Conforming approximations for Kirchhoff plates and shells

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.

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

Chapter 8 - Basic Equations and Solution Procedure

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

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.

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

A distressed material can be considered as a two-phase composite material with the distressed areas (volumes) as one phase and the original material as the other phase. A parallel diffusion model was developed in which the damaged phase and the original phase are arranged parallel to the direction of diffusion. The parallel model is associated with the isogradient principle in that the concentration gradients in the damaged phase and in the original phase are the same. A serial diffusion model was developed in which the two phases are arranged perpendicular to the direction of diffusion. The serial model is associated with the isoflux principle in that the fluxes in the two phases are the same. The two-phase composite models were further extended into general multiphase composite models to evaluate the effect of multilevel damage on the effective diffusivity of distressed materials. A principle of minimum complementary “chemical-flux energy” and a principle of minimum “potential chemical energy” were established similar to the commonly used principles of minimum complementary energy and minimum potential energy. Based on the two new minimum energy principles, it was shown that the parallel model is the lower bound and the serial model is the upper bound for effective diffusivity of distressed materials. Diffusivity of distressed concrete was used as an example. The experimental data agreed quite well with predictions of the theoretical models.

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.

Based on the magnetic dipole interaction theory, a theoretical model was proposed to describe the magneto-mechanical coupling behaviors of magnetorheological elastomers with the principle of minimum potential energy. In this model, the fully coupled interaction among all particles and chains was considered according to the micro-structure of magnetorheological elastomers. The energy equations of magnetic interaction and the elastic potential energy equations based on the Mooney-Rivlin model were derived respectively. Then a theoretical model was developed to describe the stress-strain relationship of magnetorheological elastomers under uniaxial load. This model agrees well with existing experimental data and can be used to explain the micro-mechanism of magneto-induced stress. The results show that the mechanism of magneto-induced stress is closely related to the inner micro-structure, and the nonlinear property of magneto-induced stress mainly depends on the interaction among both particles and chains.

Usually, the governing equations of the numerical manifold method (NMM) are derived from the minimum potential energy principle. For many applied problems it is difficult to derive in general outset the functional forms of the governing equations. This obviously strongly restricts the implementation of the minimum potential energy principle or other variational principles in NMM. In fact, the governing equations of NMM can be derived from a more general method of weighted residuals. By choosing suitable weight functions, the derivation of the governing equations of the NMM from the weighted residual method leads to the same result as that derived from the minimum potential energy principle. This is demonstrated in the paper by deriving the governing equations of the NMM for linear elasticity problems, and also for Laplace’s equation for which the governing equations of the NMM cannot be derived from the minimum potential energy principle. The performance of the method is illustrated by three numerical examples.