside that artefact, and the plateau between roughly \(0.25\%\) and \(0.65\%\) sits outside it.

That is the honest statement of the problem, and the honest response is not to discard the window. It is to fix the measurement, which means more pixels across the gauge, prestress inside the clause 9.4 limit and the speed clause 9.6 asks for. Until that is done we cannot tell how much of this is the standard and how much is us.

An experiment: a window that scales with the material

In the meantime we have been testing an alternative, and it is reported here as an experiment, not as a recommendation. Instead of fixed strain limits, take the window as a fraction of the strain at which the specimen reaches its ultimate stress,

\[ 0.10\,\varepsilon_u \le \varepsilon \le 0.50\,\varepsilon_u \]

which lands on the same part of the curve whatever the material, and scales itself to something brittle as readily as to something ductile. On the record above it selects \(0.19\%\) to \(0.94\%\), which is the plateau in Figure 8.6.5.

The test of any window is not whether it looks right on one record but whether it gives the same answer on specimens that should be the same. Four were printed and pulled together.

records = ['122259', '123728', '124746', '125231']

def evaluate(rec, lo, hi, by='strain'):
    p = f'testData/UTM_Test_20260830_{rec}.csv'
    n = sum(1 for line in open(p) if line.startswith('#'))
    d = np.genfromtxt(p, delimiter=',', skip_header=n, names=True)
    keep = (d['DIC_Blobs'] == 2) & (d['DIC_Cauchy'] != 0.0)
    e, s = d['DIC_Cauchy'][keep], d['Stress_MPa'][keep]
    peak = np.argmax(s)
    if by == 'strain':      m = (e >= lo) & (e <= hi)
    elif by == 'eps_u':     m = (e >= lo*e[peak]) & (e <= hi*e[peak])
    else:                                              # fraction of the ultimate stress
        m = (s >= lo*s[peak]) & (s <= hi*s[peak]); m[peak+1:] = False
    return np.polyfit(e[m], s[m], 1)[0], s[peak]

methods = [(r"\text{ISO 527, } 0.05\text{ to }0.25\,\%", 0.0005, 0.0025, 'strain'),
           (r"\text{fixed, } 0.25\text{ to }1.00\,\%",   0.0025, 0.0100, 'strain'),
           (r"0.30\,R_m \text{ to } 0.70\,R_m",          0.30,   0.70,   'stress'),
           (r"0.10\,\varepsilon_u \text{ to } 0.50\,\varepsilon_u", 0.10, 0.50, 'eps_u')]

rows = []
for label, lo, hi, by in methods:
    v = np.array([evaluate(r, lo, hi, by)[0] for r in records])
    rows.append(r"%s & %s & %.0f & %.1f\,\%% \\" % (label, ' & '.join(f'{x:.0f}' for x in v),
                                               v.mean(), 100*v.std(ddof=1)/v.mean()))
Rm = np.array([evaluate(r, 0.10, 0.50, 'eps_u')[1] for r in records])
rows.append(r"\hline \text{ultimate strength } R_m & %s & %.1f & %.1f\,\%% \\"
            % (' & '.join(f'{x:.1f}' for x in Rm), Rm.mean(), 100*Rm.std(ddof=1)/Rm.mean()))

ltx(r"\begin{array}{l r r r r r r} \text{fit window} & " +
    ' & '.join(records) + r" & \text{mean} & \text{CV} \\ \hline " +
    ' '.join(rows) + r"\end{array}")

The four specimens agree on ultimate strength to about two percent. They agree on the modulus to between six and twelve percent depending only on how it was evaluated, and the standard’s window is the worst of the four on this data.

The last column is the coefficient of variation, the standard deviation divided by the mean. It is the natural measure here because the question is not how large the modulus is but how repeatable the procedure that produced it.

What to take from this

Strength is easy to measure and stiffness is hard, and a large part of the difficulty sits in the evaluation and not in the specimen. That is worth carrying into any test you run.

For the laboratory, use the ISO 527 window, and report it alongside the number together with the strain source, the test speed, the prestress and how many specimens were behind it. A modulus without those cannot be checked by anyone, and one measured a different way cannot be compared with anything.

The experiment above is not yet a method. It rests on four specimens of one material on one machine, it has not been tried on a material with a very different \(\varepsilon_u\), and a brittle specimen that fractures at its maximum stress makes \(\varepsilon_u\) mean something different from what it means for a specimen that yields first. It also, being wider, is more exposed to gaps in the strain record than the narrow window it replaces. Treat it as an open question this department is working on, not as an answer.

What is settled is smaller and more useful. There is no true modulus here waiting to be recovered by a sufficiently careful experiment. There is a curve, and there are conventions for reducing it to a number, and on a printed specimen the conventions disagree with one another by more than the specimens disagree among themselves. Following a standard does not make that go away. It does mean that everyone disagrees in the same direction, which is what makes comparison possible at all, and it is why we follow it.