R.D. Bradshaw

Fiber waviness in nanotube-reinforced polymer composites—I: Modulus predictions using effective nanotube properties

Results in the literature demonstrate that substantial improvements in the mechanical behavior of polymers have been attained through the addition of small amounts of carbon nanotubes as a reinforcing phase. This suggests the possibility of new, extremely lightweight carbon nanotube-reinforced polymers with mechanical properties comparable to those of traditional carbon-fiber composites. Motivated by micrographs showing that embedded nanotubes often exhibit significant curvature within the polymer, we have developed a model combining finite element results and micromechanical methods to determine the effective reinforcing modulus of a wavy embedded nanotube. This effective reinforcing modulus (ERM) is then used within a multiphase micromechanics model to predict the effective modulus of a polymer reinforced with a distribution of wavy nanotubes. We found that even slight nanotube curvature significantly reduces the effective reinforcement when compared to straight nanotubes. These results suggest that nanotube waviness may be an additional mechanism limiting the modulus enhancement of nanotube-reinforced polymers. # 2003 Elsevier Ltd. All rights reserved.

Effects of nanotube waviness on the modulus of nanotube-reinforced polymers

Recent experimental results demonstrate that substantial improvements in the mechanical behavior of polymers can be obtained using very small amounts of carbon nanotubes as a reinforcing phase. Here, a method is developed to incorporate the typically observed curvature of the embedded nanotubes into traditional micromechanical methods for determination of the effective modulus of the nanotube-reinforced polymer. Using a combined finite element and micromechanical approach, it was determined that the nanotube curvature significantly reduces the effective reinforcement when compared to straight nanotubes. This model suggests that nanotube waviness may be an additional mechanism limiting the modulus enhancement of nanotube-reinforced polymers.

Fiber waviness in nanotube-reinforced polymer composites—II: modeling via numerical approximation of the dilute strain concentration tensor

Nanotube-reinforced polymers offer significant potential improvements over the pure polymer with regard to mechanical, electrical and thermal properties. This article investigates the degree to which the characteristic waviness of nanotubes embedded in polymers can impact the effective stiffness of these materials. A 3D finite element model of a single infinitely long sinusoidal fiber within an infinite matrix is used to numerically compute the dilute strain concentration tensor. A Mori–Tanaka model utilizes this tensor to predict the effective modulus of the material with aligned or randomly oriented inclusions. This hybrid finite elementmicromechanical modeling technique is a powerful extension of general micromechanics modeling and can be applied to any composite microstructure containing non-ellipsoidal inclusions. The results demonstrate that nanotube waviness results in a reduction of the effective modulus of the composite relative to straight nanotube reinforcement. The degree of reduction is dependent on the ratio of the sinusoidal wavelength to the nanotube diameter. As this wavelength ratio increases, the effective stiffness of a composite with randomly oriented wavy nanotubes converges to the result obtained with straight nanotube inclusions. The approach developed in this paper can also be utilized in the analysis of other problems involving nanotube-reinforced polymers, including alternate nanotube representations, viscoelastic response, assessing the effect of low matrix-NT bond strength and in the determination of thermal and electrical conductivity. # 2003 Elsevier Ltd. All rights reserved.

A Sign Control Method for Fitting and Interconverting Material Functions for Linearly Viscoelastic Solids

The multidata method was originally proposed to fit a Prony exponentialseries function to experimental viscoelastic modulus and compliance data;this was accomplished by the application of a linear least squares solver.This paper considers a similar approach, but extended in two key ways.First, it has been applied to the solution of convolution integralequations; specifically those used for material function interconversion.Second, it has been modified to force the signs of the Prony seriescoefficients to be positive; this is an essential criterion for the properphysical interpretation of a Prony series material function, which istypically not satisfied by multidata method solutions. Sign control isimplemented by an iterative Levenberg–Marquardt solution algorithm withan appropriate constraint, and can be used for both fitting experimentaldata and interconversion. To use the method, a Prony series must becapable of adequately representing the applicable functions. The method isdemonstrated by first fitting and converting experimental modulus data.Formulation for a composite lamina is also shown, in which a system ofintegral equations is reduced to a single integral equation; this can thenbe solved using the new method. Finally, application of the new method tofrequency domain transformations is demonstrated, along with comparisonsto other techniques.

Mechanical response of linear viscoelastic composite laminates incorporating non-isothermal physical aging effects

Abstract This paper presents a method of predicting the mechanical response of composite laminates including the effects of linear viscoelasticity and physical aging. Effective-time theory has been used to characterize the physical aging behavior of each linear viscoelastic lamina. In accordance with experimental findings, the aging behavior of each lamina is allowed to differ in the shear and transverse directions. The mechanical loading is restricted to the linear range, which decouples the aging and load behavior. A recursive algorithm has been used to solve the hereditary convolution integral that governs the response of each ply. Classical thin-laminate theory is then used to assemble the individual ply response equations and determine the overall laminate response to general in-plane force and moment loading. The method automatically recovers the ply-level stresses and strains, which are often critical to strength and durability predictions. The model can use either lamina compliance or modulus properties as its basis. Several illustrative examples of long-term laminate response to variable loading are presented and the impact of physical aging is explored. It is shown that for multidirectional laminates, the stiffness of the lamina in the fiber directions can allow simplifications of the model.

Recovering nonisothermal physical aging shift factors via continuous test data: Theory and experimental results

For isothermal physical aging, a few simple tests to characterize the aging shift factors allow reasonable prediction of the mechanical response. In this paper, a new technique is developed to extract aging shift factors from creep data during a nonisothermal history. Previous methods have generated discrete experimental shift factors by a series of short-term creep tests, in which the load portion alone is used for evaluation; this is particularly time consuming for nonisothermal histories, since many data points ( requiring several tests) may be needed for an adequate characterization of the response. This paper presents a new continuous shift factor (CSF) method, based on the validity of effective time theory, which generates a continuous experimental shift factor curve from a single test. Results are presented for this method when applied to a polyimide/carbon fiber composite material tested in shear under temperature jump conditions; this nonisothermal aging data for a polymer matrix composite is shown to exhibit similar response to that of homogeneous polymers. The new CSF technique will be useful in the development of models to predict the shift factor due to coupled aging and thermal history.

Analysis of variable stress history on polymeric composite materials with physical aging

Abstract The long-term viscoelastic behavior of polymeric materials below their glass transition temperatures is greatly affected by physical aging. This effect also extends to polymeric matrix composites. The understanding of the effect of physical aging is necessary to allow designers to predict the long-term performance of composites structures. This paper combines an integral-type constitutive law with lamination theory to describe the viscoelastic response of a composite material. The effect of physical aging on the response is included via a reduced time parameter in the integration. Results of composite response to variable load history are examined and aging effects illustrated.