Viscoelasticity

Viscoelasticity is time-dependent mechanical behaviour that combines the characteristics of viscous fluids and elastic solids. A purely elastic material returns to its original shape the moment it is unloaded; a viscoelastic material shows time-dependent strain and stress responses instead.

Two components. The elastic component stores energy and provides immediate, reversible deformation. The viscous component dissipates energy and causes time-dependent, irreversible flow.

Why it matters. All biological tissues behave viscoelastically. Their viscoelastic properties determine how they respond to physiological and traumatic loads, how implants interact with them, and how injuries occur.

The ideal solid. An elastic material obeys Hooke's law: stress is proportional to strain. Deformation is instantaneous and completely reversible, the energy is stored, and there is no time dependence.

The ideal fluid. A viscous material obeys Newton's law of viscosity: stress is proportional to strain rate. Deformation is permanent, the energy is dissipated as heat, and the behaviour is rate-dependent.

The viscoelastic material. It combines the two, so its response depends on both the magnitude and the rate of strain. Deformation has an immediate and a time-dependent component, and some of the energy is stored while some is dissipated.

How the time dependence shows. The response depends on how fast the load is applied. The material goes on deforming after the load has stopped changing, and when it is held at a constant deformation the force required falls with time.

Temperature. Viscoelastic properties are temperature-dependent: higher temperatures increase viscous behaviour and lower temperatures increase elastic behaviour. This is relevant to cryopreserved tissues and to temperature effects on implants.

Creep

Creep is a progressive increase in deformation under constant load. The viscous component allows the material to keep deforming although the load does not change. Deformation is rapid at first, then slows asymptotically toward equilibrium.

The three phases. A creep curve shows primary and secondary creep, and sometimes tertiary creep:

1. Primary creep: rapid initial deformation at a decreasing rate
2. Secondary creep: a constant, steady-state rate of deformation
3. Tertiary creep: accelerating deformation leading to failure

Where it shows. Creep lies behind the daily loss of disc height, the loss of a fracture reduction held in a cast or external fixator, the elongation of ligament grafts under constant tension and the deformation of polyethylene, each taken up with its tissue or implant below. It is also how a scoliosis brace works, applying a constant force that produces gradual correction, and it is why total disc replacement devices may subside into the endplates.

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Stress Relaxation

Stress relaxation is a progressive decrease in the force required to maintain a constant deformation. At constant strain the internal stresses redistribute and decrease, because the viscous component allows molecular rearrangement.

Time course. The decay is exponential. Most of the relaxation occurs in the first minutes to hours, and it continues at a slower rate for days to weeks.

Where it shows. Casts become loose as the initial swelling subsides and the tissues relax. Only the second of these is stress relaxation, which is why the distinguishing clue in the table further down is tension falling while limb girth is unchanged. A fracture reduction maintained by ligamentotaxis loses tension over time; graft, plate and fixator-wire tension are covered under clinical implications.

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Hysteresis

Hysteresis is the energy dissipated when a material is loaded and then unloaded. The energy absorbed during loading is not completely recovered during unloading, so the two curves do not overlap: the loading curve lies above the unloading curve and the difference is dissipated as heat. The area of the loop on the stress-strain curve is the energy lost, and a larger area means more dissipation.

Why it matters. Hysteresis is a protective mechanism: tissues absorb impact energy through it, which protects deeper structures, and it affects the damping of vibrations and impacts. Ligaments and intervertebral discs show significant hysteresis. Repeated loading causes cumulative energy dissipation, which may contribute to tissue heating during repetitive loading.

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Strain-Rate Sensitivity

The mechanical properties depend on the rate at which load is applied. Materials are stiffer and stronger at higher strain rates and more compliant at lower rates. At high rates the viscous component has less time to flow and the material behaves more elastically; at low rates viscous flow contributes more.

How large is the effect? It depends on the tissue, and the answer is smaller than usually taught. In bone, Carter and Hayes found that strength and modulus scale with strain rate to the power of 0.06: real, but modest, since a thousandfold increase in rate raises them by roughly half. In ligament the effect can be smaller still.

The ligament data. Dorlot tested 38 canine ACLs across strain rates spanning more than three orders of magnitude (0.12 to 220% per second) and found the mechanical behaviour not sensitive to strain rate within that range. Human studies show only modest effects within physiological rates. The dramatic rate effects appear mainly at extreme impact rates and at the bone-ligament junction, where failure shifts between avulsion and mid-substance with rate. Both the bone and the canine ligament papers are carded in the Evidence Base.

The safe position for an examination. Rate-dependence is a genuine property of every biological tissue, its magnitude varies by tissue and is often modest, and the common claim that slow bench testing greatly underestimates in-vivo strength is better supported for bone than for ligament. That applies equally to the teaching that static tests underestimate tissue strength during dynamic activities and that ligaments tear in sport at higher loads than in laboratory testing. Impact testing still gives different results from quasi-static testing, and physiological loading rates should be used when testing.

The models build tissue behaviour from two ideal elements: a spring, the elastic element, and a dashpot, the viscous element.

Maxwell model. Spring and dashpot in series. It models stress relaxation well: under constant strain, stress decreases exponentially, with a relaxation time equal to viscosity divided by elastic modulus. Under constant load it predicts instantaneous elastic deformation followed by continued viscous flow, that is, infinite deformation, so it does not model creep accurately.

Kelvin-Voigt model. Spring and dashpot in parallel. It models creep well, giving a delayed elastic response: deformation under constant load is time-dependent but ultimately reaches the elastic equilibrium value. It predicts no instantaneous deformation, which is unrealistic, and it cannot model stress relaxation.

Standard linear solid. A three-parameter combination of Maxwell and Kelvin-Voigt elements that models both creep and stress relaxation: an instantaneous elastic response, followed by a time-dependent response approaching equilibrium. It is the most realistic simple model for biological tissues, approximates actual tissue behaviour better, and is used in finite element modelling.

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Beyond three elements. A generalised Maxwell model places multiple Maxwell elements in parallel and fits stress relaxation data better; a generalised Kelvin-Voigt model places multiple Kelvin-Voigt elements in series and fits creep data better. Quasi-linear viscoelasticity (QLV), proposed by Fung, is an advanced model for soft tissues that separates the time-dependent and strain-dependent responses.

The solid phase. Collagen fibres form the structural framework and provide elastic stiffness; their crimp pattern allows initial compliance, and uncrimping contributes to the toe region. Proteoglycans hydrate the matrix, trap water within it and resist fluid flow, contributing to the time-dependent response. Elastin provides elastic recoil.

The fluid phase. Interstitial water, and its flow through the matrix, contributes to the viscous behaviour. Water content, and with it the degree of viscoelasticity, varies by tissue.

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Bone. Bone is moderately viscoelastic, more viscous when wet, and rate-sensitive. Its creep is minimal compared with soft tissues and matters in long-term loading. At impact rates it is much stronger and stiffer and fails more brittly, so high-energy fractures have more comminution; cortical bone Young's modulus increases approximately 30% at physiological strain rates compared with quasi-static loading. Bone density measurement may vary with loading rate.

Articular cartilage. Cartilage is highly viscoelastic because fluid flows through its solid matrix, so a biphasic model (solid matrix plus fluid) is required. Under a constant load fluid exudes and cartilage thickness decreases with sustained loading, recovering when the load is removed. Joint loading patterns affect cartilage health: cyclic loading may be protective, and prolonged static loading may be harmful.

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Ligaments and tendons. These are strongly viscoelastic and exhibit all four phenomena. Their hysteresis dissipates energy and protects the joint during impact, while repeated loading causes progressive damage. Graft creep and relaxation are covered under clinical implications, and preconditioning has its own section.

Intervertebral disc. The disc is highly viscoelastic through fluid movement in the nucleus pulposus and annulus fibrosus. Under body weight fluid exudes and disc height decreases during the day, recovering overnight when the spine is unloaded in the recumbent position; young adults may lose up to 20 mm of height during a typical day. When the spine is held in a fixed position, intradiscal pressure decreases.

The disc in practice. Diurnal variation affects spinal mechanics, and morning stiffness relates to overnight fluid accumulation. Disc degeneration alters the viscoelastic properties.

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Meniscus. The meniscus is moderately viscoelastic, and collagen fibre orientation affects its behaviour. Viscoelastic damping absorbs impact and protects the articular cartilage from shock loading. Meniscal tears alter the viscoelastic properties, and meniscectomy removes the shock-absorbing function.

Fracture fixation. Alignment maintained by a cast or external fixator may worsen over time through creep of the soft tissues, and serial radiographs monitor for loss of reduction. Compression plates lose some compression force over weeks, while dynamic compression continues as bone resorbs and remodels. External fixator and circular frame wires lose tension over days and may require retensioning.

Ligament reconstruction. The initial tension applied during ACL reconstruction decreases through stress relaxation. Some surgeons pretension grafts cyclically; others retension, or account for the expected relaxation. Viscoelastic creep may contribute to residual laxity after reconstruction, so proper initial tension is important.

Spine. The disc's viscoelastic properties allow it to share load with the facets over time and affect intradiscal pressure measurements. Viscoelastic settling affects construct behaviour after instrumentation, and loss of disc height after fusion affects adjacent segments.

Arthroplasty. Ultra-high molecular weight polyethylene is viscoelastic: creep contributes to its deformation and wear, and highly crosslinked polyethylene reduces creep. In soft-tissue balancing, ligament stress relaxation affects stability, and an initially tight balance may loosen.

Injury. High-energy trauma produces different injury patterns from low-energy trauma, with different bone fragments and different soft-tissue damage. Dashboard injuries differ from gradual compression injuries.

Storage modulus, loss modulus and phase lag are the numbers examiners use to turn the qualitative idea of hysteresis into a measurable material property.

The phase lag. When a sinusoidal strain is applied, stress and strain are out of phase by a phase angle delta, and this lag is the fingerprint of viscoelasticity. The angle places the material between the two ideals:

• Purely elastic solid: zero degrees; stress and strain exactly in phase, all energy stored and returned
• Purely viscous fluid: ninety degrees; stress leads strain by a quarter cycle, all energy dissipated
• Viscoelastic material: between zero and ninety degrees; the closer to ninety, the more viscous (dissipative) the behaviour

The complex modulus. The complex (dynamic) modulus is written E* = E' + iE''. The storage modulus (E') is the in-phase component: the elastic energy stored and recovered each cycle, the solid-like stiffness contribution. The loss modulus (E'') is the out-of-phase component: the energy dissipated as heat each cycle, the viscous damping contribution.

The loss tangent. tan delta = E'' / E' is the damping coefficient, a single dimensionless number for how lossy a material is. A larger tan delta means more energy dissipated per cycle and a larger hysteresis loop, so it is the frequency-domain counterpart of loop area. Because viscous flow is rate-dependent, E', E'' and tan delta all vary with loading frequency, which is exactly why the same tissue shows strain-rate sensitivity.

Clinical relevance. Damping, a high tan delta, is how the meniscus, intervertebral disc, articular cartilage and heel fat pad attenuate impact and protect the underlying bone and cartilage. Degeneration and ageing reduce tissue tan delta, so more shock is transmitted. Implant liners, shock-absorbing insoles and prosthetic feet are selected partly on their damping behaviour.

What it is. Preconditioning is the application of several cycles of loading and unloading, to a fixed strain or load, before any measurement is recorded. During the first few cycles the response is not reproducible: the stress-strain curve shifts, the peak stress at a given strain falls, and the hysteresis loop shrinks from cycle to cycle. After roughly the first several to ten cycles the loops superimpose and the tissue reaches a repeatable, steady-state ("preconditioned") response.

Why it happens. The tissue is reorganising internally toward a stable configuration. Crimped collagen fibres are progressively recruited and aligned, interstitial fluid and proteoglycan-bound water redistribute, and weak, labile inter-fibre bonds break; once these have equilibrated, successive cycles look identical.

In the laboratory. Mechanical properties must be measured from the preconditioned steady state, otherwise the data are irreproducible and cannot be compared between specimens. Standard biomechanical protocols precondition specimens, commonly for around ten cycles, before recording. Woo et al (1986) showed that hysteresis area was significantly reduced only in the first few loading cycles, the signature of preconditioning.

In the body. An athletic warm-up is preconditioning of living tissue. Cyclic sub-maximal loading stabilises the viscoelastic response, slightly lowers stiffness and dissipates the large first-cycle energy loss, which is one biomechanical rationale for warming up before sport. The very first loading cycle of the day (the first squat, the first pitch, the first steps out of bed) shows the largest hysteresis and the greatest tissue lengthening.

Not fatigue. Preconditioning is largely recoverable with rest. That distinguishes it from fatigue damage, in which microdamage accumulates and is permanent.

The commonest exam error is confusing the four phenomena, or confusing viscoelastic creep with plastic deformation. The controlled variable tells them apart.

Viscoelasticity is a basic-science principle rather than a disease, so there are no disease-specific society guidelines. Its translation into practice runs through implant-testing standards, registry-derived wear/loosening data, and curriculum frameworks that are shared worldwide.

Standards and Testing Frameworks

Registry Signals

Arthroplasty registries do not measure viscoelasticity directly, but its consequences appear as outcomes. The NJR (UK), AOANJRR (Australia), AJRR (US) and SHAR/Swedish registries consistently show lower revision for wear/osteolysis with highly-crosslinked polyethylene than with conventional UHMWPE, consistent with reduced creep and wear of crosslinked material. Registry data therefore serve as the large-scale clinical correlate of the bench creep findings (Williams et al, 2003).

High- vs Limited-Resource Practice

Education: Viscoelasticity is core basic-science curriculum in orthopaedic training worldwide, where candidates are expected to define creep, stress relaxation, hysteresis and rate-dependence and apply them to fixation, grafting and bearing surfaces.

Related pages: Articular Cartilage Structure holds the biphasic composition that makes cartilage time-dependent, and it is fluid flow through the solid matrix rather than intrinsic polymer viscosity that produces most of the behaviour described here; Intervertebral Disc Biology is the tissue in which creep is most visible clinically, through diurnal stature loss and the foraminal narrowing the Pollintine card measures; Ligament Biology and Tendon Healing carry the crimp-and-collagen structure behind the toe region, hysteresis and the stress relaxation that loosens a graft after fixation; Polyethylene UHMWPE and XLPE and Highly Crosslinked Vitamin E Polyethylene cover the bearing material whose creep must be distinguished from true wear - a distinction that confounds every gravimetric simulator study including the one carded here; Tribology and Wear and Wear Mechanisms separate those two processes formally; Implant and Fracture Biomechanics applies stress relaxation to construct stability, and Ilizarov External Fixation is where it bites hardest, since wire tension falls measurably in the days after frame application; and Joint Biomechanics: Spine, Hip and Knee place these tissue properties into whole-joint loading.

Is cartilage time-dependence viscoelasticity or fluid flow? Biphasic theory (Mak, Lai and Mow, 1987; Setton, Zhu and Mow, 1993) shows that cartilage creep is dominated by flow-dependent drag of interstitial fluid through a porous matrix, not by intrinsic solid-matrix viscoelasticity. Lumping both together as "viscoelastic" is a simplification examiners may probe.

Disc or vertebra in spinal creep? It is widely taught that the disc drives diurnal height loss. Pollintine et al (2010) showed that the vertebral bodies deform more elastically and that the disc contributes only about half of creep, with the balance from the vertebrae and neural-arch impaction.

Does graft pretensioning improve outcomes? Cyclic pretensioning reduces in-vitro stress relaxation. High-quality clinical evidence that it lowers residual laxity or improves patient-reported outcomes after ACL reconstruction remains limited and inconsistent.

Polyethylene creep or wear? Early polyethylene penetration on radiographs reflects a mix of creep ("bedding-in") and true wear. Separating the two is non-trivial and affects how early radiographic penetration is interpreted (Williams et al, 2003).