Rank one plus a null-Lagrangian is an inherited property of two-dimensional compliance tensors under homogenisation

1998 ◽  
Vol 128 (2) ◽  
pp. 283-299 ◽  
Author(s):  
Yury Grabovsky ◽  
Graeme W. Milton
Keyword(s):  
Shear Modulus ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Pure Crystal ◽  
Compliance Tensor ◽  
Rank One ◽  
Two Space Dimensions ◽  
Local Compliance ◽  
Null Lagrangian ◽  
Surprising Discovery

Assume that the local compliance tensor of an elastic composite in two space dimensions is equal to a rank-one tensor plus a null-Lagrangian (there is only one symmetric one in two dimensions). The purpose of this paper is to prove that the effective compliance tensor has the same representation: rank-one plus the null-Lagrangian. This statement generalises the wellknown result of Hill that a composite of isotropic phases with a common shear modulus is necessarily elastically isotropic and shares the same shear modulus. It also generalises the surprising discovery of Avellaneda et al. that under a certain condition on the pure crystal moduli the shear modulus of an isotropic polycrystal is uniquely determined. The present paper sheds light on this effect by placing it in a more general framework and using some elliptic PDE theory rather than the translation method. Our results allow us to calculate the polycrystalline G-closure of the special class of crystals under consideration. Our analysis is contrasted with a two-dimensional model problem for shape-memory polycrystals. We show that the two problems can be thought of as ‘elastic percolation’ problems, one elliptic, one hyperbolic.

scholarly journals Solutions to Benjamin-Bona-Mahony Equation in Two Space Dimensions by Various Methods

2018 ◽  
Author(s):  
Alper Korkmaz
Keyword(s):  
Gordon Equation ◽  
Expansion Method ◽  
Algebraic Method ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Sine Gordon Equation ◽  
Hyperbolic Tangent ◽  
Real Solutions ◽  
Two Space Dimensions ◽  
Bbm Equation

Four methods in two different families have been constructed to derive the exact solutions to Benjamin-Bona-Mahony equation in two space dimensions. Simply defined hyperbolic tangent, hyperbolic secant and hyperbolic cosecant ansatzes and the expansion method based on the Sine-Gordon equation in two dimensions are directly substituted into the governing ODE reduced from the two dimensional BBM equation. Classical algebraic method is used to find the relations among the target parameters representing the nonzero coefficients in the predicted solutions and the wave transform parameters. Some complex and real solutions have been constructed in explicit forms.

Effective moduli of plane polycrystals with a rank-one local compliance tensor

1998 ◽  
Vol 128 (6) ◽  
pp. 1325-1354
Author(s):  
L. V. Gibiansky
Keyword(s):  
Boundary Conditions ◽  
Effective Properties ◽  
Positive Definite ◽  
Effective Moduli ◽  
Compliance Tensor ◽  
Rank One ◽  
Effective Compliance ◽  
Local Compliance ◽  
The Given ◽  
Linear Invariants

Recently, Grabovsky and Milton have studied effective properties of plane polycrystals made of a crystal with a rank-one local compliance tensor and a positive definite ‘weak’ direction. This paper continues their investigation. By using several complementary approaches, it is proved that the effective compliance tensor of such a polycrystal has exactly one weak direction. The new proofs clarify the link of the degenerate polycrystal problem with previously obtained results on the polycrystal properties. Investigation of the phase boundary conditions has allowed us to find the L-closure set of the effective properties of all polycrystals that can be constructed by sequential laminations of the original degenerate crystals. Bounds on the G-closure, i.e. a set of the effective properties of all the polycrystals that can be built from the given set of degenerate crystals, are found. They are given by a polygonal set in the plane of two linear invariants of the effective compliance tensors. The effective moduli of the polycrystals made from crystals with non-definite ‘weak’ directions are also studied.

Invariant properties of the stress in plane elasticity and equivalence classes of composites

1992 ◽  
Vol 438 (1904) ◽  
pp. 519-529 ◽  
Keyword(s):  
Stress Field ◽  
Functional Dependence ◽  
Matrix Material ◽  
Plane Elasticity ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Shear Moduli ◽  
Isotropic Composite ◽  
Compliance Tensor ◽  
The Matrix

Attention is drawn to the invariance of the stress field in a two-dimensional body loaded at the boundary by fixed forces when the compliance tensor S(x) is shifted uniformly by S 1 (λ, - λ), where λ is an arbitrary constant and S 1 ( k,u )is the compliance tensor of a isotropic material with two-dimensional bulk and shear moduli k and μ . This invariance is explained from two simple observations: first, That in two dimensions the tensor S (1/2, -1/2) acts to locally rotate the stress by 90° and the second that this rotated field is the symmetrized gradient of a vector field and therefore can be treated as a strain. For composite materials the invariance of the stress field implies that the effective compliance tensor S * also gets shifted by S 1 l( (λ, - λ) when the constituent moduli are each shifted by S (λ, - λ). This imposes constraints on the functional dependence of S * on the material moduli of the components. Applied to an isotropic composite of two isotropic components it implies that when the inverse bulk modulus is shifted by the constant 1/λ and the inverse shear modulus is shifted by — 1/λ, then the inverse effective bulk and shear moduli undergo precisely the same shifts. In particular it explains why the effective Young’s modulus of a two-dimensional media with holes does not depend on the Poisson’s ratio of the matrix material.

Urbanicity, City Delegations, and Two-Dimensional Liberalism

2018 ◽  
Author(s):  
Thomas K. Ogorzalek
Keyword(s):  
National Level ◽  
Two Dimensions ◽  
Political Order ◽  
Two Dimensional ◽  
Local Institutions ◽  
Horizontal Integration ◽  
National Politics ◽  
United Front ◽  
Strategies For Success

This theoretical chapter develops the argument that the conditions of cities—large, densely populated, heterogeneous communities—generate distinctive governance demands supporting (1) market interventions and (2) group pluralism. Together, these positions constitute the two dimensions of progressive liberalism. Because of the nature of federalism, such policies are often best pursued at higher levels of government, which means that cities must present a united front in support of city-friendly politics. Such unity is far from assured on the national level, however, because of deep divisions between and within cities that undermine cohesive representation. Strategies for success are enhanced by local institutions of horizontal integration developed to address the governance demands of urbanicity, the effects of which are felt both locally and nationally in the development of cohesive city delegations and a unified urban political order capable of contending with other interests and geographical constituencies in national politics.

scholarly journals Sharp Rank-One Convexity Conditions in Planar Isotropic Elasticity for the Additive Volumetric-Isochoric Split

2021 ◽  
Vol 143 (2) ◽  
pp. 301-335
Author(s):  
Jendrik Voss ◽  
Ionel-Dumitrel Ghiba ◽  
Robert J. Martin ◽  
Patrizio Neff
Keyword(s):  
Differential Inequalities ◽  
Two Dimensional ◽  
Ellipticity Condition ◽  
One Dimensional ◽  
Suitable Sense ◽  
Precise Analysis ◽  
Rank One ◽  
Isotropic Elasticity ◽  
Convexity Conditions

AbstractWe consider the volumetric-isochoric split in planar isotropic hyperelasticity and give a precise analysis of rank-one convexity criteria for this case, showing that the Legendre-Hadamard ellipticity condition separates and simplifies in a suitable sense. Starting from the classical two-dimensional criterion by Knowles and Sternberg, we can reduce the conditions for rank-one convexity to a family of one-dimensional coupled differential inequalities. In particular, this allows us to derive a simple rank-one convexity classification for generalized Hadamard energies of the type $W(F)=\frac{\mu }{2} \hspace{0.07em} \frac{\lVert F \rVert ^{2}}{\det F}+f(\det F)$ W ( F ) = μ 2 ∥ F ∥ 2 det F + f ( det F ) ; such an energy is rank-one convex if and only if the function $f$ f is convex.

scholarly journals Interface Roughening in Two Dimensions

2021 ◽  
Vol 182 (3) ◽  
Author(s):  
Gernot Münster ◽  
Manuel Cañizares Guerrero
Keyword(s):  
Field Theory ◽  
Monte Carlo ◽  
Capillary Wave ◽  
System Size ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Wave Approximation ◽  
Statistical Field ◽  
Statistical Field Theory ◽  
Interface Roughening

AbstractRoughening of interfaces implies the divergence of the interface width w with the system size L. For two-dimensional systems the divergence of $$w^2$$ w 2 is linear in L. In the framework of a detailed capillary wave approximation and of statistical field theory we derive an expression for the asymptotic behaviour of $$w^2$$ w 2 , which differs from results in the literature. It is confirmed by Monte Carlo simulations.

scholarly journals Is contact angle a cause or an effect? – A cautionary tale

2020 ◽  
Vol 146 ◽  
pp. 03004
Author(s):  
Douglas Ruth
Keyword(s):  
Contact Angle ◽  
Solid Surface ◽  
Contact Angles ◽  
Two Dimensions ◽  
Experimental Results ◽  
Flow In Porous Media ◽  
Two Dimensional ◽  
Wide Range ◽  
Fluid Drop ◽  
Surrounding Fluid

The most influential parameter on the behavior of two-component flow in porous media is “wettability”. When wettability is being characterized, the most frequently used parameter is the “contact angle”. When a fluid-drop is placed on a solid surface, in the presence of a second, surrounding fluid, the fluid-fluid surface contacts the solid-surface at an angle that is typically measured through the fluid-drop. If this angle is less than 90°, the fluid in the drop is said to “wet” the surface. If this angle is greater than 90°, the surrounding fluid is said to “wet” the surface. This definition is universally accepted and appears to be scientifically justifiable, at least for a static situation where the solid surface is horizontal. Recently, this concept has been extended to characterize wettability in non-static situations using high-resolution, two-dimensional digital images of multi-component systems. Using simple thought experiments and published experimental results, many of them decades old, it will be demonstrated that contact angles are not primary parameters – their values depend on many other parameters. Using these arguments, it will be demonstrated that contact angles are not the cause of wettability behavior but the effect of wettability behavior and other parameters. The result of this is that the contact angle cannot be used as a primary indicator of wettability except in very restricted situations. Furthermore, it will be demonstrated that even for the simple case of a capillary interface in a vertical tube, attempting to use simply a two-dimensional image to determine the contact angle can result in a wide range of measured values. This observation is consistent with some published experimental results. It follows that contact angles measured in two-dimensions cannot be trusted to provide accurate values and these values should not be used to characterize the wettability of the system.

Fluctuations and Shear Modulus of a Classical Two-Dimensional Electron Solid: Experiment

1982 ◽  
Vol 49 (3) ◽  
pp. 212-215 ◽  
Author(s):  
F. Gallet ◽  
G. Deville ◽  
A. Valdès ◽  
F. I. B. Williams
Keyword(s):  
Shear Modulus ◽  
Two Dimensional ◽  
Dimensional Electron
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