scholarly journals Wrinkle surface instability of an inhomogeneous elastic block with graded stiffness

2017 ◽  
Vol 473 (2200) ◽  
pp. 20160882 ◽  
Author(s):  
Shengyou Yang ◽  
Yi-chao Chen
Keyword(s):  
Necessary Conditions ◽  
Energy Functional ◽  
Surface Instability ◽  
Structural Geometry ◽  
First Variation ◽  
Material Inhomogeneity ◽  
The First Variation ◽  
Homogeneous Block ◽  
Potential Energy Functional ◽  
Film Substrate

Surface instabilities have been studied extensively for both homogeneous materials and film/substrate structures but relatively less for materials with continuously varying properties. This paper studies wrinkle surface instability of a graded neo-Hookean block with exponentially varying modulus under plane strain by using the linear bifurcation analysis. We derive the first variation condition for minimizing the potential energy functional and solve the linearized equations of equilibrium to find the necessary conditions for surface instability. It is found that for a homogeneous block or an inhomogeneous block with increasing modulus from the surface, the critical stretch for surface instability is 0.544 (0.456 strain), which is independent of the geometry and the elastic modulus on the surface of the block. This critical stretch coincides with that reported by Biot (1963 Appl. Sci. Res. 12 , 168–182. ( doi:10.1007/BF03184638 )) 53 years ago for the onset of wrinkle instabilities in a half-space of homogeneous neo-Hookean materials. On the other hand, for an inhomogeneous block with decreasing modulus from the surface, the critical stretch for surface instability ranges from 0.544 to 1 (0–0.456 strain), depending on the modulus gradient, and the length and height of the block. This sheds light on the effects of the material inhomogeneity and structural geometry on surface instability.

scholarly journals Existence Results for the Conformal Dirac–Einstein System

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Chiara Guidi ◽  
Ali Maalaoui ◽  
Vittorio Martino
Keyword(s):  
Perturbation Methods ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Existence Results ◽  
First Variation ◽  
The First Variation

AbstractWe consider the coupled system given by the first variation of the conformal Dirac–Einstein functional. We will show existence of solutions by means of perturbation methods.

scholarly journals On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

2020 ◽  
Vol 51 (4) ◽  
pp. 313-332
Author(s):  
Firooz Pashaie
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Principal Curvatures ◽  
First Variation ◽  
Chen’S Conjecture ◽  
The First Variation ◽  
Linearized Operator ◽  
Euclidean Submanifolds

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

scholarly journals On solving the plateau problem in parametric form

1983 ◽  
Vol 6 (2) ◽  
pp. 341-361
Author(s):  
Baruch cahlon ◽  
Alan D. Solomon ◽  
Louis J. Nachman
Keyword(s):  
Minimal Surface ◽  
Numerical Method ◽  
Minimal Surfaces ◽  
Plateau Problem ◽  
Parametric Form ◽  
First Variation ◽  
Plateau’S Problem ◽  
Numerical Example ◽  
Plateau's Problem ◽  
The First Variation

This paper presents a numerical method for finding the solution of Plateau's problem in parametric form. Using the properties of minimal surfaces we succeded in transferring the problem of finding the minimal surface to a problem of minimizing a functional over a class of scalar functions. A numerical method of minimizing a functional using the first variation is presented and convergence is proven. A numerical example is given.

Boundary Perturbations due to the Presence of Small Cracks

2015 ◽  
Author(s):  
Habib Ammari ◽  
Elie Bretin ◽  
Josselin Garnier ◽  
Hyeonbae Kang ◽  
Hyundae Lee ◽  
...  
Keyword(s):  
Asymptotic Formula ◽  
Neumann Boundary ◽  
Representation Formula ◽  
Fundamental Solutions ◽  
Energy Functional ◽  
Finite Hilbert Transform ◽  
Time Harmonic ◽  
Elastic Potential Energy ◽  
Potential Energy Functional ◽  
Boundary Perturbations

This chapter considers the perturbations of the displacement (or traction) vector that are due to the presence of a small crack with homogeneous Neumann boundary conditions in an elastic medium. It derives an asymptotic formula for the boundary perturbations of the displacement as the length of the crack tends to zero. Using analytical results for the finite Hilbert transform, the chapter derives an asymptotic expansion of the effect of a small Neumann crack on the boundary values of the solution. It also derives the topological derivative of the elastic potential energy functional and proves a useful representation formula for the Kelvin matrix of the fundamental solutions of Lamé system. Finally, it gives an asymptotic formula for the effect of a small linear crack in the time-harmonic regime.

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.

scholarly journals IV. On the discrimination of maxima and minima solutions in the calculus of variations

1887 ◽  
Vol 178 ◽  
pp. 95-129 ◽  
Keyword(s):  
Differential Equations ◽  
Calculus Of Variations ◽  
General Rule ◽  
System Of Differential Equations ◽  
First Variation ◽  
Minimum Value ◽  
Fonctions Analytiques ◽  
Independent Variable ◽  
The First Variation ◽  
Made In

The criteria for distinguishing between the maximum and minimum values of integrals have been investigated by many eminent mathematicians. In 1786 Legendre gave an imperfect discussion for the case where the function to be made a maximum is ʃ f (x,y, dy / dx ) dx . Nothing further seems to have been done till 1797, when Lagrange pointed out, in his ‘Théorie des Fonctions Analytiques,' published in 1797, that Legendre had supplied no means of showing th at the operations required for his process were not invalid through some of the multipliers becoming zero or infinite, and he gives an example to show that Legendre’s criterion, though necessary, was not sufficient. In 1806 Brunacci, an Italian mathematician, gave an investigation which has the important advantage of being short, easily compiehensible, and perfectly general in character, but which is open to the same objection as that brought against Legendre’s method. The next advance was made in 1836 by the illustrious Jacobi, who treats only of functions containing one dependent and one independent variable. Jacobi says (Todhunter, Art. 219, p. 243): “I have succeeded in supplying a great deficiency in the Calculus of Variations. In problems on maxima and minima which depend on this calculus no general rule is known for deciding whether a solution really gives a maximum or a minimum, or neither. It has, indeed, been shown that the question amounts to determining whether the integrals of a certain system of differential equations remain finite throughout the limits of the integral which is to have a maximum or a minimum value. But the integrals of these differential equations were not known, nor had any other method been discovered for ascertaining whether they remain finite throughout the required interval. I have, however, discovered that these integrals can be immediately obtained when We have integrated the differential equations which must be satisfied in order that the first variation may vanish.” Jacobi then proceeds to state the result of his transformation for the cases where the function to be integrated contains x, y, dy / dx , and x, y, dy / dx 2 , and in this solution the analysis appears free from all objection, though, where he proceeds to consider the general case, the investigation does not appear to be quite satisfactory in form, inasmuch as higher and higher differential coefficients of By are successively introduced into the discussion (see Art. 5). Jacobi’s analysis is much more complicated than Brunacci's, its advantage being that the coefficients used in the transformation could be easily determined; hence it supplied the means of ascertaining whether they became infinite or not.

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