Principles of minimum potential energy and complementary energy

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

Involutory transformations and variational principles with multi-varoables in thin plate bending problems

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.

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.

Nonlinear electromechanical bending of bi-modular piezoelectric laminated beams

ABSTRACTA variable kinematic 1D finite element (FE) method is presented for 3D thermoelastic analysis of rotating disks with variable thickness. The principle of minimum potential energy is used to derive general governing equations of the disks subjected to body forces, surface forces, concentrated forces, and thermal loads. To solve the equations, the 1D Carrera unified formulation (CUF), which enables to go beyond the kinematic assumptions of classical beam theories, is employed. Based on the 1D CUF, the disk is considered as a beam, which can be discretized into a finite number of 1D elements along its axis. The displacement field over the beam’s cross section is approximated by Lagrange expansions. This methodology leads to an FE formulation that is invariant with respect to the order of expansions used over the cross sections, and thus the 3D problem reduces to a 1D problem. The effect of the cross section discretization on displacement and stress fields is investigated. Results obtained from this method are in good agreement with the reference analytical and finite difference solutions. The proposed innovative method can be very effective in the thermoelastic analysis of rotating disks.

A maximum entropy (MaxEnt) based probabilistic approach is developed to model mechanical systems characterized by symmetric positive-definite matrices bounded from below and above. These matrices are typically encountered in the constitutive modeling of heterogeneous materials, where the bounds are deduced by employing the principles of minimum complementary energy and minimum potential energy. Current random matrix approaches in the context of computational stochastic mechanics are adapted only to the Wishart or matrix-variate Gamma probability model supported over the entire space of symmetric positive-definite matrices and therefore unable to exploit additional information available through the lower and upper bounds when appropriate. Specifically, for a given material, the constitutive matrix is construed as a random matrix. A probability measure that reflects the constraints consistent with the energy-based bounds is constructed, and an associated sampling scheme is described to synthesize realizations of this random matrix. An additional constraint of the mean matrix is also considered, and an appropriate probability model and sampling scheme are developed and illustrated numerically for this case. The present work also develops a new algorithm to compute the parameters of the Wishart or matrix-variate Gamma probability model. This new algorithmic scheme is more consistent with the MaxEnt proposition.

A note on the contributions of Cotterill, Castigliano, Crotti and Engesser to an energy principle of structures

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.

Damaged mosaic laminate model of woven fabric composites with transverse yarn cracking and interface debonding

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

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.

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.

Stocky thin- or thick-walled beams: Theory and analysis