Residual stress equations

Using the equilibrium equations of the elasticity theory enables one to determine the stress tensor component (Tjj normal to the plane of translumination. The other stress components can be determined using additional measurements or additional information. We assume that there exists a temperature field T, the so-called fictitious temperature, which causes a stress field, equal to the residual stress pattern. In this paper we formulate the boundary-value problem for determining all components of the residual stresses from the results of the translumination of the specimen in a system of parallel planes. Theory of the fictitious temperature has been successfully used in the case of plane strain [2]. The aim of this paper is to show how this method can be applied in the general case. 

Equation 7 predicts the correct yield pressure only if the material is isotropic, the cylinder free from residual stress prior to the appHcation of pressure, and sufftcientiy long, eg, more than five diameters, for there to be no end effects. 

As can be seen from the above equations, the standard deviation of the strength increases significantly with the number of processes used in manufacture that are adding the residual stresses. This may be the reason for the apparent reluctance of suppliers to give precise statistical data about their product (Carter, 1997). 

The divergence of (4.21) yields a Poisson equation for p. However, the residual stress tensor r6 is unknown because it involves unresolved SGS terms (i.e., UfiJfi). Closure of the residual stress tensor is thus a major challenge in LES modeling of turbulent flows. 

The residual thermal stress was investigated with a Plexus stress analyzer. The residual stress, was calculated from the radii of wafer curvatures before and after polyimide film deposition by the following equation ... 

Note that if el = e2 then the curvatures are zero from Equations 8.24 and 8.26 and the midplane strain is just ex from equation (8.28). In this case the residual stresses are identically zero. 

The temperature dependence of the relaxation times introduced into mechanical equations by means of Eq. (2.93) was used to calculate the residual stresses in cooling amorphous polymers. 

The stress intensity due to residual stresses is given by equation (7.7) [10, 21] ... 

Equation (12.61) is also true for residual stresses in metal. The latter kind of stress usually will have tensile and compressive stress fields associated with it. As far as the solubility of hydrogen is concerned, the effect of the tensile stress field (which increases solubility) overwhelms the counter-effect due to the compressive stress field (which tends only to decrease the already small solubility). Therefore, the larger the lattice strain or distortion, the larger the concentration of hydrogen (Fig. 12.74). All imperfections in crystals are regions of distortion or strain. Hence, absorbed hydrogen finds its way to, and concentrates at, such imperfections. 

Equation (16-4) shows that knowledge of is required. If the specimen contains only applied stress, d can be obtained from a measurement on the unloaded specimen. (Such a stress measurement is rarely made, and then only for certain research purposes it is far easier to measure applied stress with an electric-resistance gauge.) If the specimen contains residual stress, d must be measured on a small stress-free portion cut out of the specimen the method then becomes destructive and of no interest. 

Equation (16-4) is therefore not a practical basis for the measurement of residual stress. We will see later that two measurements of plane spacing on the stressed specimen are required for a nondestructive determination of stress. 

The opposite viewpoint, advocated by Boris et al. [16], is that no explicit filtering should be performed and no explicit residual stress model should be used (Oj = 0). Instead, an appropriate numerical method is used to attempt to solve the Navier-Stokes equation for v(r, f). Because the grid is not fine enough to resolve the solution to the Navier-Stokes equation, significant numerical stresses arise. Thus, filtering and residual-stress modeling are performed implicitly by the numerical method. 

The above equations describe the large-scale motion. Lij represents the interactions among the large scales. The effect of small scales appears through the residual stress tensors (i.e., Cij and Rij). In particular, Cij represents the interactions between the large and small scales, and Rij reflects the interactions between subgrid scales. The tensors Lij, Cij and Rij are known as the Leonard stress, cross-term stress and the residual Reynolds stress, respectively. 

In practice, the Leonard stress is often dominated by the numerical errors inherent in the finite difference (and finite volume) representation and is thus neglected or lumped into the deviatoric stress tensor (e.g., [97] [106] [186], p. 325). Consequently, as the box filter is applied to the Navier-Stokes equation, the residual stresses assume the form of sub-grid scale stresses ... 

To circumvent the drawbacks of the sin2 P-technique including the problem of a non-linearity of the modulus of elasticity, attempts have been made to apply other tests to obtain estimates of the residual stresses. The curvature measurement is probably one of the most widely used method for determining residual stress and involves measuring the bending of the coated sample in response to both quenching and thermal stresses. From the measured radius of curvature, the stress can be calculated according to the Stoney equation (Stoney, 1909) as... 

Using special data reduction relationships, the principal residual stresses and their angular orientation are calculated from the measured strains according to the equations shown earlier. 

This is the equation for the maximum reversible work of adhesion for two solids in contact and is the counterpart of Equation 4 for the case of a liquid in contact with a solid. Our definition of the final solid-solid state as an equilibrium state implies that there are no residual stresses in either solid produced by the solidification of L2. 

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