Nonlinear creep behavior

The stress dependence of the creep of PET/0.6PHB was also investigated in [36] at the room temperature of 20 1 C. Experimental creep curves were measured for 2h at nine stress levels in the range from 10 to 50 MPa at 0.5 MPa intervals. It should be noted that the maximal value 

In the general case, the nonlinear creep behavior could be described by a multiple integral representation using several kernels. The practical application of such equations is limited by the absence of a clearly defined strategy of creep kernel determination as well as relations between the kernel and resolvent. Various simplified versions of this approach have also been proposed. Usually it is assumed that the creep kernel and relaxation time are independent of the stress value. At the same time, it is known that a good approximation to nonlinear creep may be obtained by using the following equation  

It was demonstrated in several papers [45-49] that increasing the stress leads to a shift of the time relaxation spectrum in a manner similar to increasing the temperature. Thus we have correspondence between time, temperature, frequency and also stress. This means, among other things, that the stress dependence of kernel K in equation (12.4) may be described by a function analogous to the temperature shift factor aj. Then the kernel K expressed as a sum of exponential functions may 

Using equation (12.4) in conjunction with (12.5), we obtain the following expression for the compliance D t) = e t)/(j when a = constant  

The last equation implies that the compliance curves at different stress levels differ only by the time scale, that is D(log f, a) = D(logf — loga ) = D[log(f/ )]. 

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The material behaviors considered will include linear elasticity plus linear or nonlinear creep behavior. The nonlinear case will be restricted to power-law rheologies. In some cases the elasticity will be idealized as rigid. In ceramics, it is commonly the case that creep occurs by mass transport on the grain boundaries.1 This usually leads to a linear rheology. In the models considered,... 

As the stress-strain linearity limit of most thermoplastics and their blends is very low, nonlinear viscoelastic behavior of heterogeneous blends needs to be considered in most cases. The nonlinearity is at least partly ascribed to the fact that the strain-induced expansion of materials with Poisson s ratio smaller than 0.5 markedly enhances the fractional free volume (240). Consequently, the retardation times are perpetually shortened in the course of a tensile creep in proportion to the achieved strain. Thus, knowledge of creep behavior over appropriate intervals of time and stress is of great practical importance. The handling and storage of the compliance curves D (t,a) in a graphical form is impractical, so numerous empirical functions have been proposed (241), eg. 

Transient Response Creep. The creep behavior of the polsrmeric fluid in the nonlinear viscoelastic regime has some different features from what were found with the linear response regime. First, there are no ready means of relating the creep compliance to the relaxation modulus as was done in the linear viscoelastic case. In fact, the relationship between the relaxation properties and the creep properties depends entirely on the exact constitutive relationship chosen for the response of the material, and numerical inversion of the specific constitutive law is ordinarily necessary to predict creep response from the relaxation... 

An additional nonlinear effect which appears in extension under high stresses, attributable to the volume expansion associated with the fact that Poisson s ratio is less than i, will be discussed in Chapter 18 it is manifested by a decrease in all the relaxation and retardation times. This effect is especially prominent in more complicated stress patterns such as combined tension and torsion, as studied by Sternstein. ° Nonllnear creep behavior under combined tension and torsion with the additional complication of changing temperature during the experiment has been studied by Mark and Findley. ... 

Finally, it is worth mentioning another approach used to describe nonlinear viscoelastic solids nonlinear differential viscoelasticity [49, 178, 179]. This theory has been successfully applied to model finite amplitude waves propagation [180-182]. It is the generalization to the three-dimensional nonlinear case of the rheological element composed by a dashpot in series with a spring. Thus in the simplest case, the stress depends upon the current values of strain and strain rate rally. In this sense, it can account for the nonlinear short-term response and the creep behavior, but it fails to reproduce the long-term material response (e.g., relaxation tests). The so-called Mooney-Rivlin viscoelastic material [183] and the incompressible version of the model proposed by Landau and Lifshitz [184] belraig to this class. 

The finite element description of the nonlinear viscoelastic behavior of technical fabric was presented by Klosowski et al. [65]. The technical fabric called Panama used in this model was made of two polyester thread families woven perpendicularly to each other with the 2/2 weave. The long term uniaxial creep laboratory tests in directions were conducted at five different constant stress levels. The dense net model [66] together with the Schapery one-integral viscoelastic constitutive model [67] was assumed for the fabric behavior characterization and the least square method in the Levenberg-Marquardt variant was used for the parameters identification. 

The nonlinear viscoelastic behavior of the fabric which is assumed to be the most adequate for fabric response analysis under the exploitation loading was adopted for the study. The model described the behavior of technical fabric with respect to relations obtained for the warp and weft. The uniaxial test in both thread directions was satisfactory for physical description of the fabric. A certain number of creep tests applied at different levels were... 

The comparison of calculated and experimental data of the creep curves showed a good correlation. After comparing the calculated results it can be concluded that the viscoelastic behavior of the technical fabric can be described by the one-integral model. In warp and weft directions the numerical curve fitting resulted in a difference of 3.5-9.9% and 0.3-2.6% respectively. Therefore, the Schapery model with the power function characterized more accurate creep behavior in weft direction than warp direction. Also the power function described the strain evaluation better than the exponential function. This research concluded that both the linear and nonlinear viscoelastic identifications based on different material models can be brought together and the results of linear characterization can be applied to the nonlinear description of the material. 

So-called sandwich rheometers have sometimes been used in the study of rubber elasticity and melt viscoelasticity [141]. In a sandwich rheometer, twin sample plaques are placed in gaps formed by a central steel plate and two outer plates that are part of the same frame. These instruments are difficult to load and clean, and there is no direct control of the gap. Sliding plate melt rheometers were developed to make measurements of nonlinear viscoelastic behavior under conditions under which cone-plate flow is unstable, ie. in large, rapid deformations [ 142 ]. The sample is placed between two rectangular plates, one of which translates relative to the other, generating, in principle, an ideal rectilinear simple shear deformation. Creep tests can be carried out either by use of a feedback loop that generates a plate displacement that gives rise to a constant stress, or by use of a pneumatic drive, as in Laun s sandwich rheometer [143]. 

On the other hand, for aircraft and spacecraft structures, real laminate behavior is pretty typically linear. Laminate behavior is reasonably linear even with some 45° layers which you would expect to contribute their nonlinear shear deformation characteristic to the overall laminate and degrade its relative performance. If you go beyond the behavior of a laminate and look at a large structure, typically the load-response characteristics are linear. Even around a cutout, linear behavior exists. Beyond that apparent linear performance of many laminates, you might not like to operate in some kind of a nonlinear response regime. Certainly not when in a fatigue environment and probably not in a creep environment either would you like to operate in a nonlinear behavior range. 

Linear viscoelasticity Linear viscoelastic theory and its application to static stress analysis is now developed. According to this theory, material is linearly viscoelastic if, when it is stressed below some limiting stress (about half the short-time yield stress), small strains are at any time almost linearly proportional to the imposed stresses. Portions of the creep data typify such behavior and furnish the basis for fairly accurate predictions concerning the deformation of plastics when subjected to loads over long periods of time. It should be noted that linear behavior, as defined, does not always persist throughout the time span over which the data are acquired i.e., the theory is not valid in nonlinear regions and other prediction methods must be used in such cases. 

FEA is applicable in several types of analyses. The most common one is static analysis to solve for deflections, strains, and stresses in a structure that is under a constant set of applied loads. In FEA material is generally assumed to be linear elastic, but nonlinear behavior such as plastic deformation, creep, and large deflections also are capable of being analyzed. The designer must be aware that as the degree of anisotropy increases the number of constants or moduli required to describe the material increases. 

For elastomers, factorizability holds out to large strains (57,58). For glassy and crystalline polymers the data confirm what would be expected from stress relaxation—beyond the linear range the creep depends on the stress level. In some cases, factorizability holds over only limited ranges of stress or time scale. One way of describing this nonlinear behavior in uniaxial tensile creep, especially for high modulus/low creep polymers, is by a power... 

Often times concentrated polymeric solutions cannot be treated as Newtonian fluids, however, and this tends to offset the simplifications which result from the creeping flow approximation and the fact that the boundaries are well defined. The complex rheological behavior of polymeric solutions and melts requires that nonlinear constitutive equations, such as Eqs. (l)-(5), be used (White and Metzner, 1963) ... 

First, and most important, nonlinear dynamics provides an intellectual framework to pursue the consequences of nonlinear behavior of transport systems, which is simply not possible in an intellectual environment that is based upon a linear mentality, characterized by well-behaved, regular solutions of idealized problems. One example that illustrates the point is the phenomenon of hydrodynamic dispersion in creeping flows of nondilute suspensions. It is well known that Stokes flows are exactly reversible in the sense that the particle trajectories are precisely retraced when the direction of the mean flow is reversed. Nevertheless, the lack of reversibility that characterizes hydrodynamic dispersion in such suspensions has been recently measured experimentally [17] and simulated numerically [18], Although this was initially attributed to the influence of nonhydrodynamic interactions among the particles [17], the numerical simulation [18] specifically excludes such effects. A more general view is that the dispersion observed is a consequence of (1) deterministic chaos that causes infinitesimal uncertainties in particle position (due to arbitrarily weak disturbances of any kind—... 

This review is intended to focus on ceramic matrix composite materials. However, the creep models which exist and which will be discussed are generic in the sense that they can apply to materials with polymer, metal or ceramic matrices. Only a case-by-case distinction between linear and nonlinear behavior separates the materials into classes of response. The temperature-dependent issue of whether the fibers creep or do not creep permits further classification. Therefore, in the review of the models, it is more attractive to use a classification scheme which accords with the nature of the material response rather than one which identifies the materials per se. Thus, this review could apply to polymer, metal or ceramic matrix materials equally well. 

Figure 5.10 Diagrams showing the transition from linear to nonlinear behavior in shear creep experiments. Note that the data are taken from creep experiments at different deformations.

Normally a logarithmic time scale is used to plot the creep curve, as shown in Figure 3.9b, so that the time dependence of strain after long periods can be included. If a material is linearly viscoelastic (Equation 3.17), then at any selected time each line in a family of creep curves (with equally spaced stress levels) should be offset along the strain axis by the same amount. Although this type of behavior maybe observed for plastics at low strains and short times, in most cases the behavior is nonlinear, as indicated in Figure 3.9c. 

Creep and Stress Relaxations As reviewed, viscoelasticity can be related to designing (Chapter 3). In general, this is tractable only if the mechanical behavior is linear, although methods for nonlinear behavior... 

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