Session: 10-05-01: Applied Mechanics, Dynamic Systems, Experimental and Computational Methods, Modeling and Virtual Simulations of Dynamic Structures, Advanced Materials and Testing

Paper Number: 163920

Effect of Temperature Gradient on Stress Distribution in a Rod Under Centric Axial Body Force 

Thermal Stresses, in the absence of mechanical loads, could cause significant stresses in a solid body which, in turn, could lead to failure of mechanical and structural parts. In a typical undergraduate mechanical engineering curriculum, students are introduced to the concept of thermally induced stresses by studying the formation of these stresses in a centric axially loaded bar subject to a uniform temperature change and in the presence of geometrical constraints. Fourier law of conduction indicates that the transfer of the heat in solids is due to the existence of temperature gradient in the solid. Thereby the notion of uniform temperature change in solids, in thermal exchange with its surroundings, is not realistic.Moreover, this simplistic assumption would not render correct stresses in bodies of interest. Realizing this short coming in pedagogy and teaching of this important concept, we preent this paper as a model and instrument to bridge this gap and introduce students to the subject in a sensibly simple way in terms of an exercise and a project.

Treatment of thermally induced stresses in a solid demands the acquisition of the temperature field in the solid first by applying the heat diffusion equation in conjunction with its boundary and initial conditions. This activity is then followed by applying equilibrium concepts and Hook’s law of elasticity to render stresses and strains in the body. As it might be deduced from the above statements, this short student project is best suited for mechanical engineering students at their senior year. This is because students at senior year have had a course in heat transfer already, and the topic of thermal stresses could be introduced to them in either that course or machine design class. The authors, however, encourage the exposure of sophomore level students to this important topic of design in their first solid mechanics course by providing the true temperature field in the solid as a known function of space and time without bothering to solve for the temperature field. This is closer to reality in which there exists a temperature gradient in the solid rather than assuming a uniform increase of temperature in the body.

To this end, a numerical example is provided by considering a solid bar subject to a body force and in thermal communication with its surroundings. The body force inconsideration could be either the centrifugal force in a simplified model of a gas turbine blade or the gravity load of a suspended bar from a support. The temperature field in the bar, due to its heat transfer with the surroundings, is obtained first. Then, it is followed by the resulting strains and stresses. Furthermore, a case where the bar is subject to geometrical constraints at its ends and subject to both body force and thermal loadings is deliberated. Comparison is made between the results of each of the above cases on the one hand, and the results obtained from the same bar and the same body force loading, however, under the fictitious uniform temperature change in the bar, on the other.

 

Presenting Author: Salim Haidar Grand Valley State University

Presenting Author Biography: Salim M. Haidar holds a Ph.D.in Applied Mathematics from Carnegie-Mellon University and is a Professor of Mathematics at Grand Valley State University. His research interests are in the areas of Nonlinear Elasticity (regularity of equilibria and material instabilities); Variational Calculus (field theory, regularity of minimizers, relaxed formulations); and PDEs (compensated compactness and homogenization methods in solving nonlinear PDE’s)

Authors:

Alireza Mohammadzadeh Grand Valley State University
Salim M. Haidar Grand Valley State University