Tensile Test Simulator: Stress-Strain Curve, Yield, UTS and Necking

Free tensile test simulator: pull steel, aluminium, cast iron, glass, plastics, rubber and more until they break and watch the stress-strain curve being drawn: yield, 0.2% offset, Lüders plateau, UTS, necking, unloading, true stress, resilience and toughness. Compare three materials.

Material

Units

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True stress–strain Shade areas Plateau serrations

Speed

Yield point or smooth yielding?

Mild and structural steels (A36, 1020) have a yield point. Carbon and nitrogen atoms pin the dislocations, so the stress climbs a little past the yield strength until a band of metal breaks free: that is the upper yield point. The force then drops to the lower yield point and the bar stretches at nearly constant stress while Lüders bands spread along it (the Lüders strain, about 1 to 3% for A36), with small serrations as each band starts. Only then does strain hardening begin, up to the tensile strength (58 to 80 ksi, 400 to 550 MPa for A36, at about 15 to 20% strain) and necking. The upper yield is usually only a few % to about 10% above the lower one, and its height depends on the test: a stiff machine, a fast pull and a smooth, well-aligned specimen give a taller, sharper peak; a soft machine or a slightly bent bar can hide it. The line also bends a little just before the peak (some grains yield early), so the top is rounded rather than a sharp corner. Codes therefore use the lower yield (or 0.2% offset) as the yield strength: 250 MPa (36 ksi) for A36.

Aluminium alloys (2014-T6, 6061-T6), copper, stainless steel and most other metals yield continuously: no peak, no drop, no plateau. The curve just bends away from the straight line, so the yield strength is defined by the 0.2% offset rule: draw a line with slope E starting at 0.2% strain, and where it meets the curve is the yield strength (414 MPa, 60 ksi, for 2014-T6).

Engineering stress\(\displaystyle \sigma = \frac{F}{A_0},\)\(\displaystyle \varepsilon = \frac{L - L_0}{L_0}\)

True stress\(\displaystyle \sigma_t = \frac{F}{A} = \sigma\,(1+\varepsilon),\)\(\displaystyle \varepsilon_t = \ln(1+\varepsilon)\)

Hardening\(\displaystyle \sigma_t = K\,\varepsilon_p^{\,n}\)

Necking starts\(\displaystyle \frac{d\sigma_t}{d\varepsilon_t} = \sigma_t \;\Rightarrow\; \varepsilon_t \approx n\)

Resilience\(\displaystyle U_r = \frac{\sigma_y^2}{2E}\)

Toughness\(\displaystyle U_t = \int_0^{\varepsilon_f} \sigma\,d\varepsilon\)

The curves are a model fitted to each material's typical Young's modulus, yield strength, tensile strength and elongation (Hollomon / Ramberg–Osgood hardening, a Lüders plateau for mild steel). Specimen: a standard round bar, 12.5 mm across with a 50 mm gauge length (0.500 in and 2.000 in in US units). Real curves also depend on the test speed, temperature and the exact alloy.

Supports

Loads drag them along the beam; drag a handle to change the size

Cross-section

Deflection limit

Bending stress (flexure formula)\(\displaystyle \sigma = \frac{M\,c}{I} = \frac{M}{S},\)\(\displaystyle S = \frac{I}{c}\)

Shear stress\(\displaystyle \tau = \frac{V\,Q}{I\,t}\)

Deflection curve\(\displaystyle E I\,\frac{d^2 y}{dx^2} = M(x),\)\(\displaystyle \frac{dM}{dx} = V\)

Rectangle\(\displaystyle I = \frac{b\,h^3}{12},\)\(\displaystyle S = \frac{b\,h^2}{6}\)

Simply supported, load P at the middle\(\displaystyle M_{max} = \frac{P L}{4},\)\(\displaystyle \delta = \frac{P L^3}{48 E I}\)

Simply supported, spread load w\(\displaystyle M_{max} = \frac{w L^2}{8},\)\(\displaystyle \delta = \frac{5 w L^4}{384 E I}\)

Cantilever, load P at the end\(\displaystyle M_{max} = P L,\)\(\displaystyle \delta = \frac{P L^3}{3 E I}\)

Fixed both ends, spread load w\(\displaystyle M_{ends} = \frac{w L^2}{12},\)\(\displaystyle \delta = \frac{w L^4}{384 E I}\)

The beam is solved exactly for any mix of loads (singularity functions: equilibrium plus the deflection conditions at the supports), so the fixed and propped cases, which statics alone cannot solve, come out right too. It assumes a straight, elastic beam with small deflections (Euler–Bernoulli), bending about the strong axis, no buckling sideways (long, slender I-beams without bracing can twist out of the way: lateral-torsional buckling) and no self-weight unless you add it as a spread load.

Words used here

Bending moment M: How much the loads try to bend the beam at a point: force × distance, summed over everything on one side. Sagging (smiling, top squeezed) is drawn upward here; hogging (frowning, top stretched, over supports) downward.
Shear force V: The force trying to slide one part of the beam past the next. It jumps at every point load and support, and the slope of the moment diagram equals it, so M is largest where V crosses zero.
Neutral axis: The line through the section that is neither stretched nor squeezed. For these symmetric sections it sits at mid-depth; the stress grows in a straight line away from it.
Second moment of area I: How the area is spread away from the neutral axis (m⁴, in⁴). Depth counts cubed, which is why I-beams put their metal in the flanges and why a plank is far stiffer on edge.
c and section modulus S = I/c: c is the distance from the neutral axis to the outermost fibre, where the stress is largest. S rolls I and c into one number: σmax = M/S.
Flexure formula σ = Mc/I: The largest bending stress in the section, valid while the material stays elastic.
Q and τ = VQ/(It): Q is the first moment of the area above the neutral axis and t the width there. The shear stress is largest at the neutral axis; in an I-beam the thin web carries nearly all of it.
Deflection δ and L/360: How far the beam sags. Codes limit it as a fraction of the span (L/360 for floors under live load, L/240 or L/180 for roofs) so floors do not feel bouncy and plaster does not crack.
Simply supported, cantilever, fixed, propped, overhang: A pin and a roller at the ends; built into a wall at one end only; built in at both ends; built in at one end and resting on a support at the other; resting on two supports with part of the beam sticking out past one of them.
Point load, spread load (UDL), moment: A force at one spot (kN, kip); a force spread evenly along a length (kN/m, kip/ft), such as a floor's weight; a twisting couple applied at one spot (kN·m, kip·ft).
EI (flexural rigidity): Young's modulus times I: how hard the beam is to bend. Deflection is inversely proportional to it.

For learning and first estimates. Real parts need certified material values, the design code that applies (AISC, Eurocode, ASME…) and a check by a qualified engineer.

Tensile test simulator: the stress-strain curve, step by step

A tensile test pulls a standard bar (here a round bar 12.5 mm across with a 50 mm gauge length, like ASTM E8 / ISO 6892-1) until it breaks, and plots the force against the stretch as a stress-strain curve. Press Start and watch the bar stretch in the grips while the curve is drawn point by point. Drag the slider to go back and forth, press Unload here to take the load off at any point, compare up to three materials on one chart, switch on the true stress-strain curve and see the resilience and toughness shaded. Steels, aluminium, titanium, copper, cast iron, glass, wood, plastics, rubber, carbon fibre and bone are in the list, or type your own material.

What the curve shows

Elastic part: a straight line whose slope is Young's modulus, \(E = \sigma / \varepsilon\). Unload here and the bar goes back to its old length.
Yield: most metals bend over smoothly, so the yield strength is taken where a line parallel to the elastic part, starting at 0.2% strain, meets the curve (the 0.2% offset rule, drawn dashed). Mild steel is different: it has an upper yield point, drops to a lower yield point and stretches at an almost constant force (the Lüders plateau) before it hardens.
Strain hardening: the metal gets stronger as it is stretched, roughly \(\sigma_t = K\,\varepsilon_p^{\,n}\) in true stress and plastic strain (Hollomon). Unload from here and the line comes down with slope E, leaving a permanent set; load it again and it stays elastic up to the higher stress.
Tensile strength (UTS) and necking: the highest point of the engineering curve. After it the bar thins in one place, so the force falls even though the metal in the neck keeps hardening. Necking starts when the hardening can no longer keep up with the loss of area, \(d\sigma_t / d\varepsilon_t = \sigma_t\) (Considère), which for the Hollomon law is at a true strain of about n.
Fracture: ductile metals break with a cup-and-cone surface after a long neck; brittle materials (cast iron, glass, concrete, wood along the grain, carbon fibre) break with no yield and no neck.

Resilience and toughness

The area under the curve is energy per unit volume. The green triangle is the modulus of resilience, the energy the bar can store and give back: \(U_r = \sigma_y^2 / (2E)\). For A36 steel that is 250² ÷ (2 × 200,000) = 0.156 MJ/m³. The blue area is the toughness, the energy needed to break it, \(U_t = \int \sigma\,d\varepsilon\): about 80 MJ/m³ for A36, five hundred times the resilience. A quick estimate for a ductile metal is the average of the yield and tensile strengths times the elongation: (250 + 400) ÷ 2 × 0.23 ≈ 75 MJ/m³.

Engineering and true stress

Engineering stress divides the force by the original area, \(\sigma = F / A_0\); true stress divides it by the area the bar has now, \(\sigma_t = \sigma\,(1 + \varepsilon)\), with the true strain \(\varepsilon_t = \ln(1 + \varepsilon)\). The two formulas hold while the bar stretches evenly; after necking the true stress is worked out from the area of the neck (dashed).

How the curves are made

Each curve is fitted to the material's typical Young's modulus, yield strength, tensile strength and elongation at break, so the simulator hits those numbers exactly: the hardening exponent n and strength coefficient K come out of the fit and are shown. Real tests also depend on the test speed, temperature, the exact alloy and its heat treatment, so treat the curves as a typical shape, not a certificate.

Questions

What is the difference between yield strength and tensile strength?

Yield strength is where permanent stretching starts; tensile strength is the highest stress the bar takes before it necks and breaks. Parts are designed to stay below yield, with a safety factor.

Why does the curve go down before it breaks?

Because engineering stress uses the original area. Once the neck forms, the area there shrinks fast, so the force needed to keep stretching falls. The true stress in the neck keeps rising.

What is elongation at break?

How much longer the gauge length is when the pieces are fitted together after the break, in percent. It measures ductility: about 20 to 25% for structural steel, 12% for 6061-T6 aluminium, under 1% for cast iron. The elastic part springs back after the break, so it is a little less than the strain at the moment of fracture.

Why do plastics look so different?

Many plastics reach a yield peak, then a neck forms and travels along the bar at a steady force (cold drawing) until the whole bar is drawn out; HDPE can stretch to several times its length. Rubber has no yield at all: it is soft at first and stiffens as its chains straighten, and it springs back completely.

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