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3 Mechanical Aspects of Biosystems
3.9 Viscoelastic Materials
If the stress energy in a material is transformed into heat through plastic flow, the
material shows viscous behavior. If the stress energy is stored as potential energy by
the distortion of molecular bonds, the material is acting elastically. Some materials
show both properties. Such a material is called viscoelastic. The transition from
elastic to plastic behavior depends on the material, on the rate of buildup of stress, on
temperature, and on the background pressure. Viscoelastic materials can be modeled
by allowing the stress-strain relation to include temporal rates and time integrals of
the stress or strain.
If a material has no static limit in response to a stress, the material is a fluid,
for which the strain tensor depends on the relative displacement per unit time rather
than just relative displacements alone.
The dissipative effects of macromolecules in biological systems are those that
cause a loss of mechanical energy into heat. For example, a vibrating chain molecule
might lose energy by bumping into adjacent molecules. We say dissipation ‘damps’
mechanical systems. ‘Dampers’ are devices which introduce forces which oppose
any motion, while dissipating the work-energy of the forces causing the motion.
Mechanically, they can be represented by ‘dashpots’, a cylindrical device holding a
fluid with a piston at one end of the cylinder, but the piston allows a small amount of
fluid to flow past to relieve any pressure in the fluid. On the suspension of cars, they
are called ‘shock absorbers’. Magnetic dashpots use a magnet on an element which
moves relative to a nearby good conductor. The induced Faraday currents oppose
the motion of the magnet, and any object attached. In fluid motion, friction both on
the surface of fluid elements and by squeezing the volume of those elements can
cause viscous heating.
Even the protoplasm inside every living cell has viscoelastic behavior, since the
microtubules and microfilaments (‘cytoskeleton’) within tend to give the material
some elastic behavior for small stress over short times, but protoplasm can change
from a gel like system to a sol when stress is exerted over longer times.
We can model viscoelastic materials with combinations of springs and dampers
(as in Fig. 3.9). For such models, each spring has a restoring force in proportion to
the stretch and compression of the spring, while each damper has an opposing force
in proportion to the stretch or compressional speed of the damper.
3.9.1 Kelvin-Voigt Model
A ‘Kelvin-Voigt’ viscoelastic material acts as a spring in parallel with a damper,
i.e. the material is elastic for slow changes in stress, and viscous for fast changes.
By balancing forces, if the Kelvin material is stretched by δx, then there will be an
opposing force given by δF = −kδx − ηd(δx)/dt. We see that without the spring
(k = 0), the material acts as a viscous fluid, and that without the damper (η = 0),
the material acts as an elastic spring. As a stress/strain relation, we have, for the
Kelvin-Voigt case,
3 Mechanical Aspects of Biosystems
3.9 Viscoelastic Materials
If the stress energy in a material is transformed into heat through plastic flow, the
material shows viscous behavior. If the stress energy is stored as potential energy by
the distortion of molecular bonds, the material is acting elastically. Some materials
show both properties. Such a material is called viscoelastic. The transition from
elastic to plastic behavior depends on the material, on the rate of buildup of stress, on
temperature, and on the background pressure. Viscoelastic materials can be modeled
by allowing the stress-strain relation to include temporal rates and time integrals of
the stress or strain.
If a material has no static limit in response to a stress, the material is a fluid,
for which the strain tensor depends on the relative displacement per unit time rather
than just relative displacements alone.
The dissipative effects of macromolecules in biological systems are those that
cause a loss of mechanical energy into heat. For example, a vibrating chain molecule
might lose energy by bumping into adjacent molecules. We say dissipation ‘damps’
mechanical systems. ‘Dampers’ are devices which introduce forces which oppose
any motion, while dissipating the work-energy of the forces causing the motion.
Mechanically, they can be represented by ‘dashpots’, a cylindrical device holding a
fluid with a piston at one end of the cylinder, but the piston allows a small amount of
fluid to flow past to relieve any pressure in the fluid. On the suspension of cars, they
are called ‘shock absorbers’. Magnetic dashpots use a magnet on an element which
moves relative to a nearby good conductor. The induced Faraday currents oppose
the motion of the magnet, and any object attached. In fluid motion, friction both on
the surface of fluid elements and by squeezing the volume of those elements can
cause viscous heating.
Even the protoplasm inside every living cell has viscoelastic behavior, since the
microtubules and microfilaments (‘cytoskeleton’) within tend to give the material
some elastic behavior for small stress over short times, but protoplasm can change
from a gel like system to a sol when stress is exerted over longer times.
We can model viscoelastic materials with combinations of springs and dampers
(as in Fig. 3.9). For such models, each spring has a restoring force in proportion to
the stretch and compression of the spring, while each damper has an opposing force
in proportion to the stretch or compressional speed of the damper.
3.9.1 Kelvin-Voigt Model
A ‘Kelvin-Voigt’ viscoelastic material acts as a spring in parallel with a damper,
i.e. the material is elastic for slow changes in stress, and viscous for fast changes.
By balancing forces, if the Kelvin material is stretched by δx, then there will be an
opposing force given by δF = −kδx − ηd(δx)/dt. We see that without the spring
(k = 0), the material acts as a viscous fluid, and that without the damper (η = 0),
the material acts as an elastic spring. As a stress/strain relation, we have, for the
Kelvin-Voigt case,