Thermoelastic stress

Derive the thermoelastic stress-strain relations for an orthotropic lamina under plane stress, Equation (4.102), from the anisotropic thermoelastic stress-strain relations in three dimensions. Equation (4.101) [or from Equation (4.100)]. 

Several issues must be addressed. First, the heat-transfer environment must yield a well-controlled temperature field in the crystal and melt near the melt-crystal interface so that the crystallization rate, the shape of the solidification interface, and the thermoelastic stresses in the crystal can be controlled. Low dislocation and defect densities occur when the temperature gradients in the crystal are low. This point will become an underlying theme of this chapter and has manifestations in the analysis of many of the transport processes described here. 

Jordan et al. (100) refined the thermoelastic stress calculation for the analysis of the spatial distribution of dislocations in GaAs grown with the LEC method. Their analysis was based on a two-dimensional model for the... 

Figure 15. Comparison of measured dislocation density in a GaAs wafer grown by the LEC method (top) with the thermoelastic stress calculation by Jordan et al. (bottom) (100). The high dislocation density around the periphery is predicted by the calculations.

The explosive character of the photoinduced solid-state chlorination reaction of MCH was first described in ref. 31, the phenomenon being interpreted on the assumption of a decrease in the chain-growth activation energy due to the thermoelastic stresses induced in the sample. A possible role of brittle fracture was not considered in that case. However, it would be of interest also to take account of that effect under the conditions used in ref. 31, the more so in that the evaluated values of stresses required to reduce the activation energy markedly are far above the thresholds of brittle fracture of the corresponding matrices (for details, see Section XII). 

Thermoelastic stresses are generated during a change from the deposition temperature to another temperature Tj. The difference between the thermal expansion coefficients of the film (f) and substrate (s) cause deformation to occur in the film plane. This is constant throughout the thickness of the film, such that ... 

In this relation, a and a, are the thermal expansion coefficients of the substrate and film. These depend on the temperature T. If the film is homogeneous and elastically isotropic, the in plane thermoelastic stress is expressed by ... 

Ej and are respectively Young s modulus and Poisson s coefficient for the film. Furthermore, assuming that the thermal expansion coefficients are independent of the temperature, the thermoelastic stress increases linearly with the film deposition temperature. 

The overall residual stress in the film is equal to the sum of the thermoelastic component and intrinsic component. In the case of low deposition temperatures, the intrinsic stress is still the main contributing factor to the overall residual stress. In contrast, in the case of high deposition temperatures, the thermoelastic stress predominates. Therefore, while the signs of these stresses are identical, there is an intermediate deposition temperature for which the residual stress is minimised (Figure 2). 

Fig. 2 Schematic representation of changes in thermoelastic stress intrinsic stress a. and overall residual stress o, in a thin film as a function of deposition temperature...

Other types of damage may be produced through thermomechanical effects. For example, when being annealed at 450°C a CVD aluminum film on a Si substrate is subjected to compressive thermoelastic stresses owing to the considerable difference between the thermal expansion coefficients of aluminum (a = 23 x 10 °C 0 and the silicon substrate (a. = 3.5 x 10 °C 0-When cooling, the film may therefore contract by as much as 1%. Due to the combined action... 

Analytical solutions for thermoelastic stress distributions within moving material, irradiated with two-dimensional CW Gaussian beams (P 1 = 0), have also been obtained [24], For a material characterized by k = 50.2 W/mK, p = 7880 kg/m3, c = 502 J/kgK, PI2r = 105 W/m, 7 = 4 mm/s, P = 10 5 K-1, v = 0.3, and p = 105 MPa (the material shear modulus), the dimensionless surface stress component varies with Pe as shown in Fig. 18.9. Here, Pe was varied by changing the beam radius, and the beam moves relative to the surface in the positive x direction. At large Pe, stresses are relatively uniform, while, at extremely small Pe, stress gradients... 

Thermoelastic stresses arise either when a constrained body is cooled or heated or when an unconstrained body is heated inhomogeneously. These stresses are computed from elastic constants and thermal expansivity [ ]. 

Solid-state elastic constants fill many needs. Engineering design calculations require them for estimating load-deflection and thermoelastic stress. Derived from fundamental interatomic forces, elastic constants index both cohesion and strength. They relate to other physical properties such as specific heat and thermal expansion, all of which help define a solid s equation of state. 

Plate AIB.6 Stresses around penetrations Thermoelastic stresses... 

Earl JS, Dulieu-Barton JM, Shenoi RA (2003) Determination of hygrothermal ageing effects in sandwich construction joints using thermoelastic stress analysis. Compos Sci Technol 63 (2) 211-223... 

Although it is not immediately obvious, Eqs. (4) and (5) have the same general dependency on the elastic modulus and strength. When the thermoelastic stress is expressed in simple form for total linear restraint in one dimension, it is of the form... 

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