Boundary traction

Microstructures are generally too complex for exact models. In a polycrystalline microstructure, grain-boundary tractions will be distributed with respect to an applied load. Microstructures of porous bodies include isolated pores as well as pores attached to grain boundaries and triple junctions. Nevertheless, there are several simple representative geometries that illustrate general coupled phenomena and serve as good models for subsets of more complex structures. 

The developed inverse stress reconstruction technique is applied to transfer the heat induced stresses from a 3D model (Examine3D) and a heat flow model (JobFem2D) into FRACOD. Since most stress concentration is anticipated inside pillar region, only pillar region is considered to reduce numerical efforts. Two 2D sections have been modelled, 1.5 m and 0.5 m below the tunnel floor. The equivalent boundary tractions of a 2D BEM... 

All stress fields are reconstructed for the simulation of fracture process. The equivalent boundary tractions were directly inputted in FRACOD models to consider effect of excavation and heating in the model. 

Bezerra L, Saigal S, 1995. Inverse Boundary Traction Reconstruction with the BEM. Int J Solids Struct 32(10) 1417-1431. 

Figure 1. (Left) Effect of temperature on boundary traction distribution. (Right) Effect of loading...

Previously the authors proposed a model for thermally-induced microcracks This model is refined and adapted for integration with the constitutive model proposed in a subsequent section. The premise of this model is that temperature change from some stress-free temperature causes the standard deviation of the grain boundary tractions to change, but the mean remains unchanged. The relationship between the standard deviation of the distribution and the temperature below the stress free temperature is assumed to be linear such that... 

As cracking proceeds the load is carried by a smaller area. Additionally, even in an uncracked state, the load carried by boundaries is magnified due to porosity in the material. The relationship between porosity P and mean boundary traction is not simple, as it will depend on the morphology of the pores. Thus, (r)will depend on porosity and crack density in the form... 

The integral equation for the elastic boundary tractions and displacements is solved by numerical methods. The boundary is divided Into N finite length elements. In this paper the surface tractions and displacements are assumed to change linearly over each of the boundary elements. Figure 2 shows a typical boundary the surface tractions are prescribed on part of the boundary and the displacements are prescribed on the remaining part of the boundary. At each node point on the boundary there are two components of traction and two components of displacement. Thus, for N elements and N nodes there are 2N unknowns in the discretized system. The boundary Integral equation for the elasticity problem Is rewritten as below ... 

The components of the stress tensor were determined from the neo-Hookean model, eqs. 1.5.3and 1.5.4,andthearbitrarypressure p was eliminated by using the boundary tractions tz = ts = 0. Then the unknown traction ti was determined from t) = TuXi. 

We can see that if instead of specifying the constitutive equation in this problem, we specify the boundary traction ti, we can... 

The probes are assumed to be of contact type but are otherwise quite arbitrary. To model the probe the traction beneath it is prescribed and the resulting boundary value problem is first solved exactly by way of a double Fourier transform. To get managable expressions a far field approximation is then performed using the stationary phase method. As to not be too restrictive the probe is if necessary divided into elements which are each treated separately. Keeping the elements small enough the far field restriction becomes very week so that it is in fact enough if the separation between the probe and defect is one or two wavelengths. As each element can be controlled separately it is possible to have phased arrays and also point or line focussed probes. 

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. 

The basis for the determination of an upper bound on the apparent Young s modulus is the principle of minimum potential energy which can be stated as Let the displacements be specified over the surface of the body except where the corresponding traction is 2ero. Let e, Tjy, be any compatible state of strain that satisfies the specified displacement boundary conditions, l.e., an admissible-strain tieldr Let U be the strain energy of the strain state TetcTby use of the stress-strain relations... 

The boundary conditions require that at the free surface each component of traction should vanish. The traction can be found from Hooke s law (6.30), with the strain components obtained from (6.46) using (6.23), and the dilation from (6.47). The normal component of the traction is... 

Of the six boundary conditions (continuity across the boundary of three components of displacement and three components of traction), those concerned with displacement and stress in the /-direction are not relevant, nor is displacement in the x-direction since the fluid can slide freely. Hence, the boundary conditions are continuity of displacement and traction normal to the surface and zero traction parallel to the surface. 

At the grain boundaries, the condition Fv = 0 should also hold. The boundaries will be under a traction, ann = hT -tr-n, and when an atom is inserted, the tractions will be displaced as the grain expands by the volume For the case in Fig. 3.10, the boundary is oriented so that its normal is parallel to the z-axis and therefore (7nn = zz- This displacement contributes work, oyiuFIa = o zz a, and reduces the potential energy of the system by a corresponding amount. This term must be added to the chemical term, fi°A, and therefore the diffusion potential along the... 

A conscious choice of such elements can be made but in general the equilibrium distribution of stress cannot be found except for particular geometries. The assumptions of uniform strain throughout the assembly or of uniform stress were respectively made by Voigt and by Reuss. Returning to the structures actually perceivable in polymers one may consider the spherulite in a semi crystalline polymer as being unsuitable as a RVE because the boundary is not included. However, an assembly of spherulites would be acceptable, since it would contain sufficient to make it entirely typical of the bulk and because such an assembly would have moduli independent of the surface tractions and displacements. The linear size of such a representative volume element of spherulites would be perhaps several hundred microns. 

Where a melt-crystallized polymer has been processed by drawing, rolling or other means to produce an aligned structure in which lamellae as well as polymer chains have discernible order, a pseudocrystalline unit cell is present. Provided that this unit cell contains elements of the crystals as well as the boundaries between crystals and that it is entirely typical of the material as a whole then it could be considered as a RVE within the meaning defined above. The lamella crystal itself sometimes considered as embedded in an amorphous matrix would not seem to be an acceptable RVE for reasons similar to those advanced against the Takayanagi model, namely that its modulus is dependent upon the surface tractions. The boundaries between lamella crystals in the matrix must be included in an acceptable RVE. 

Here, Tyn, Ty12, iy22, u, v 2 andpJ represent the nodal values of the stress, velocity and pressure. Finally, the right hand side of the momentum equations contain the contribution of the body forces and the tractions imposed at the boundary ... 

Similar to scalar problems, the first step of the BEM is to discretize the boundary into a series of elements over which the velocity and traction are assumed to vary according to some interpolation functions. 

Consequently, there are 3(N + NIP) velocity unknowns and 3N traction unknowns. This makes eqn. (10.95) a system of 3(N+NIP) equations with 3(N+NIP) + 3Nunknowns. Each boundary nodal point has either traction or velocity specified for each direction as a boundary condition, thus the system in eqn. (10.95) can ultimately be arranged into a solvable system of linear algebraic equations as... 

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. 

Boundary element methods can be used for particulate flows where direct1 formulations can be used. The surface tractions on the solids are integrated to compute the hydrodynamic force and torque on those particles, which for suspended particles must be zero. 

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