Micromechanical continuum models

Sofronis P, Robertson IM. TEM observations and micromechanical/continuum models for the effect of hydrogen on the mechanical behavior of metals. Philos. Mag. A 2002 82 3405-3413. 

CNTs have extremely high stiffness and strength, and are regarded as perfect reinforcing fibers for developing a new class of nanocomposites. The use of atomistic or molecular dynamics (MD) simulations is inevitable for the analysis of such nanomaterials in order to study the local load transfers, interface properties, or failure modes at the nanoscale. Meanwhile, continuum models based on micromechan-ics have been shown in several recent studies to be useful in the global analysis for characterizing such nanomaterials at the micro- or macro-scale. 

There have been some attempts to overcome the analysis of compaction problems, mostly by introducing numerical modeling approaches. The modeling approaches often used in compaction analysis are (a) phenomenological continuum models, (b) micromechanically based continuum models, and (c) discrete-element models. The parameters that should be analyzed when tableting is under development are as follows ... 

In Odegard s study [48], a method has been presented for finking atomistic simulations of nano-structured materials to continuum models of tfie corresponding bulk material. For a polymer composite system reinforced with SWCNTs, the method provides the steps whereby the nanotube, the local polymer near the nanotube, and the nanotube/ polymer interface can be modeled as an effective continuum fiber by using an equivalent-continuum model. The effective fiber retains the local molecular stractuie and bonding information, as defined by MD, and serves as a means for finking tfie eqniv-alent-continuum and micromechanics models. The micromechanics method is then available for the prediction of bulk mechanical properties of SWCNT/polymer com-... 

Mechanical properties of polymer nanocomposites can be predicted by using analytical models and numerical simulations at a wide range of time- and length scales, for example, from molecular scale (e.g., MD) to microscale (e.g., Halpin-Tsai), to macroscale (e.g., FEM), and their combinations. MD simulations can study the local load transfers, interface properties, or failure modes at the nanoscale. Micromechanical models and continuum models may provide a simple and rapid way to predict the global mechanical properties of nanocomposites and correlate them with the key factors (e.g., particle volume fraction, particle geometry and orientation, and property ratio between particle and matrix). Recently, some of these models have been applied to polymer nanocomposites to predict their thermal-mechanical properties. Young s modulus, and reinforcement efficiency and to examine the effects of the nature of individual nanopartides (e.g., aspect ratio, shape, orientation, clustering, and the modulus ratio of nanopartide to polymer matrix). 

Nevertheless, as response data have accumulated and the nature of the porous deformation problems has crystallized, it has become apparent that the study of such solids has forced overt attention to issues such as lack of thermodynamic equilibrium, heterogeneous deformation, anisotrophic deformation, and inhomogeneous composition—all processes that are present in micromechanical effects in solid density samples but are submerged due to continuum approaches to mechanical deformation models. 

When the experimentalist set an ambitious objective to evaluate micromechanical properties quantitatively, he will predictably encounter a few fundamental problems. At first, the continuum description which is usually used in contact mechanics might be not applicable for contact areas as small as 1 -10 nm [116,117]. Secondly, since most of the polymers demonstrate a combination of elastic and viscous behaviour, an appropriate model is required to derive the contact area and the stress field upon indentation a viscoelastic and adhesive sample [116,120]. In this case, the duration of the contact and the scanning rate are not unimportant parameters. Moreover, bending of the cantilever results in a complicated motion of the tip including compression, shear and friction effects [131,132]. Third, plastic or inelastic deformation has to be taken into account in data interpretation. Concerning experimental conditions, the most important is to perform a set of calibrations procedures which includes the (x,y,z) calibration of the piezoelectric transducers, the determination of the spring constants of the cantilever, and the evaluation of the tip shape. The experimentalist has to eliminate surface contamination s and be certain about the chemical composition of the tip and the sample. 

The earliest works of trying to model different length scales of damage in composites were probably those of Halpin [235, 236] and Hahn and Tsai [237]. In these models, they tried to deal with polymer cracking, fiber breakage, and interface debonding between the fiber and polymer matrix, and delamination between ply layers. Each of these different failure modes was represented by a length scale failure criterion formulated within a continuum. As such, this was an early form of a hierarchical multiscale method. Later, Halpin and Kardos [238] described the relations of the Halpin-Tsai equations with that of self-consistent methods and the micromechanics of Hill [29],... 

R. Talreja Further developments in continuum damage modeling of composites aided by micromechanics. ASME App. Mech. Div. 150, 103 (1993)... 

Due to the complexity of the formation of interphases, a completely satisfying microscopic interpretation of these effects cannot be given today, especially since the process of the interphase formation is not yet understood in detail. Therefore, a micromechanical model cannot be devised for calculating the global effective properties of a thin polymer film including the above-mentioned size effects governed by the interphases. On the other hand, a classical continuum-based model is not able to include any kind of size effect. An alternative to the above-mentioned classical continuum or the microscopical model is the formulation of an extended continuum mechanical model which, on the one hand, makes it possible to capture the size effect but, on the other hand, does not need all the complex details of the underlying microstmcture of the polymer network. 

Since the assumption of uniformity in continuum mechanics may not hold at the microscale level, micromechanics methods are used to express the continuum quantities associated with an infinitesimal material element in terms of structure and properties of the micro constituents. Thus, a central theme of micromechanics models is the development of a representative volume element (RVE) to statistically represent the local continuum properties. The RVE is constracted to ensure that the length scale is consistent with the smallest constituent that has a first-order effect on the macroscopic behavior. The RVE is then used in a repeating or periodic nature in the full-scale model. The micromechanics method can account for interfaces between constituents, discontinuities, and coupled mechanical and non-mechanical properties. Their purpose is to review the micromechanics methods used for polymer nanocomposites. Thus, we only discuss here some important concepts of micromechanics as well as the Halpin-Tsai model and Mori-Tanaka model. 

The stiffness of polymer-based composite systems depends on numerous factors such as the stiffness of constituents, the volume fraction of each component, and the size, shape and orientation of reinforcements. As a whole there are three distinct types of polymer composites continuous fibre-reinforced polymer composites, short fibre-reinforced polymer composites, and polymer nanocomposites. Theoretical models based on micromechanical models are well developed and provide an adequate representation of composite stiffness. These micromechanical models are formulated based on assumptions of continuum mechanics. However, for nanocomposite materials, with fillers of size approximately 1 nm compared to the typical carbon fibre diameter of 50 tm, the rules and requirements for continuum... 

Govindjee and Simo [52] combined micromechanical and phenomenological approaches to develop a continuum damage model for carbon black filled elastomers, where the softening effect is considered as the detachment of carbon particles from the elastomer matrix. 

Computational techniques have extensively been used to study the interfacial mechanics and nature of bonding in CNT-polymer composites. The computational studies can be broadly classified as atomistic simulations and continuum methods. The atomistic simulations are primarily based on molecular dynamic simulations (MD) and density functional theory (DFT) [105], [106-110] (some references). The main focus of these techrriques was to understand and study the effect of bonding between the polymer and nanotube (covalent, electrostatic or van der Waals forces) and the effect of fiiction on the interface. The continuum methods extend the continuum theories of micromechanics modeling and fiber-reirrforced composites (elaborated in the next section) to CNT/polymer composites [111-114] and explain the behavior of the composite from a mechanics point of view. 

Researchers have performed experiments on CNT/polymer bulk composites at the macroscale and observed the enhancements in mechanical properties (like elastic modulus and tensile strength) and tried to correlate the experimental results and phenomena with continuum theories like micromechanics of composites or Kelly Tyson shear lag model [105,115-120]... 

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