Anelastic material

In this section, pedagogical models for the time dependence of mechanical response are developed. Elastic stress and strain are rank-two tensors, and the compliance (or stiffness) are rank-four material property tensors that connect them. In this section, a simple spring and dashpot analog is used to model the mechanical response of anelastic materials. Scalar forces in the spring and dashpot model become analogs for a more complex stress tensor in materials. To enforce this analogy, we use the terms stress and strain below, but we do not treat them as tensors. 

Anelastic material is a special case of viscoelasticity in the sense that this kind of materials do not fully recover its original state on the removal of load. 

Finally, Fig. 8.3 shows a third form of elastic behaviour found in certain materials. This is called anelasfic behaviour. All solids are anelastic to a small extent even in the regime where they are nominally elastic, the loading curve does not exactly follow the unloading curve, and energy is dissipated (equal to the shaded area) when the solid is cycled. Sometimes this is useful - if you wish to damp out vibrations or noise, for example you... 

For a Hookian material, the concept of minimum strain energy states that a material fails, for example cell wall disruption occurs, when the total strain energy per unit volume attains a critical value. Such an approach has been used in the past to describe a number of experimental observations on the breakage of filamentous micro-organisms [78,79]. Unfortunately, little direct experimental data are available on the Young s modulus of elasticity, E, or shear modulus of elasticity G representing the wall properties of biomaterial. Few (natural) materials behave in an ideal Hookian manner and in the absence of any other information, it is not unreasonable to assume that the mechanical properties of the external walls of biomaterials will be anisotropic and anelastic. 

This apparent time dependent cell disruption is caused because of the statistically random distribution of the orientation of the cells within a flow field and the random changes in that distribution as a function of time, the latter is caused as the cells spin in the flow field in response to the forces that act on them. In the present discussion this is referred to as apparent time dependency in order to distinguish it from true time-dependent disruption arising from anelastic behaviour of the cell walls. Anelastic behaviour, or time-dependent elasticity, is thought to arise from a restructuring of the fabric of the cell wall material at a molecular level. Anelasticity is stress induced and requires energy which is dissipated as heat, and if it is excessive it can weaken the structure and cause its breakage. 

If the material is anelastic, the stress and the strain will not coincide. The strain will lag behind by an amount which is determined by the phase angle, ( ). Thus ... 

Anelasticity therefore affects the mechanical properties of materials. As seen below, its study yields unique information about a number of kinetic processes in materials, such as diffusion coefficients, especially at relatively low temperatures. 

In such cases W will be a function of position as well as of temperature and the coordinates of the deformation gradient tensor. Finally, most materials, in particular polymers, are anelastic. Energy is dissipated in them during a deformation and the stored energy function W cannot be defined. It is still of value, however, to consider ideal materials in which W does exist and to seek its form since such ideal materials may approximate quite closely to the real ones. 

Viscoelastic materials are those which exhibit both viscous and elastic characterists. Viscoelasticity is also known as anelasticity, which is present in systems when undergoing deformation. Viscous materials, like honey, polymer melt etc, resist shear flow (shear flow is in a solid body, the gradient of a shear stress force through the body) and strain, i.e. the deformation of materials caused by stress, is linearly with time when a stress is applied [1-4]. Shear stress is a stress state where the stress is parallel or tangencial to a face of the material, as opposed to normal stress when the stress is perpendicular to the face. The variable used to denote shear stress is r which is defined as ... 

Viscoelastic materials can follow at least three different behaviors, i.e. linear vis-coleasticity, nonlinear viscoelasticity and anelastic behaviour. 

Viscoelasticity is the phenomenon of time-dependent strain. Often, it is also referied to as anelasticity. Glassy materials, above the glass transition temperature show Newtonian viscosity, i.e. the stress is proportional to the strain rate. This property is exploited in the drawing of fiber and sheet forms. We can write, in terms of normal stresses and strains. 

In an ideal elastic solid, a one-to-one relationship between stress and strain is expected. In practice, however, there are often small deviations. These are termed anelastic effects and result from internal friction in the material. Part of the strain develops over a period of time. One source of anelasticity is thermoelasticity, in which the volume of a body can be changed by both temperature and applied stress. The interaction will depend on whether a material has time to equilibrate with the surroundings. For example, if a body is rapidly dilated, the sudden... 

In many materials, the mechanical response can show both elastic and viscous types of behavior the combination is known as viscoelasticity. In elastic solids, the strain and stress are considered to occur simultaneously, whereas viscosity leads to time-dependent strain effects. Viscoelastic effects are exhibited in many different forms and for a variety of structural reasons. For example, the thermoelastic effect was shown earlier to give rise to a delayed strain, though recovery of the strain was complete on unloading. This delayed elasticity is termed anelastic-ity and can result from various time-dependent mechanisms (internal friction). Figure 5.9 shows an example of the behavior that occurs for a material that has a combination of elastic and anelastic behavior. The material is subjected to a constant stress for a time, t. The elastic strain occurs instantaneously but, then, an additional time-dependent strain appears. On unloading, the elastic strain is recovered immediately but the anelastic strain takes some time before it disappears. Viscoelasticity is also important in creep but, in this case, the time-dependent strain becomes permanent (Fig. 5.10). In other cases, a strain can be applied to a material and a viscous flow process allows stress relaxation (Fig. 5.11). 

Figure 5.9 Strain response for a material that exhibits a combination of elastic and anelastic behavior when a constant stress is applied for a time t.

Time-dependent hysteresis effects can also occur in crystalline materials and these lead to mechanical damping. Models, such as the SLS and the generalized Voigt model, have been used extensively to describe anelastic behavior of ceramics. It is, thus, useful to describe the sources of internal friction in these materials that lead to anelasticity. The models discussed in the last section are also capable of describing permanent deformation processes produced by creep or densification in crystalline materials. For polycrystalline ceramics, creep is usually considered from a different perspective and this will be discussed further in Chapter 7. 

Some materials demonstrate anelastic behavior and this is often modeled by a spring in series with a parallel spring and dashpot unit (standard linear solid, SLS). 

Anelasticity in a material gives rise to permanent deformation. True or False ... 

Anelastic Mechanical behavior in which the stress and strain are not single-valued functions of each other. This occurs particularly when a periodic stress is applied due to internal friction in a viscoelastic material. 

Anelasticity a-n l-as- ti-s3-te (1947) n. The dependence of elastic strain on both stress and time, resulting in a lag of strain behind stress. In materials subjected to cyclic stress, the anelastic effect causes damping. 

When a material is loaded under a fixed stress, after a certain time the strain continues to increase at a rate depending on the type of material. This slow continuing deformation of a material when subjected to a constant stress is called the creep mechanism, which is a typical anelastic behavior. The rate at which the strain change occurs is called the strain rate and is denoted ds/dt and expressed in s . For each material loaded under a constant stress it is... 

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