磁流变弹性体的力-磁耦合模型

With the displacement field taken as the only fundamental unknown field in a mixed-boundary-value problem for linear elastostatics, the principle of minimum potential energy asserts that a potential energy functional, which is defined as the difference between the free energy of the body and the work done by the prescribed surface tractions and the body forces --- assumes a smaller value for the actual solution of the mixed problem than for any other kinematically admissible displacement field which satisfies the displacement boundary condition. This principle provides a weak or variational method for solving mixed boundary-value-problems of elastostatics. In particular, instead of solving the governing Navier form of the partial differential equations of equilibrium, one can search for a displacement field such that the first variation of the potential energy functional vanishes. A similar principle of minimum complementary energy, which is phrased in terms of statically admissible stress fields which satisfy the equilibrium equation and the traction boundary condition, is also discussed. The principles of minimum potential energy and minimum complementary energy can also be applied to derive specialized principles which are particularly well-suited to solving structural problems; in this context the celebrated theorems of Castigliano are discussed.

Chapter 8 - Basic Equations and Solution Procedure

The solution of elastostatic problems and the principles of minimum potential energy, minimum complementary work, under fuzzy boundary conditions

The aim of this research is to analyze buckling and bending behavior of a sandwich Reddy beam with porous core and composite face sheets reinforced by boron nitride nanotubes (BNNTs) and shape memory alloy (SMA) wires resting on Vlasov\'s foundation. To this end, first, displacement field\'s equations are written based on the higher-order shear deformation theory (HSDT). And also, to model the SMA wire properties, constitutive equation of Brinson is used. Then, by utilizing the principle of minimum potential energy, the governing equations are derived and also, Navier\'s analytical solution is applied to solve the governing equations of the sandwich beam. The effect of some important parameters such as SMA temperature, the volume fraction of SMA, the coefficient of porosity, different patterns of BNNTs and porous distributions on the behavior of buckling and bending of the sandwich beam are investigated. The obtained results show that when SMA wires are in martensite phase, the maximum deflection of the sandwich beam decreases and the critical buckling load increases significantly. Furthermore, the porosity coefficient plays an important role in the maximum deflection and the critical buckling load. It is concluded that increasing porosity coefficient, regardless of porous distribution, leads to an increase in the critical buckling load and a decrease in the maximum deflection of the sandwich beam.

Static bending of a twisted beam subjected to distributed transverse loads is established by using the Timoshenko beam theory and applying the principle of minimum potential energy. The closed-form solution for the static deflection of cantilever twisted beam is then derived. The effects of the twist angle, thickness ratio and shear deformation on the static bending characteristics of twist beams are investigated and discussed.

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

Solving contact forces with the competition between potential energy and entropy in elastic mechanics

Finite element interpolations of arbitrary polynomial degree which are exactly and minimally C1 continuous across interelement boundaries are studied. A perspective view of the techniques proposed by other authors is presented and some new possibilities are considered. The problem is discussed with reference to the plate and shell finite element approximations, based on the principle of minimum potential energy. The approach also permits identification of sufficient conditions for guaranteeing full rank of stiffness matrices based on modified principles of potential energy. The results of this analysis can be used to investigate the linear dependencies among nodal variables which also occur in the formulation of other fourth‐order problems.

Mechanical adjustment and prediction of metal-composite reconfigurable tubes

Structures are defined usually with respect to tolerable/permissible limiting conditions of displacement, deformation and stresses etc, which enables continuous load application and static equilibrium, otherwise the equilibrium becomes dynamical resulting in unstable structural system. This study aims to evaluate, effect of vibrations and damping on structural stability, and identified that vibration is mechanical phenomenon involving action of impact forces that produces oscillatory motion on the structure and characterized with oscillations, displacement, and frequency (f = w/2Л) about a static mean position of rest (ie, F = ma = 0). The research study entails review of literatures and work on structural vibration, stability and damping mechanism to restraint effect within permissible limits. Dampers are commonly used to constraint structures to infinitesimal displacement during load application, also the paper identify that vibratory systems are means of storing potential energy (mass), kinetic energy (spring) and means by which energy is gradually dissipated through oscillations (ie, F = ma). The motion can be optimized using principle of minimum potential energy and virtual work” expressing workdone on a system undergoing virtual displacement (W = Fδx = 0, because δx = 0). Vibration damping is an influence upon a system that prevents or reduces its oscillation, and is implemented by processes that dissipate energy stored in oscillations. The damping ratio describe the system parameters which varies from undamped (ξ = 0), underdamped (ξ < 1), critically damped (ξ =1) and overdamped (ξ > 1). Static equilibrium requires damping of structures between critically damped and overdamped to ensure minimal oscillatory amplitude as expected for stability and functional performance. Similarly, structural evaluation is implemented using virtual work method,analytically defined as total work done, W = F δs = 0 (ie, δs → 0 or negligible). In conclusion, the paper identify that dynamical tendency is characterized with instability while structural performance is identified with infinitesimal or limit state deformation and displacement; hence corresponding vibratory displacement and oscillations must be minimal using appropriate damping mechanism to reduce the cumulative effect on structural system and to provide stability and safety of structural systems.

Unified static equilibrium modeling and analysis of elastic rods with large deformations for complex constraints

Semi-numerical analysis of rectangular plates in bending

On a solution strategy for energy-based mesh optimization in finite hyperelastostatics

Both total potential energy and buckling equation of two-span continuous beam with lateral elastic brace under uniform load are deduced, based on energy variation method and the principle of minimum potential energy. Buckling of H-beams is simulated by ANSYS software, then compared to theoretical value, validated its rationality. High precision buckling moment formula is regressed using 1stOpt which is a famous mathematical optimization analysis software in China. The relationship between lateral brace stiffness and buckling moment is obtained. Results: with lateral brace stiffness increases, critical bending moment of beam increases within up-limit, e.g. when lateral brace stiffness increases to certain extent, buckling moment no longer increases.

Variable angle tow steering technology is capable of manufacturing complex aviation parts with a trajectory of intricate curvature planned based on stress or profile characteristics, which greatly improves the forming efficiency, design flexibility and mechanical properties of composite structures. In view of the forming defects such as buckling and wrinkles caused by the lateral bending of fiber prepreg tow, a theoretical buckling model based on the Rayleigh Ritz method, the principle of minimum potential energy and the viscoelastic foundation is established, in which the adhesion coefficient is characterized by the degree of intimate contact to introduce process parameters. On the basis of the contact mechanics analysis, the distribution of the compaction pressure and bending stress is studied to improve the theoretical model, and the critical buckling load and the minimum radius of the tow under the normal and tangential contact conditions are determined precisely. Finally, the finite element models of compaction and variable angle steering placement are proposed, and the theoretical model and simulation model are verified by corresponding trials. It is demonstrated that defects can be effectively suppressed through optimizing process parameters.