scholarly journals Integrable structure in discrete shell membrane theory

2014 ◽  
Vol 470 (2165) ◽  
pp. 20130757 ◽  
Author(s):  
W. K. Schief
Keyword(s):  
Barycentric Coordinates ◽  
Internal Stresses ◽  
Discrete Version ◽  
Membrane Theory ◽  
Equilibrium Equations ◽  
Surface Theory ◽  
Integrable Structure ◽  
Discrete Analogues ◽  
Generalized Barycentric Coordinates ◽  
External Forces

We present natural discrete analogues of two integrable classes of shell membranes. By construction, these discrete shell membranes are in equilibrium with respect to suitably chosen internal stresses and external forces. The integrability of the underlying equilibrium equations is proved by relating the geometry of the discrete shell membranes to discrete O surface theory. We establish connections with generalized barycentric coordinates and nine-point centres and identify a discrete version of the classical Gauss equation of surface theory.

Discrete Mechanics and Optimal Control for Constrained Multibody Dynamics

2007 ◽  
Author(s):  
Sigrid Leyendecker ◽  
Sina Ober-Blo¨baum ◽  
Jerrold E. Marsden ◽  
Michael Ortiz
Keyword(s):  
Optimal Control ◽  
Rigid Body ◽  
Null Space ◽  
Numerical Dissipation ◽  
Discrete Version ◽  
Dynamical Equations ◽  
Time Stepping ◽  
Minimal Dimension ◽  
Energy And Momentum ◽  
External Forces

This paper formulates the dynamical equations of mechanics subject to holonomic constraints in terms of the states and controls using a constrained version of the Lagrange-d’Alembert principle. Based on a discrete version of this principle, a structure preserving time-stepping scheme is derived. It is shown that this respect for the mechanical structure (such as a reliable computation of the energy and momentum budget, without numerical dissipation) is retained when the system is reduced to its minimal dimension by the discrete null space method. Together with initial and final conditions on the configuration and conjugate momentum, the reduced time-stepping equations serve as nonlinear equality constraints for the minimisation of a given cost functional. The algorithm yields a sequence of discrete configurations together with a sequence of actuating forces, optimally guiding the system from the initial to the desired final state. The resulting discrete optimal control algorithm is shown to have excellent energy and momentum properties, which are illustrated by two specific examples, namely reorientation and repositioning of a rigid body subject to external forces and the reorientation of a rigid body with internal momentum wheels.

scholarly journals Effects of crystallization and plastic flow on the thermoluminescence of ice

1964 ◽  
Vol 5 (39) ◽  
pp. 333-337 ◽  
Author(s):  
L. B. Ronca ◽  
E. J. Zeller
Keyword(s):  
Crystal Lattice ◽  
Plastic Flow ◽  
Peak Height ◽  
Internal Stresses ◽  
Peak Ratio ◽  
Lattice Effect ◽  
External Forces ◽  
Made In

AbstractGlow-curves of ice, made in the laboratory, have a distribution of peak-height ratios, with the maximum at approximately 0.75. The distribution was probably caused by varying conditions of crystallization, impossible to control with the present equipment. Glow-curves of ice which has been made to flow have a distribution of peak ratios with the maximum at approximately 1.75.It is concluded that dislocations in the crystal lattice effect the peak ratio. Dislocations are produced during crystallization by internal stresses, and to a greater extent, by external forces during the flow.

On Some Classes of Correct Problems in the Membrane Theory of Convex Shells

2020 ◽  
pp. 25-29
Author(s):  
E.V. Tyurikov
Keyword(s):  
Smooth Boundary ◽  
Hilbert Problem ◽  
Boundary Value ◽  
Geometric Description ◽  
Membrane Theory ◽  
Equilibrium Equations ◽  
Wide Family ◽  
Boundary Solutions ◽  
System Of Equilibrium Equations ◽  
Riemann Hilbert Problem

On the basis of the theory of the modified Riemann-Hilbert problem for generalized analytic functions, a geometric description is given of a fairly wide family of correct by I. N. Vekua of boundary value problems of the membrane theory of convex hulls with a piecewise smooth boundary. Solutions to the corresponding Riemann-Hilbert problem for an elliptic system of equilibrium equations are found in the classes of N.I. Muskhelishvili and realize a state of tense equilibrium under the condition of stress concentration in corner points. An effective formula is given for calculating the index of the boundary condition, which allows us to formulate the results in a visible form. Families of shells are found for which the solvability picture of the main boundary-value problem coincides with the solvability picture of the Vekua problem for shells with a smooth border.

scholarly journals The bending of an elastic beam by a liquid drop: a variational approach

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130066 ◽  
Author(s):  
Sébastien Neukirch ◽  
Arnaud Antkowiak ◽  
Jean-Jacques Marigo
Keyword(s):  
Contact Line ◽  
Variational Approach ◽  
Liquid Drop ◽  
Elastic Beam ◽  
Triple Line ◽  
Equilibrium Equations ◽  
External Forces ◽  
Contact Line Pinning ◽  
Three Phases

We study the interaction of a liquid drop with an elastic beam in the case where bending effects dominate. We use a variational approach to derive equilibrium equations for the system in the presence of gravity and in the presence or absence of contact line pinning. We show that the derived equilibrium equations for the beam subsystem reveal the external forces applied on the beam by the liquid and vapour phases. Among these, the force applied at the triple line (the curve where the three phases meet) is found to lie along the liquid–vapour interface.

New Model for Shear Failure of RC Beam-Column Interior Joints with Different Depth Beams

2011 ◽  
Vol 243-249 ◽  
pp. 573-577
Author(s):  
Hong Li ◽  
Shuang Gao ◽  
Tao Wu ◽  
Bo Quan Liu
Keyword(s):  
Mathematical Model ◽  
Shear Failure ◽  
Failure Process ◽  
Seismic Capacity ◽  
Equilibrium Equations ◽  
Beam Shear ◽  
Shear Beam ◽  
Core Height ◽  
External Forces ◽  
Flexural Resistance

On the basis of the experiment of 12 abnormal joints with different depth beams, failure process, seismic capacity and construction measure of this kind of joints are analyzed. Shear strength and failure model of joints with equivalent core height are analyzed by using mathematical model based on quadruple flexural resistance. The model satisfies equilibrium equations relating magnitude of external forces such as column shear, beam shear and axial force in column, to the magnitude of stress resultants. The results show that the model can reasonably explain the reasons for joints failure, have explicit mathematical model and good agreement between experimental results and calculated results of this model is achieved.

scholarly journals New perspectives on polygonal and polyhedral finite element methods

2014 ◽  
Vol 24 (08) ◽  
pp. 1665-1699 ◽  
Author(s):  
Gianmarco Manzini ◽  
Alessandro Russo ◽  
N. Sukumar
Keyword(s):  
Finite Element ◽  
Finite Elements ◽  
Finite Element Methods ◽  
Rates Of Convergence ◽  
Explicit Construction ◽  
Barycentric Coordinates ◽  
Finite Difference Schemes ◽  
Stability Matrix ◽  
Generalized Barycentric Coordinates ◽  
Element Method

Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.

Choosing a Laser-Cutting Technology to be Applied to a Deep-Drawn Product

2007 ◽  
Vol 537-538 ◽  
pp. 639-646
Author(s):  
Janez Grum ◽  
Tomaž Kek
Keyword(s):  
Residual Stresses ◽  
Cross Section ◽  
Laser Cutting ◽  
Complex Form ◽  
Hole Drilling ◽  
Component Part ◽  
Internal Stresses ◽  
Cutting Out ◽  
Free Body ◽  
External Forces

Residual stresses are an inevitable result of mechanical and heat treatments of steel. The stress field generated at the surface and through the cross section of a product is usually non-uniform, and there can be important stress gradients. In general residual stresses can be defined as self-balanced internal stresses existing in a free body on which no external forces are acting. In the present study the focus was on the residual stresses occurring in a steel deep-drawn product of a complex form. The residual stresses occur mainly because of different degrees of plastic deformation at different locations. The residual stresses occurring in the product as a result of the manufacturing process of deep drawing can be comparatively strong. In slitting or cutting a useful product out of a deepdrawn part by laser cutting, a portion of the residual stresses will get relieved and produce distortion of the deep-drawn product. The studies made showed that the distortion of the deep-drawn product depended on the mode of cutting-out. Also the state of the residual stresses after slitting or cuttingout of the deep-drawn product will depend on the mode of cutting-out, i.e., on the sequence of individual cuts chosen. The product used in the analysis made was a deep-drawn steel mud-guard, which makes a component part of a working machine. The various modes of cutting-out a useful product of 1260x800x370 mm in size were performed with laser cutting using cutting oxygen. The residual stresses in the mud-guard were determined with a semi-destructive hole drilling strain gage method.

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