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Buckling of Beams and Plates
Published in L.M.B.C. Campos, Higher-Order Differential Equations and Elasticity, 2019
involving two arbitrary functions (6.342a) analytic in the complex plane; (v) the last term is a biharmonic non-harmonic function, that is, satisfies a biharmonic equation (6.343b) but not a Laplace (6.340c) equation, because (subsection II.4.6.1) the general integral of the biharmonic equation (6.343b) is of the form (II.4.184c) ≡(6.343c): f,g,h,j∈|C:∇4Θ=0⇔Θx,y=Refz+z*hz+gz*+zjz*,
Mathematical Theory for Maxwell Equations
Published in K.T. Chau, Applications of Differential Equations in Engineering and Mechanics, 2019
Mathematically, Chau (2018) showed that there is close resemblance between three-dimensional dynamic elasticity and the Maxwell equations. Both of them can be solved in terms of a nonhomogeneous wave equation. In three-dimensional elasticity, Helmholtz decomposition reduces the problem mathematically to solving biharmonic equations of a vector potential and a scalar potential (see Section 4.2.1 of Chau, 2018). This is very similar to the Lorenz gauge situation that we encountered in an earlier section. Alternatively, in three-dimensional elasticity, the Galerkin vector reduces the problem to solving a biharmonic equation of a single vector function (see Section 4.2.3 of Chau, 2013). In fact, instead of using A and ρ, in electrodynamics we can also employ a single vector function, and it is called the Hertz vector (having a role similar to the Galerkin vector in three-dimensional elasticity). We will consider the Hertz vector in this section. The beginning of Hertz’s career coincided with the death of Maxwell, so they never had correspondence. As we mentioned in the introduction, it was Hertz who first conceived and conducted experiments that showed the existence of electromagnetic waves as predicted by Maxwell, and this led to the general acceptance of Maxwell equations. Hertz was awarded the Rumford Medal of the Royal Society because of this. Unfortunately, Maxwell did not live to see this. Theoretically, the electric and magnetic Hertz vector approach was proposed by Hertz, and is considered next.
Plane Thermoelastic Problems
Published in Naotake Noda, Richard B. Hetnarski, Yoshinobu Tanigawa, Thermal Stresses, 2018
Naotake Noda, Richard B. Hetnarski, Yoshinobu Tanigawa
Next, let us show the general solutions of the harmonic equation and biharmonic equation in the polar coordinate system (r, θ). Referring to the general solution (4.82) of the harmonic equation in the cylindrical coordinate system (r, θ, z), the harmonic functions take forms () (1lnr)(1θ),(rnr−n)(cosnθsinnθ)
A theoretical study on ground surface settlement induced by a braced deep excavation
Published in European Journal of Environmental and Civil Engineering, 2022
Haohua Chen, Jingpei Li, Changyi Yang, Ce Feng
In this paper, a feasible analytical approach is proposed to predict the excavation induced GSS under different displacement modes of retaining wall. The problem considered is reduced to solving a biharmonic equation with mixed boundary conditions. Based on the elastic superposition principle, the mixed boundary is decomposed into displacement boundary and stress boundary, with which the governing equation is solved accordingly by the Separation of Variable Method and the Fourier Transform Method. To determine the unknown coefficients in the solution, a least-squares based method is developed and proposed to transform the in-site scatter data of wall deflection to a continuous displacement function in the form of Fourier series. The proposed approach is feasible enough to give a reliable prediction for the excavation induced GSS.
Modified nonlocal boundary value problem method for an ill-posed problem for the biharmonic equation
Published in Inverse Problems in Science and Engineering, 2019
Problems of modelling of thin plates vibrations with various conditions of fixation on different sides (edges) of the plate are the most interesting cases. Frequently in practice there arise problems of modelling of vibrations of plates, boundaries of which consist of finite number of smooth arcs with part of them being clamped (embedded), and the rest of the arcs being in free leaning. Such conditions are permissible and the problems arising in their modelling are investigated rather detailed. In the present paper we consider the mathematical model, which arises when one of the sides of the flat plate is free. The mathematical modelling leads to the problem for the homogeneous biharmonic equation with different boundary conditions on opposite boundaries.