Elastic response constants

The transition from ideal elastic to plastic behaviour is described by the change in relaxation time as shown by the stress relaxation in Fig. 66. The immediate or plastic decrease of the stress after an initial stress cr0 is described by a relaxation time equal to zero, whereas a pure elastic response corresponds with an infinite relaxation time. The relaxation time becomes suddenly very short as the shear stress increases to a value equal to ry. Thus, in an experiment at a constant stress rate, all transitions occur almost immediately at the shear yield stress. This critical behaviour closely resembles the ideal plastic behaviour. This can be expected for a polymer well below the glass transition temperature where the mobility of the chains is low. At a high temperature the transition is a... 

In addition to the tensile and shear moduli, a compressive modulus, or modulus of compressibility, K, exists to describe the elastic response to compressive stresses (see Fignre 5.7). The compressive modulus is also sometimes called the bulk modulus. It is the proportionality constant between the compressive stress, CTc, and the bulk strain, represented by the relative change in bulk volume, AV/Vo-... 

FIG. 15.52 Elastic responses due to the deformation of the director field Frank elastic constants. Kindly provided by Prof. SJ. Picken (2003). 

Think about what happens when, say, an elastomer is under tensile stress. The elastic constants, s and c, cannot be scalar quantities, otherwise Eqs. 10.5 and 10.6 would not completely describe the elastic response. When the elastomer is stretched, a contraction... 

With elastically anisotropic materials the elastic behavior varies with the crystallographic axes. The elastic properties of these materials are completely characterized only by the specification of several elastic constants. For example, it can be seen from Table 10.3 that for a cubic monocrystal, the highest symmetry class, there are three independent elastic-stiffness constants, namely, Cn, C12, and C44. By contrast, polycrystalline aggregates, with random or perfectly disordered crystallite orientation and amorphous solids, are elastically isotropic, as a whole, and only two independent elastic-stiffness coefficients, C44 and C12, need be specified to fully describe their elastic response. In other words, the fourth-order elastic modulus tensor for an isotropic body has only two independent constants. These are often referred to as the Lame constants, /r and A, named after French mathematician Gabriel Lame (1795-1870) ... 

It is important to realize that this type of behavior is not just a simple addition of linear elastic and viscous responses. An ideal elastic solid would display an instantaneous elastic response to an applied (non-destructive) stress (top of Figure 13-74). The strain would then stay constant until the stress was removed. On the other hand, if we place a Newtonian viscous fluid between two plates and apply a shear stress, then the strain increases continuously and linearly with time (bottom of Figure 13-74). After the stress is removed the plates stay where they are, there is no elastic force to restore them to their original position, as all the energy imparted to the liquid has been dissipated in flow. 

If we now perform a creep experiment, applying a constant stress, a0 at time t = 0 and removing it after a time f, then the strain/ time plot shown at the top of Figure 13-89 is obtained. First, the elastic component of the model (spring) deforms instantaneously a certain amount, then the viscous component (dashpot) deforms linearly with time. When the stress is removed only the elastic part of the deformation is regained. Mathematically, we can take Maxwell s equation (Equation 13-85) and impose the creep experiment condition of constant stress da/dt = 0, which gives us Equation 13-84. In other words, the Maxwell model predicts that creep should be constant with time, which it isn t Creep is characterized by a retarded elastic response. 

But Just like the Maxwell model, the Voigt model is seriously flawed. It is also a single relaxation (or retardation) time model, and we know that real materials are characterized by a spectrum of relaxation times. Furthermore, just as the Maxwell model cannot describe the retarded elastic response characteristic of creep, the Voigt model cannot model stress relaxation—-under a constant load the Voigt element doesn t relax (look at the model and think about it ) However, just as we will show that the form of the equation we obtained for the relaxation modulus from... 

A unique set of interfaciai stiffness constants suffices to characterize the macroscopic elastic response of an interface between two rough contact surfaces regardless of the direction of incidence of the ultrasonic wave [35], In addition, the stiffness constants of a double interface can be successfully recovered by combining ultrasonic spectroscopy and pulse-echo measurements with the theoretical procedures available for a single imperfect interface. 

Equation (5.23) simulates the elastic response or sudden overshoot when a constant shear rate is applied. The peak shear stress is reached at a time /max given by ... 

The same parameters can also be determined by applying a constant shear stress to the interface and measuring the resulting shear strain as a function of time (see fig. 3.40), so-called interfacial creep tests. At t = 0, a shear stress is suddenly applied, and kept constant thereafter. For ideally viscous monolayers a steady increase of the shear strain with t will be observed, while for an elastic material the observed strain will be instantaneous and constcmt in time. For a viscoelastic material, as in fig. 3.40, there is first am Instantaneous increase AB in the strain, the elastic response followed by a delayed elastic response BC and a viscous... 

Elastic response In this case, the compliance J shows a small increase at t = 0 and this remains almost constant for the whole period t. When the stress is removed, J changes sign and reaches 0 after some time t that is, complete creep recovery occurs. 

Deformation is measured by a quantity known as strain (strain is a relative extension or contraction of dimension). Strain is similarly a tensor of the second rank having nine components (3x3 matrix). The relation between stress and strain in the elastic regime is given by the classical Hooke s law. It is therefore obvious that the Hooke s proportionality constant, known as the elastic modulus, is a tensor of 4 rank and is represented by a (9 x 9) matrix. Before further discussion we note the following. The stress tensor consists of 9 elements of which stability conditions require cjxy=(jyx, stress components in the symmetric stress matrix are only six. Similarly there are only six independent strain components. Therefore there can only be six stress and six strain components for an elastic body which has unequal elastic responses in x, y and z directions as in a completely anisotropic solid. The representation of elastic properties become simple by following the well known Einstein convention. The subscript xx, yy, zz, yz, zx and xy are respectively represented by 1, 2, 3, 4, 5 and 6. Therefore Hooke s law may now be written in a generalized form as. 

For isotropic elastic solids there are only two independent elastic constants, or compliances. While Young s modulus E and the shear modulus // are the most widely used, we shall choose as the two physically independent pair of moduli the shear modulus /i and the bulk modulus K, where the first gauges the shear response and the second the bulk or volumetric response. However, in stating the linear elastic response in the equations below we still choose the more compact pair of E and //. Thus, for the six strain elements we have... 

Non-Newtonian fluids exhibit complex flow behavior, typically because they contain additives such as polymers or particles that alter the response of the fluid. Unlike Newtonian fluids, which have constant viscosity (resistance to flow), the viscosity of a non-Newtonian fluid is not constant. The viscosity depends oti the shear rate and the amount of strain accumulated, as well as the nature of the additives, including the concentration of additives, the polymer molecular weight, and the particle size. In addition, flexible polymers stretch in a flow, leading to an elastic response. Development of microfluidic devices has centered on miniaturizing assays to analyze the biological, physical, and chemical properties of DNA, proteins, and biopolymers in solution, as well as suspensimis of cells and bioparticles. Since the analysis is typically performed in the... 

The experiment we introduced at the beginning of the previous subsection is also called the creep experiment. A small stress of Gq is imposed on a solid sample for a time period of to at a constant temperature after the stop of stress, the strain of changing with the time period of t monitors the relaxatirMi curve. There are four typical responses separately corresponding to viscous, elastic, anelastic and viscoelastic responses, as illustrated in Fig. 6.8. The creep curve of polymer viscoelasticity exhibits both instant and retarded elastic responses upon imposing and removal of the stress, and eventually reaches the permanent deformation. 

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