constitutive relationships in biology
— The cytoplasm may behave like a viscous fluid over time
— The nucleus resists external forces elastically
— The actin cortex or the membrane may combine elastic and viscous effects
— Some tissues stiffen when stretched : nonlinear behavior.
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| Term | Definition |
|---|---|
| constitutive relationships in biology | — The cytoplasm may behave like a viscous fluid over time — The nucleus resists external forces elastically — The actin cortex or the membrane may combine elastic and viscous effects — Some tissues stiffen when stretched : nonlinear behavior. |
| linear elastic material (hooke's law) | An elastic material returns to its initial shape after deformation. The stress–strain relationship is linear: σ = E ε where E is Young’s modulus (stiffness). This is a reversible behavior. |
| linear elasticity applies when | deformations are small and proportional to the applied stresses |
| during a simple tensile test, the stress of a spring is defined as? | σ = F/S0 with S0 the initial cross-sectional area. |
| biological examples of linear elasticity | — Cytoskeleton under small deformation — Stretched cell membrane |
| poisson's ratio | When a solid is subjected to tension, it elongates along the direction of the force but contracts in the perpendicular directions. This transverse contraction is characterized by Poisson’s ratio ν, defined as : ∆a/a0= ∆b/b0 = −ν x ∆L/L0 = −νε11 |
| stress-strain curve | During a tensile test, a typical curve is obtained with three phases: — Phase OA (elastic) : stress is proportional to strain, reversible. — Phase AB to C (plastic) : irreversible deformation after exceeding the elastic limit. — Point C (fracture) : material failure. |
| elastic deformation | — A material is elastic if it returns to its shape after unloading. — For small deformations, the response is linear : σ = Eε — Transverse contraction is characterized by the Poisson’s ratio ν. — Young’s modulus E measures stiffness (slope of the linear phase). |
| viscous material (newtonian fluid) | A viscous material flows when stress is applied. The stress is proportional to the strain rate : σ = η ε ̇ where η is the viscosity and ε ̇ the shear rate. This is a dissipative behavior (energy loss). |
| biological examples of viscous material | — Cytoplasm ; — Fluid extracellular medium ; — Nucleoplasm (over long timescales). |
| perfectly plastic material | A plastic material undergoes irreversible deformation beyond a threshold (yield limit). This model is rare in biology but can be useful in extreme cases. |
| behavior of perfectly plastic material | — Before the threshold: elastic response ; — Beyond the threshold: permanent deformation. |
| biological example of perfectly plastic material | — Rupture of an intercellular junction ; — Irreversible nuclear creep under prolonged stress. |
| elastic vs. viscous vs. plastic materials | — Elastic : σ = Eε – Reversible, rigid (e.g., cell cortex) — Viscous : σ = ηε ̇ – Fluid, dissipative (e.g., cytoplasm) — Plastic : irreversible deformation beyond a threshold (e.g., membrane rupture) |
| linear viscoelastic laws | using combinations of springs (elastic elements) and dashpots (viscous elements). many biological materials exhibit both elastic and viscous properties |
| maxwell model -> viscous + elastic (in series) | The constitutive equation of the model is : dσ/dt + σ/τ = E * dε/dt, with τ = η/E This equation links the rate of change of stress and strain. It can be solved for common cases like stress relaxation. If a constant strain ε = ε0 is suddenly applied at t = 0, the solution is : σ(t) = E ε0 e^−t/τ |
| characteristics of the maxwell model | — Good model for viscoelastic fluids. — Reproduces exponential decay of stress under constant strain. — Characteristic time τ = η/E controls relaxation rate. |
| characteristics of the kelvin-voigt model | — Creep — No stress relaxation: stress remains constant if strain is imposed. — Good model for elastic and viscous structures in parallel, such as the cell cortex or soft tissues. |
| biological interpretation of kelvin-voigt model | In an indentation test on a cell (AFM, micro-needle), one would observe a progressive indentation increase, reflecting the combined viscous + elastic behavior of the cortex. |
| creep | the gradual increase in strain under constant stress |
| creep time equation | τ = η/E quantifies how quickly deformation evolves toward equilibrium |
| tension/compression test | stretching or compressing a cell or tissue to extract Young’s modulus. |
| indentation | pressing a probe (AFM tip or bead) into a cell to measure local stiffness. |
| micropipette aspiration | suction of a membrane fragment to evaluate its tension and viscoelasticity. |
| passive microrheology | tracking the Brownian motion of microbeads in the cytoplasm. |
| active microrheology | applying forces using optical or magnetic tweezers to probe local mechanical properties. |
| quantities measured by mechanical characterization experiments | — Young’s modulus E : measure of the material’s elasticity (stiffness). — Viscosity η : resistance to flow or slow deformation. — Creep or relaxation time : temporal characteristics of viscoelastic responses. |
| biological examples that can be measured with mech. characterization experiments | — Tumor cells: often softer than healthy cells, detectable by AFM or magnetic beads. — Embryology: stiffness of embryonic tissues evolves during development, influencing morphogenesis. — Aging or differentiation: cell mechanical properties change with physiological state. |
| cell mechanical properties like stiffness, viscosity ___ | are accessible via well-established experiments (tension, AFM, microrheology). they help distinguish cellular states (normal vs. tumor, differentiated vs undifferentiated) and shed light on biological processes |