Semi-inverse solution for Saint-Venant's problem in the theory of porous elastic materials

The purpose of this research is to study the Saint-Venant's problem for right cylinders with general cross-section made of inhomogeneous anisotropic elastic materials with voids. We reformulate the quasi-static equilibrium equations with the axial variable playing the role of a parameter. Two classes of semi-inverse solutions to Saint-Venant's problem are described in terms of five generalized plane strain problems. These classes are used in order to obtain a semi-inverse solution for the relaxed Saint-Venant's problem. An application of this results in the study of extension, bending, torsion and flexure of right circular cylinders in the case of isotropic materials is presented.     

Stress functions for a heterogeneous section of a tree

Consider a tree that is composed of wood, which is orthotropic with respect to the cylindrical coordinate axes, where the z-axis is directed up the center of the tree. When a section of a tree is considered as a Relaxed Saint-Venant's Problem the stresses in the plane of a transverse cross section will equal zero. By allowing some of, or all of the compliance coefficients to be functions of the radial coordinate r, the structure of the stress functions can be fundamentally altered. If the compliance coefficient in the z-direction (S₃₃₃₃) is allowed to remain constant, the S_rz and S_θz shear stresses will be functions of the coordinates, dimensions, applied loads, and compliance coefficients, while the other stresses will only be functions of the coordinates, dimensions and applied loads. If S₃₃₃₃ is also allowed to be a function of r, then all the nonzero stresses will be functions of the coordinates, dimensions, applied loads, and compliance coefficients.     

Shear stresses in elastic beams: an intrinsic approach

The theory of non-uniform flexure and torsion of Saint-Venant's beam with arbitrary multiply connected cross section is revisited in a coordinate-free form to provide a computationally convenient context. Numerical implementations, by Matlab, are performed to evaluate the maximum elastic shear stresses in beams with rectangular cross sections for different Poisson's ratios. The deviations between the maximum and mean stresses are then diagrammed to adjust the results provided by Jourawski's method.     

A higher order model for thin-walled structures with deformable cross-sections

A higher order model for the analysis of linear, prismatic thin-walled structures that considers the cross-section warping together with the cross-section in-plane flexural deformation is presented in this paper. The use of a one-dimentional model for the analysis of thin-walled structures, which have an inherent complex three-dimensional (3D) behaviour, can only be successful and competitive when compared with shell finite element models if it fulfills a twofold objective: (i) an enrichment of the model in order to as accurately as possible reproduce its 3D elasticity equations and (ii) the definition of a consistent criterion for uncoupling the beam equations, allowing to identify structural deformation modes.The displacement field is approximated through a linear combination of products between a set of linear independent functions defined over the cross-section and the associated weights only dependent on the beam axis; this approximation is not constrained by any ab initio kinematic assumptions. Towards an efficient application of the approximation procedure, the cross-section is discretized into thin-walled elements, being the displacement field approximated for each element independently of the displacement direction. The approximation is thus hp refined enhancing the “capture” of the 3D structural mechanics of thin-walled structures. The beam model governing equations are obtained through the integration over the cross-section of the corresponding elasticity equations weighted by the cross-section global approximation functions.A criterion for uncoupling the beam governing equations is established, allowing to (i) retrieve the classic equations of the thin-walled beam theory both for open and closed sections and (ii) derive a set of uncoupled deformation modes representing higher order effects. The criterion is based on the solution of the polynomial eigenvalue problem associated with the beam differential equations, allowing to quantify the Saint-Venant principle for thin-walled structures. In fact, the solution of the non linear eigenvalue problem yields a twelve fold null eigenvalue (representing polynomial solutions) that are verified to represent beam classic solutions and sets of pairs and quadruplets of non-null eigenvalues corresponding to higher order modes of deformation.     

Saint-Venant End Effects in Anti-plane Shear for Classes of Linear Piezoelectric Materials

This paper is concerned with further investigation of the effect of mechanical/electrical coupling on the decay of Saint-Venant end effects in linear piezoelectricity. Saint-Venant's principle and related results for elasticity theory have received considerable attention in the literature but relatively little is known about analogous issues in piezoelectricity. The current rapidly developing smart structures technology provides motivation for the investigation of such problems. The decay of Saint-Venant end effects is investigated in the context of anti-plane shear deformations for linear homogeneous piezoelectric solids. For a rather general class of anisotropic piezoelectric materials, the governing partial differential equations of equilibrium are a coupled system of second-order partial differential equations for the mechanical displacement u and electric potential ?. The traction boundary-value problem with prescribed surface charge can be formulated as an oblique derivative boundary-value problem for this elliptic system. Energy-decay estimates using differential inequality methods are used to study the axial decay of solutions on a semi-infinite strip subjected to non-zero boundary conditions only at the near end. This analysis is carried out for a rather general class of materials (the tetragonal ${\bar 4}$ crystal class). The boundary-value problem involves a full coupling of mechanical and electrical effects. There are four independent material constants appearing in the problem. An explicit estimated decay rate (a lower bound for the actual decay rate) is obtained in terms of two dimensionless piezoelectric parameters d ₀,r, the first of which provides a measure of the degree of piezoelectric coupling. The estimated decay rate is shown to be monotone decreasing with increasing values of the coupling parameter d ₀. In the limit as d ₀→0, we recover the exact decay rate for the purely mechanical case. Thus, for the tetragonal ${\bar 4}$ class of materials, piezoelectric end effects are predicted to penetrate further into the strip than their elastic counterparts, confirming recent results obtained in other contexts in linear piezoelectricity.     

Free vibration analysis of thin rotating cylindrical shells using wave propagation approach

The wave propagation approach is extended to study the frequency characteristics of thin rotating cylindrical shells. Based on Sanders’ shell theory, the governing equations of motion, which take into account the effects of centrifugal and Coriolis forces as well as the initial hoop tension due to rotation, are derived. And, the displacement field is expressed in the form of wave propagation associated with an axial wavenumber k _m and circumferential wavenumber n. Using the wavenumber of an equivalent beam with similar boundary conditions as the cylindrical shell, the axial wavenumber k _m is determined approximately. Then, the relation between the natural frequency with the axial wavenumber and circumferential wavenumber is established, and the traveling wave frequencies corresponding to a certain rotating speed are calculated numerically. To validate the results, comparisons are carried out with some available results of previous studies, and good agreements are observed. Finally, the relative errors induced by the approximation using the axial wavenumber of an equivalent beam are evaluated with respect to different circumferential wavenumbers, length-to-radius ratios as well as thickness-to-radius ratios, and the conditions under which the analysis presented in this paper will be accurate are discussed.     

A Theory of General Solutions of Plane Problems in Two-Dimensional Octagonal Quasicrystals

A theory of general solutions of plane problems is developed for the coupled equations in plane elasticity of two-dimensional octagonal quasicrystals. In virtue of the operator method, the general solutions of the antiplane and inplane problems are given constructively with two displacement functions. The introduced displacement functions have to satisfy higher order partial differential equations, and therefore it is difficult to obtain rigorous analytic solutions directly and is not applicable in most cases. In this case, a decomposition and superposition procedure is employed to replace the higher order displacement functions with some lower order displacement functions, and accordingly the general solutions are further simplified in terms of these functions. In consideration of different cases of characteristic roots, the general solution of the antiplane problem involves two cases, and the general solution of the inplane problem takes three cases, but all are in simple forms that are convenient to be applied. Furthermore, it is noted that the general solutions obtained here are complete in x ₃-convex domains.      

The paraxial approximation for a dense electron beam

Approximate equations are derived for a narrow (paraxial) electron beam with a three-dimensional axis; these equations generalize the familiar equations [1] to the case in which the field of the charge in the beam is important. A class of beams is distinguished for which the problem reduces to ordinary differential equations. The paraxial approximation for a narrow beam of electron trajectories in a given field is well known for the case in which the characteristic dimension L_* of the irregularities is much greater than a_*, the characteristic width of the beam. The field of a three-dimensional beam can be included in the equations [1] if the beam density ρ is nonuniform over lengths L_*. The self-field of the beam was not actually taken into account in[1]. In this paper we attempt to remedy this deficiency, with partial success for an axisymmetric beam with a rectilinear axis [2]. Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, Vol. 9, No. 5, pp. 3–10, 1968.     

Some exact solutions of the equations of a stationary monoenergetic beam of charged particles

We consider the class of invariant solutions which can describe only vortex flows (curl P 0, P is the generalized momentum) and show that they contain solutions corresponding to flows from a plane or cylindrical emitter with a linear voltage drop across it (direct heating) in the temperature-limited regime^*. The solution is obtained in analytic form for emission from a plane in a uniform magnetic field perpendicular to the flow plane. It also (for=0) defines a plane magnetron in the T-regime. The solution of the problem for a cylindrical emitter reduces to considering equations describing a cylindrical diode or magnetron in the T-regime, where the shape of the collector is given by the potential distribution curve for these cases. We can extend the results to a relativistic beam if restrictions are imposed on its relative dimensions which permit us to ignore the magnetic self-field. Brillouin type flows (including irrotational ones) are studied in which particles move without intersecting the equipotential surfaces along three-dimensional spirals on the surface of cones. An analytic solution is given for relativistic Brillouin flow in a conical diode when strict allowance is mede for the magnetic self-field.     

On a general treatment for coupled thermal stress problems in a cylindrical coordinate system

Summary In this paper, we study the three-dimensional coupled thermoelastic problem governed by the cylindrical coordinate system under the consideration of the thermo-mechanical coupling effect. The basic relations for the three-dimensional displacement field and the corresponding stress components are derived from the field equations of motion. These expressions enable us to determine both the temperature and the stress distributions simultaneously. As an illustration, numerical calculations are carried out for an axisymmetrical coupled thermoelastic problem. From the numerical results it is shown that the thermomechanical coupling term appearing in the corrected heat conduction equation gives a fairly large difference between the coupled theory and the uncoupled one.

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Zur allgemeinen Behandlung gekoppelter Wärmespannungsprobleme in Zylinderkoordinaten

Übersicht Es wird das dreidimensionale Wärmespannungsproblem unter Berücksichtigung der thermomechanischen Kopplung untersucht. Ausgehend von den Bewegungsgleichungen werden Temperatur und Spannungen angegeben. Zur Illustration der Methode wird ein axialsymmetrisches Problem numerisch behandelt und der Einfluß der thermo-meehanischen Kopplung auf die Wärmeleitgleichung gezeigt.

     

Closed-form solution of a shear deformable,extensional ring in contact between two rigid surfaces

Contact of a circular ring with a flat, rigid ground is considered using curved beam theory and analytical methods. Applications include tires, springs, and stiffeners, among others. The governing differential equations are derived using the principle of virtual work and the formulation includes deformations due to bending, transverse shear and circumferential extension. The three associated stiffness quantities, EI, GA and EA, respectively, remain as independent parameters in the differential equations. This allows the special cases such as an inextensible Timoshenko beam (EI and GA) or an extensible Euler beam (EI and EA) to be obtained directly by the appropriate limits. The effect of these three stiffness parameters on the contact pressure solution is studied, which shows how those fundamental parameters can be selected for the purpose of the application. Although the formulation is for small displacement theory, both radial and circumferential distributed loads are considered, which allows the pressure in the deformed state to be vertical rather than radial, which is shown to be important. Closed form expressions for all force and displacement quantities are obtained in terms of the angular location of the edge of contact, which must be determined numerically. Extensibility complicates the analytical expressions within the contact region, and a series solution is proposed in this case. A two-term asymptotic expression for the stiffness of the ring is determined analytically. Finally, all solutions are validated using the commercial finite element software ABAQUS, with attention to non-linear behavior and the range of validity of these solutions.     

Saint-Venant end effects for plane deformations of linear piezoelectric solids

The recent developments in smart structures technology have stimulated renewed interest in the fundamental theory and applications of linear piezoelectricity. In this paper, we investigate the decay of Saint-Venant end effects for plane deformations of a piezoelectric semi-infinite strip. First of all, we develop the theory of plane deformations for a general anisotropic linear piezoelectric solid. Just as in the mechanical case, not all linear homogeneous anisotropic piezoelectric cylindrical solids will sustain a non-trivial state of plane deformation. The governing system of four second-order partial differential equations for the two in-plane displacements and electric potential are overdetermined in general. Sufficient conditions on the elastic and piezoelectric constants are established that do allow for a state of plane deformation. The resulting traction boundary-value problem with prescribed surface charge is an oblique derivative boundary-value problem for a coupled elliptic system of three second-order partial differential equations. The special case of a piezoelectric material transversely isotropic about the poling axis is then considered. Thus the results are valid for the hexagonal crystal class 6mm. The geometry is then specialized to be a two-dimensional semi-infinite strip and the poling axis is the axis transverse to the longitudinal direction. We consider such a strip with sides traction-free, subject to zero surface charge and self-equilibrated conditions at the end and with tractions and surface charge assumed to decay to zero as the axial variable tends to infinity. A formulation of the problem in terms of an Airy-type stress function and an induction function is adopted. The governing partial differential equations are a coupled system of a fourth and third-order equation for these two functions. On seeking solutions that exponentially decay in the axial direction one obtains an eigenvalue problem for a coupled system of fourth and second-order ordinary differential equations. This problem is the piezoelectric analog of the well-known eigenvalue problem arising in the case of an anisotropic elastic strip. It is shown that the problem can be uncoupled to an eigenvalue problem for a single sixth-order ordinary differential equation with complex eigenvalues characterized as roots of transcendental equations governing symmetric and anti-symmetric deformations and electric fields. Assuming completeness of the eigenfunctions, the rate of decay of end effects is then given by the real part of the eigenvalue with smallest positive real part. Numerical results are given for PZT-5H, PZT-5, PZT-4 and Ceramic-B. It is shown that end effects for plane deformations of these piezoceramics penetrate further into the strip than their counterparts for purely elastic isotropic materials.     

A point heat source on the apex of a transversely isotropic piezothermoelastic cone

We use the compact harmonic general solutions of transversely isotropic piezothermoelastic materials to construct the three-dimensional solutions of a steady point heat source on the apex of a transversely isotropic piezothermoelastic cone by four newly introduced harmonic functions. All components of coupled field are expressed in terms of elementary functions and are convenient to use. When the apex angle 2α equals to π, the solutions reduce to those of the semi-infinite body problem, which is called Green's function. Numerical results are given graphically by contours.     

Study on laminar film condensation of saturated steam on a vertical flat plate for consideration of various physical factors including variable thermophysical properties

The extended theory of the steady state laminar film condensation process of pure saturated vapour at atmospheric pressure on an isothermal vertical flat plate is established. Its equations provide a complete account of the physical process for consideration of various physical factors including variable thermophysical properties, except for surface tension at the liquid-vapour film interface. First, similarity considerations are proposed to transform the governing system of partial differential equations and its boundary conditions into the corresponding dimensionless system. Then, the dimensionless new system is computed numerically in two steps: First neglecting shear force at the interface, so that the initial values of the boundary conditionsW _{xl, s} andW _{yl, s} are obtained. Then, the calculations of a problem of the three-point boundary-value for coupling the equations of liquid film with those of vapour film are carried out. Furthermore, the correlations for heat transfer coefficient and mass flow rate are proposed by analysis of heat and mass transfer and it is found that the heat transfer coefficient is function of dimensionless temperature gradient $\dot L$ , and that the condensate mass flow rate is function of the mass flow rate parameter (η _lδ W _{xl, s} ? 4W _{yl, s} )of liquid. In addition, the corresponding heat and mass transfer correlations expressed by subcooled temperature Δt are developed. According to Nusselt's theory four different assumptions are set up for an investigation of the effects of the film condensation of saturated vapour, so that the validity of Nusselt's theory can be further clarified. Quantitative comparisons from the results of the heat transfer coefficient and mass flow rate of the condensate indicate that the effect of variable thermophysical properties on the heat and mass transfer is appreciable. The effect of thermal convection in the condensate film is obviously larger than those of shear force at liquid-vapour interface, and the effect of the inertia in the condensate film is very small. Finally, it is also shown that Nusselt's theory, in using Drew reference temperature, will decrease the heat transfer coefficient by at most 5.11%, and will increase the mass flow rate of the condensate by at most 2.45%, provided that the effect of the surface tension is not taken into account.     

Stability of curvilinear Euler-Bernoulli beams in temperature fields

In this work, stability of thin flexible Bernoulli-Euler beams is investigated taking into account the geometric non-linearity as well as a type and intensity of the temperature field. The applied temperature field T(x,z) is yielded by a solution to the 2D Laplace equation solved for five kinds of thermal boundary conditions, and there are no restrictions put on the temperature distribution along the beam thickness. Action of the temperature field on the beam dynamics is studied with the help of the Duhamel theory, whereas the motion of the beam subjected to the thermal load is yielded employing the variational principles.The heat transfer (Laplace equation) is solved with the use of the finite difference method (FDM) of the third-order accuracy, while the integrals along the beam thickness defining the thermal stress and moments are computed using Simpson's method. Partial differential equations governing the beam motion are reduced to the Cauchy problem by means of application of FDM of the second-order accuracy. The obtained ordinary differential equations are solved with the use of the fourth-order Runge-Kutta method.The problem of numerical results convergence versus a number of beam partitions is investigated. A static solution for a flexible Bernoulli-Euler beam is obtained using the dynamic approach based on employment of the relaxation/set-up method.Novel stability loss phenomena of a beam under the thermal field are reported for different beam geometric parameters, boundary conditions, and the temperature intensity. In particular, it has been shown that stability of the flexible beam during heating the beam surface essentially depends on the beam thickness.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

Improved torsional analysis of laminated box beams

The improved torsional analysis of the laminated box beams with single- and double-celled sections subjected to a torsional moment is performed by introducing 14 displacement parameters. For this, a thin-walled laminated box beam theory considering the effects of shear and elastic couplings is presented. The governing equations and the force-displacement relations are derived from the variation of the strain energy. The system of linear algebraic equations with non-symmetric matrix is constructed by introducing the displacement parameters and by transforming the higher order simultaneous differential equations into first order ones. This numerical technique determines eigenmodes corresponding to 12 zero and 2 non-zero eigenvalues and derives displacement functions for displacement parameters based on the undetermined parameter method. Finally, the element stiffness matrix is determined using the member force-displacement relations. The theory developed by this study is validated by comparing several torsional responses from the present approach with those from the finite element beam model using the Lagrangian interpolation polynomials and three-dimensional analysis results using the shell elements of ABAQUS for coupled laminated beams with single- and double-celled sections.     

A Hamiltonian State Space Approach for 3D Analysis of Circular Cantilevers

A 3D exact analysis of extension, torsion and bending of a cantilever of a circular cross section is studied with emphasis on the fixed-end effect. Through Hamiltonian variational formulation, the basic equations of elasticity in cylindrical coordinates and the boundary conditions of the problem are formulated into the state space setting in which the state vector comprises the displacement vector and the conjugate stress vector as the dual variables. Upon delineating the Hamiltonian characteristics of the system, 3D solutions for transversely isotropic circular cantilevers subjected to an axial force, a torque, terminal couples and transverse forces are determined, thereby, the fixed-end effects and applicability of the solutions of generalized plane strains and the elementary theory of bending of beams are evaluated.     

Comprehensive form of solution for Lamb＇s dynamic problem expressed by Green＇s functions＊

By the analysis for the vectors of a wave field in the cylindrical coordinate and Sommerfeld＇s identity as well as Green＇s functions of Stokes＇ solution pertaining the conventional elastic dynamic equation, the results of Green＇s function in an infinite space of an axisymmetric coordinate are shown in this paper. After employing a supplementary influence field and the boundary conditions in the free surface of a senti-space, the authors obtain the solutions of Green＇s function for Lamb＇s dynamic problem. Besides, the vertical displacement uzz and the radial displacement urz can match Lamb＇s previous results, and the solutions of the linear expansion source u~r and the linear torsional source uee are also given in the paper. The authors reveal that Green＇s function of Stokes＇ solution in the semi-space is a comprehensive form of solution expressing the dynamic Lamb＇s problem for various situations. It may benefit the investigation of deepening and development of Lamb＇s problems and solution for pertinent dynamic problems conveniently.