Anelasticity

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 is a behaviour differing from elastic behaviour.

Definition and elasticity

Considering first an ideal elastic material, Hooke's law defines the relation between stress and strain as:
The constant is called the modulus of elasticity while its reciprocal is called the modulus of compliance.
There are three postulates that define the ideal elastic behaviour:

the strain response to each level of applied stress has a unique equilibrium value;
the equilibrium response is achieved instantaneously;
the response is linear.

These conditions may be lifted in various combinations to describe different types of behaviour, summarized in the following table:

	Unique equilibrium relationship	Instantaneous	Linear
Ideal elasticity	Yes	Yes	Yes
Nonlinear elasticity	Yes	Yes	No
Instantaneous plasticity	No	Yes	No
Anelasticity	Yes	No	Yes
Linear viscoelasticity	No	No	Yes

Anelasticity is therefore by the existence of a part of time dependent reaction, in addition to the elastic one in the material considered. It is also usually a very small fraction of the total response and so, in this sense, the usual meaning of "anelasticity" as "without elasticity" is improper in a physical sense.
The formal definition of linearity is: "If a given stress history produces the strain, and if a stress gives rise to, then the stress will give rise to the strain." The postulate of linearity is used because of its practical usefulness. The theory would become much more complicated otherwise, but in cases of materials under low stress this postulate can be considered true.
In general, the change of an external variable of a thermodynamic system causes a response from the system called thermal relaxation that leads it to a new equilibrium state. In the case of mechanical changes, the response is known as anelastic relaxation, and in the same formal way can be also described for example dielectric or magnetic relaxation. The internal values are coupled to stress and strain through kinetic processes such as diffusion. So that the external manifestation of the internal relaxation behaviours is the stress strain relation, which in this case is time dependant.

Static response functions

s can be made where either the stress or strain is held constant for a certain time. These are called quasi-static, and in this case, anelastic materials exhibit creep, elastic aftereffect, and stress relaxation.
In these experiments a stress applied and held constant while the strain is observed as a function of time. This response function is called creep defined by and characterizes the properties of the solid. The initial value of is called the unrelaxed compliance, the equilibrium value is called relaxed compliance and their difference is called the relaxation of the compliance.
After a creep experiment has been run for a while, when stress is released the elastic spring-back is in general followed by a time dependent decay of the strain. This effect is called the elastic aftereffect or “creep recovery”. The ideal elastic solid returns to zero strain immediately, without any after-effect, while in the case of anelasticity total recovery takes time, and that is the aftereffect. The linear viscoelastic solid only recovers partially, because the viscous contribution to strain cannot be recovered.
In a stress relaxation experiment the stress σ is observed as a function of time while keeping a constant strain and defining a stress relaxation function similarly to the creep function, with unrelaxed and relaxed modulus M_U and M_R.
At equilibrium,, and at a short timescale, when the material behaves as if ideally elastic, also holds.

Dynamic response functions and loss angle

To get information about the behaviour of a material over short periods of time dynamic experiments are needed. In this kind of experiment a periodic stress is imposed on the system, and the phase lag of the strain is determined.
The stress can be written as a complex number where is the amplitude and the frequency of vibration. Then the strain is periodic with the same frequency where is the strain amplitude and is the angle by which the strain lags, called loss angle. For ideal elasticity. For the anelastic case is in general not zero, so the ratio is complex. This quantity is called the complex compliance. Thus,
where, the absolute value of, is called the absolute dynamic compliance, given by.
This way two real dynamic response functions are defined, and. Two other real response functions can also be introduced by writing the previous equation in another notation:
where the real part is called "storage compliance" and the imaginary part is called "loss compliance".
J₁ and J₂ being called "storage compliance" and "loss compliance" respectively is significant, because calculating the energy stored and the energy dissipated in a cycle of vibration gives following equations:
where is the energy dissipated in a full cycle per unit of volume while the maximum stored energy per unit volume is given by:
The ratio of the energy dissipated to the maximum stored energy is called the "specific damping capacity”. This ratio can be written as a function of the loss angle by.
This shows that the loss angle gives a measure of the fraction of energy lost per cycle due to anelastic behaviour, and so it is known as the internal friction of the material.

Resonant and wave propagation methods

The dynamic response functions can only be measured in an experiment at frequencies below any resonance of the system used. While theoretically easy to do, in practice the angle is difficult to measure when very small, for example in crystalline materials. Therefore, subresonant methods are not generally used. Instead, methods where the inertia of the system is considered are used. These can be divided into two categories:

methods employing resonant systems at a natural frequency
wave propagation methods
Forced vibrations

The response of a system in a forced-vibration experiment with a periodic force has a maximum of the displacement at a certain frequency of the force. This is known as resonance, and the resonant frequency. The resonance equation is simplified in the case of. In this case the dependence of on frequency is plotted as a Lorentzian curve. If the two values and are the ones at which falls to half maximum value, then:
The loss angle that measures the internal friction can be obtained directly from the plot, since it is the width of the resonance peak at half-maximum. With this and the resonant frequency it is then possible to obtain the primary response functions. By changing the inertia of the sample the resonant frequency changes, and so can the response functions at different frequencies can be obtained.

Free vibrations

The more common way of obtaining the anelastic response is measuring the damping of the free vibrations of a sample. Solving the equation of motion for this case includes the constant called logarithmic decrement. Its value is constant and is. It represents the natural logarithm of the ratio of successive vibrations' amplitudes:
It is a convenient and direct way of measuring the damping, as it is directly related to the internal friction.

Wave propagation

Wave propagation methods utilize a wave traveling down the specimen in one direction at a time to avoid any interference effects. If the specimen is long enough and the damping high enough, this can be done by continuous wave propagation. More commonly, for crystalline materials with low damping, a pulse propagation method is used. This method employs a wave packet whose length is small compared to the specimen. The pulse is produced by a transducer at one end of the sample, and the velocity of the pulse is determined either by the time it takes to reach the end of the sample, or the time it takes to come back after a reflection at the end. The attenuation of the pulse is determined by the decrease in amplitude after successive reflections.

Boltzmann superposition principle

Each response function constitutes a complete representation of the anelastic properties of the solid. Therefore, any one of the response functions can be used to completely describe the anelastic behaviour of the solid, and every other response function can be derived from the chosen one.
The Boltzmann superposition principle states that every stress applied at a different time deforms the material as it if were the only one. This can be written generally for a series of stresses that are applied at successive times. In this situation, the total strain will be:
or in the integral form, is the stress is varied continuously:
The controlled variable can always be changed, expressing the stress as a function of time in a similar way:
These integral expressions are a generalization of Hooke's law in the case of anelasticity, and they show that material acts almost as they have a memory of their history of stress and strain. These two of equations imply that there is a relation between the J and M. To obtain it the method of Laplace transforms can be used, or they can be related implicitly by:
In this way though they are correlated in a complicated manner and it is not easy to evaluate one of these functions knowing the other. Hover it is still possible in principle to derive the stress relaxation function from the creep function and vice versa thanks to the Boltzamann principle.