Continuum mechanics models

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

The measurable linear viscoelastic functions are defined either in the time domain or in the frequency domain. The interrelations between functions in the firequenpy domain are pxirely algebraic. The interrelations between functions in the time domain are convolution integrals. The interrelations between functions in the time and frequency domain are Carson-Laplace or inverse Carson-Laplace transforms. Some of these interrelations will be given below, and a general scheme of these interrelations may be found in [1]. These interrelations derive directly from the mathematical theory of linear viscoelasticity and do not imply any molecular or continuum mechanics modelling. 

CONTINUUM MECHANICS MODELS OF STABLE ISOTOPE TRANSPORT IN HIGH TEMPERATURE CRUSTAL SYSTEMS... 

Applications of f/r calculations to continuum mechanics models. The fluid-rock ratio equations yield molar ratios, //r, of oxygen or carbon in the fluid and the rock. The relationship between the //r values and the continuum mechanics calculations are best illustrated by re-arranging Equation (54) in terms of the dimensionless concentration. The equation for the multi-pass fluid-rock ratio (in mole equivalents of the isotope under consideration) can be recast in terms of the dimensionless concentration of the rock. 

The continuum mechanics modeled by JAS3D are based on two fundamental governing equations. The kinematics is based on the conservation of momentum equation, which can be solved either for quasi-static or dynamic conditions (a quasistatic procedure was used for these analyses). The stress-strain relationships are posed in terms of the conventional Cauchy stress. JAS3D includes at least 30 different material models. 

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. 

In the present study an extended continuum mechanical model is derived which is able to predict either weak or stiff boundary layers in thin films. As a possible application, the formation of interphases in polymer films is investigated. In this case it was shown [7, 24, 37] that the local stiffness in the polymer depends on the combination of polymer and substrate. 

Peksen, M., Peters, R., Blum, L., and Stolten, D. (2010) Component design in SOFC technology 3D computational continuum mechanics modelling and experimental validation of a planar type Pre-heater, presented at the 7th Symposium on Fuel Cell Modelling and Experimental Validation, Morges (Lausanne), Switzerland. 

Nanoscopic particles, dispersed in a block copolymer, have dimensions that are appropriate for Brownian dynamics simulations (268). Clay composites have a range of length scales, but if the gallery spacing between the layers is not large, MD methods can be used (269) with periodicity in the directions parallel to the clay platelets. However, continuum mechanical models need to be invoked for the description of exfoliated clay systems (270). These materials have so much interfacial area that adhesion properties are very important (271). Traditional continuum bounds methods (130) usually ignore the interphases on the grounds that they comprise a very small volume fraction of the total material, and so are not expected to be very accurate for exfoliated clay systems. 

Continuum mechanics models for the adhesion of perfectly elastic spheres under the action of reversible surface forces are well develops. The essential features of the JKR model are shown in Figure la. The surface traction acting on a contact area of radius a comprises two terms (i) a Hertz pressurepi(r), caused by the compressive force Pi, which flattens the spherical surfaces and (ii) an adhesive tension pa( ) hich gives rise to the adhesive force Pa. The net contact force P can be expressed ... 

In the case of micro-composites, specific filler smface area of common fillers is usually less than the value 10 m /g. In these systems, a very small portion of polymei- molecules is in direct interaction with the filler surface. Moreover, dimensions of polymer chains and volumes chai acteristic of microscopic relaxar tion modes are orders of magnitude smaller compared to the dimensions of filler particles (see Table 6.1). Thus, continuum mechanics approaches, such as micromechauical models can be successfully used for description of their mechanical behavior. The continuum mechanics model, well usable for prediction of the mechanical contribution (i) to the modulus of elasticity of polymer composites, is the simple Kerner-Nielsen (K-N) model [66] ... 

In this chapter an overview of conceptually different fracture theories is presented which have in common that they do not make explicite reference to the characteristic properties of the molecular chains, their configurational and super-molecular order and their thermal and mechanical interaction. This will be seen to apply to the classical failure criteria and general continuum mechanical models. Rate process fracture theories take into consideration the viscoelastic behavior of polymeric materials but do not derive their fracture criteria from detailed morphological analysis. These basic theories are invaluable, however, to elucidate statistical, non-morphological, or continuum mechanical aspects of the fracture process. 

At the largest length scales, field theoretic and continuum mechanics models are able to predict the equiUbriirm stmcture of multicomponent systems and macroscopic flow response of a polymer system. However, these models do not contain molecular detail. They are based either on a phenomenological description of the free energy of the system, such as the Flory-Hu ns or Landau free energies, or on actual hydrodynamic parameters such as viscosity. 

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