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 8.6.4 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.

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.

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.

What that costs the standard

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.

A standard written for one is not automatically wrong for the other, but it stops being self-evidently right, and the parts of it that encode assumptions about the material have to be checked rather than followed. The strain window prescribed for fitting the modulus is exactly such a part. The end of this chapter checks it against our own measurements and rejects it.

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 does not settle the fit-window question, which is why the rest of this chapter is needed, but it is the right place 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.

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.

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 one defined as a fraction of the strain at which the specimen reaches its ultimate stress,

\[ 0.10\,\varepsilon_u \le \varepsilon \le 0.50\,\varepsilon_u \tag{8.6.4}\]

which lands on the same part of the curve whatever the material, and scales itself to something brittle as readily as to something ductile. Why this window rather than the one the testing standard prescribes, and what difference the choice makes, is taken up at the end of the chapter.

Finding \(\varepsilon_u\) comes first, because the window depends on it.

i_u = np.argmax(sigma)                     # the ultimate point
R_m, eps_u = sigma[i_u], eps[i_u]          # ultimate strength and the strain there

window = (eps >= 0.10*eps_u) & (eps <= 0.50*eps_u)
E_fit, intercept = np.polyfit(eps[window], sigma[window], 1)

ltx(r"R_m &=", R_m, r"~\text{MPa}",
    r"\\ \varepsilon_u &=", 100*eps_u, r"~\%",
    r"\\ \text{fit window} &=", 100*0.10*eps_u, r"~\%\text{ to }", 100*0.50*eps_u, r"~\%",
    r"\\ E &=", E_fit, r"~\text{MPa}",
    r"\\ \text{points used} &=", int(window.sum()), 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, and that accounts for about a quarter of the intercept.

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. The shift needed is the intercept divided by the slope.

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 \(0.002\), that is \(\sigma = E(\varepsilon - 0.002)\). We want the stress where the measurement crosses it. Rather than solving anything, we look at the sign of the difference between the two curves, find where it changes, and interpolate linearly across that one interval. 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] > 0.50*eps_u][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 \(3068~\text{MPa}\), inside the range quoted for PLA, an offset yield strength of \(38.3~\text{MPa}\) and an ultimate strength of \(40.9~\text{MPa}\).

Two of those numbers deserve comment. The yield strength sits at \(93\%\) 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 almost at its peak. And the last strain we can report is \(3.0\%\), 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 8.6.3 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, $0.10\,\varepsilon_u$ to $0.50\,\varepsilon_u$')
axz.plot(100*eps_corrected, E_fit*eps_corrected, 'g:', lw=1.4, label='fitted elastic slope')
axz.set_xlim(0, 120*0.50*eps_u)
axz.set_ylim(0, 1.15*sigma[window].max())
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.

What the standards prescribe, and why we do not follow it

ISO 527-1 removes the decision by fixing the window at \(0.05\%\) to \(0.25\%\) strain [1], and ASTM D3039 does the same for composites at \(0.1\%\) to \(0.3\%\) [5]. Fixing it is the right instinct, and we keep it: everyone who follows a rule gets the same number from the same data, which is the entire purpose of a testing standard.

What we do not keep is the particular window, because it rests on an assumption our specimens do not satisfy. It supposes that the material has a straight portion and that the portion lies between those two strains. Figure 8.6.4 shows what is there instead.

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. The standard’s window sits inside that artefact. The window used above, \(0.19\%\) to \(0.94\%\) for this specimen, sits on the plateau.

What difference it makes

The test of a 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. A rule that sorts nominally identical specimens by twelve percent is measuring something other than the material.

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 most of the difficulty sits in the evaluation rather than in the specimen. That is worth carrying into any test you run.

The deeper point is the one this chapter opened with. 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 the conventions disagree with one another by more than the specimens disagree among themselves. Following a standard does not escape that. It only means the convention was chosen by a committee, for a material made a different way, and then not stated in the report because everyone assumed it.

For the laboratory, use the window of 8.6.4 and report it alongside the number. A modulus quoted without the window it was fitted over, the strain source and the number of specimens behind it cannot be checked by anyone else.

For honesty, note that the fixed \(0.25\%\) to \(1.00\%\) window scores similarly on these four specimens, and four specimens cannot separate them. We prefer the normalised form because it needs no adjustment when the material changes, not because the evidence here is decisive. Whether it survives more specimens, more materials and a second testing machine is an open question that this department is working on.