Galerkin-Eshelby Meshless Approach to Multiple Inclusion Problem

2017 ◽  
Vol 1144 ◽  
pp. 167-171
Author(s):  
Stanislav Šulc ◽  
Jan Novák
Keyword(s):  
Analytical Solution ◽  
Heterogeneous Material ◽  
Inclusion Problem ◽  
Shape Functions ◽  
Minimum Potential Energy ◽  
Multiple Inclusion ◽  
Single Inclusion ◽  
Homogeneous Matrix ◽  
Minimum Potential ◽  
The Galerkin Method

This paper presents a method of computing perturbation fields in a heterogeneous material composed of multiple ellipsoidal inclusions embedded in homogeneous matrix. The work rests on the renowned Eshelby analytical solution to the single inclusion problem. The strain perturbations, arising from this solution, are used as the shape functions in the Galerkin method, which finds the solution as a combination of these functions with minimum potential energy.

scholarly journals A Trigonometric Analytical Solution of Simply Supported Horizontally Curved Composite I-Beam considering Tangential Slips

2016 ◽  
Vol 2016 ◽  
pp. 1-12
Author(s):  
Qin Xu-xi ◽  
Liu Han-bing ◽  
Wu Chun-li ◽  
Gu Zheng-wei
Keyword(s):  
Analytical Solution ◽  
Trigonometric Series ◽  
Beam Theory ◽  
Tangential Direction ◽  
Governing Equations ◽  
Minimum Potential Energy ◽  
Simply Supported ◽  
Horizontally Curved ◽  
Partial Interaction ◽  
Minimum Potential

This paper presents an analytical solution of the simply supported horizontally composite curved I-beam by trigonometric series considering the effect of partial interaction in the tangential direction. Governing equations and boundary conditions are obtained by using the Vlasov curved beam theory and the principle of minimum potential energy. The deflection functions and the Lagrange multiplier functions are expressed as trigonometric series to satisfy the governing equations and the simply supported constraints at both ends. The numerical results of deflections and forces which are obtained by this method are compared with both FEM results and experimental results, and the inaccuracy between the analytical solutions in this paper and the FEM results is small and reasonable.

A study of mixed-mode composite delamination using enriched interface elements

2013 ◽  
Vol 117 (1195) ◽  
pp. 959-967
Author(s):  
I. Guiamatsia ◽  
J. K. Ankersen ◽  
L. Iannucci
Keyword(s):  
Finite Elements ◽  
Analytical Solution ◽  
Elastic Foundation ◽  
Displacement Field ◽  
Mixed Mode ◽  
Crack Tip ◽  
Shape Functions ◽  
Interface Elements ◽  
Minimum Element ◽  
Element Size

Abstract This paper examines the performance of enriching the shape functions of interface finite elements in the prediction of mixed-mode delamination. Enriching second-order interface and solid elements with the analytical solution of a beam on elastic foundation problem yields the correct displacement field ahead of the crack tip. Despite the enrichment being fixed at elements nodes, resulting in non-traceability of the crack tip location, the strategy is shown to perform consistently well, increasing the minimum element size from the typical 0·5mm to 5mm, for a range of classical mixed-mode bending (MMB) specimens.

Principles of minimum potential energy and complementary energy

2020 ◽  
pp. 228-248
Author(s):  
Lallit Anand ◽  
Sanjay Govindjee
Keyword(s):  
Boundary Condition ◽  
Potential Energy ◽  
Displacement Field ◽  
Mixed Boundary ◽  
Energy Functional ◽  
The Body ◽  
Minimum Potential Energy ◽  
Complementary Energy ◽  
Minimum Potential ◽  
Potential Energy Functional

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.

Optimization for Minimum Potential Energy, Maximum Rigidity and Minimum Deformability

1984 ◽  
pp. 9-87
Author(s):  
Andrej M. Brandt ◽  
Wojciech Dzieniszewski ◽  
Stefan Jendo ◽  
Wojciech Marks ◽  
Stefan Owczarek ◽  
...  
Keyword(s):  
Potential Energy ◽  
Minimum Potential Energy ◽  
Minimum Potential ◽  
Energy Maximum

A Global Extremum Principle in Mixed Form for Equilibrium Analysis With Elastic/Stiffening Materials (a Generalized Minimum Potential Energy Principle)

1994 ◽  
Vol 61 (4) ◽  
pp. 914-918 ◽  
Author(s):  
J. E. Taylor
Keyword(s):  
Potential Energy ◽  
Problem Formulation ◽  
Extremum Problem ◽  
Equilibrium Analysis ◽  
Minimum Potential Energy ◽  
Problem Statement ◽  
Energy Principle ◽  
Potential Energy Principle ◽  
Minimum Potential ◽  
Minimum Potential Energy Principle

An extremum problem formulation is presented for the equilibrium mechanics of continuum systems made of a generalized form of elastic/stiffening material. Properties of the material are represented via a series composition of elastic/locking constituents. This construction provides a means to incorporate a general model for nonlinear composites of stiffening type into a convex problem statement for the global equilibrium analysis. The problem statement is expressed in mixed “stress and deformation” form. Narrower statements such as the classical minimum potential energy principle, and the earlier (Prager) model for elastic/locking material are imbedded within the general formulation. An extremum problem formulation in mixed form for linearly elastic structures is available as a special case as well.

Buckling Analysis of a Bi-Directional Strain-Gradient Euler–Bernoulli Nano-Beams

2020 ◽  
Vol 20 (11) ◽  
pp. 2050114
Author(s):  
Murat Çelik ◽  
Reha Artan
Keyword(s):  
Potential Energy ◽  
Gradient Elasticity ◽  
Buckling Analysis ◽  
Functionally Graded ◽  
Minimum Potential Energy ◽  
End Conditions ◽  
Graded Material ◽  
Minimum Potential ◽  
Gradient Elasticity Theory ◽  
Basic Equations

Investigated herein is the buckling of Euler–Bernoulli nano-beams made of bi-directional functionally graded material with the method of initial values in the frame of gradient elasticity. Since the transport matrix cannot be calculated analytically, the problem was examined with the help of an approximate transport matrix (matricant). This method can be easily applied with buckling analysis of arbitrary two-directional functionally graded Euler–Bernoulli nano-beams based on gradient elasticity theory. Basic equations and boundary conditions are derived by using the principle of minimum potential energy. The diagrams and tables of the solutions for different end conditions and various values of the parameters are given and the results are discussed.

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