Continuum stress equations

We use the convected derivative D/Dt to remind us that this equation is to be applied to the ensemble of particles/macromolecules belonging to a fixed material point. The second term represents the effect of the flow and contains p [Eq. (2-101)], whereas the last term represents the rotational diffusion process. Finally, we require an expression relating the bulk (i.e., continuum) stress to the orientation distribution. For a dilute suspension, we... 

The factor 2 in front of the second integral accounts for the fact that two surfaces, each with surface energy Us, are created from the surface S in the reference configuration. Continuum stress and deformation fields near a geometrical discontinuity such as a crack edge are potentially singular. In such a case, the differentiability requirements on these fields for interpretation of the continuum equilibrium equation or the divergence theorem may not be satisfied consequently, a special interpretation is required (Freund 1990). 

The correspondence between the continuum elasticity equations developed for the stress and strain tensors and the strain energy density, and the general relations between the force F and time derivative of the position r, or the spatial derivative of the potential energy as given by classical mechanics. The last two equations correspond to the special case of a harmonic potential energy, which is implicit in the linear relation between stress and strain. 

The functions and depend on the collision function model using gas density and temperature, and should satisfy the moment equation. The above-mentioned equation is substituted for the Boltzmann s equation, and a set of inhomogeneous linear equations is obtained by equating terms of equal order. The use of distribution functionsand so on leads to the determination of transport terms needed to close the continuum equations appropriate to the particular level of approximation. The continuum stress tensor and heat flux vector can be written in terms of the distribution function (f >). This can be further simplified in terms of macroscopic velocity and temperature derivatives. 

Continuum theory has also been applied to analyse tire dynamics of flow of nematics [77, 80, 81 and 82]. The equations provide tire time-dependent velocity, director and pressure fields. These can be detennined from equations for tire fluid acceleration (in tenns of tire total stress tensor split into reversible and viscous parts), tire rate of change of director in tenns of tire velocity gradients and tire molecular field and tire incompressibility condition [20]. 

It has been shown that the thermodynamic foundations of plasticity may be considered within the framework of the continuum mechanics of materials with memory. A nonlinear material with memory is defined by a system of constitutive equations in which some state functions such as the stress tension or the internal energy, the heat flux, etc., are determined as functionals of a function which represents the time history of the local configuration of a material particle. 

When a fluid flows past a solid surface, the velocity of the fluid in contact with the wall is zero, as must be the case if the fluid is to be treated as a continuum. If the velocity at the solid boundary were not zero, the velocity gradient there would be infinite and by Newton s law of viscosity, equation 1.44, the shear stress would have to be infinite. If a turbulent stream of fluid flows past an isolated surface, such as an aircraft wing in a large wind tunnel, the velocity of the fluid is zero at the surface but rises with increasing distance from the surface and eventually approaches the velocity of the bulk of the stream. It is found that almost all the change in velocity occurs in a very thin layer of fluid adjacent to the solid surface ... 

When considering boundary conditions, a useful dimensionless hydrodynamic number is the Knudsen number, Kn = X/L, the ratio of the mean free path length to the characteristic dimension of the flow. In the case of a small Knudsen number, continuum mechanics will apply, and the no-slip boundary condition assumption is valid. In this formulation of classical fluid dynamics, the fluid velocity vanishes at the wall, so fluid particles directly adjacent to the wall are stationary, with respect to the wall. This also ensures that there is a continuity of stress across the boundary (i.e., the stress at the lower surface—the wall—is equal to the stress in the surface-adjacent liquid). Although this is an approximation, it is valid in many cases, and greatly simplifies the solution of the equations of motion. Additionally, it eliminates the need to include an extra parameter, which must be determined on a theoretical or experimental basis. 

Lu et al. [7] extended the mass-spring model of the interface to include a dashpot, modeling the interface as viscoelastic, as shown in Fig. 3. The continuous boundary conditions for displacement and shear stress were replaced by the equations of motion of contacting molecules. The interaction forces between the contacting molecules are modeled as a viscoelastic fluid, which results in a complex shear modulus for the interface, G = G + mG", where G is the storage modulus and G" is the loss modulus. G is a continuum molecular interaction between liquid and surface particles, representing the force between particles for a unit shear displacement. The authors also determined a relationship for the slip parameter Eq. (18) in terms of bulk and molecular parameters [7, 43] ... 

The axisymmetric nature of conical hoppers results in es = 0 and, according to Eq. (2.20), cre = (compatibility requirement, i.e., the relationship of strains. This relation, with the aid of constitutive relations between stress and strain (e.g., Hooke s law), provides an additional equation for stress so that the problem can be closed. However, the compatibility relation for a continuum solid may not be extendable to the cases of powders. Thus, additional assumptions or models are needed to yield another equation for stresses in powders. 

As was demonstrated by Pyshnograi (1994), the last term in (6.7) can be written in symmetric form, if the continuum of Brownian particles is considered incompressible. In equation (6.7), the sum is evaluated over the particles in a given macromolecule. The monomolecular approximation ensures that the stress tensor of the system is the sum of the contributions of all the macromolecules. In this form, the expression for the stresses is valid for any dynamics of the chain. One can consider the system to be a dilute polymer solution or a concentrated solution and melt of polymers. In any case the system is considered as a suspension of interacting Brownian particles. 

In equation (6.33), the stresses in the moving viscoelastic liquid (6.31) are added to the stresses in the continuum of Brownian particles. When the equations of motion are formulated, we have to take into account the presence of the two interacting and interpenetrating continuous media formed by the viscoelastic liquid carrier and the interacting Brownian particles that model the macromolecules. However, the contribution of the carrier in the case of a concentrated solution is slight, and we shall ignore it henceforth. 

The modulus and yield kinetic parameters of the block polymer B can be related to those of the homopolymer in terms of a microcomposite model in which the silicone domains are assumed capable of bearing no shear load. Following Nielsen (10) we successfully applied the Halpin-Tsai equations to calculate the ratio of moduli for the two materials. This ratio of 2 is the same as the ratio of the apparent activation volumes. Our interpretation is that the silicone microdomains introduce shear stress concentrations on the micro scale that cause the polycarbonate block continuum to yield at a macroscopic stress that is half as large as that for the homopolymer. The fact that the activation energies are the same however indicates that aside from this geometric effect the rubber domains have little influence on the yield mechanism. 

The non-linear response of elastomers to stress can also be handled by abandoning molecular theories and using continuum mechanics. In this approach, the restrictions imposed by Hooke s law are eliminated and the derivation proceeds through the strain energy using something called strain invariants (you don t want to know ). The result, called the Mooney-Rivlin equation, can be written (for uniaxial extension)—Equation 13-60 ... 

We have been able to interpret this experimental data by means of a very simple continuum model. Assuming the cluster to be plastic with a uniform internal stress, the relationship between this internal stress and the radius of curvature of the cluster s free surface is given by the Young equation... 

Derivations of conservation equations from the viewpoint of kinetic theory usually do not exhibit explicitly the diffusion terms, such as diffusion stresses, that appear on the right-hand sides of equations (49), (50), and (51), since it is unnecessary to introduce quantities such as afj specifically in these derivations. Kinetic-theory developments work directly with the left-hand sides of equations (49), (50), and (51). Transport coefficients (Appendix E) are defined only in terms of these kinetic-theory quantities because prescriptions for calculating the individual continua transports, afj and qf, are unduly complex. Moreover, measurement of diffusion stresses is feasible only by direct measurement of diffusion velocities, followed by use of equation (24). Therefore, it has not been fruitful to study the diffusion terms which, in a sense, may be viewed as artifacts of the continuum approach. 

Generally, for any dimension therefore, if a crack of length I already exists in an infinite elastic continuum, subject to uniform tensile stress a perpendicular to the length of the crack, then for the onset of brittle fracture, Griffith equates (the differentials of) the elastic energy E with the surface energy E ... 

For numerical investigations of stress localizations in laminates, the discretizational effort can be reduced significantly if only the boundary needs to be discretized, as it is for e -ample the case in the classical boundary element method (BEM). But in this method a fundamental solution is needed which is in many cases difficult to achieve or even unknown. The Boundary Finite Element Method (BFEM) to be presented here does not require such a fundamental solution, because the element formulation is based on the finite element method (FEM), Thus the BFEM can be characterized to be a finite element based boundary discretization method. This method was originally developed from Wolf and Song [10] under the name Consistent Finite Element Cell Method for time-dependent problems in soil-mechanics. The basic assumption of this method is that a stiffness matrix describing the force-displacement relation at discrete degrees of freedom at the boundary of the continuum is scalable with respect to one point in three-dimensional space, the so-called similarity center, if similar contours within the continuum are considered. In contrast to this, the current work deals with the case of equivalent cross-sectional properties, i.e., that cross-sections parallel to the boundary can be described by the same stiffness matrix, which is the appropriate formulation for the case of the free-edge effect and the matrix crack problem. The boundary stiffness matrix results from a Matrix-Riccati equation. The field quantities inside of the continuum can be calculated from an ordinary differential equation. 

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