scholarly journals Asymptotic variational analysis of incompressible elastic strings

2020 ◽  
pp. 1-28
Author(s):  
Dominik Engl ◽  
Carolin Kreisbeck
Keyword(s):  
Cross Sections ◽  
Three Dimensional ◽  
Differential Constraint ◽  
Energy Functionals ◽  
Elastic Strings ◽  
Non Linear ◽  
Applied Forces ◽  
Linear Differential ◽  
Γ Convergence ◽  
External Forces

Starting from three-dimensional non-linear elasticity under the restriction of incompressibility, we derive reduced models to capture the behaviour of strings in response to external forces. Our Γ-convergence analysis of the constrained energy functionals in the limit of shrinking cross-sections gives rise to explicit one-dimensional limit energies. The latter depend on the scaling of the applied forces. The effect of local volume preservation is reflected either in their energy densities through a constrained minimization over the cross-section variables or in the class of admissible deformations. Interestingly, all scaling regimes allow for compression and/or stretching of the string. The main difficulty in the proof of the Γ-limit is to establish recovery sequences that accommodate the non-linear differential constraint imposed by the incompressibility. To this end, we modify classical constructions in the unconstrained case with the help of an inner perturbation argument tailored for 3d-1d dimension reduction problems.

Soret and Dufour Effects on Three Dimensional Upper-Convected Maxwell Fluid with Chemical Reaction and Non-Linear Radiative Heat Flux

2017 ◽  
Vol 15 (3) ◽  
Author(s):  
M. Ramzan ◽  
M. Bilal ◽  
Jae Dong Chung
Keyword(s):  
Heat Flux ◽  
Differential Equations ◽  
Chemical Reaction ◽  
Radiative Heat ◽  
Three Dimensional ◽  
Maxwell Fluid ◽  
Radiative Heat Flux ◽  
Soret And Dufour Effects ◽  
Non Linear ◽  
Linear Differential

Abstract Three dimensional chemically reactive upper-convected Maxwell (UCM) fluid flow over a stretching surface is considered to examine Soret and Dufour effects on heat and mass transfer. During the formulation of energy equation, non-linear radiative heat flux is considered. Similarity transformation reduces the partial differential equations of flow problem into ordinary differential equations. These non-linear differential equations are then solved by using bvp4c MATLAB built-in function. A comparison of the present results with the published work is also included. Effects of some prominent parameters such as Soret and Dufour number, chemical reaction parameter, Prandtl number, Schmidt number and thermal radiation on velocity, temperature and concentration are discussed graphically and numerically. A comparison with the previously published work is also included in a tabular form.

A justification of the Timoshenko beam model through Γ-convergence

2017 ◽  
Vol 15 (02) ◽  
pp. 261-277 ◽  
Author(s):  
Lior Falach ◽  
Roberto Paroni ◽  
Paolo Podio-Guidugli
Keyword(s):  
Linear Elasticity ◽  
Timoshenko Beam ◽  
Three Dimensional ◽  
Convergence Theory ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Energy Functionals ◽  
Γ Convergence ◽  
Suitable Sequence

We validate the Timoshenko beam model as an approximation of the linear-elasticity model of a three-dimensional beam-like body. Our validation is achieved within the framework of [Formula: see text]-convergence theory, in two steps: firstly, we construct a suitable sequence of energy functionals; secondly, we show that this sequence [Formula: see text]-converges to a functional representing the energy of a Timoshenko beam.

Stiffness and design for strength of trapezoidal Belleville springs

2011 ◽  
Vol 46 (8) ◽  
pp. 825-836 ◽  
Author(s):  
N L Pedersen ◽  
P Pedersen
Keyword(s):  
Finite Element ◽  
Stress Distribution ◽  
Cross Sections ◽  
Poisson Ratio ◽  
Three Dimensional ◽  
Displacement Curve ◽  
Structural Behaviour ◽  
Design Standards ◽  
Element Analysis ◽  
Non Linear

Belleville springs or coned disc springs are commonly used in machine design. The geometric dimensions of the spring and the determination of non-linear force–displacement curve are regulated by different standards. However, the theory behind Belleville spring design standards is founded on a study published in 1936. Furthermore, the common spring design with cross-sections of uniform thickness poses problems in terms of non-uniformity of stress distribution. In view of this, non-linear three-dimensional finite element analyses of spring designs including uniform or variable thickness are carried out in this paper. Finite element results are compared with analytical predictions and critically analysed in terms of the effect of Poisson ratio, overall stiffness, and stress distribution in the spring. This is done in order to verify the range of validity of design standards. Finite element analysis emerges as a powerful and computationally cheap approach to assess the structural behaviour of Belleville springs regardless of their geometry and level of non-linearity.

scholarly journals Rigorous derivation of active plate models for thin sheets of nematic elastomers

2017 ◽  
Vol 25 (10) ◽  
pp. 1804-1830 ◽  
Author(s):  
Virginia Agostiniani ◽  
Antonio DeSimone
Keyword(s):  
Finite Elasticity ◽  
Three Dimensional ◽  
Energy Functional ◽  
Variable Degree ◽  
Energy Functionals ◽  
Nematic Order ◽  
Reduction Techniques ◽  
Nematic Elastomers ◽  
Plate Models ◽  
Γ Convergence

In the context of finite elasticity, we propose plate models describing the spontaneous bending of nematic elastomer thin films due to variations along the thickness of the nematic order parameters. Reduced energy functionals are deduced from a three-dimensional description of the system using rigorous dimension reduction techniques, based on the theory of Γ-convergence. The two-dimensional models are non-linear plate theories, in which deviations from a characteristic target curvature tensor cost elastic energy. Moreover, the stored energy functional cannot be minimised to zero, thus revealing the presence of residual stresses, as observed in numerical simulations. Three nematic textures are considered: splay-bend and twisted orientations of the nematic director, and a uniform director perpendicular to the mid-plane of the film, with variable degree of nematic order along the thickness. These three textures realise three very different structural models: one with only one stable spontaneously bent configuration, a bistable model with two oppositely curved configurations of minimal energy, and a shell with zero stiffness to twisting.

Michell truss type theories as a Γ-limit of optimal design in linear elasticity

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Heiner Olbermann
Keyword(s):  
Optimal Design ◽  
Linear Elasticity ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Michell Trusses ◽  
Design Problems ◽  
Michell Truss ◽  
Applied Forces ◽  
Γ Convergence

AbstractWe show how to derive (variants of) Michell truss theory in two and three dimensions rigorously as the vanishing weight limit of optimal design problems in linear elasticity in the sense of Γ-convergence. We improve our results from [H. Olbermann, Michell trusses in two dimensions as a Γ-limit of optimal design problems in linear elasticity, Calc. Var. Partial Differential Equations 56 2017, 6, Article ID 166] in that our treatment here includes the three-dimensional case and that we allow for more general boundary conditions and applied forces.

scholarly journals Die ruimtelike vorm van 'n onelastiese, buigbare, geankerde kabel

1992 ◽  
Vol 11 (3) ◽  
pp. 94-101 ◽  
Author(s):  
T. P. Dreyer
Keyword(s):  
Unit Length ◽  
Linear Equations ◽  
Numerical Procedure ◽  
Continuous Model ◽  
The Other ◽  
End Point ◽  
Non Linear ◽  
Linear Differential ◽  
The One ◽  
External Forces

Consider an inelastic, perfectly flexible cable with given external forces acting on the total length of the cable. The one end-point is fixed in the origin and the other end-point is anchored at a given point (a;b;c) in space. The resulting configuration of the cable in space can be modelled by a system of non-linear differential equations. In this article it is shown that this continuous model of the cable can always be solved in terms of an integral. In the special case of a constant (i.e. independent of the position on the cable) external force per unit length the solution is given explicitly in terms of three constants that describe the tension at the origin. These three constants are determined by the boundary values a, b and c at the other end-point, and must be calculated in general by a numerical procedure from the three resulting simultaneous non-linear equations. A few applications of this method are shown.

scholarly journals Derivation of the nonlinear bending–torsion model for a junction of elastic rods

2012 ◽  
Vol 142 (3) ◽  
pp. 633-664 ◽  
Author(s):  
Josip Tambača ◽  
Igor Velčić
Keyword(s):  
Cross Sections ◽  
Minimization Problem ◽  
Three Dimensional ◽  
Contact Forces ◽  
Elastic Rods ◽  
Nonlinear Bending ◽  
One Dimensional ◽  
Starting Point ◽  
Γ Convergence ◽  
The One

We derive the one-dimensional bending–torsion equilibrium model for the junction of straight rods. The starting point is a three-dimensional nonlinear elasticity equilibrium problem written as a minimization problem for a union of thin, rod-like bodies. By taking the limit as the thickness of the three-dimensional rods tends to zero, and by using ideas from the theory of Γ-convergence, we find that the resulting model consists of the union of the usual one-dimensional nonlinear bending–torsion rod models which satisfy the following transmission conditions at the junction point: continuity of displacement and rotation of the cross-sections; balance of contact forces and contact couples.

The weighted energy-dissipation principle and evolutionary Γ-convergence for doubly nonlinear problems

2019 ◽  
Vol 25 ◽  
pp. 36 ◽  
Author(s):  
Matthias Liero ◽  
Stefano Melchionna
Keyword(s):  
Energy Dissipation ◽  
Evolution Equations ◽  
Regularization Parameter ◽  
Nonlinear Problems ◽  
Nonlinear Evolution Equations ◽  
Energy Functionals ◽  
Weighted Energy ◽  
Doubly Nonlinear ◽  
Γ Convergence ◽  
External Forces

We consider a family of doubly nonlinear evolution equations that is given by families of convex dissipation potentials, nonconvex energy functionals, and external forces parametrized by a small parameter ε. For each of these problems, we introduce the so-called weighted energy-dissipation (WED) functional, whose minimizers correspond to solutions of an elliptic-in-time regularization of the target problems with regularization parameter δ. We investigate the relation between the Γ-convergence of the WED functionals and evolutionary Γ-convergence of the associated systems. More precisely, we deal with the limits δ → 0, ε → 0, as well as δ + ε → 0 either in the sense of Γ-convergence of functionals or in the sense of evolutionary Γ-convergence of functional-driven evolution problems, or both. Additionally, we provide some quantitative estimates on the rate of convergence for the limit ε → 0, in the case of quadratic dissipation potentials and uniformly λ-convex energy functionals. Finally, we discuss a homogenization and a dimension reduction problem as examples of application.

Geological Field Sketches and Illustrations

2019 ◽  
Author(s):  
Matthew J. Genge
Keyword(s):  
Cross Sections ◽  
Earth Science ◽  
Three Dimensional ◽  
Worked Examples ◽  
Scientific Publications ◽  
Thin Sections ◽  
Geological Features ◽  
Different Types ◽  
The Subject

Drawings, illustrations, and field sketches play an important role in Earth Science since they are used to record field observations, develop interpretations, and communicate results in reports and scientific publications. Drawing geology in the field furthermore facilitates observation and maximizes the value of fieldwork. Every geologist, whether a student, academic, professional, or amateur enthusiast, will benefit from the ability to draw geological features accurately. This book describes how and what to draw in geology. Essential drawing techniques, together with practical advice in creating high quality diagrams, are described the opening chapters. How to draw different types of geology, including faults, folds, metamorphic rocks, sedimentary rocks, igneous rocks, and fossils, are the subjects of separate chapters, and include descriptions of what are the important features to draw and describe. Different types of sketch, such as drawings of three-dimensional outcrops, landscapes, thin-sections, and hand-specimens of rocks, crystals, and minerals, are discussed. The methods used to create technical diagrams such as geological maps and cross-sections are also covered. Finally, modern techniques in the acquisition and recording of field data, including photogrammetry and aerial surveys, and digital methods of illustration, are the subject of the final chapter of the book. Throughout, worked examples of field sketches and illustrations are provided as well as descriptions of the common mistakes to be avoided.

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