Instantaneous deformation

The function iKt-t ) may be interpreted as a memory function having a form as shown in Figure 3.14. For an elastic solid, iff has the value unity at all times, while for a purely viscous liquid iff has the value unity at thfe current time but zero at all other times. Thus, a solid behaves as if it remembers the whole of its deformation history, while a purely viscous liquid responds only to its instantaneous deformation rate and is uninfluenced by its history. The viscoelastic fluid is intermediate, behaving as if it had a memory that fades exponentially with time. The purely elastic solid and the purely viscous fluid are just extreme cases of viscoelastic behaviour. 

The phenomenological approach does not preclude a consideration of the molecular origins of the characteristic timescales within the material. It is these timescales that determine whether the observation you make is one which sees the material as elastic, viscous or viscoelastic. There are great differences between timescales and length scales for atomic, molecular and macromolecular materials. When an instantaneous deformation is applied to a body the particles forming the body are displaced from their normal positions. They diffuse from these positions with time and gradually dissipate the stress. The diffusion coefficient relates the distance diffused to the timescale characteristic of this motion. The form of the diffusion coefficient depends on the extent of ordering within the material. 

Stresses in viscoelastic materials "remember" deformation prehistory and so are not an unambiguous function of instantaneous deformations however, they may be expressed by a functional. For a linear viscoelastic material, the relationship between stresses and deformations... 

Therefore under a constant stress, the modeled material will instantaneously deform to some strain, which is the elastic portion of the strain, and after that it will continue to deform and asynptotically approach a steady-state strain. This last portion is the viscous part of the strain. Although the Standard Linear Solid Model is more accurate than the Maxwell and Kelvin-Voigt models in predicting material responses, mathematically it returns inaccurate results for strain under specific loading conditions and is rather difficult to calculate. 

A parallel array of E and h gives a Kelvin-Voigt element. This model does not allow an instantaneous deformation (the stress on the dashpot would be infinite), and it does not show stress relaxation. At a constant stress it exhibits creep at time t its strain is ( ) the stress in the spring then is ... 

Both models, the Maxwell element and the Kelvin-Voigt element, are limited in their representation of the actual viscoelastic behaviour the former is able to describe stress relaxation, but only irreversible flow the latter can represent creep, but without instantaneous deformation, and it cannot account for stress relaxation. A combination of both elements, the Burgers model, offers more possibilities. It is well suited for a qualitative description of creep. We can think it as composed of a spring Ei, in series with a Kelvin-Voigt element with 2 and 772. and with a dashpot, 771... 

Figure 7.8. Typical creep curve for a plastic fat. A instantaneous deformation upon loading B instantaneous sample recovery upon unloading C time-dependent recovery of sample D permanent sample deformation (adapted from deMan and Beers, 1987).

It is clear that the determination of such a modulus temperature curve takes an awful lot of time. Moreover, the transitions in the glassy region are difficult to determine, because the time needed for such a transition will be very small it may be of the order of or even much faster than the time in practice to apply an instantaneous deformation. For that reason in general use is made of dynamic mechanical measurements as a function of frequency to elucidate the modulus temperature curves, in particular in the glassy region. An additional advantage is that elastic and viscous forces are separated in this kind of measurements. 

In a stress relaxation experiment the Maxwell-element is subjected to an instantaneous deformation sQ which is held constant. It means that ... 

In the case of real substances, the response to the shear stress involves an instantaneous deformation of the Hookean type followed by gradual increase of the shear strain with time. If the strain at long times approaches a limiting value e q, the substance is considered a solid. However, if at long times the strain is a linear function of time, the substance is considered a liquid. Schematic representations of these responses are given in Figures 5.5a and 5.5b for real solids and liquids. 

Instantaneously deformed high molar mass polymer melts (long polymer chains in their liquid state) behave at intermediate times as networks with well-defined values of shear modulus, called the plateau modulus Ge, which is independent of molar mass for long-chain polymers. This rubbery plateau is seen for all polymer melts with... 

In accordance with Hooke s law, the spring will undergo instantaneous deformation (strain, y) following application of a shear stress (a) ... 

Note Actually, instantaneous deformation cannot occur in practice the deformation rate corresponds to the sound velocity in the material, often of the order of a km s 1. 

This property is an important consideration in the design of parts from polytetrafluoroethylene. PTFE deforms substantially overtime when it is subjected to load. Metals similarly deform at elevated temperatures. Creep is defined as the total deformation under stress after a period of time, beyond the instantaneous deformation upon load application. Significant variables that affect creep are load, time under load, and temperature. Creep data under various conditions in tensile, compressive, and torsional modes can be found in Figs. 3.12 through 3.19. 

A possible regime characteristic to specific mechanical behavior of such model involves fast (instantaneous) deformation to the net strain of y0... 

On removal of the applied stress, the material experiences creep recovery. Figure 14.5 shows the creep and the creep recovery curves of the Maxwell element. It shows that the instantaneous application of a constant stress, Oo, is initially followed by an instantaneous deformation due to the response of the spring by an amount Oq/E. With the sustained application of this stress, the dashpot flows to relieve the stress. The dashpot deforms linearly with time as long as the stress is maintained. On the removal of the applied stress, the spring contracts instantaneously by an amount equal to its extension. However, the deformation due to the viscous flow of the dashpot is retained as permanent set. Thus the Maxwell element predicts that in a creep/creep recovery experiment, the response includes elastic strain and strain recovery, creep and permanent set. While the predicted response is indeed observed in real materials, the demarcations are nevertheless not as sharp. 

Instantaneous deformation occurs when load is removed. 

A sample subjected to tensile or compressive stress (Figure 40.27) exhibits the instantaneous deformation (elastic part) at first, which then increases with time (viscoelastic creep). After cycling the moisture content to and from a higher moisture content, the creep has been significantly greater due to mechanosorptive action. 

Subjected to an applied force, an elastic solid instantaneously deforms. The deformation is fully recovered with the removal of the force. In contrast, the deformation of a viscous fluid increases with time when a force is applied. With the removal of the force, a viscous fluid ceases to deform further, but any prior deformation remains. A viscoelastic material exhibits both elasticity and viscosity. Subjected to an applied force, it deforms and its deformation increases with time, i.e., it creeps. When the force is removed, only partial deformation is recovered instantaneously. It recovers more, but not all, of its deformation as time progresses. Depending on the time scale of interest, a viscoelastic material could behave solid like or fluid like or a combination of both. 

Figure 11.10 shows the strain with time of constant stress for a viscous and elastic material. The stress is applied at t and removed at tf. The elastic model shows an instantaneous deformation when stress is applied at f, a constant deformation with time, and then a return to its original length when the load is removed. Therefore, the elastic solid does not creep. The viscous (dashpot) model deforms continuously (creeps) horn f, to and remains permanently deformed after removal of the load. 

Hie change is calculated using the Smoluchowski equation (8.15). For the instantaneous deformation, the velodty gradient = dsapldt (dr being the duration time of the deformation) dominates the time evolution of W, so that... 

A—instantaneous deformation when loaded, X—creep exponent,... 

Drop test performance requirements, (i) Instantaneous deformation due to impact of the sphere shall not enter the protected zone as illustrated in Figures W-25, W-26, and W-28. 

The linearisation of the creep curves, by representing in double logarithmic co-ordinates the variation of deformation with time can be done applying the method developed by B.W. Chery and D.F. Kindles [887], i.e. of overlapping of the stress vs. time, for the prediction of the polymers behaviour to creep for long periods of solicitation. In the case of the vitreous polymers, the creep rate can depend, in certain conditions, only by the applied stress and instantaneous deformation. 

Approach to the first problem is based on the fact that we cannot define series of poses, but can define operators to act on the coupled points beforehand. Therefore, operators are regarded as invariants and p>oses as variable. Poses are generated by a set of initial pose and series of operators. Here, the role of operators and states are reversed. For gel robots system, operators are electric fields defined by typical sets of applied voltages to the electrodes. Instantaneous deformation response of the gel is unique when the following three elements are settled form of gel, surrounding electric field, and relative position and orientation of the gel and the electric fields. Series of input of electric fields causes... 

The characterization of materials behaviour is of importance in many research fields. From an analytical point of view, the mechanical behaviour of materials could be described either by using empirical laws based on experimental observations or by using a framework to develop constitutive laws. In the latter case, the thermodynamic of irreversible processes could be used as the framework. These constitutive laws are generally dedicated to complex problems and are thus developed in a three-dimensional context. Materials behave in different ways under loading but, under specific conditions, they all will generally exhibit instantaneous and time-dep>endant deformations. Instantaneous deformation could be elastic, plastic and so forth, while time-dependant deformation generally refers to the viscosity of the material. 

For solids, the elastic behaviour is related to the instantaneous deformation of a material and is expressed using Young s modulus. This is generally measured by two different methods. 

Figure 15.7 gives a typical creep curve for bovine serum albumin films, showing an initial, instantaneous, deformation, characteristic of an elastic body, followed by a non-linear flow that gradually declines and approaches the steady flow behaviour of a viscous body. 

---