Micro-mechanics modeling

The interdiffusion of polymer chains occurs by two basic processes. When the joint is first made chain loops between entanglements cross the interface but this motion is restricted by the entanglements and independent of molecular weight. Whole chains also start to cross the interface by reptation, but this is a rather slower process and requires that the diffusion of the chain across the interface is led by a chain end. The initial rate of this process is thus strongly influenced by the distribution of the chain ends close to the interface. Although these diffusion processes are fairly well understood, it is clear from the discussion above on immiscible polymers that the relationships between the failure stress of the interface and the interface structure are less understood. The most common assumptions used have been that the interface can bear a stress that is either proportional to the length of chain that has reptated across the interface or proportional to some measure of the density of cross interface entanglements or loops. Each of these criteria can be used with the micro-mechanical models but it is unclear which, if either, assumption is correct. 

Hounslow, M.J., Mumtaz, H.S., Collier, A.P., Barrick, J.P. and Bramley, A.S., 2001. A micro-mechanical model for the rate of aggregation during precipitation from solution. Chemical Engineering Science, 56, 2543-2552. 

The above interpretations of the Mullins effect of stress softening ignore the important results of Haarwood et al. [73, 74], who showed that a plot of stress in second extension vs ratio between strain and pre-strain of natural rubber filled with a variety of carbon blacks yields a single master curve [60, 73]. This demonstrates that stress softening is related to hydrodynamic strain amplification due to the presence of the filler. Based on this observation a micro-mechanical model of stress softening has been developed by referring to hydrodynamic reinforcement of the rubber matrix by rigid filler... 

In the last part of the chapter (Sect. 5), a micro-mechanical model of rubber reinforcement by flexible filler clusters is developed that allows for a... 

Micro-mechanics models for foam deformation are simplifications of the real structure. Figure 4.23 shows a repeating element of the Kelvin foam cell of Fig. 4.22, prior to deformation. The flat surface at the front is a mirror symmetry plane through the polymer structure, as is the hidden flat surface... 

Micro-mechanical modeling and available experimental evidence indicates that the composite toughness, Kic (composite), can be described as the sum ofthe matrix toughness, Kic (matrix), and a contribution due to whiskertoughening, AKjc (whisker reinforcement). In other words. 

Glaessgen EH, Griffin OH. Finite element based micro-mechanics modeling of textile composites. NASA Conference Pubhcation 3311, Part 2. In Poe CC, Harris CE, editors. Mechanics of textile composites conference. Langley Researeh Centre 1994. p. 555-87. 

The main results of this micro-mechanical model in the quasi-static regime have been compared with experimental results obtained by placing polystyrene (PS)-polyvinyl pyridine (PVP) diblock copolymers with a short PVP block between PS and PVP homopolymers. The fracture toughness was found to increase linearly with Z from that of the bare PS/PVP interface, while the slope of the line increased with the degree of polymerization of the block being pulled out. If the data for the different copolymers were plotted as AGc vs. (where... 

A number of micro-mechanical models have been developed over the years to predict the mechanical behavior of particulate composites [23-2. Halpin-Tsai model has received special attention owing to better prediction of the properties for a variety of reinforcement geometries. The relative tensile modulus is expressed as... 

M. Kluppel and J. Schramm, in An Advanced Micro-Mechanical Model of Hyperelasticity and Stress Softening of Reinforced Rubbers, Proceedings of Constitutive Models for Rubber, Balkema, Rotterdam, 1999. 

Ivanov, I. and Tabiei, A. (2001), Three-dimensional computational micro-mechanical model for woven-fabric composites . Composites Structures, 54, 489-496. 

Qi J H and Boyce M C (2004) Constitutive model for stretch-induced softening of the stress-stretch behavior of elastomeric materials, J Mech Phys Solids 52 2187-2205. Drozdov A A and Dorfmann A (2003) A micro-mechanical model for the response of filled elastomers at finite strains, Int J Plasticity 19 1037-1067. 

In addition, when considering the molecular nature of the reinforcement at the nano-scale, the limits of validity of size independent continuum mechanics should be determined, since the discrete nature of the matter at the nano-scale prohibits simple scaling down of the micro-mechanics models [31-38]. In this contribution, the phenomena related to the behavior of chains in the immediate vicinity of nano-sized rigid inclusions residing in the polymer matrix are review-... 

Janssen D et al. (2008) Micro-mechanical modeling of the cement-bone interface The effect of friction, morphology and material properties on the micromechanical response. Journal of Biomechanics 41 3158-3163... 

The direct use of micromechanical models for nanocomposites is however doubtfid due to the significant scale difference between nanoparticles and macro-partides. As such, two methods have recently been proposed for modeling the mechanical behavior of polymer nanocomposites equivalent continuum approach and self-similar approach. In equivalent continuum approach, molecular dynamics (MD) simulation is first used to model the molecular interaction between nanopartide and polymer. Then, a homogeneous equivalent continuum reinforcing element (i.e., an effective nanopartide) is constmcted. Finally, micro-mechanical models are used to determine the effective bulk properties of a... 

Masonry Modeling, Fig. 10 Proposed micro-mechanical model (a) elementary cell (b) subdivision in layers along thickness and subdivision of each layer in sub-domains (c) imposition of internal equilibrium, equilibrium on interfaces and anti-periodicity... 

G. Chanvillard and PC. Aitcin, Micro-mechanical modeling of the pull-out behavior of corrugated wire drawn steel fibres from cementitious matrices , in Proc. Materials Research Society Symp., Vol. 211, Materials Research Society, Pittsburgh, PA, 1991, 107-202. 

The above equations correspond to the intuitive understanding that in the former case, the mechanical response of the laminar composite is essentially dominated by the performance of the fibers, thus giving the upper-boimd reinforcing effect, while in the latter case, the matrix (and the fibers-to-matrix adhesion) is playing the key role and dictates the lower-bound reinforcement. As we will see, some micro-mechanical models for short fibers composites clearly emphasize such upper and lower limits, and it follows that what can really be achieved when manufacturing a short fiber-reinforced polymer object is indeed between those two extremes. 

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