Linear viscoelastic behavior finite

The relations between stress, strain, and their time dependences are in general described by a constitutive equation or rheological equation of state. If strains and/or rates of strain are finite, the constitutive equation may be quite complicated. If they are infinitesimal, however, corresponding to linear viscoelastic behavior, the constitutive equation is relatively simple, and most of the phenomena described in this book fall under its jurisdiction. 

For positive exponent values, the symbol m with m > 0 is used. The spectrum has the same format as in Eq. 8-1, H X) = H0(X/X0)m, however, the positive exponent results in a completely different behavior. One important difference is that the upper limit of the spectrum, 2U, has to be finite in order to avoid divergence of the linear viscoelastic material functions. This prevents the use of approximate solutions of the above type, Eqs. 8-2 to 8-4. 

The present discussion has a twofold objective First, to review the literature in the stress analysis of adhesive joints using the finite-element method. Second, to present a finite-element computational procedure for adhesive joints experiencing two-dimensional deformation and stress fields. The adherends are linear elastic and can undergo large deformations, and the adhesive experiences linear strains but nonlinear viscoelastic behavior. Following these general comments, a review of the literature is presented. The technical discussion given in the subsequent sections comes essentially from the authors research(i 2> conducted for the Oifice of Naval Research. 

Ghoneim and Chen(33) developed a viscoelastic-viscoplastic law based on the assumption that the total strain rate tensor can be decomposed into a viscoelastic and a viscoplastic component. A linear viscoelasticity model is used in conjunction with a modified plasticity model in which hardening is assumed to be a function of viscoplastic strains as well as the total strain rate. The resulting finite-element algorithm is then used to analyze the strain rate and pressure effects on the mechanical behavior of a viscoelastic-viscoplastic material. 

This section considers the behavior of polymeric liquids in steady, simple shear flows - the shear-rate dependence of viscosity and the development of differences in normal stress. Also considered in this section is an elastic-recoil phenomenon, called die swell, that is important in melt processing. These properties belong to the realm of nonlinear viscoelastic behavior. In contrast to linear viscoelasticity, neither strain nor strain rate is always small, Boltzmann superposition no longer applies, and, as illustrated in Fig. 3.16, the chains are displaced significantly from their equilibrium conformations. The large-scale organization of the chains (i.e. the physical structure of the liquid, so to speak) is altered by the flow. The effects of finite strain appear, much as they do when a polymer network is deformed appreciably. 

However, these simple empirical expressions are far from universal, and fail to account for effects specific to nonlinear behavior, such as the appearance of finite first and second normal stress differences (Tyy = Ni(y) and <7yy — steady shear flow. (For a linear viscoelastic material in shear, ctxx, Cyy and a-zz are equal to the applied pressure, usually atmospheric pressure.) TTiese may be linked to the development of molecular anisotropy in polymer melts subject to flow, and are responsible for the Weissenberg effect, which refers to the tendency for a nonlinear viscoelastic fluid to climb a rotating rod inserted into it, as well as practically important phenomena such as die swell [20]. 

A comprehensive analytical model for predicting long term durability of resins and of fibre reinforced plastics (FRP) taking into account viscoelastic/viscoplastic creep, hygrothermal effects and the effects of physical and chemical aging on polymer response has been presented. An analytical tool consisting of a specialized test-bed finite element code, NOVA-3D, was used for the solution of complex stress analysis problems, including interactions between non-linear material constitutive behavior and environmental effects. 

A review of the theoretical basis, finite-element model, and sample applications of the program NOVA are presented. The updated incremental Lagrangian formulation is used to account for geometric nonlinearity (i.e., small strains and moderately large rotations), the nonlinear viscoelastic model of Schapery is used to account for the nonlinear constitutive behavior of the adhesive, and the nonlinear Fickean diffusion model in which the diffusion coefficient is assumed to depend on the temperature, penetrant concentration, and dilational strain is used. Several geometrically nonlinear, linear and nonlinear viscoelastic and moisture... 

Figure 10 outlines the dynamic moduli as function of the frequency o) for different values of static prestrain e q [71]. At lower frequencies (ca O) the storage modulus tends to a finite nonzero value with a nonzero derivative. This behavior cannot be described by (linear or nonUnear) standard viscoelastic constitutive equations. The data collated by [71] suggest a non-monotonic dependence of the storage modulus upon the static prestrain e q from 0 = 0-65 to G 0 = 0.75, the storage modulus S considerably decreases, but it increases again at e 0 = 0.95. A similar, but less accentuated, trend is shown by the loss modulus. Experiments collated in [72, 73] and more recently in [74] are in agreement with Lee and Kim s results. 

Particulate materials, such as clay or particulate gels of the type discussed in Chapter 4, may be plasticy rather than viscoelastic. Two simple types of plastic behavior are illustrated in Fig. 18b a perfectly plastic material is elastic up to t)xt yield stress, Oy > but it deforms without limit if a higher stress is applied in a linearly hardening material there is a finite slope after the yield stress. In real plastic materials, the stress-strain relations are likely to be curved, rather than linear. If the stress is raised above Oy and then released, the elastic strain is recovered but the plastic strain is not. This differs from viscous behavior in its time-dependence if the stress on a linearly hardening plastic material is raised to Oh and held constant, the strain remains at a viscoelastic material would continue to deform at a rate proportional to Ou/ri-... 

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