Hysteresivity
Encyclopedia
Hysteresivity derives from “hysteresis
”, meaning “lag”. It is the tendency to react slowly to an outside force, or to not return completely to its original state. Whereas the area within a hysteresis loop represents energy dissipated to heat and is an extensive quantity with units of energy, the hysteresivity represents the fraction of the elastic energy that is lost to heat, and is an intensive
property that is dimensionless.
. Most often, frictional stress is described as being analogous to the stress that results from the flow of a viscous fluid
, but in many engineering materials, in soft biological tissue
s, and in living cells, the concept that friction arises only from a viscous stress is now known to be erroneous. For example, Bayliss and Robertson
and Hildebrandt demonstrated that frictional stress in lung tissue
is dependent upon the amount of lung expansion but not the rate of expansion, findings that are fundamentally incompatible with the notion of friction being caused by a viscous stress. If not by a viscous stress, how then does friction arise, and how is it properly described?
In many inert and living materials, the relationship between elastic and frictional stresses turns out to be very nearly invariant
(something unaltered by a transformation). In lung tissues, for example, the frictional stress is almost invariably between 0.1 and 0.2 of the elastic stress, where this fraction is called the hysteresivity, h, or, equivalently, the structural damping coefficient. It is a simple phenomenological fact, therefore, that for each unit of peak elastic strain energy that is stored during a cyclic deformation, 10 to 20 % of that elastic energy is taxed as friction and lost irreversibly to heat. This fixed relationship holds at the level of the whole lung
, isolated lung parenchyma
l tissue strips, isolated smooth muscle
strips, and even isolated living cells.
This close relationship between frictional and elastic stresses is called the structural damping law or, sometimes, the constant phase model. The structural damping law implies that frictional losses are coupled tightly to elastic stresses rather than to viscous stresses, but the precise molecular mechanical origin
of this phenomenon remains unknown.
In material science, the complex elastic modulus of a material, G*(f), at frequency of oscillatory deformation, f, is given by,
where:
This relationship can be rewritten as,
where:
In systems conforming to the structural damping law, the hysteresivity "h" is constant with or insensitive to changes in oscillatory frequency
, and the loss modulus G” (= h G’) becomes a constant fraction of the elastic modulus.
Hysteresis
Hysteresis is the dependence of a system not just on its current environment but also on its past. This dependence arises because the system can be in more than one internal state. To predict its future evolution, either its internal state or its history must be known. If a given input alternately...
”, meaning “lag”. It is the tendency to react slowly to an outside force, or to not return completely to its original state. Whereas the area within a hysteresis loop represents energy dissipated to heat and is an extensive quantity with units of energy, the hysteresivity represents the fraction of the elastic energy that is lost to heat, and is an intensive
Intensive
In grammar, an intensive word form is one which denotes stronger or more forceful action relative to the root on which the intensive is built. Intensives are usually lexical formations, but there may be a regular process for forming intensives from a root...
property that is dimensionless.
Overview
When a force deforms a material it generates elastic stresses and internal frictional stressesFriction
Friction is the force resisting the relative motion of solid surfaces, fluid layers, and/or material elements sliding against each other. There are several types of friction:...
. Most often, frictional stress is described as being analogous to the stress that results from the flow of a viscous fluid
Viscosity
Viscosity is a measure of the resistance of a fluid which is being deformed by either shear or tensile stress. In everyday terms , viscosity is "thickness" or "internal friction". Thus, water is "thin", having a lower viscosity, while honey is "thick", having a higher viscosity...
, but in many engineering materials, in soft biological tissue
Biological tissue
Tissue is a cellular organizational level intermediate between cells and a complete organism. A tissue is an ensemble of cells, not necessarily identical, but from the same origin, that together carry out a specific function. These are called tissues because of their identical functioning...
s, and in living cells, the concept that friction arises only from a viscous stress is now known to be erroneous. For example, Bayliss and Robertson
and Hildebrandt demonstrated that frictional stress in lung tissue
Lung
The lung is the essential respiration organ in many air-breathing animals, including most tetrapods, a few fish and a few snails. In mammals and the more complex life forms, the two lungs are located near the backbone on either side of the heart...
is dependent upon the amount of lung expansion but not the rate of expansion, findings that are fundamentally incompatible with the notion of friction being caused by a viscous stress. If not by a viscous stress, how then does friction arise, and how is it properly described?
In many inert and living materials, the relationship between elastic and frictional stresses turns out to be very nearly invariant
Invariant (physics)
In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.-Examples:In the current era, the immobility of polaris under the diurnal motion of the celestial sphere is a classical illustration of physical invariance.Another...
(something unaltered by a transformation). In lung tissues, for example, the frictional stress is almost invariably between 0.1 and 0.2 of the elastic stress, where this fraction is called the hysteresivity, h, or, equivalently, the structural damping coefficient. It is a simple phenomenological fact, therefore, that for each unit of peak elastic strain energy that is stored during a cyclic deformation, 10 to 20 % of that elastic energy is taxed as friction and lost irreversibly to heat. This fixed relationship holds at the level of the whole lung
, isolated lung parenchyma
Parenchyma
Parenchyma is a term used to describe a bulk of a substance. It is used in different ways in animals and in plants.The term is New Latin, f. Greek παρέγχυμα - parenkhuma, "visceral flesh", f. παρεγχεῖν - parenkhein, "to pour in" f. para-, "beside" + en-, "in" + khein, "to pour"...
l tissue strips, isolated smooth muscle
Smooth muscle
Smooth muscle is an involuntary non-striated muscle. It is divided into two sub-groups; the single-unit and multiunit smooth muscle. Within single-unit smooth muscle tissues, the autonomic nervous system innervates a single cell within a sheet or bundle and the action potential is propagated by...
strips, and even isolated living cells.
This close relationship between frictional and elastic stresses is called the structural damping law or, sometimes, the constant phase model. The structural damping law implies that frictional losses are coupled tightly to elastic stresses rather than to viscous stresses, but the precise molecular mechanical origin
Molecular mechanics
Molecular mechanics uses Newtonian mechanics to model molecular systems. The potential energy of all systems in molecular mechanics is calculated using force fields...
of this phenomenon remains unknown.
In material science, the complex elastic modulus of a material, G*(f), at frequency of oscillatory deformation, f, is given by,
where:
- G*(f)= complex elastic modulus at frequencyFrequencyFrequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
of oscillatory deformation, f - G’ = the elastic modulusElastic modulusAn elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance's tendency to be deformed elastically when a force is applied to it...
- G” = the loss modulus
- j 2 = -1
This relationship can be rewritten as,
where:
- h = G”/G’.
In systems conforming to the structural damping law, the hysteresivity "h" is constant with or insensitive to changes in oscillatory frequency
Oscillation
Oscillation is the repetitive variation, typically in time, of some measure about a central value or between two or more different states. Familiar examples include a swinging pendulum and AC power. The term vibration is sometimes used more narrowly to mean a mechanical oscillation but sometimes...
, and the loss modulus G” (= h G’) becomes a constant fraction of the elastic modulus.
Further reading
- Bayliss L and Robertson G. The visco-elastic properties of the lungs. QJ Experimental Physiology (journal) 29, 1939.
- Bursac P, Lenormand G, Fabry B, Oliver M, Weitz DA, Viasnoff V, Butler JP, and Fredberg JJ. Cytoskeletal remodelling and slow dynamics in the living cell. Nat Mater 4: 557-571, 2005.
- Crandall SH. The role of damping in vibration theory. J Sound and Vibration 11: 3-18, 1970.
- Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, and Fredberg JJ. Scaling the microrheologyRheologyRheology is the study of the flow of matter, primarily in the liquid state, but also as 'soft solids' or solids under conditions in which they respond with plastic flow rather than deforming elastically in response to an applied force....
of living cells. Phys Rev Lett 87: 148102, 2001. - Fabry B, Maksym GN, Butler JP, Glogauer M, Navajas D, Taback NA, Millet EJ, and Fredberg JJ. Time scale and other invariants of integrative mechanical behavior in living cells. Phys Rev E Stat Nonlin Soft Matter Phys 68: 041914, 2003.
- Fabry B, Maksym GN, Shore SA, Moore PE, Panettieri RA, Jr., Butler JP, and Fredberg JJ. Time course and heterogeneity of contractile responses in cultured human airway smooth muscle cells. J Appl Physiol 91: 986-994., 2001.
- Fredberg JJ, Bunk D, Ingenito E, and Shore SA. Tissue resistance and the contractile state of lung parenchyma. J Appl Physiol 74: 1387-1397, 1993.
- Fredberg JJ, Jones KA, Nathan M, Raboudi S, Prakash YS, Shore SA, Butler JP, and Sieck GC. Friction in airway smooth muscle: mechanism, latch, and implications in asthma. J Appl Physiol 81: 2703-2712, 1996.
- Fredberg JJ and Stamenovic D. On the imperfect elasticity of lung tissue. J Appl Physiol 67: 2408-2419, 1989.
- Fung Y. BiomechanicsBiomechanicsBiomechanics is the application of mechanical principles to biological systems, such as humans, animals, plants, organs, and cells. Perhaps one of the best definitions was provided by Herbert Hatze in 1974: "Biomechanics is the study of the structure and function of biological systems by means of...
: Mechanical Properties of Living Tissues. New York:: Springer-Verlag, 1988. - Hantos Z, Daroczy B, Suki B, Nagy S, and Fredberg JJ. Input impedanceElectrical impedanceElectrical impedance, or simply impedance, is the measure of the opposition that an electrical circuit presents to the passage of a current when a voltage is applied. In quantitative terms, it is the complex ratio of the voltage to the current in an alternating current circuit...
and peripheral inhomogeneity of dog lungs. J Appl Physiol 72: 168-178, 1992. - Hildebrandt J. Comparison of mathematical models for cat lung and viscoelastic balloon derived by Laplace transform methods from pressure-volume data. Bull Math Biophys 31: 651-667, 1969.
- Hubmayr RD. Biology lessons from oscillatory cell mechanics. J Appl Physiol 89: 1617-1618, 2000.
- Jensen A, Atileh H, Suki B, Ingenito EP, and Lutchen KR. Airway caliber in healthy and asthmatic subjects: effects of bronchial challenge and deep inspirations. J Appl Physiol 91: 506-515; discussion 504-505, 2001.
- Kaczka DW, Ingenito EP, Suki B, Lutchen KR. Partitioning airway and lung tissue resistances in humans: effects of bronchoconstriction. J Appl Physiol 82: 1531-1541, 1997.