Viscoelastic Analysis

The nonlinear constitutive law due to Schapery may be linearized by assuming that the nonlinearizing parameters 8 y d g2 have a value of unity. In addition, the stress-dependent part of the exponent in the definition of the shift function is set to zero. Consequently, the constitutive law reduces to the hereditary integral form commonly used to describe a linear viscoelastic material. 

FIGURE 2. Variation of the normalized peel stress along the bond centerline of a lap joint. 

Two test cases are used to validate the linear viscoelastic analysis capability implemented in the present finite-element program named NOVA. In the first case, the tensile creep strain in a single eight-noded quadrilateral element was computed for both the plane-stress and plane-strain cases using the program NOVA. The results were then compared to the analytical solution for the plane-strain case presented in Reference 49. A uniform uniaxial tensile load of 13.79 MPa was applied on the test specimen. A three-parameter solid model was used to represent the tensile compliance of the adhesive. The Poisson s ratio was assumed to remain constant with time. The following time-dependent functions were used in Reference 49 to represent the tensile compliance for FM-73M at 72 °C  

By approximating Poisson s ratio with the elasticity relation, we obtain 

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Eracture mechanics concepts can also be appHed to fatigue crack growth under a constant static load, but in this case the material behavior is nonlinear and time-dependent (29,30). Slow, stable crack growth data can be presented in terms of the crack growth rate per unit of time against the appHed R or J, if the nonlinearity is not too great. Eor extensive nonlinearity a viscoelastic analysis can become very complex (11) and a number of schemes based on the time rate of change of/have been proposed (31,32). 

This book is intended primarily for students in the various fields of engineering but it is felt that students in other disciplines will welcome and benefit from the engineering approach. Since the book has been written as a general introduction to the quantitative aspects of the properties and processing of plastics, the depth of coverage is not as great as may be found in other texts on the physics, chemistry and stress analysis of viscoelastic materials, this has been done deliberately because it is felt that once the material described here has been studied and understood the reader will be in a better position to decide if he requires the more detailed viscoelastic analysis provided by the advanced texts. 

This is indeed a system of three second-order differential equations. The tensor elements Cyki may be complex-valued in case of viscoelasticity. Analysis shows that the propagation can be split into three orthogonally polarized planar waves propagating along a wave vector k. Those three waves may have different propagating celerities. Phase celerity and polarization ilj are connected through Christoffel equation ... 

The quasielastic method as developed by Schapery [26] is used in the development of the viscoelastic residual stress model. The use of the quasielastic method is motivated by the fact that the relaxation moduli are required in the viscoelastic analysis of residual stresses, whereas the experimental characterization of composite materials is usually in terms of the creep compliances. An excellent account of the development of the quasielastic method is given in [27]. The underlying restriction in the application of the quasielastic method is that the compliance response of the material shows little curvature when plotted versus log time [28]. Harper [27] shows excellent agreement between the quasielastic method and direct inversion for AS4/3510-6 graphite/epoxy composite. For most graphite/thermoset systems, the restrictions imposed by the quasielastic method are satisfied. 

Based on the results of the characterization studies, the viscoelastic residual moments of the cross-ply specimens were predicted by the viscoelastic analysis using Equation 8.39 together with the curvature-moment relations in Equation 8.50 for the intermittent cure specimens. 

The viscoelastic analysis of poly(cycloheptyl methacrylate) (PCHpM), poly (cycloheptylmethyl methacrylate) (PCHpMM) and poly(cyclooctylmethacrylate) (PCOcM) (see Scheme 2.4) is a good example of the relaxational behavior of polymers containing saturated rings in the side chain. 

Models of mechanical behavior of tissues have been difficult to develop primarily because of the time dependence of the viscoelasticity. Analysis of viscoelastic behavior of even simple polymers at strains greater than a few percent is not accurate. In addition, most tissues undergo strains larger than a few percent, which makes the analysis require an understanding of the elongation behavior. In this chapter we focus on using modeling techniques to analyze the physical basis for determination of the tensile behavior of ECMs found in connective tissue. 

In the solid state deformation, the nonlinear viscoelastic effect is most clearly shown in the yield behavior. The type of stresses applied to a system has little effect on the linear viscoelastic relaxation, but becomes very important as the stress level increases. At high stress levels, the contribution from the external work done on a lattice cell has to be included in the nonlinear viscoelastic analysis. By taking into account the long range cooperative interaction, the external work can,... 

Saphiannikova M, Geue T M, Hennenberg O, Morawetz K, Pietsch U. (2004) Linear viscoelastic analysis of formation and relaxation of azobenzene polymer gratings. J Chem Phys 120 4039 045... 

The beauty of the linear viscoelastic analysis lies in the fact that once a viscoelastic function is known, the rest of the functions can be determined. For example, if one measures the comphance function J t), the values of the components of the complex compliance function can in principle be determined from J(t) by using Fourier transforms [Eqs. (6.30)]. On the other hand, the components of the complex relaxation moduh can be obtained from those of / (co) by using Eq. (6.50). Even more, the real components of both the complex relaxation modulus and the complex compliance function can be determined from the respective imaginary components, and vice versa, by using the Kronig-Kramers relations. Moreover, the inverse of the Fourier transform of G (m) and/or G"(co) [/ (co) and/or /"(co)] allows the determination of the shear relaxation modulus (shear creep compliance). Finally, the convolution integrals of Eq. (5.57) allow the determination of J t) and G t) by an efficient method of numerical calculation outlined by Hopkins and Hamming (13). 

Dynamic mechanical tests have been widely applied in the viscoelastic analysis of polymers and other materials. The reason for this has been the technical simplicity of the method and the low tensions and deformations used. The response of materials to dynamic perturbation fields provides information concerning the moduli and the compliances for storage and loss. Dynamic properties are of considerable interest when they are analyzed as a function of both frequency and temperature. They permit the evaluation of the energy dissipated per cycle and also provide information concerning the structure of the material, phase transitions, chemical reactions, and other technical properties, such as fatigue or the resistance to impact. Of particular relevance are the applications in the field of the isolation of vibrations in mechanical engineering. The dynamic measurements are a... 

The viscoelastic counterpart of these equations can easily be obtained by using the differential operators as in the previous section. However, we postpone the viscoelastic analysis for a forthcoming paragraph. 

Wapner, P. G., and Forsman, W. C., Fourier transform method in linear viscoelastic analysis the vibrating reed, Trans. Soc. Rheol., 15, 603-626 (1971). 

Johannsmann D (1999) Viscoelastic analysis of organic thin films on quartz resonators. Macromolecular Chemistry and Physics 200 501-516. 

Verbeeten WMH, Peters GWM, Baaijens FPT (2001) Differential constitutive equations for polymer melts The extended Pom-Pom model. J Rheol 45 823-843 Verbeeten WMH, Peters GWM, Baaijens FPT (2002) Viscoelastic analysis of complex polymer melt flows using the extended Pom-Pom model. J Non-Newtonian Fluid Mech 108 301-326 Verleye V, Dupret F (1993) Prediction of fiber orientation in complex injection molded parts. 

In this chapter, we present a viscoelastic analysis of the IXS spectra collected on liquid Cs at 493 K and four pressure points, below and above the transition (Fig. 1). We show that the viscoelastic parameters are either pressure independent or smoothly dependent, except for that corresponding to the microscopic relaxation process this latter indeed doubles or even triples at the transition, while it stays... 

Schaffer and Adams< 2) carried out a nonlinear viscoelastic analysis of a unidirectional composite laminate using the finite-element method. The nonlinear viscoelastic constitutive law proposed by Schapery<26) was used in conjunction with elastoplastic constitutive relations to model the composite response beyond the elastic limit. 

In this section results of a number of linear elastic, linear viscoelastic, and nonlinear viscoelastic analyses are discussed in light of available experimental or analytical results. All results are obtained using NOVA on an IBM 3090 computer in double precision arithmetic. First, the results of geometric nonlinear analysis are presented and compared with those obtained by other finite-element programs. Then linear and nonlinear viscoelastic analysis... 

A constant value for the Poisson ratio is assumed for the adhesive. The results from a linear viscoelastic analysis are also presented for comparison. 

S. Roy and J. N. Reddy, Nonlinear Viscoelastic Analysis of Adhesively Bonded Joints, Report No. VPI-E-86.28, ONR, Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, VA (November 1986). 

F. Delale and F. Erdogan, Viscoelastic analysis of adhesively bonded joint, J. Appl. Mech. 48, 331-338 (1981). 

Y R. Nagaraja and R. S. Alwar, Viscoelastic Analysis of an Adhesive-Bonded Plane Lap Joint, Comput. Struct. 6, 621-627 (1980). 

S. Yadagiri and C. Papi Reddy, Viscoelastic analysis of nearly incompressible solids, Comput. Struct. 20, 817-825 (1985). 

The viscoelastic analysis for DMA requires that the sample be in the linear viscoelastic range. In practice, this means that the strain/stress behavior is independent of the strain/stress level. Unmodified polymers, such as PMMA and PC, which are amorphous, are not likely to exhibit strain-dependent behavior as long as the strain amplitude is kept below about 0.3%. However, certain filled materials, especially carbon black or sUica-filled rubbers, may... 

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