The deformation index is a parameter that specifies the mode of control under which time-varying deformation or loading processes occur in a solid. It is useful for evaluating the interaction of elastic stiffness with viscoelastic [1] or fatigue behavior. [2]
If deformation is maintained constant while load is varied, the process is said to be deformation controlled. Similarly, if load is held constant while deformation is varied, the process is said to be load controlled. Between the extremes of deformation and load control, there is a spectrum of intermediate modes of control including energy control.
For example, between two rubber compounds with the same viscoelastic behavior but different stiffnesses, which compound is preferred for a given application? In a strain controlled application, the lower stiffness rubber would operate at smaller stress and therefore produce less viscous heating. But in a stress controlled application, the higher stiffness rubber would operate at small strains thereby producing less viscous heating. In an energy controlled application, the two compounds might give the same amount of viscous heating. The right selection for minimizing viscous heating therefore depends on the mode of control.
Futamura's deformation index can be defined as follows. is the parameter whose value is controlled (ie held constant). is Young's modulus of linear elasticity. is the strain. is the stress.
Particular choices of yield particular modes of control and determine the units of . For , we get strain control: . For , we get energy control: . For , we get stress control: .
The parameter was originally proposed by Shingo Futamura, who won the Melvin Mooney Distinguished Technology Award in recognition of this development. Futamura was concerned with predicting how changes in viscoelastic dissipation were affected by changes to compound stiffness. Later, he extended applicability of the approach to simplify finite element calculations of the coupling of thermal and mechanical behavior in a tire. [3] William Mars adapted Futamura's concept for application in fatigue analysis.
Given that the deformation index may be written in a similar algebraic form, it may be said that the deformation index is in a certain sense analogous to the polytropic index for a polytropic process.
A composite material is a material which is produced from two or more constituent materials. These constituent materials have notably dissimilar chemical or physical properties and are merged to create a material with properties unlike the individual elements. Within the finished structure, the individual elements remain separate and distinct, distinguishing composites from mixtures and solid solutions.
In physics, Hooke's law is an empirical law which states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance—that is, Fs = kx, where k is a constant factor characteristic of the spring, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke. He first stated the law in 1676 as a Latin anagram. He published the solution of his anagram in 1678 as: ut tensio, sic vis. Hooke states in the 1678 work that he was aware of the law since 1660.
In engineering, deformation refers to the change in size or shape of an object. Displacements are the absolute change in position of a point on the object. Deflection is the relative change in external displacements on an object. Strain is the relative internal change in shape of an infinitesimally small cube of material and can be expressed as a non-dimensional change in length or angle of distortion of the cube. Strains are related to the forces acting on the cube, which are known as stress, by a stress-strain curve. The relationship between stress and strain is generally linear and reversible up until the yield point and the deformation is elastic. The linear relationship for a material is known as Young's modulus. Above the yield point, some degree of permanent distortion remains after unloading and is termed plastic deformation. The determination of the stress and strain throughout a solid object is given by the field of strength of materials and for a structure by structural analysis.
In materials science and solid mechanics, Poisson's ratio (nu) is a measure of the Poisson effect, the deformation of a material in directions perpendicular to the specific direction of loading. The value of Poisson's ratio is the negative of the ratio of transverse strain to axial strain. For small values of these changes, is the amount of transversal elongation divided by the amount of axial compression. Most materials have Poisson's ratio values ranging between 0.0 and 0.5. For soft materials, such as rubber, where the bulk modulus is much higher than the shear modulus, Poisson's ratio is near 0.5. For open-cell polymer foams, Poisson's ratio is near zero, since the cells tend to collapse in compression. Many typical solids have Poisson's ratios in the range of 0.2–0.3. The ratio is named after the French mathematician and physicist Siméon Poisson.
Dynamic mechanical analysis is a technique used to study and characterize materials. It is most useful for studying the viscoelastic behavior of polymers. A sinusoidal stress is applied and the strain in the material is measured, allowing one to determine the complex modulus. The temperature of the sample or the frequency of the stress are often varied, leading to variations in the complex modulus; this approach can be used to locate the glass transition temperature of the material, as well as to identify transitions corresponding to other molecular motions.
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to plasticity, in which the object fails to do so and instead remains in its deformed state.
A Maxwell material is the most simple model viscoelastic material showing properties of a typical liquid. It shows viscous flow on the long timescale, but additional elastic resistance to fast deformations. It is named for James Clerk Maxwell who proposed the model in 1867. It is also known as a Maxwell fluid.
Stress–strain analysis is an engineering discipline that uses many methods to determine the stresses and strains in materials and structures subjected to forces. In continuum mechanics, stress is a physical quantity that expresses the internal forces that neighboring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material.
In materials science and continuum mechanics, viscoelasticity is the property of materials that exhibit both viscous and elastic characteristics when undergoing deformation. Viscous materials, like water, resist shear flow and strain linearly with time when a stress is applied. Elastic materials strain when stretched and immediately return to their original state once the stress is removed.
In materials science, work hardening, also known as strain hardening, is the strengthening of a metal or polymer by plastic deformation. Work hardening may be desirable, undesirable, or inconsequential, depending on the context.
A Kelvin-Voigt material, also called a Voigt material, is the most simple model viscoelastic material showing typical rubbery properties. It is purely elastic on long timescales, but shows additional resistance to fast deformation. It is named after the British physicist and engineer Lord Kelvin and German physicist Woldemar Voigt.
A Burgers material is a viscoelastic material having the properties both of elasticity and viscosity. It is named after the Dutch physicist Johannes Martinus Burgers.
Dynamic modulus is the ratio of stress to strain under vibratory conditions. It is a property of viscoelastic materials.
Nanoindentation, also called instrumented indentation testing, is a variety of indentation hardness tests applied to small volumes. Indentation is perhaps the most commonly applied means of testing the mechanical properties of materials. The nanoindentation technique was developed in the mid-1970s to measure the hardness of small volumes of material.
In materials science, stress relaxation is the observed decrease in stress in response to strain generated in the structure. This is primarily due to keeping the structure in a strained condition for some finite interval of time hence causing some amount of plastic strain. This should not be confused with creep, which is a constant state of stress with an increasing amount of strain.
The standard linear solid (SLS), also known as the Zener model, is a method of modeling the behavior of a viscoelastic material using a linear combination of springs and dashpots to represent elastic and viscous components, respectively. Often, the simpler Maxwell model and the Kelvin–Voigt model are used. These models often prove insufficient, however; the Maxwell model does not describe creep or recovery, and the Kelvin–Voigt model does not describe stress relaxation. SLS is the simplest model that predicts both phenomena.
Damage mechanics is concerned with the representation, or modeling, of damage of materials that is suitable for making engineering predictions about the initiation, propagation, and fracture of materials without resorting to a microscopic description that would be too complex for practical engineering analysis.
Viscoplasticity is a theory in continuum mechanics that describes the rate-dependent inelastic behavior of solids. Rate-dependence in this context means that the deformation of the material depends on the rate at which loads are applied. The inelastic behavior that is the subject of viscoplasticity is plastic deformation which means that the material undergoes unrecoverable deformations when a load level is reached. Rate-dependent plasticity is important for transient plasticity calculations. The main difference between rate-independent plastic and viscoplastic material models is that the latter exhibit not only permanent deformations after the application of loads but continue to undergo a creep flow as a function of time under the influence of the applied load.
Plasticity theory for rocks is concerned with the response of rocks to loads beyond the elastic limit. Historically, conventional wisdom has it that rock is brittle and fails by fracture while plasticity is identified with ductile materials. In field scale rock masses, structural discontinuities exist in the rock indicating that failure has taken place. Since the rock has not fallen apart, contrary to expectation of brittle behavior, clearly elasticity theory is not the last work.
Anelasticity is a property of materials that describes their behaviour when undergoing deformation. Its formal definition does not include the physical or atomistic mechanisms but still interprets the anelastic behaviour as a manifestation of internal relaxation processes. It's a special case of elastic behaviour.
