Mechanical Properties of Materials
Relating stress to strain by experiment: the stress–strain diagram, Hooke's law, and the key material properties used in design.
Chapter Objectives
- Describe the tension/compression test and the stress–strain diagram it produces.
- Identify the elastic, yielding, strain-hardening, and necking regions, and the associated material properties.
- Distinguish ductile from brittle behavior; apply the offset method for yield strength.
- Define strain energy density, modulus of resilience, modulus of toughness, Poisson's ratio, and the shear modulus.
3.1The Tension and Compression Test
Why Test?
A material's strength — its ability to sustain load without excessive deformation or failure — is inherent in the material itself and must be found by experiment. The tension or compression test is the most important such test, yielding the stress–strain relationship for metals, ceramics, polymers, and composites.
The Standard Test
A specimen of standard shape — a constant circular cross section with enlarged ends, typically initial diameter \(d_0 = 13\) mm (0.5 in.) and gage length \(L_0 = 51\) mm (2 in.) between two punch marks — is stretched at a very slow, constant rate until failure while the machine records the load \(P\). The elongation \(\delta = L - L_0\) is measured with an extensometer, or the strain is read directly with an electrical-resistance strain gage cemented to the specimen: the gage deforms with the material, and the change in its electrical resistance is calibrated to strain.
3.2The Stress–Strain Diagram
Plotting the test data as stress (vertical) versus strain (horizontal) gives the stress–strain diagram, a description of the material valid for a specimen of any size. No two diagrams for one material are identical — composition, microscopic imperfections, manufacturing, load rate, and temperature all influence the result.
- \(A_0\), \(L_0\) = original cross-sectional area and gage length — held constant in the calculation.
- Plotting these values gives the conventional stress–strain diagram.
Elastic Behavior
The initial region of the curve is a straight line up to the proportional limit \(\sigma_{pl}\); slightly beyond, the curve bends until the elastic limit is reached (the two points are usually nearly indistinguishable). The defining feature: if the load is removed anywhere in this region, the specimen recovers its original shape — no damage is done.
- \(E\) = modulus of elasticity (Young's modulus) — the slope of the straight-line portion of the diagram.
- Since strain is dimensionless, \(E\) carries units of stress (Pa, psi, ksi). Discovered by Hooke (1676); quantified by Young (1807).
Yielding
Stress slightly above the elastic limit causes the material to deform permanently — plastic deformation — at the yield stress (yield point) \(\sigma_Y\). Hot-rolled low-carbon steels show an upper then a lower yield point. During yielding the specimen elongates without any increase in load; such behavior is termed perfectly plastic.
Strain Hardening and Necking
After yielding, the specimen can again carry increasing stress along a curve that flattens as it rises to a maximum: the ultimate stress \(\sigma_u\) — this rise is strain hardening. Beyond \(\sigma_u\), the cross-section begins to shrink in one localized region, forming a neck; the engineering curve bends downward until fracture at the fracture stress \(\sigma_f\).
True vs. Conventional Diagram
Using the actual instantaneous area \(A\) and length \(L\) gives true stress and true strain. The two diagrams coincide for small strain and diverge in strain hardening: the conventional curve drops in necking only because it divides by the constant \(A_0\) — the true curve shows the material actually sustains increasing stress until fracture. The distinction is usually unimportant: design is done in the elastic range, where the error of engineering values is only about 0.1 %.
3.3Stress–Strain Behavior of Ductile and Brittle Materials
Ductile Materials
A material that sustains large strains before fracture (e.g. mild steel, brass, zinc). Ductile materials are favored in design because they absorb shock energy and give visible warning — large deformation — before failing.
- \(L_f\), \(A_f\) = gage length at fracture and neck area at fracture. Mild steel: ≈ 38 % elongation, ≈ 60 % area reduction.
The Offset Method (Yield Strength)
Metals such as aluminum have no well-defined yield point. A yield strength is then defined graphically: from a 0.2 % strain offset (ε = 0.002) on the strain axis, draw a line parallel to the initial elastic slope; its intersection with the curve is \(\sigma_{YS}\). The yield strength is not a true physical property — it is the stress producing a chosen permanent strain — but in practice the yield strength, yield point, elastic limit, and proportional limit are assumed to coincide unless stated otherwise. Natural rubber is an exception with no proportional limit at all: a polymer with nonlinear elastic behavior. Wood is moderately ductile, highly variable, and much weaker in tension perpendicular to its grain, so tensile loads are applied parallel to the grain.
Brittle Materials
Materials showing little or no yielding before failure (e.g. gray cast iron, concrete). Tensile fracture initiates at a microscopic crack and spreads suddenly, so the tensile fracture stress is scattered and reported as an average. Brittle materials resist compression far better than tension — cracks close up under compression and the specimen bulges. Concrete's compressive strength is roughly 12.5 times its tensile strength, which is why it is reinforced with steel wherever tension occurs. Note also that material behavior depends on temperature: many materials become harder and more brittle when cold, softer and more ductile when warm.
3.4Strain Energy
As a load deforms a material it does external work, which (neglecting heat loss) is stored internally as strain energy. For a small element under uniaxial stress, the force grows uniformly from zero, so the work is the average force times the displacement — giving \(U = \tfrac{1}{2}\sigma\epsilon\,V\) for element volume \(V\).
- \(u\) = strain energy per unit volume (J/m³); the second form holds for linear elastic behavior.
- The triangular area under the elastic portion of the diagram: the most energy per unit volume the material can absorb without permanent damage. Important for bumpers and shock absorbers.
Modulus of Toughness
\(u_t\) = the entire area under the stress–strain diagram: the maximum strain energy per unit volume the material can absorb just before fracture — critical for members that may be accidentally overloaded. Alloying (e.g. varying carbon content in steel) trades off resilience, toughness, strength, and ductility: hard high-carbon steels are strong but less tough; soft low-carbon steels are tough but weaker.
3.5Poisson's Ratio
Lateral Contraction
A bar pulled in tension not only elongates axially (\(\epsilon_{long} = \delta/L\)) but also contracts laterally (\(\epsilon_{lat} = \delta'/r\)); under compression the reverse occurs. Poisson observed that within the elastic range the ratio of these strains is a constant of the material, since both displacements are proportional to the same force.
- \(\nu\) (nu) is dimensionless and unique for a homogeneous, isotropic material; \(0 \le \nu \le 0.5\), with most nonporous solids between 0.25 and 0.355.
- The minus sign makes \(\nu\) positive: axial elongation (+) accompanies lateral contraction (−). The lateral strain arises with no lateral force acting.
3.6The Shear Stress–Strain Diagram
Pure shear distorts a homogeneous, isotropic element uniformly, producing shear strain γ. To measure shear behavior, a thin tube is twisted; the recorded torque and angle of twist (Chapter 5 methods) yield the shear stress–strain diagram, which shows the same character as the tension test: a linear elastic region up to a proportional limit \(\tau_{pl}\), strain hardening to an ultimate shear stress \(\tau_u\), and fracture at \(\tau_f\).
- \(G\) = shear modulus of elasticity (modulus of rigidity) — the slope of the elastic line on the τ–γ diagram, \(G = \tau_{pl}/\gamma_{pl}\).
- Units of \(G\) are those of stress (Pa, psi), since γ is in radians.
- The three elastic constants are not independent: knowing any two gives the third, so \(\nu\) can be computed rather than measured.
3.7Failure of Materials Due to Creep and Fatigue
The properties above describe static loading at constant temperature. Two other failure modes matter when loads are sustained for long periods at elevated temperature, or are repeated and cycled.
Creep
Time-dependent permanent deformation under sustained load. Creep governs metals and ceramics at high temperature, and can occur in polymers, composites, wood, and concrete from long-term loading alone (a rubber band held stretched for a long time never fully recovers). Design uses the creep strength: the highest stress the material can sustain for a specified time, temperature, and allowable creep strain — found by testing multiple specimens at different stresses and plotting stress versus time-to-strain (a steel guideline: 0.1 % creep strain per year for bolts and piping).
Fatigue
Fracture under repeated or cyclic stress — often at stress levels well below the static yield stress. A microscopic crack initiates at a point of high stress concentration and grows with each cycle until sudden, brittle-like fracture. Design uses the stress–cycles (S–N) diagram: the stress that a specimen can endure for a specified number of cycles. For steels there is an endurance (fatigue) limit — a stress below which fatigue failure never occurs regardless of the number of cycles; aluminum has no such limit, so a finite design life is specified instead.
Physical meaning
A stress–strain curve is a material response, not a force diagram: its slope describes stiffness, its area describes energy per volume, and its limits identify recoverable versus permanent strain.
Problem-solving strategy
- Classify the requested response and identify whether the supplied data are elastic, inelastic, thermal, or geometric.
- Draw the specimen or member and its free-body diagram; state a tension-positive axial convention.
- Choose the applicable constitutive relation and, where needed, write equilibrium and compatibility independently.
- Keep force, stress, strain, and displacement distinct while carrying one consistent unit system.
- Audit yield, instability, serviceability, and the limits of the idealized result before using it.
Applications and material judgment
Recognition strategy
The load and extensometer reading invite engineering stress and strain, followed by Hooke’s law because the stated unloading is linear.
Given information
Use original geometry, \(P=18.0\ \text{kN}\), \(A_0=120\ \text{mm}^2\), \(L_0=50.0\ \text{mm}\), and \(\delta=0.0300\ \text{mm}\).
Model and interpretation
Treat the coupon as uniform away from grips, with uniaxial tension and tension-positive stress; this is an average elastic result.
Verification
Units: Because \(\text{MPa}\) divided by dimensionless strain remains \(\text{MPa}\), the reported modulus has stress units.
Signs and direction: Tensile load, positive extension, and positive strain give positive stress and modulus under the declared convention.
Plausibility: The value \(250\ \text{GPa}\) is a plausible stiff-metal modulus; it is not a yield strength or an allowable stress.
Engineering interpretation
The slope quantifies elastic stiffness of this tested material condition. A component may still yield, buckle, or deflect excessively at a stress far below a simple modulus-based calculation suggests.
Recognition strategy
All supplied values describe a uniaxial linear-elastic state, so Hooke’s law, Poisson’s ratio, and triangular stress–strain area apply.
Given information
Take \(\sigma=220\ \text{MPa}\), \(E=110\ \text{GPa}\), and \(\nu=0.34\); no transverse restraint or transverse stress is present.
Model and interpretation
Use an ideal homogeneous isotropic element, with axial tension positive and energy density based on recoverable loading from zero stress.
Verification
Units: \(\text{Pa}^2/\text{Pa}=\text{Pa}=\text{J/m}^3\), so the energy-density units agree with stored energy per volume.
Signs and direction: The negative lateral strain is contraction caused by positive axial tension, not evidence of an applied lateral force.
Plausibility: The strain is small and the assumed stress is elastic; if loading passed yield, this triangular energy expression would understate total work.
Engineering interpretation
This energy is recoverable only while the supplied elastic limit remains valid. Energy capacity alone does not establish impact safety because geometry, rate effects, and fracture resistance matter.
Recognition strategy
The word alternating and the notch factor signal fatigue awareness; calculate the local idealized stress, then separate static yielding from cyclic failure.
Given information
Use \(\sigma_a=105\ \text{MPa}\), \(K_f=1.8\), and \(S_y=420\ \text{MPa}\); the mean stress and S–N data are not supplied.
Model and interpretation
This is a linear-elastic local stress estimate at the fillet, not an average force result and not a complete fatigue-life model.
Verification
Units: A dimensionless \(K_f\) leaves the stress in \(\text{MPa}\), so the local alternating result carries the correct units.
Signs and direction: Alternating tensile stress changes sign through a cycle; reporting its amplitude as positive does not mean the entire cycle is tension only.
Plausibility: A local amplitude below yield is plausible but cannot prove infinite life, since cracks can initiate below static yield.
Engineering interpretation
A professional review would use validated fatigue data and mean-stress correction, then inspect the fillet and manufacturing quality. Static strength, ductility, and fatigue resistance are distinct material decisions.
Chapter recap
- Use the stress–strain curve to choose an elastic relation only within its demonstrated range, and distinguish strength, stiffness, ductility, and energy absorption.
- Before moving on: identify assumptions, set signs and units, compute the requested response, and check yielding, serviceability, and the relevant failure mode.