A constitutive model for the nonlinear viscoelastic viscoplastic behavior of glassy polymers

Two features of the glassy state of an amorphous polymer, which play a key role in determining its mechanical properties, are the distributed nature of the microstructural state and the thermally activated (temporal) evolution of this state. In this work, we have sought to capture these features in a mechanistically motivated constitutive model by considering a distribution in the activation energy barrier to deformation in a thermally activated model of the deformation process. We thus model what is traditionally termed the nonlinear viscoelastic behavior as an elastic-inelastic transition, where the energetically distributed nature of inelastic events and their evolution with straining is taken into account. The thermoreversible nature of inelastic deformation is modeled by invoking the notion of strain energy stored by localized inelastic shear transformations. The model results are compared to experimental data for constant true strain rate uniaxial compression tests (nonmonotonic) at different rates and temperatures; its predictive capabilities are further tested by comparison with compressive creep tests at different stress levels and temperatures.     

The onset of nonlinear viscoelasticity in multiaxial creep of glassy polymers: A constitutive model and its application to PMMA

A physically based, isostructural, constitutive model is described for simulating the onset of nonlinear viscoelasticity in multiaxial creep of glassy polymers, as needed in stress analyses of load-bearing components. In the linear viscoelastic limit, shear response reduces to that of a generalized Maxwell model, while hydrostatic response is Hookean. Nonlinearity enters through Eyring-type rate process kinetics. The equations of the model are solved numerically using a pseudo-linear approximation through each time step, leading to an incremental equation for stress that would be convenient for use in finite element analyses. The model and its assumptions were tested using tension, shear and combined tension/shear creep experiments on well-aged poly(methyl methacrylate) at 70°C. Reproducibility tests confirmed the assumption of constant glass structure for strains up to ～ 1.5 × 10^?2. Shear and pressure activation volumes were obtained by fitting the dependence of the shear compliance on stress invariants. The data showed unequivocally that shear activation volumes vary with log(relaxation time), and excellent agreement was obtained for a linear variation, consistent with the “compensation rule” of polymer thermo-viscoelasticity. The activation volumes are large (many monome units), indicating the cooperative nature of the elementary flow process. Interestingly, they are of the same order as those applying to yield and plastic flow. Although the model finds success in simulating creep, it fails to describe so accurately the strain recovery on unloading. Possible explanations are suggested.     

Fracture of porous viscoelastic materials under multiaxial state of stress

Crack growth in porous and viscoelastic materials has been studied. The decrease of strength of hardened cement paste and concrete under high sustained load can be described with this theoretical approach. Creep as well as time dependence of elastic modulus and strength enter the calculation. Stress relaxation in the material near cracktips can be taken into account by introducing a function m(t,τ). Results are the same in the case of uniaxial and biaxial state of stress. [It could be shown that the probability for shear cracks in a cylindrical sample increases rapidly with increasing confining pressure.] Good agreement with experimental data was found.     

A thermodynamically consistent, nonlinear viscoelastic approach for modeling glassy polymers

A thermodynamically consistent nonlinear viscoelastic constitutive theory is derived to capture the wide range of behavior observed in glassy polymers, including such phenomena as yield, stress/volume/enthalpy relaxation, nonlinear stress-strain behavior in complex loading histories, and physical aging. The Helmholtz free energy for an isotropic, thermorheologically simple, viscoelastic material is constructed, and quantities such as the stress and entropy are determined from the Helmholtz potential using Rational Mechanics. The constitutive theory employs a generalized strain measure and a material clock, where the rate of relaxation is controlled by the internal energy that is likewise determined consistently from the viscoelastic Helmholtz potential. This is perhaps the simplest model consistent with the basic requirements of continuum physics, where the rate of relaxation depends upon the thermodynamic state of the polymer. The predictions of the model are compared with extensive experimental data in the following companion paper.     

Fourier transform mechanical analysis for determining the nonlinear viscoelastic properties of polymers

Fourier Transform Mechanical Analysis (FTMA) has been used to study the strain amplitude dependent nonlinear dynamic mechanical properties of two elastomer compounds, nitrile rubber (NBR) and Neoprene, at various frequencies up to 750 Hz. Basic theory and experimental results are presented for one-dimensional isothermal single frequency simple shear deformation. The Green-Rivlin constitutive equation was used to model the observed behavior. The energy dissipation mechanism and a physical meaning for the material functions in the Green-Rivlin representation have also been examined. The stress Fourier spectrum contained terms at the input frequency and its higher harmonics. It can qualitatively indicate the type of mathematical mode that best describes the observed behavior. The first harmonic storage and loss moduli showed strong dependence on the strain amplitude and frequency. The FTMA methodology presented can be used to systematically conduct nonlinear dynamic mechanical studies on any polymer. It can provide enough insight which, along with a knowledge of the molecular structure, may indicate a path for developing a better representative continuum constitutive model of these complex materials.     

Cyclic deformation behavior of an epoxy polymer. Part II: Predictions of viscoelastic constitutive models

The experimental results presented in Part I of this study were used to evaluate the predictive capabilities of two viscoelastic constitutive models. One of the models, developed by Xia and Ellyin, is in a differential form. The other, which is a modified Schapery model by Lai and Bakker, is in an integral form. The results of the comparison indicate that the Xia‐Ellyin constitutive model simulated the experimental observations well. This was attributed to the existence of a general rule that delineates the loading and unloading parts of the cyclic response. The modified Schapery model was able to predict the general trends of the deformation behavior; however, it was unable to correctly simulate the unloading behavior. This difference became more pronounced when the applied cyclic stress/strain was high. At high applied loads, the material response became more nonlinear. POLYM. ENG. SCI. 45:103–113, 2005. © 2004 Society of Plastics Engineers     

Extensive validation of a thermodynamically consistent, nonlinear viscoelastic model for glassy polymers

The nonlinear thermoviscoelastic formalism presented in the preceding paper is validated with four amorphous polymer systems. Validation is performed over a broad range of relaxation phenomena in the glass transition region, including the temperature and rate-dependence of the stress-strain behavior through yield, volume and enthalpy relaxation, and stress relaxation during multi-step loading histories. The objective is to obtain quantitative agreement between the constitutive theory and all experimental results using one set of model parameters for each material system. The nonlinear viscoelastic formalism is shown to predict the wide range of behavior observed experimentally, indicating that the formalism does capture the essential physics of glassy polymers. Moreover, the material parameters required in the constitutive formalism can be readily obtained from independent experiments and are relatively insensitive to how these parameters are determined experimentally from the various characterization techniques.     

Analysis of multiaxial impact behavior of polymers

Impact performance is a primary concern in many applications of polymers. In this paper, finite element analysis (FEA) and ABAQUS/Explicit are used to simulate the deformation and failure of polymers in the standard ASTM D3763 multiaxial impact test. The specimen geometry and loading mode in this multiaxial impact test provides a close correlation with practical impact conditions. A previously developed constitutive model (“DSGZ” model) for polymers under monotonic compressive loading is generalized and extended for any loading mode and takes into account the different behavior of polymers in uniaxial tensile and compression tests. The phenomenon of thermomechanical coupling during plastic deformation is also included in the analysis. This generalized DSGZ model, along with thermomechanical coupling and a failure criterion based on maximum plastic strain, is incorporated in the FEA model as a coupled‐field user material subroutine to produce a unique tool for the prediction of the impact behavior of polymeric materials. Load‐displacement curves from FEA simulations are compared with experimental data for two glassy polymers, ABS‐1 and ABS‐2. The simulations and experimental data are in excellent agreement up to the maximum impact load. It is shown that not accounting for the different behavior of the polymer in uniaxial tensile and compression tests and thermomechanical coupling effects leads to an overestimation of the load and impact energy, especially at large displacements and plastic deformations. Friction also plays an important role in the impact behavior. If one neglects the friction between the striker and polymer disk, the predicted impact loads are lower as compared with experimental data at large displacements.     

A non-linear viscoelastic material constitutive model for polyurea

A non-linear viscoelastic constitutive model for polyurea by assuming a separable time and strain dependent material behaviour was proposed in this study. The strain dependent function was described by a nine-parameter Mooney–Rivlin model whereas the time function was assumed to follow a Prony series. A method based on the finite time-increment formulation was used to calculate the material parameters using a simple fitting routine. The effectiveness and accuracy of the model was validated using existing compression and tension data from the literature. The proposed model was found to be very efficient in capturing the behaviour of polyurea under both compressive and tensile loadings for a wide range of strain rates.     

Predictions of limiting mechanical performance for anisotropic crystalline polymers

A four-stage synthesis of molecular, micromechanical, and macromechanical models is used to predict the dependence of the longitudinal and transverse Young's moduli and the axial and transverse shear moduli of anisotropic polyethylene on percent crystallinity and the state of molecular orientation. Variational methods are employed to establish the upper and lower limits for anisotropic elastic response. The difference between lower and upper bound limits is interpreted as the potential for improving mechanical performance. A modified form of the Tsai-Halpin equation is used to examine parametric ranging (via a contiguity factor, ξ) between the lower and Tupper bound limits. In this application, the contiguity factor is interpreted as a characteristic of the internal stress-strain distribution which is dependent upon the size, shape, packing geometry, and elastic properties of the crystalline and amorphous regions. The potential for achieving high modulus polymeric materials is illustrated by treating percent crystallinity, molecular orientation, and contiguity as materials design variables subject to control by processing conditions. Optimum property trade-offs, necessary for balancing the over all mechanical behavior of anisotropic materials, are illustrated through the control of orientation and contiguity, The theoretical predictions for the moduli of anisotropic polyethylene are in good agreement with values reported for material processed by traditional procedures as well as ultra-oriented polyethylene.     

Predicting the behavior of nonlinear viscoelastic materials under spring loading

Structural parts made of plastics are usually tested under creep loading conditions, i.e., the stress is applied almost suddenly and then kept constant, while defomation is measured. In practice, however it happens that such a structural part is loaded by an elastic member (for example, by a spring). In this case, the acting force is no longer constant; it decreases in the course of time, while the deformation of the specimen increases (and, obviously, that of the spring decreases). The present paper describes a numerical approach for the solution of this problem, based on the assumption that the creep behavior of the material is known. An example is presented.     

The viscoelastic extension of polymer fibres: complex loadings

The response of oriented polymer fibres to complex loading patterns is investigated. It is shown that the creep and stress relaxation is non-linear with the applied stress. The ratio of the creep rate and the stress-relaxation rate is given by the local slope of the tensile curve and not by the elastic modulus as predicted by linear viscoelastic theory. A consequence of this observation is that viscoelastic and yield deformations are coupled. By analysing the results of the step-creep and the strain-relaxation-strain experiments performed on poly(p-phenylene terephthalamide) fibres, it is shown that the linear superposition principle does not apply to the tensile deformation of polymer fibres above the yield point. Finally the various components of the tensile deformation that should be covered by a constitutive equation for polymer fibres are discussed.     

The viscoelastic properties of modified thermoplastic impregnated multiaxial aramid fabrics

This study reports the manufacture of new fabric forms from the preparation of hybrid laminated multiaxial composites with enhanced thermo‐mechanical properties. Thermal and dynamic mechanical analysis of polymer matrix films and fabricated hybrid composites were used to determine the optimal material composition and reinforcement content for composites with improved viscoelastic properties. The introduction of 5 wt% silica nanoparticles in a composite of p‐aramid–poly(vinyl butyral) led to significant improvements in the mechanical properties, and the addition of silane coupling agents yielded maximal values of the storage modulus for hybrid nanocomposites. The introduction of silane led to a better dispersion and deagglomeration of SiO₂ particles, and to the formation of chemical bonds between organic and inorganic constituents, or p‐aramid–poly(vinyl butyral) composites. In this way, the mobility of macromolecules was reduced, which can be seen from the decreasing value of damping factor for the p‐aramid–poly(vinyl butyral) composite. Analysis of the glass transition temperature of the composite with amino‐functionalized silica nanoparticles revealed improved thermal stability in addition to the aforementioned mechanical properties of the tested materials. POLYM. COMPOS., 2012. © 2011 Society of Plastics Engineers     

A constitutive model for large multiaxial deformations of solid polypropylene at high temperature

Experiments on a blow‐molding grade of polypropylene have been performed at 135°C using a biaxial testing machine. Both simultaneous and sequential equibiaxial tests were performed at strain rates relevant to solid phase processing regimes. A constitutive model has been developed that includes a single Eyring process and two Edwards‐Vilgis networks. The effectiveness of this model for predicting the observed stress‐strain behavior is explored. Predictions of simultaneous stretching and the first stretch in sequential experiments are excellent. The second stretch in sequential experiments is less well predicted, but the model's performance is useful overall. The model is incorporated into a commercial finite element code and its practicality is demonstrated. POLYM. ENG. SCI., 2009. © 2009 Society of Plastics Engineers     

Constitutive relations for compressible glassy polymers

Constitutive equations are derived for the isothermal viscoelastic response of glassy polymers at small strains. The model employs the concept of temporary polymeric networks for shear relaxation and the diffusion mechanism for volume recovery of compressible materials. Adjustable parameters are found by fitting experimental data for poly(methyl methacrylate) and polycarbonate in uniaxial tensile tests. It is demonstrated that the stress-strain relations correctly predict nonmonotonic changes in the specific volume observed in creep tests.     

A model for the non-isothermal viscoelastic behavior of polymers

A new constitutive model is derived for the viscoelastic behavior of polymers under non-isothermal loading. The model extends the Tobolsky concept of adaptive links (entanglements) between polymeric molecules to thermoviscoelastic media. Employing this model, we calculate residual stresses built up in an epoxy resin plate under cooling from the rubber-glass transition temperature, analyze the effect of material parameters on residual stresses, and compare results of numerical simulation with experimental data.     

Dynamic viscoelastic measurements of photosensitive polymers

A new type of rheometer for measuring the dynamic viscoelastic properties of photosensitive polymers was developed. When the photosensitive polymer is exposed to ultraviolet (UV) irradiation, the dynamic modulus and dynamic viscosity increase with time due to network polymerization. Since the curing is completed within a few seconds, the apparatus is designed to follow the rapid changes in rheological properties. The most significant advantages of this apparatus are as follows: (1) The required amount of sample is extremely small (about 0.01 mg). (2) Although the measurements are carried out with liquid polymer, a cell for sample is not necessary. (3) Since the sample thickness is about 10 μm, the temperature can be easily controlled. (4) By measuring the thickness before and after UV irradiation, the volume contraction of the sample can be determined. The curing behavior is measured for urethane acrylate and epoxy acrylate prepolymers, and the effect of acetophenone and benzoin which act as photoinitiators, is examined. Acetophenone ensures more efficient absorption of UV light than benzoin.     

Macroscopic failure of multilayered viscoelastic polymers

This study considers creep deformation (macroscopic failure) of laminated polymer materials. A closed form analytical model is developed to characterize creep in laminates loaded in the linear viscoelastic region. For nonlinear viscoelastic loading, a numerical solution is developed. The models are based on an application of laminated plate theory to materials with viscoelastic layers. Both approaches are able to predict laminate behavior with input of appropriate material properties for each layer. Experimental data are presented, which confirm the validity of the analytical model for laminates constructed from nylon66 and liquid crystalline polymer films. The validated macroscale failure models are capable of characterizing a variety of materials. A case study is included to demonstrate the influence of layer properties and orientation on overall laminate performance. POLYM. ENG. SCI., 2008. © 2007 Society of Plastics Engineers