The stress–strain curve and the material properties of orthopaedic biomaterials

Stress and strain

A load applied to an object produces a deformation that depends on how big the object is. To describe the material rather than the object, both quantities are normalised. Stress is the force divided by the cross-sectional area over which it acts, measured in newtons per square metre, or pascals. Strain is the change in length divided by the original length; it is a ratio, so it has no units and is often quoted as a percentage. Plotting one against the other for a specimen pulled to failure gives the stress–strain curve, and almost every material property examined in the written paper can be read from it.

Reading the curve

Elastic region

The initial straight part of the curve. Stress is proportional to strain (Hooke’s law), and the material returns to its original length when the load is removed.

Young’s modulus (elastic modulus)

The slope of the elastic region: stress divided by strain. It describes how stiff the material is, and it is a property of the material itself. A steep slope means a stiff material.

Yield point

The stress at which the material begins to deform plastically. Beyond it, some deformation is permanent. Where the transition is gradual, the yield strength is defined by a 0.2% strain offset. Bending a plate to contour it takes the plate past its yield point.

Plastic region

From the yield point to failure. Deformation here is not recovered on unloading. In metals the stress needed to continue deforming rises for a time — strain hardening.

Ultimate tensile strength

The highest stress on the curve: the maximum the material can sustain. Beyond it a ductile specimen necks and the engineering stress falls until it breaks.

Failure (breaking) point

Where the specimen fractures. In a brittle material it lies at or just beyond the end of the elastic region; in a ductile material it lies well beyond the yield point.

Toughness

The area under the whole curve up to failure: the energy per unit volume the material absorbs before it breaks. The area under the elastic region alone is resilience — the energy that can be absorbed and recovered.

Ductility and brittleness

A ductile material undergoes a large plastic deformation before failure; a brittle one fails with little or none. Metals are ductile; ceramics and bone cement are brittle.

Material properties and structural properties

Young’s modulus, yield strength and ultimate strength belong to the material. How a nail, a plate or a bone behaves also depends on its shape, and those are structural properties: stiffness, or rigidity, is the product of the modulus and a geometric term.

• Bending rigidity is the modulus multiplied by the second moment of area. For a solid cylinder the second moment of area is proportional to the fourth power of the radius, so a small increase in the diameter of an intramedullary nail produces a large increase in rigidity.
• For a plate, the second moment of area is proportional to the width and to the cube of the thickness, so doubling the thickness of a plate makes it eight times as rigid in bending in that plane.
• Torsional rigidity depends in the same way on the polar moment of inertia, which also rises with the fourth power of the radius. A slotted (open-section) nail is far less rigid in torsion than a closed one.

This is why a statement such as ‘titanium is less stiff than steel’ is true of the materials but not necessarily of two implants: a larger titanium nail can be more rigid than a smaller steel one.

Isotropy, anisotropy and viscoelasticity

An isotropic material has the same mechanical properties in every direction; metals and ceramics approximate to this. Bone, tendon, ligament and cartilage are anisotropic: their properties depend on the direction of loading. Cortical bone is strongest in compression, weaker in tension and weakest in shear.

Biological tissues and polymers are also viscoelastic: their response depends on time and on the rate of loading, and four behaviours follow.

Creep

Progressive deformation under a constant load. It is the basis of serial casting and tissue expansion, and the reason polyethylene cold-flows in service.

Stress relaxation

A fall in stress with time when the material is held at a constant deformation. A tensioned ligament graft loses tension after fixation for this reason.

Hysteresis

The loading and unloading curves do not coincide; the area between them is energy dissipated, mostly as heat.

Rate dependence

Stiffness and strength increase with the rate of loading. Bone loaded rapidly is stiffer and stores more energy before it fails, which is why high-energy fractures are comminuted.

Fatigue

Most implants do not fail because a single load exceeds their ultimate strength. They fail by fatigue: the growth of a crack under repeated loads that are each well below that strength. Fatigue behaviour is described by an S–N curve, which plots the stress applied against the number of cycles to failure. The endurance limit is the stress below which the material survives an effectively unlimited number of cycles — conventionally ten million.

Fatigue cracks start at stress risers — screw holes, sharp changes in section, scratches and notches. Titanium alloys are particularly notch-sensitive, so a scratch on a titanium implant matters more than the same scratch on steel. Clinically the lesson is that fracture fixation is a race between union and fatigue failure of the implant: a plate over an ununited fracture will eventually break.

The materials compared

Metals

All three implant alloys resist corrosion by forming a passive oxide layer on their surface — chromium oxide on stainless steel and cobalt–chrome, titanium oxide on titanium. Corrosion occurs when that layer is breached or cannot re-form: galvanic corrosion between dissimilar metals in contact, crevice corrosion in a confined space starved of oxygen, fretting corrosion from micromotion at a junction such as a modular taper, and pitting.

Stiffness has a biological cost. An implant much stiffer than bone carries load that the bone would otherwise carry, and the unloaded bone resorbs, as Wolff’s law predicts. This stress shielding is the argument for titanium, whose modulus is closer to that of bone, in uncemented stems and in plates. Titanium’s weakness is its surface: it wears and scratches readily, so it is not used as a bearing.

Polyethylene

Ultra-high-molecular-weight polyethylene is the bearing in most joint replacements. Irradiation breaks its chains and creates free radicals; the radicals can combine to cross-link neighbouring chains, which markedly reduces wear, or react with oxygen, which makes the material brittle. Highly cross-linked polyethylene is therefore irradiated and then heat-treated to deal with the residual free radicals. Remelting, above the melting point, removes them but reduces crystallinity and with it the mechanical properties; annealing, below the melting point, preserves the mechanical properties but leaves some free radicals behind. Blending with vitamin E as an antioxidant is a third approach. The trade-off to remember is that cross-linking buys wear resistance at the expense of fatigue strength and fracture toughness.

Bone cement

Polymethylmethacrylate is a grout, not a glue: it has no adhesive properties and works by mechanical interlock with cancellous bone and with the surface of the implant. Its modulus lies between that of the metal stem and that of cancellous bone, so it transmits load gradually between them. It is strongest in compression and weak in tension and shear, it is brittle, and it is viscoelastic — it creeps, which some stem designs rely on. Polymerisation is exothermic.

Ceramics

Alumina and zirconia-toughened alumina are extremely hard and can be polished to a very smooth, wettable surface, so they produce the lowest wear of the available bearings. The price is brittleness: a ceramic has a high modulus and very little capacity for plastic deformation, so it fails by fracture rather than by yielding. Squeaking is a recognised complication of ceramic-on-ceramic bearings.