Unit 3: Stress and strain

BTY730 — Biomechanics 7 min read

The mechanical response of any tissue, implant or structural member begins with a single question: how does a body deform when force is applied to it? This unit builds the vocabulary — force, deformation, stress, strain — and the experimental and mathematical tools that connect them, culminating in the parameters (stiffness, strength, ductility) that let a biomechanist select bone, cartilage, metal or polymer for a load-bearing role.

Defining ideas the unit relies on:

Continuum assumption: Material is treated as continuously distributed, so force and deformation can be expressed per unit area and per unit length rather than atom by atom.
Intensity, not total: Stress and strain normalise force and displacement by geometry, making them properties of the material rather than of the specimen size.
Small-deformation elasticity: Unless stated, deformations are small and initially reversible, so linear relations apply.
Sign convention: Tension and elongation are positive; compression and shortening are negative.
SI units: Force in newtons (N), area in m², stress in pascals (Pa = N/m²), strain dimensionless.

II. From Force to Deformation

How external loading produces internal resistance and shape change, and how both are measured intensively.

A. Applied forces and deformations

An external load sets up internal forces and a corresponding change in shape or size that the material must accommodate.

Applied force (load): External action P transmitted to a body, measured in newtons; e.g. body weight transmitted through the femoral head.
Deformation: The resulting change in dimension — total elongation δ (m) in tension, contraction in compression, angular change in shear.
Internal reaction: By equilibrium, an internal force equal and opposite to P acts across any imagined cut section, distributed over the cross-section.
Dependence on geometry: The same load P produces larger deformation in a longer or thinner member — motivating normalisation.

B. Stress and strain

Stress and strain remove geometry from the picture by expressing force per area and deformation per original length.

Normal stress:
TEXT
```
σ = P / A
```
where σ = stress (Pa), P = axial force (N), A = cross-sectional area (m²).
Normal strain:
TEXT
```
ε = δ / L₀
```
where ε = strain (dimensionless), δ = elongation (m), L₀ = original length (m).
Shear stress and strain: τ = V / A (V = force parallel to the surface); shear strain γ is the angular distortion in radians.
Worked example: A tendon of area A = 20 mm² = 20×10⁻⁶ m² carrying P = 400 N sustains σ = 400 / 20×10⁻⁶ = 20 MPa.
Why intensive: Because σ and ε are per-unit quantities, a stress-strain curve characterises the material and predicts behaviour at any specimen size.

III. Loading Configurations and the Tension Test

The standard ways force is applied, and the experiment that turns them into data.

A. Basic loading configurations

Every complex load decomposes into a small set of elementary modes, each with a characteristic stress.

Axial (tension / compression): Force along the member's axis; produces σ = P/A uniformly. Tension elongates (a ligament pulled taut); compression shortens (a vertebral body under body weight).
Shear: Force parallel to the section; τ = V/A; e.g. a pin in a hip fixation plate.
- Bending: Transverse load creates a moment M; stress varies linearly across depth, σ = M·y / I, tensile on one face, compressive on the other (a long bone under a fall).
- Torsion: Twisting moment T produces shear that grows with radius, τ = T·r / J (a tibia in a rotational skiing injury).
- Combined loading: Real joints experience several modes at once, superposed while behaviour stays linear.

B. Uniaxial tension test

The tension test applies a controlled, single-axis pull to a standardised specimen to record its intrinsic force-deformation response.

Purpose: Isolate one loading mode so measured behaviour reflects the material, not the geometry.
Specimen: A "dogbone" with a reduced gauge length L₀ and uniform gauge area A₀, gripped at wide ends to localise deformation.
Procedure: A testing machine extends the specimen at a constant rate; a load cell reads P, an extensometer reads δ over L₀.
Controlled variables: Temperature, strain rate and hydration matter for biological tissue, which is viscoelastic and stiffens at higher strain rates.
Output: Paired (δ, P) data, later converted to (ε, σ).

C. Load-elongation diagrams

Plotting raw load against elongation gives the specimen's response before any normalisation.

Axes: Load P (N) versus elongation δ (mm) — both extensive, so the curve depends on A₀ and L₀.
Initial linear region: Slope equals the specimen stiffness k = P/δ (N/m), the structural analogue of a spring constant.
Peak and failure: The maximum load is the specimen's ultimate capacity; the curve terminates at fracture.
Area under the curve: Total work done on the specimen (energy to failure), in joules.
Conversion to stress-strain: Dividing P by A₀ and δ by L₀ rescales the axes into the size-independent stress-strain diagram, which is the intrinsic material record.

IV. Elastic Response and Material Characterisation

The law governing initial behaviour, the properties read off the curve, the models used to idealise it, and the resulting material parameters.

A. Hooke's law

Within the initial elastic range, stress is directly proportional to strain.

Statement (uniaxial):
TEXT
```
σ = E · ε
```
where E = Young's modulus (Pa), the slope of the linear region.
Meaning of E: Intrinsic stiffness — cortical bone ≈ 17 GPa, tendon ≈ 1 GPa, steel ≈ 200 GPa; a higher E means less strain for a given stress.
Shear form: τ = G·γ, with G the shear modulus.
Poisson coupling: Axial strain induces transverse strain, ν = −ε_lateral / ε_axial, typically 0.3 for metals, near 0.5 for near-incompressible soft tissue.
Validity: Only up to the proportional limit; beyond it the relation is no longer linear.

B. Properties based on stress-strain diagrams

The stress-strain curve is read like a map, each landmark defining a design-relevant property.

Proportional limit: Highest stress at which σ = Eε still holds.
Elastic limit / yield strength: Stress beyond which deformation becomes permanent; when yield is gradual, the 0.2% offset yield is used — a line of slope E drawn from ε = 0.002.
Ultimate tensile strength (UTS): Maximum stress the material sustains, the curve's peak.
Fracture stress: Stress at the break point.
Ductility: Percent elongation at fracture, (L_f − L₀)/L₀ × 100; high for mild steel, low for cortical bone (brittle).
Resilience: Elastic energy stored to the elastic limit — area under the linear portion.
Toughness: Total energy absorbed to fracture — the entire area under the curve.

C. Idealized model for material behavior

Idealised models replace the real curve with simplified segments that make analysis tractable.

Linear elastic model: A single straight line σ = Eε; fully reversible, no permanent set — the default for small-deformation problems.
Elastic-perfectly-plastic model: Elastic to the yield stress σ_y, then a horizontal plateau at constant σ_y; captures metals that flow at fixed stress.
- Elastic-plastic with hardening: Adds a sloped post-yield line to represent strain hardening.
- Rigid-plastic model: Ignores elastic strain entirely; used in metal-forming analysis where plastic strain dominates.
- Viscoelastic model: For tissue and polymers, stress depends on strain and strain rate, modelled by springs and dashpots (Maxwell, Kelvin-Voigt) to capture creep and stress relaxation.
- Choice rule: Use the simplest model whose assumptions hold over the loading range of interest.

D. Mechanical properties of materials

The material constants extracted above classify a material and drive selection.

Stiffness properties: E, G, and ν — govern deflection under load; three constants fully describe an isotropic linear material via G = E / [2(1+ν)].
Strength properties: Yield strength σ_y, UTS, fracture stress — set the safe load ceiling.
Ductile vs brittle:
- Ductile: Large plastic strain before fracture, clear yield and necking (mild steel, ductile bone under slow load) — fails with warning.
- Brittle: Little plastic strain, fracture near the elastic limit (glass, dry cortical bone, ceramics) — fails abruptly.
Energy properties: Resilience and toughness set the capacity to absorb impact — critical for helmets and implants.
Anisotropy in biological tissue: Bone and tendon have direction-dependent E, being stiffer along the grain of collagen or osteons than across it.
Safety factor: Design stress is kept below σ_y by a factor n = σ_y / σ_allowable, accommodating variability in living tissue and loading.