Some remarks concerning Gao–Strang's complementary gap function

Dual minimum principles for displacements and stresses are well established for linear variational problems and also for nonlinear (and monotone) constitutive laws. This paper studies the problem of geometric nonlinearity. By introducing a gap function , we recover complementary variational principles in the equilibrium problems of mathematical physics. When the gap function is nonnegative those become minimum principles. The theory is based on convex analysis, and the applications made here are to nonlinear mechanics.

A minimum principle of complementary energy is established for cable networks involving only the stress components as variables with geometrical nonlinearities and nonlinear elastic materials. The minimization problem of total potential energy is reformulated as a variational problem with a convex objective functional and an infinite number of second-order cone constraints; its Fenchel dual problem is shown to coincide with the minimization problem of the complementary energy. It is of interest to note that the obtained complementary energy attains always its minimum value at the equilibrium state irrespective of the stability of the cable networks, contrary to the fact that only stationary principles have been presented for elastic trusses and continua, even in the case of a stable equilibrium state.

Summary This work addresses the development of some alternative hybrid and mixed variational formulations for the geometrically-exact three-dimensional first-order-shear shell boundary value problem [1,2]. In the framework of the complementaryenergy-based formulations, a Legendre transformation is used to introduce the complementary energy density in the variational statements as a function of the cross-sectional resultants only. The corresponding variational principles are shown to feature stationarity within the framework of the boundary-value-problem. The main features of the principles are highlighted, giving special attention to their relationships from both theoretical and numerical point of view. Variational principles constitute the core of the development of numerical methods in Solid Mechanics. The utility of such principles is two-fold: first, they provide a very convenient method for the derivation of the governing equations and natural boundary conditions of the boundary value problem and, second, they provide the mathematical foundation required to produce consistent numerical approximations. In this second role, the variational methods have been most useful in Computational Solid Mechanics [3]. Many different variational principles can be constructed depending on the equations enforced in the weak form. Perhaps the most remarkable variational principle is the Principle of Stationary Total Potential Energy, which states that, among all kinematically admissible displacement fields, those that satisfy the equilibrium conditions in the domain and at the boundary, lead to a stationary value of the total potential energy functional. In this paper, after the presentation of many mixed variational principles for the nonlinear analysis of shells, we derive a hybrid form of the Two-field Principle of Total Complementary Potential Energy, which is used on a totally consistent derivation of equilibrated co-rotational finite elements [4].

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.

Trusses composed of bars with specially formulated bilinear force-deformation characteristics proposed by Logo and Taylor are treated. Mixed extremum principles are derived, based on potential and complementary potential energy, to form the basis for a load history analysis of trusses. Through the choice of appropriate parameters of force-deformation relationships, the classical principles of linearly elastic and perfectly plastic structures can be obtained and the material and/or geometric nonlinearity, postbuckling behavior, and limited deformation capacity of the bars can be taken into consideration. The principles are stated in the form of a constrained, nonsmooth, nonlinear mathematical programming problem and provide a convenient tool for numerical analysis. Their application is illustrated by two numerical examples.

In this paper, static bending analysis of functionally graded plates with piezoelectric layers has been carried out considering geometrical nonlinearity in different sets of mechanical and electrical loadings. Only the geometrical nonlinearity has been taken into account. The governing equations are obtained using potential energy and Hamilton’s principle. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect by using higher order elements. The present finite element used displacement and electric potential as nodal degrees of freedom. Results are presented for two constituent FGM plate under different mechanical boundary conditions. Numerical results for FGM plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response of the plate are also studied.

The paper deals with a cohesive crack model in which the cohesive (crack-bridging) stress is a specified decreasing function of the crack-opening displacement. Under the assumption that no part of the crack undergoes unloading, the complementary energy and potential energy of an elastic structure which has a cohesive crack and is loaded by a flexible elastic frame is formulated using continuous influence functions representing compliances or stiffnesses relating various points along the crack. By variational analysis, in which the derivatives of the compliance or stiffness functions with respect to the crack length are related to the crack-tip stress intensity factors due to various unit loads, it is shown that the minimizing conditions reduce to the usual compatibility or equilibrium equations for the cohesive cracks. The variational equations obtained can be used as a basis for approximate solutions. Furthermore, the conditions of stability loss of a structure with a growing cohesive crack are obtained from the condition of vanishing of the second variation of the complementary energy or the potential energy. They have the form of a homogeneous Fredholm integral equation for the derivatives of the cohesive stresses or crack opening displacements with respect to the crack length. Loadings with displacement control, load control, or through a flexible loading frame are considered. Extension to the analysis of size effect on the maximum load or maximum displacement are left to a subsequent companion paper.

This book covers the essential elements of engineering mechanics for mechanical elements in tension-compression, torsion, and bending. Its approach emphasizes a fundamental bottom-up approach to the subject for a concise and uncluttered presentation. Each problem class is firmly rooted in its underlying kinematic assumption, and the concepts of equilibrium and material response are carefully separated. The technical theory is connected to the three-dimensional theory with a careful emphasis on kinematic assumptions and the selected use of material models for plastic response, as well as composite or inhomogeneous material distributions. Buckling phenomena and the principles of stationary potential energy and stationary complementary potential energy are covered in detail for both exact as well as approximate solutions. The principles of virtual work – virtual displacements and virtual forces – are treated, and their relation to potential energy methods as well as the basic elements of kinematics, equilibrium, and material constitutive response are fully discussed.

Nowadays the phase controlled switching is a preferred method for capacitive switching. Thus the prestrike characteristic is an important issue for the control strategy. The objective of this paper is to understand the prestrike characteristics of vacuum interrupters (VIs) in the capacitive making operations of 40.5 kV vacuum circuit breakers (VCBs). In the experiment, three VIs with CuCr50 (50 wt% of Cr) contact material and three VIs with CuCr40 (40 wt% of Cr) contact material with contact diameter of 50 millimeters were prepared. The CuCr50 and CuCr40 contact materials are made by using infiltration and arc-melted technology, respectively. An L-C oscillating circuit generated an inrush current with a frequency of 207 Hz and with a peak of 6.48 kA, while the applied voltage was 46.3 kV which is equivalent to the peak voltage for a 35 kV three-phase network whose capacitive voltage factor is 1.4. The experimental results show that the complementary cumulative distribution function (CCDF) of the prestrike gaps dpre follows a 3-parameter Weibull distribution. The prestrike gap below which prestrike would always occur is similar for all six test VIs which is 2.4–3.1 mm. However, the average values of d\0, ?50 and ?90 for infiltration CuCr50 contact material are 57.5%, 45.3% and 28.9% higher than the ones in arc-melted CuCr40 contact material, respectively. During the inrush current flowing, at each zero-crossing of the inrush current the prestrike arc tries to extinguish with the occurrence of three types of contact gap breakdowns. Furthermore, if the contact gap breakdowns are withstood at the inrush current zero-crossing, inrush current interruption phenomenon happens, which proves a common phenomenon in capacitive making operation. The inrush current interruption phenomenon is mainly determined by the prestrike distance and not by the material quality. The experimental results imply that the field emission dominated breakdown and the micro-particle induced breakdown both take effect in the prestriking and contact gap breakdown process.

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

A general theory of piecewise linear plasticity based on maximum shear

Micromechanically based stochastic finite elements: length scales and anisotropy

Atmospheric water vapour content is a key variable that controls the development of deep convective storms and rainfall extremes over the central Andes. Direct measurements of water vapour are challenging; however, recent developments in microwave processing allow the use of phase delays from L-band radar to measure the water vapour content throughout the atmosphere: Global Navigation Satellite System (GNSS)-based integrated water vapour (IWV) monitoring shows promising results to measure vertically integrated water vapour at high temporal resolutions. Previous works also identified convective available potential energy (CAPE) as a key climatic variable for the formation of deep convective storms and rainfall in the central Andes. Our analysis relies on GNSS data from the Argentine Continuous Satellite Monitoring Network, Red Argentina de Monitoreo Satelital Continuo (RAMSAC) network from 1999 to 2013. CAPE is derived from version 2.0 of the ECMWF’s (European Centre for Medium-Range Weather Forecasts) Re-Analysis (ERA-interim) and rainfall from the TRMM (Tropical Rainfall Measuring Mission) product. In this study, we first analyse the rainfall characteristics of two GNSS-IWV stations by comparing their complementary cumulative distribution function (CCDF). Second, we separately derive the relation between rainfall vs. CAPE and GNSS-IWV. Based on our distribution fitting analysis, we observe an exponential relation of rainfall to GNSS-IWV. In contrast, we report a power-law relationship between the daily mean value of rainfall and CAPE at the GNSS-IWV station locations in the eastern central Andes that is close to the theoretical relationship based on parcel theory. Third, we generate a joint regression model through a multivariable regression analysis using CAPE and GNSS-IWV to explain the contribution of both variables in the presence of each other to extreme rainfall during the austral summer season. We found that rainfall can be characterised with a higher statistical significance for higher rainfall quantiles, e.g., the 0.9 quantile based on goodness-of-fit criterion for quantile regression. We observed different contributions of CAPE and GNSS-IWV to rainfall for each station for the 0.9 quantile. Fourth, we identify the temporal relation between extreme rainfall (the 90th, 95th, and 99th percentiles) and both GNSS-IWV and CAPE at 6 h time steps. We observed an increase before the rainfall event and at the time of peak rainfall—both for GNSS-integrated water vapour and CAPE. We show higher values of CAPE and GNSS-IWV for higher rainfall percentiles (99th and 95th percentiles) compared to the 90th percentile at a 6-h temporal scale. Based on our correlation analyses and the dynamics of the time series, we show that both GNSS-IWV and CAPE had comparable magnitudes, and we argue to consider both climatic variables when investigating their effect on rainfall extremes.

The classical linear stability equations for columns take the form of equilibrium equations in terms of displacements. From a variational point of view these equations emerge as conditions of extremum from an energy or virtual work functionals. In principle it should be possible to arrive at the governing equations for stability of columns from a complementary energy or complementary virtual work functional involving force quantities alone. This is rarely done. Both the energy and the complementary energy approaches depend upon the relevant constitutive equations. In this study a formulation is employed wherein focus is maintained on the constitutive equations. It is shown that under certain admissibility conditions a least squares functional, forcing the satisfaction of the constitutive equations, yields the energy and the so-called pure complementary energy functionals as subsets.

This chapter introduces the calculus of variations and associated mathematical operators on functionals, which are used to formulate boundary value problems in structural mechanics. Weak forms of equilibrium are derived in terms of the principle of virtual work and the theorem of minimum potential energy. These are used to establish the governing field equations (Euler–Lagrange differential equations) and associated geometric, natural, or mixed boundary conditions. Weak forms of kinematic compatibility are derived in terms of the principle of complementary virtual Work and the theorem of minimum complementary potential energy. Examples are provided for each.