Modified Couple Stress Thermoelastic Diffusive Beam Due to Ramp Type Heating

Abstract

This chapter represents the thermoelastic diffusion beam in modified couple stress theory due to ramp type heating. The Euler-Bernoulli beam theory, Laplace transform technique, and eigenvalue approach is applied to solve the problem in the non-dimensional form. Euler-Bernoulli beam theory and the Laplace transform technique are used to write the basic equations in the form of a vector-matrix differential equation, which is solved by the eigenvalue approach. The transformed components of displacement, lateral deflection, temperature, axial stress, chemical potential, and concentration are obtained. A numerical inversion technique has been adopted to recover the solutions in the physical domain. The analytical results have been calculated numerically with the help of MATLAB software. Copper material is used for mathematical modeling. Numerically simulated results for displacement, lateral deflection, temperature, axial stress, chemical potential, and concentration are depicted graphically for various values of the time. Comparisons have been made for different values of the time. Some particular cases are discussed in the problem.