M027 · Solid mechanics

Stress–Strain Material Tester

What is left in a material after you load it and then take the load off? That depends on the material. An elastic specimen returns to its original length, a plastic one keeps a permanent stretch, and a viscoelastic one can hold an internal stress that slowly relaxes. Returning to zero extension does not prove that no stress is stored inside.

Stretch an elastic, plastic, brittle or viscoelastic specimen, hold it, unload it, compress it and bring it back to zero strain. Watch how its internal state and energy change, then estimate the loading slope from the measured force and gauge extension.

Selected protocol time $\text{Ready}$

Engineering strain $\text{Ready}$

Engineering stress $\text{Ready}$

Signed tensile force $\text{Ready}$

Permanent plastic strain $\text{Ready}$

Stored recoverable and hardening energy $\text{Ready}$

Signed external work $\text{Ready}$

Dissipated or fracture-loss energy $\text{Ready}$

Work–energy closure $\text{Ready}$

Virgin loading slope from raw record $\text{Ready}$

Acquired stations $\text{Ready}$

Model state $\text{Ready}$

Set the geometry and load history, then record

What is left in a material after you load it and then take the load off?

Original gauge length

L_0

0.10\,m

Original specimen breadth

b_0

10.00\,mm

Original specimen thickness

h_0

2.00\,mm

Instantaneous Young modulus

E_0

70.00\,GPa

Initial yield stress

\sigma_y

250.00\,MPa

Isotropic hardening modulus

H

2.00\,GPa

Peak engineering strain

\varepsilon_{max}

0.01\,

Ramp strain rate

\dot\varepsilon

0.00\,s^{-1}

Each hold duration

t_h

4.00\,s

Maxwell relaxation time

\tau

3.00\,s

Equilibrium modulus fraction

E_\infty/E_0

0.30\,

Brittle rupture strain

\varepsilon_f

0.00\,

Law: elastic, plastic, brittle, viscoelastic

s

1.00\,

Protocol progress fraction

\xi

0.35\,

Sensor perturbation gain

\epsilon_n

0.00\,

A uniform rectangular gauge section is clamped between two grips. The crosshead sets the gauge extension exactly, and stress uses the original cross-section area. Orange marks show the ends of the gauge length. Strains are drawn at true scale, with no magnification of the stretch. The frame is schematic; the machine's own flexibility and the specimen's sideways contraction are not modeled.

The test pulls the specimen into tension, holds, unloads to zero strain, pushes it into compression, holds again and returns to zero strain. The progress slider replays this history from an untouched specimen and jumps to the nearest stored state; it does not set the stress on its own. The compression stage is still a uniform material test, and specimen buckling is not modeled.

Plasticity uses one-dimensional return mapping with linear isotropic hardening; the Bauschinger effect is left out. The viscoelastic material is an equilibrium spring in parallel with one Maxwell arm, updated exactly over each ramp or hold. Brittle fracture drops the stress to zero for good once the strain magnitude passes a threshold, and the released energy counts as loss. Crack growth and contact after fracture are not modeled.

An acquisition records the whole test and recovers the first loading slope from the raw force and extension and the known original dimensions. The fit uses the first 0.03 percent of tensile strain. For the viscoelastic material the result is an apparent slope that depends on loading rate. Sensor noise is repeatable: 0.02 newtons and 0.1 micrometres per unit of noise gain. The Lab uses no materials database, does not model necking, does not certify toughness and gives no confidence interval.

The specimen deforms uniformly and follows simple small-strain teaching laws. The machine sets the gauge extension exactly, and stress always uses the original area. Necking, contact after fracture, machine compliance and buckling in compression are not modeled, and no real material is certified.

Start with the essentials

Focus question: Does zero extension mean no stored stress?
One-sentence intuition: The picture of the specimen, the readouts and the energy record all describe the same physical state.

Core mathematical model

Engineering strain and stress

\begin{aligned}\varepsilon&=\frac{\Delta L}{L_0}\\\sigma&=\frac{F}{A_0}\end{aligned}

Both use the original length and area. The Lab does not compute true stress or follow the shrinking area of a neck.

Plasticity with isotropic hardening

\begin{aligned}\sigma&=E(\varepsilon-\varepsilon_p)\\|\sigma|&\leq\sigma_y+H\alpha\\d\alpha&=|d\varepsilon_p|\end{aligned}

Permanent strain can be positive or negative, depending on the direction of plastic flow. Accumulated plastic strain and the yield radius can only grow.

A spring in parallel with one Maxwell arm

\begin{aligned}\sigma&=E_\infty\varepsilon+z\\\dot z+\frac{z}{\tau}&=(E_0-E_\infty)\dot\varepsilon\end{aligned}

During a hold, the stress in the transient spring relaxes away. Each constant-rate segment is updated exactly, so a stiff time step cannot distort the result.

Work and energy balance

\begin{aligned}W&=V_0\int\sigma\,d\varepsilon\\W&=\Psi+D\end{aligned}

The external work, which can be positive or negative, equals the energy stored elastically and by hardening plus the energy dissipated irreversibly. Energy released at brittle fracture counts as loss.

Loading slope from raw data

\begin{aligned}\frac{F_i}{A_0}&\simeq c+\widehat E\frac{\Delta L_i}{L_0}\\0&<\frac{\Delta L_i}{L_0}\leq3\times10^{-4}\end{aligned}

The fit uses only data from the first tensile ramp of a fresh specimen, plus its original geometry. For a viscoelastic specimen the slope depends on the loading rate.

Common difficulties

Where the model stops working

Typical misconceptionIf the curve on screen looks smooth, the physics must be right.

Better mental modelCheck whether the state lies inside the range where the model holds, and look at the recorded data. These teaching models do not certify any real specimen.

Run the experiment

01
Unload after yielding
Choose Yield and unload, then drag the progress slider through the whole protocol.
What to observe: Bringing the total strain back to zero can take a nonzero stress, because permanent strain remains in the material.
02
Hold extension fixed
Choose Viscoelastic hold, then change the hold duration and the ramp rate.
What to observe: While the extension is held fixed, the stress relaxes, but the equilibrium spring still carries some tension.
03
Follow the energy released at fracture
Choose Brittle rupture and look at the energy row before and after the rupture threshold.
What to observe: The stress cutoff cannot be undone. The stored energy it releases shows up as loss, so the energy still adds up.
04
Estimate the loading slope
Acquire the full protocol, look at the raw channels, export the CSV and change the noise level.
What to observe: The fit on the fresh specimen uses only raw force and extension. For a viscoelastic specimen, the apparent slope changes with loading rate.

Stress–Strain Material Tester

History inside a material

Core mathematical model

Engineering strain and stress

Plasticity with isotropic hardening

A spring in parallel with one Maxwell arm

Work and energy balance

Loading slope from raw data

Common difficulties

Where the model stops working

Run the experiment

Unload after yielding

Hold extension fixed

Follow the energy released at fracture

Estimate the loading slope

Related experiments and articles

Fracture mechanics from Griffith’s energy balance to branching cracks

Relaxation, stretching, extension and elastic instability