Traction boundary condition

To avoid imposition of unrealistic exit boundary conditions in flow models Taylor et al. (1985) developed a method called traction boundary conditions . In this method starting from an initial guess, outflow condition is updated in an iterative procedure which ensures its consistency with the flow regime immediately upstream. This method is successfully applied to solve a number of turbulent flow problems. 

Tanner, R.I. 2000. Engineering Rheology, 2nd edti, Oxford University Press, Oxford. Taylor, C., Ranee, J. and Medwell, J. O., 1985. A note on the imposition of traction boundary conditions when using FEM for solving incompressible flow problems. Comnmn. Appl. Numer. Methods 1, 113-121. 

To find a solution to this problem using BEM, we must solve the Stokes system of equations with their corresponding equivalent integral formulation eqn. (10.82) with traction boundary conditions at the entrance and end of the tube and with no-slip boundary conditions at the tube walls. We start by creating the surface mesh and by selecting the position of the internal points where we are seeking the solution. Figure 10.19 shows a typical BEM mesh with 8-noded quadratic elements. 

A discussion of the traction boundary conditions—where the totaJ normal stress is prescribed on the inflow and outflow boundaries—for Jeffreys-type fluids is given in [31], and for Maxwell-type fluids in [32]. 

M. Renardy, Existence of steady flows of viscoelaistic fluids of Jeffreys-type with traction boundary conditions, Diff. Int. Eq., 2 (1989) 431-437. 

M. Renardy, Existence of steady flows for Maxwell fluids with traction boundary conditions on open boundaries, Z. Angew. Math. Mech. 75 (1995) 153-155. 

The final feature of the dislocation dynamics method that must be introduced so as to give such methods the possibility of examining real boundary value problems in plastic deformation is the treatment of boundary conditions. In particular, if we wish to consider the application of displacement and traction boundary conditions on finite bodies, the fields of the relevant dislocations are no longer the simple infinite body Volterra fields that have been the workhorse of our discussions throughout this book. To confront the situation presented by finite bodies, a useful scheme described in Lubarda et al. (1993) as well as van der Giessen and Needleman (1995) is to use the finite element method to solve for the amendments to the Volterra fields that need to be considered in a finite body. Denote the Volterra fields for an infinite body as In this case the fields of interest are given by... 

Specifically, because Z(z) for -a < x free condition along the crack. At x = a, the function would tend to infinity and, thereby, satisfy the required stress intensification. To additionally satisfy the remote traction boundary conditions, the following form of Z(z) is chosen to be a possible solution ... 

It is considered that the drop formation in the ink jet printing process is e governed by the timing of the electric pulse. This piezoelectrically controls the ink flow, inertia forces, viscous forces, surface tension and gravity. The final process model consists of two, one-dimensional equations originating from the kinematic boundary condition, the normal and tangential components of the traction boundary condition and the z-component of the momentum equation, thus ... 

In solving particular problems, suitable boundary conditions must be applied in particular circumstances. For the equations of motion, we distinguish between velocity boundary conditions and traction boundary conditions. 

The traction boundary condition is prescribed on the parts of boundary F, ... 

On the boundary 6Q of the domain Q, the boundary conditions can be either velocity or traction boundary conditions. 

This is known as the weak formulation. Since the boundary condition for the traction (t = a n) is directiy incorporated in the weak formulation, the traction boundary condition is sometimes called the natural boundary condition. 

The system for the one-dimensional elastic problem can be given by a governing equation (4.1a), the Dirichlet (i.e., displacement) boundary condition (4.1b) and the Neumann (i.e., traction) boundary condition (4.1c). This system is referred to as the strong form [SF]. 

By applying the divergence theorem to (4.35), the penalized strong form [PSF], which is equivalent to (4.35), is given by the following governing equation (4.36a), displacement boundary condition (4.36b) and traction boundary condition (4.36c) ... 

With these small slope approximations made explicit, the traction boundary conditions (8.132) on the evolving surface can be written in terms of surface shape and system parameters as... 

It ean be shown [15] that the linear displaeement boundary condition gives an "over-stiff response, and the constant traction boundary condition an "under-stiff response. These results thus give upper and lower bounds for the problem. 

In order to determine the effective elasticity matrix E, it is necessary in general to apply six linearly independent loadings defined by linear displacement and constant traction boundary conditions. For a material such as that considered here, which is assumed to be macroscopically isotropic, two constants completely define the macroscopic response, and these can be obtained from a single test. In particular, the two material constants used are the bulk and shear moduli, whose effective values are given by (1.39). 

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