Instantaneous elastic strain

Figure 16 (145). For an elastic material (Fig. 16a), the resulting strain is instantaneous and constant until the stress is removed, at which time the material recovers and the strain immediately drops back to 2ero. In the case of the viscous fluid (Fig. 16b), the strain increases linearly with time. When the load is removed, the strain does not recover but remains constant. Deformation is permanent. The response of the viscoelastic material (Fig. 16c) draws from both kinds of behavior. An initial instantaneous (elastic) strain is followed by a time-dependent strain. When the stress is removed, the initial strain recovery is elastic, but full recovery is delayed to longer times by the viscous component.

During aging, there are changes in most textural and physical properties of the gel. Inorganic gels are viscoelastic materials responding to a load with an instantaneous elastic strain and a continuous viscous deformation. Because the condensation reaction creates additional bridging bonds, the stiffness of the gel network increases, as does the elastic modulus, the viscosity, and the modulus of rupture. 

Background At elevated temperatures the rapid application of a sustained creep load to a fiber-reinforced ceramic typically produces an instantaneous elastic strain, followed by time-dependent creep deformation. Because the elastic constants, creep rates and stress-relaxation behavior of the fibers and matrix typically differ, a time-dependent redistribution in stress between the fibers and matrix will occur during creep. Even in the absence of an applied load, stress redistribution can occur if differences in the thermal expansion coefficients of the fibers and matrix generate residual stresses when a component is heated. For temperatures sufficient to cause the creep deformation of either constituent, this mismatch in creep resistance causes a progres-... 

The instantaneous modulus from every cycle can be determined by dividing the applied stress by the instantaneous elastic strain at the beginning of each cycle. A comparison of the instantaneous strains at the beginning of each of the cycles from 96 to 384 reveals a steady decrease in the strain magnitude. The resulting steady increase in the measured instantaneous modulus indi-... 

The parameters of the Kelvin-Voigt model and the internal are fourth order tensors, while the strain and the stress are second order tensors. An imdetermined number of Kelvin-Voigt elements give flexibility to the model without increasing the complexity of the constitutive law as it will be discussed later in this section. Assuming a virgin material, having no p>ermanent strain due to earlier where J is the instantaneous elastic strain, where... 

Instantaneous elastic strain and delayed viscous strain arise in the part only when it is released from the mould there is a change in the shape of the product, with equilibrium and relaxation of internal stresses. 

When this model is subjected to a constant stress, the response includes an instantaneous elastic strain caused by spring 1, retarded elastic strain by the Kelvin component, viscous flow by dashpot 1, instantaneous elastic strain on unloading from spring 1, retarded strain recovery from the Kelvin element and permanent deformation in dashpot 1. The multiparameter model response is shown in Figure 4.13. This model can be described as the combined response of a Hookean elastic element, a Kelvin retarded-elastic solid and a Newtonian viscous fluid. 

The four-parameter model provides at least a qualitative representation of all the phenomena generally observed in the creep of viscoelastic materials instantaneous elastic strain, retarded elastic strain, steady-state viscous flow, instantaneous elastic recovery, retarded elastic recovery, and permanent set. It also describes at least qualitatively the behavior of viscoelastic materials in other types of deformation. Of equal importance is the fact that the model parameters can be identified with the various molecular response mechanisms in polymers, and can, therefore, be used to predict the influences that changes in molecular structure will have on mechanical response. The following analogies may be drawn. 

Figure 6.21 Burger solid and its response to a constant applied stress. On loading under tq an instantaneous elastic strain yi, a delayed elastic strain ya and a viscous strain ya appear. On unloading only elastic strain y-, and delayed elastic strain ya recover.

Equation (6.58) produces an instantaneous elastic strain (first term, j i = -ro/Gi), a delayed elastic strain (second term, Y t) = (tq/G2)[1 - exp(-t/r )]) and a viscous strain (third term, y (t) =-rot/GiT ). The first two terms recover on unloading, whereas viscous flow is irreversible. Often used in the literature, anelasticity refers to the instantaneous plus delayed recoverable deformation while viscous flow, in contrast, is not recoverable. 

The creep response according to the Burgers model in Fig. 34.4 covers all elementary aspects of time-dependant viscoelastic behavior including instantaneous elastic strain, secondary steady state creep in the long-term area, and a delayed elastic strain transition behavior that can be, for example, fitted to experimental data according to the choice of the t/i, Ei, ijz, and Ez parameters. 

O Figure 34.10 represents the typical course of a creep experiment using single lap shear specimen bonded with a viscoelastic acrylic adhesive under tensile load. The curve progression can be divided into three phases of creep. At the end of phase I, instantaneous elastic strain and delayed elastic strain are completed and the creep progress in phase II is dominated by secondary creep. At the end of phase II, creep accelerates leading to failure of the specimen at the end of phase III. 

To this point, it has been assnmed that elastic deformation is time independent—that is, that an applied stress produces an instantaneous elastic strain that remains constant over the period of time the stress is maintained. It has also been assumed that upon release of the load, the strain is totally recovered—that is, that the strain immediately returns to zero. In most engineering materials, however, there will also exist a time-dependent elastic strain component—that is, elastic deformation will continue after the stress application, and upon load release, some finite time is required for complete recovery. This time-dependent elastic behavior is known as anelasticity, and it is due to time-dependent microscopic and atomistic processes that are attendant to the deformation. For metals, the anelastic component is normally small and is often neglected. However, for some polymeric materials, its magnitude is significant in this case it is termed viscoelastic behavior, which is the discussion topic of Section 15.4. 

For the intermediate viscoelastic behavior, the imposition of a stress in the manner of Figure 15.5a results in an instantaneous elastic strain, which is followed by a viscous, time-dependent strain, a form of anelasticity (Section 6.4) this behavior is illustrated in Figure 15.5c. 

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