Model micro-mechanics

The aim of this chapter is to describe the micro-mechanical processes that occur close to an interface during adhesive or cohesive failure of polymers. Emphasis will be placed on both the nature of the processes that occur and the micromechanical models that have been proposed to describe these processes. The main concern will be processes that occur at size scales ranging from nanometres (molecular dimensions) to a few micrometres. Failure is most commonly controlled by mechanical process that occur within this size range as it is these small scale processes that apply stress on the chain and cause the chain scission or pull-out that is often the basic process of fracture. The situation for elastomeric adhesives on substrates such as skin, glassy polymers or steel is different and will not be considered here but is described in a chapter on tack . Multiphase materials, such as rubber-toughened or semi-crystalline polymers, will not be considered much here as they show a whole range of different micro-mechanical processes initiated by the modulus mismatch between the phases. 

The micro-mechanical processes will be presented next, followed by the models used to describe them. The predictions of the models will then be compared with results obtained using well-defined coupling chains. Application of the models to the joining of dissimilar polymers will then be described. Finally welding of glassy polymers will be considered. 

When the stress that can be bom at the interface between two glassy polymers increases to the point that a craze can form then the toughness increases considerably as energy is now dissipated in forming and extending the craze structure. The most used model that describes the micro-mechanics of crazing failure was proposed by Brown [8] in a fairly simple and approximate form. This model has since been improved and extended by a number of authors. As the original form of the model is simple and physically intuitive it will be described first and then the improvements will be discussed. 

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... 

The presence of Z-pins as the through-the-thickness reinforcement has been shown to result in dramatic increases in the apparent resistance to crack propagation under Mode I and Mode II loading conditions, in laboratory tests on standard unidirectional (UD) beam samples [2]. The Z-pin pull-out has been identified as the dominant energy micro-mechanism responsible for the resistance to delamination under Mode I conditions. The behaviour of individual Z-pins in pull-out from a UD-laminate has been characterised and modelled and the single Z-pin pull-out curves used as input into a 2D Finite Element (FE) model of delamination under Mode I [3, 4]. 

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... 

H.W. Song, H.J. Kim, V. Saraswathy, T.H. Kim, Micro-mechanics based corrosion model for predicting the service life of reinforced concrete structures, Int. J. Electrochem. Sci. 2 (2007) 341-354. 

Where m is the shape parameter of distribution function (Tang C.A. et al. 2003). In the formula, its physical meaning reflects the homogeneity of the rock medium and it ranges from 1 to 100 the larger m-value, the more uniform rock is. Parameter a is the value of the unit mechanic property tto is the average value of all units a) is the micro mechanic non-uniformity distribution of rock primitives. The coal rock mechanics parameters of model are shown in Table 1. 

As a general rale, visco-plastic material behavior is specified for FEM simulation of chip removal. Thermo-mechanically coupled calculations are used. For describing material behavior, the use of the Johnson-Cook equation (Eq. 4) is preferred. Another semiempiiical model presented by Zerilli and Armstrong considers micro-mechanical effects in relation to the thermal activation behavior of face-centered (fee) and body-centered cubic (bcc) structures of the workpiece material. Other constitutive material laws are formulated by Oxley, Clifton Hensel-Spittel, and El-Magd, respectively, whereas the stress s is determined by different linear or exponential procedures within the equation terms. 

In MD the considered microscopic material properties and the underlying constitutive physical equations of state provide a sufficiently detailed and consistent description of the micro mechanical and thermal state of the modeled material to allow for the investigation of the local tool tip/workpiece contact dynamics at the atomic level (Hoover 1991). The description of microscopic material properties considers, e.g., microstmcture, lattice constants and orientation, chemical elements, and the atomic interactions. The following table lists the representation of material properties and physical principles in MD, which have to be described numerically in an efficient way to allow for large-scale systems, i.e., models with hundred thousands, millions, or even billions of particles (Table 1). 

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. 

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 micro-mechanics of composites or Kelly Tyson shearing model [105,115-120]. 

Govindjee, S. and Simo, J. (1991) A micro-mechanically based continuum damage model for carbon black-filled rubbers incorporating Mullins effect. Journal of the Mechanics and Physics of Solids, 39, 87-112. 

In general, SMPF is perceived as a two-phase composite material with a crystalline phase mixed with an amorphous phase. A multiscale viscoplasticity theory is developed. The amorphous phase is modeled using the Boyce model, while the crystalline phase is modeled using the Hutchinson model. Under an isostrain assumption, the micromechanics approach is used to assemble the microscale RVE. The kinematic relation is used to link the micro-mechanics constitutive relation to the macroscopic constitutive law. The proposed theory takes into account the stress induced crystallization process and the initial morphological texture, while the polymeric texture is updated based on the apphed stresses. The related computational issue is discussed. The predictabihty of the model is vahdated by comparison wifli test results. It is expected that more accurate measurement of the stress and strain in the SMPF with large deformation may further enhance the predictability of the developed model. It is also desired to reduce the number of material parameters in the model. In other words, a deeper understanding and physics based theoretical modeling are needed. 

Darabi, M.K., Abu Al-Rub, R.K., and Little, D.N. (2012) A continuum damage mechanics framework for modeling micro-damage healing. International Journal of Solids and Structures, 49, 492-513. 

Now, in the Dugdale model of craze micro-mechanics we can estimate the fracture energy as... 

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