Static inconsistencies in certain axiomatic higher-order shear deformation theories for beams, plates and shells

AbstractStatic inconsistencies that arise when modelling the flexural behaviour of beams, plates and shells with clamped boundary conditions using a certain class of axiomatic, higher-order shear deformation theory are discussed. The inconsistencies pertain to displacement-based theories that enforce conditions of vanishing shear strain at the top and bottom surfaces a priori. First it is shown that the essential boundary condition of vanishing Kirchhoff rotation perpendicular to an edge (w,x=0 or w,y=0) is physically inaccurate, as the rotation at a clamped edge may in fact be non-zero due to the presence of transverse shear rotation. As a result, the shear force derived from constitutive equations erroneously vanishes at a clamped edge. In effect, this boundary condition overconstrains the structure leading to underpredictions in transverse bending deflection and overpredictions of axial stresses compared to high-fidelity 3D finite element solutions for thick and highly orthotropic plates. Generalised higher-order theories written in the form of a power series, as in Carrera’s Unified Formulation, do not produce this inconsistency. It is shown that the condition of vanishing shear tractions at the top and bottom surfaces need not be applied a priori, as the transverse shear strains inherently vanish if the order of the theory is sufficient to capture all higher-order effects. Finally, the transverse deflection of the generalised higher-order theories is expanded in a power series of a non-dimensional parameter and used to derive a material and geometry dependent shear correction factor that provides more accurate solutions of bending deflection than the classical value of 5/6

Shape Control for Experimental Continuation

An experimental method has been developed to locate unstable equilibria of nonlinear structures quasi-statically. The technique involves loading a structure by application of either a force or a displacement at a main actuation point, while simultaneously controlling the overall shape using additional probe points. The method is applied to a shallow arch, and unstable segments of its equilibrium path are identified experimentally for the first time. Shape control is a fundamental building block for the experimental---as opposed to numerical---continuation of nonlinear structures, which will significantly expand our ability to measure their mechanical response.Comment: Updated Figure 6 experimental results with correct calibration factor for linear transducer. Updated Figure 6 finite element results with correct load multiplier for half-model. Updated paper text to reflect these changes. 5 pages, 6 figure

Virtual Testing of Experimental Continuation

We present a critical advance in experimental testing of nonlinear structures. Traditional quasi-static experimental methods control the displacement or force at one or more load-introduction points on a structure. This approach is unable to traverse limit points in the control parameter, as the immediate equilibrium beyond these points is statically unstable, causing the structure to snap to another equilibrium. As a result, unstable equilibria---observed numerically---are yet to be verified experimentally. Based on previous experimental work, and a virtual testing environment developed herein, we propose a new experimental continuation method that can path-follow along unstable equilibria and traverse limit points. To support these developments, we provide insightful analogies between a fundamental building block of our technique---shape control---and analysis concepts such as the principle of virtual work and Galerkin's method. The proposed testing method will enable the validation of an emerging class of nonlinear structures that exploit instabilities for novel functionality

Crystal structures and freezing of dipolar fluids

We investigate the crystal structure of classical systems of spherical particles with an embedded point dipole at T=0. The ferroelectric ground state energy is calculated using generalizations of the Ewald summation technique. Due to the reduced symmetry compared to the nonpolar case the crystals are never strictly cubic. For the Stockmayer (i.e., Lennard-Jones plus dipolar) interaction three phases are found upon increasing the dipole moment: hexagonal, body-centered orthorhombic, and body-centered tetragonal. An even richer phase diagram arises for dipolar soft spheres with a purely repulsive inverse power law potential $\sim r^{-n}$. A crossover between qualitatively different sequences of phases occurs near the exponent $n=12$. The results are applicable to electro- and magnetorheological fluids. In addition to the exact ground state analysis we study freezing of the Stockmayer fluid by density-functional theory.Comment: submitted to Phys. Rev.

Magnetization of ferrofluids with dipolar interactions - a Born--Mayer expansion

For ferrofluids that are described by a system of hard spheres interacting via dipolar forces we evaluate the magnetization as a function of the internal magnetic field with a Born--Mayer technique and an expansion in the dipolar coupling strength. Two different approximations are presented for the magnetization considering different contributions to a series expansion in terms of the volume fraction of the particles and the dipolar coupling strength.Comment: 19 pages, 11 figures submitted to PR

Dealing with Incomplete Household Panel Data in Inequality Research

Population surveys around the world face the problem of declining cooperation and participation rates of respondents. Not only can item nonresponse and unit nonresponse impair important outcome measures for inequality research such as total household disposable income; there is also a further case of missingness confronting household panel surveys that potentially biases results. The approach commonly used in such surveys of interviewing all adult household members and aggregating their individual incomes to yield a final outcome measure for welfare analyses often suffers from partial unit non-response (PUNR), i.e., the non-response of at least one unit, or member, of an otherwise participating household. In these cases, the aggregate income of all household members lacks at least one individual's income. These processes are typically not random and require appropriate correction. Using data from more than twenty waves of the German Socio-Economic Panel (SOEP) we evaluate four different strategies to deal with this phenomenon: (a) Ignorance, i.e., assuming the missing individual's income to be zero. (b) Adjustment of the equivalence scale to account for differences in household size and composition. (c) Elimination of all households observed to suffer PUNR, and re-weighting of households observed to be at risk of but not affected by PUNR. (d) Longitudinal imputation of the missing income components. The aim of this paper is to show how the choice of technique affects substantive results in the inequality research. We find indications of substantial bias on income inequality and poverty as well as on income mobility. These findings are obviously even more important in cross-national comparative analyses if the data providers in the individual countries deal differently with PUNR in the underlying data.Household Panel Surveys, Partial Unit Non-Response, Inequality, Mobility, Imputation, SOEP

Economic Gains from Publicly Provided Education in Germany

The aim of this paper is to estimate income advantages arising from publicly provided education and to analyse their impact on the income distribution in Germany. Using representative micro-data from the SOEP and considering regional and education-specific variation, from a cross-sectional perspective the overall result is the expected levelling effect. When estimating the effects of accumulated educational transfers over the life course within a regression framework, however, and controlling for selectivity of households with children as potential beneficiaries of educational transfers, we find evidence that social inequalities are increasing from an intergenerational perspective, reinforced in particular by public transfers for non-compulsory education, thus negating any social equalisation effects achieved within the compulsory education framework.Education, Public Transfers, Income Distribution, Economic Wellbeing, SOEP

The Impact of Home Production on Economic Inequality in Germany

Using representative income and time use-data from the German Socio-Economic Panel (SOEP), we estimate non-monetary income advantages arising from home production and analyse their impact on economic inequality. As an alternative to existing measures, we propose a predicted wage approach based on a bias-adjusted measure of hours spent on home production. Sensitivity analyses comparing results obtained from different approaches provide indications of methodological effects arising from the choice of method. Although the substantive notion of reduced inequality is stable, the degree of variation in our findings underscores the need for a harmonized approach in cross-nationally comparative research.home production, non-cash incomes, economic inequality, well-being, SOEP