Stress approximation

During pressure sintering, interiDarticle compressive stress, approximated by the externally applied stress and nonnalized by the relative density of the compact p, supplements the surface tension driving force for pore shrinkage ... 

The first finite element schemes for differential viscoelastic models that yielded numerically stable results for non-zero Weissenberg numbers appeared less than two decades ago. These schemes were later improved and shown that for some benchmark viscoelastic problems, such as flow through a two-dimensional section with an abrupt contraction (usually a width reduction of four to one), they can generate simulations that were qualitatively comparable with the experimental evidence. A notable example was the coupled scheme developed by Marchal and Crochet (1987) for the solution of Maxwell and Oldroyd constitutive equations. To achieve stability they used element subdivision for the stress approximations and applied inconsistent streamline upwinding to the stress terms in the discretized equations. In another attempt, Luo and Tanner (1989) developed a typical decoupled scheme that started with the solution of the constitutive equation for a fixed-flow field (e.g. obtained by initially assuming non-elastic fluid behaviour). The extra stress found at this step was subsequently inserted into the equation of motion as a pseudo-body force and the flow field was updated. These authors also used inconsistent streamline upwinding to maintain the stability of the scheme. 

Even though in classical lamination theory by virtue of the Kirchhoff hypothesis we assume the stresses and are zero, we can still obtain these stresses approximately by integration of the stress equilibrium equations... 

The diversity of these subcellular actin structures is remarkable and appears to be determined by the interactions of many actin-binding proteins (ABPs) as well as by changes in the concentrations of intracellular signaling molecules such as Ca and cAMP, by small GTP-binding proteins, and by signals arising from mechanical stress. Approximately 50% of the actin molecules in most animal cells are unpolymerized subunits in the cytosolic pool and exist in a state of dynamic equilibrium with labile F-actin filamentous structures (i.e., new structures are formed while existing structures are renewed) (Hall, 1994). 

Launder and Spalding [95] recognized that the relation obtained for k in the log-law layer (1.428) permits us to redefine the friction velocity, when the k value is considered known from the previous iteration or time level. Hence, a non-equilibrium boundary layer shear stress approximation is given by -<3w = pv Vk-... 

The corresponding dissipation rate term is approximated using (1.405), the Prandtl-Kolmogorov relation (1.403), the eddy viscosity hypothesis (1.380), and the non-equilibrium boundary layer shear stress approximation (1.441) ... 

Inserting a matrix layer leads to a number of problems. Its finite thickness impacts the YVF and, consequently, raises the iy-FVE. On the other hand, a very thin layer inevitably leads to highly distorted elements and poor quality of stress approximation in that region. 

Allan NL, Barron THK, Bruno JAO (1996) The zero static internal stress approximation in lattice dynamics, and the calculation of isotope effects on molar volumes. J Chem Phys 105 8300-8303 Allinger N (1977) Conformational analysis. 130. MM2. A hydrocarbon forcefteld utihzing Vi and V2 torsional terms. J Am Chem Soc 99 8127-8134... 

To determine the velocity distribution for turbulent flow at steady state inside a circular tube, we divide the fluid inside the pipe into two regions a central core where the Reynolds stress approximately equals the shear stress and a thin, viscous sublayer adjacent to the wall where the shear stress is due only to viscous shear and the turbulence effects are assumed negligible. Later we include a third region, the buffer zone, where both stresses are important. 

Figure 3-30. An example of a stress-relaxation curve with an initial applied stress approximating its yield stress.

The through-the-thickness variation of fields are assumed to be consistent with the plane stress approximation and elementary bending theory. In particular, a kinematic assumption is adopted whereby material lines which are initially straight and perpendicular to the midplane of the plate or film remain so during deformation, and the influence of tractions acting on planes parallel to the midplane is assumed to be negligibly small. 

The model used to arrive at equation (5.102) is based on a constant shear stress approximation for transfer of stress across the interface between matrix and ceramic fiber, and this is often interpreted as the flow stress of the matrix or as the friction stress at the interface. Such a model leads to the view that the average fiber stress varies linearly as a function of distance along the fiber which is proportional to shear stress. Thus the shear stress at the interface is expected to equal the frictional stress only over a specific part of the fiber and then to decrease steadily with distance along the interface. A more detailed model concentrating on the transfer of load across a frictional interface between elastic solids is reported by Dollar and Steif, which suggests that the results obtained by using equation (5.102) are approximations because the constant shear stress approximation overestimates the extent of slip. Furthermore, the error increases as the coefficient of friction in the interface increases and as the load increases. Nevertheless this hardness indentation procedure does lend itself to obtaining much useful comparative data for one type of ceramic fiber say, in a series of matrices. Thus the technique is in line with how hardness indentation methods are most commonly used. 

Another popular and useful approach for many practical engineering problems that can be reduced to two dimensional plane strain or plane stress approximations involves an auxiliary stress potential. In this approach, a bi-harmonic equation is developed based on the stresses (in terms of the potential) satisfying both the equilibrium equation and the compatibility equations. The result is that stresses derived from potentials satisfying the biharmonic equation automatically satisfy the necessary field equations and only the boundary conditions must be verified for any given problem. A rich set of problems may be solved in this manner and examples can be found in many classical texts on elasticity. In conjunction with the use of the stress potential, the principle of superposition is also often invoked to combine the solutions of several relatively simple problems to solve quite complex problems. 

Figure 2 shows the plasma corticosterone concentrations before and 15 minutes following the stress of ether (one minute) in rats that were either uninjected or had received an intraperitoneal injection of 1-tyrosine or a-MT. The results show that the stress-induced increase in plasma steroids was reduced in the a-MT treated rats. Figure 3 shows that this reduced steroid response reflects reduced ACTH secretion in response to stress. Approximately 50 inhibition of the ACTH secretion in response to the stress of ether and sham adrenalectomy was found and increasing the dose of a-MT did not inhibit this response further. 

From the early days of adhesives testing it has been recognized that in-service loading on adhesive joints frequently subjects the bondline to cleavage stresses (more-or-less concentrated tensile stresses approximately perpendicular to the bondline). Thus another of D-14 s earlier standards is D-1062, Test Method for Cleavage Strength of Metal-to-Metal Adhesive Bonds. It employs a relatively simple specimen (Fig. 16) and loading procedure. 

Kashani, M., Young, R., 2008. Hoop stress approximation in offshore design codes. Marine Structures 21, 224—239. 

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