Cohesive Models

It is useful to try to estimate the theoretical strength of materials from fundamental principles in order to understand what physical properties contribute to the strength, to find how far away we are from the maximum possible strength, and to get some insight into how to improve the performance of materials. There are several models for the theoretical strength that depend on the assumed failure mechanism of the material. We shall first consider the cohesive model which assumes failure by simply separating the bonds that hold the atoms together. 

One way of estimating the theoretical strength of a material would be to compute the maximum force/area required to separate the atoms. We assume the potential as a function of a linear displacement x can be represented by the Morse potential. The Morse potential and its derivatives can be written as 

The second derivative has a root at x, ax = o + ln2, so the maximum stress will be given by 

If there are no defects in the crystal, the elastic bonds theoretically will stretch until x reaches Xmax- Beyond this point the system energy is lowered by increasing the separation. In other words, if the stress is released at x x ax/ the system will recover elastically, but if strained beyond a max the bonds will be broken and the system will separate into two parts. Since from Equation 7.19 

The /xq must be found from Equation 7.41 using the measured values of Cn and Uq/V. Putting this into Equation 7.42 yields 

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The model experiments of Xu and Rosakis on low-speed impact over sandwich structures were simulated applying cohesive models. The simulation captures qualitatively the main experimental observations. The most relevant correspondence is in the development of the first crack at the interface between the layers, the presence of shear stresses along the interface, which renders the crack shear driven and often inter-sonic, and the transition between interlayer crack growth and intra-layer crack branching. The effects of impact speed and bond shear strength are also investigated and highly satisfactory predictions are obtained. 

An adhesive-cohesive model for protein compressibility has been proposed by Dadarlat and Post [57]. This model assumes that the compressibility is a competition between adhesive protein-water interactions and cohesive protein-protein interactions. Computer simulations suggest that the intrinsic compressibility largely accounts for the experimental compressibilities indicating that the contribution of hydration water is small. The model also accounts for the correlation between the compressibility of the native state and the change in heat capacity upon unfolding for nine single chain proteins. 

Arias, L, Serebrinsky, S., and Ortiz, M. (2006) A phenomenological cohesive model of ferroelectric fatigue. Acto Mater., 54, 975-984. 

Nevertheless, numerous key components of the necessary experimental and theoretical foundation for understanding aqueous solubility of cocrystals have been provided by Rodrfguez-Homedo and coworkers. In several seminal papers, they have brought together invaluable information on solubility products, solubility complexation, and phase-solubility diagrams in order to provide cohesive models for explaining cocrystal solubility. These models indicate, for example, that (i) the aqueous solubility of a cocrystal AB is accurately described by the solubility... 

The behavior of insoluble monolayers at the hydrocarbon-water interface has been studied to some extent. In general, a values for straight-chain acids and alcohols are greater at a given film pressure than if spread at the water-air interface. This is perhaps to be expected since the nonpolar phase should tend to reduce the cohesion between the hydrocarbon tails. See Ref. 91 for early reviews. Takenaka [92] has reported polarized resonance Raman spectra for an azo dye monolayer at the CCl4-water interface some conclusions as to orientation were possible. A mean-held theory based on Lennard-Jones potentials has been used to model an amphiphile at an oil-water interface one conclusion was that the depth of the interfacial region can be relatively large [93]. 

The permachor method is an empirical method for predicting the permeabiUties of oxygen, nitrogen, and carbon dioxide in polymers (29). In this method a numerical value is assigned to each constituent part of the polymer. An average number is derived for the polymer, and a simple equation converts the value into a permeabiUty. This method has been shown to be related to the cohesive energy density and the free volume of the polymer (2). The model has been modified to liquid permeation with some success. 

Orowan (1949) suggested a method for estimating the theoretical tensile fracture strength based on a simple model for the intermolecular potential of a solid. These calculations indicate that the theoretical tensile strength of solids is an appreciable fraction of the elastic modulus of the material. Following these ideas, a theoretical spall strength of Bq/ti, where Bq is the bulk modulus of the material, is derived through an application of the Orowan approach based on a sinusoidal representation of the cohesive force (Lawn and Wilshaw, 1975). 

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. 

Step 3. The set of fracture properties G(t) are related to the interfaee structure H(t) through suitable deformation mechanisms deduced from the micromechanics of fracture. This is the most difficult part of the problem but the analysis of the fracture process in situ can lead to valuable information on the microscopic deformation mechanisms. SEM, optical and XPS analysis of the fractured interface usually determine the mode of fracture (cohesive, adhesive or mixed) and details of the fracture micromechanics. However, considerable modeling may be required with entanglement and chain fracture mechanisms to realize useful solutions since most of the important events occur within the deformation zone before new fracture surfaces are created. We then obtain a solution to the problem. 

The simplest model is the lattice-gas or Ising model. The whole space is divided into a lattice of N sites, and on each site two different states are possible a crystalline state denoted by the variable 5, = 1 and a gaseous state by Sj = -. The variable s denotes the degree of crystalline order. The cohesion of nearest-neighboring solid atoms leads to the following interaction energy... 

In this approach, connectivity indices were used as the principle descriptor of the topology of the repeat unit of a polymer. The connectivity indices of various polymers were first correlated directly with the experimental data for six different physical properties. The six properties were Van der Waals volume (Vw), molar volume (V), heat capacity (Cp), solubility parameter (5), glass transition temperature Tfj, and cohesive energies ( coh) for the 45 different polymers. Available data were used to establish the dependence of these properties on the topological indices. All the experimental data for these properties were trained simultaneously in the proposed neural network model in order to develop an overall cause-effect relationship for all six properties. 

Figure 26 ANN model (5-8-6) training and testing results for cohesive energies of 45 different polymers.

The cohesive stress ac is assumed to be constant (Dugdale model) as in Eq. (7.5). Chan, Donald and Kramer [87] found a good agreement between the critical energy release rate GIC, as estimated by the Dugdale model and G)C as computed from the actual stress and displacement profiles in their experiments. 

The deformation zones were calculated for the polymers of Table 5.1 and Table 6.1 according to the Dugdale-Barenblatt-model. Yield stress ay from tensile tests was used instead of the cohesive stress ctc since a reasonable agreement of ay and ctc... 

The interaction between particle and surface and the interaction among atoms in the particle are modeled by the Leimard-Jones potential [26]. The parameters of the Leimard-Jones potential are set as follows pp = 0.86 eV, o-pp =2.27 A, eps = 0.43 eV, o-ps=3.0 A. The Tersoff potential [27], a classical model capable of describing a wide range of silicon structure, is employed for the interaction between silicon atoms of the surface. The particle prepared by annealing simulation from 5,000 K to 50 K, is composed of 864 atoms with cohesive energy of 5.77 eV/atom and diameter of 24 A. The silicon surface consists of 45,760 silicon atoms. The crystal orientations of [ 100], [010], [001 ] are set asx,y,z coordinate axes, respectively. So there are 40 atom layers in the z direction with a thickness of 54.3 A. Before collision, the whole system undergoes a relaxation of 5,000 fsat300 K. 

When the parameters deduced from the calculations on AI2 and AI3 are applied to bulk Al, the cohesive energy is too small and the bond length is too large. The small cohesive energy is expected because our computed AI2 at the TZ2P-CPF level is only 71% of the experimental value (42.46). The bulk values are in much better agreement with experiment if the model is parameterized using the experimental and r values for the E state. Hence, the... 

The degree of realism of these model structures can be assessed by comparison of computed properties with experimental ones. The cohesive energy is, by definition, the difference in energy per mole of substance between a parent chain in its bulk environment and the same parent chain in vacuo, i.e., when all intermolecular forces are eliminated. This difference is readily computed from the minimized... 

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