Finite Dilatation Research Articles

Fluid flow past a freely moving body in a straight or distorted channel

The focus here is on a thin solid body passing through a channel flow and interacting with the flow. Unsteady two-dimensional interactive properties from modelling, analysis and computation are presented along with comparisons. These include the effects of a finite dilation or constriction, as the body travels through, and the effects of a continuing expansion of the vessel. Finite-time clashing of the body with the channel walls is investigated as well as the means to avoid clashing. Sustained oscillations are found to be possible. Wake properties behind the body are obtained, and broad agreement in trends between full-system and reduced-system responses is found for increased body mass.Graphical abstract

Meanings in the Digital Age. Homo Singularity

The paper approaches the techno-humanitarian balance of physical (accelerating) and humanitarian (controlling) technologies. It demonstrates that the absence of the human in this ba­lance makes the idea about ensuring the socio-system’s sustainable development through the establishment of techno-humanitarian balance erroneous. The required adequate proportion of “powers” between the technologies in the techno-humanitarian balance necessitates the civilization to attempt to “harness” not only the technological singularity but the humanitarian singularity, too. It is shown that the techno-humanitarian balance in the singularity mode destroys the human physical and mental nature. The human is introduced into the binary technological balance through transforming it into a triple balance and adding the semantic technologies inherent to human beings. The triple balance is characterized by the oppositions between intelligence and thinking, information and meanings. The paper explores the triple balance and its edges in the context of the ultimate singularity. It is shown that the human being, through thinking and meanings, can correlate themselves with the semantic singularity (the infinity of the semantic field) — thus becoming Homo Singularity. These conclusions are substantiated through the V.V. Nalimov’s probabilistic model of consciousess, which also mathe­matically formalizes the process of semantic decapsulation of the personality and the personal interaction with the semantic vacuum (infinity).The paper introduces the concept of finite dilatation of the cultural semantic field and the big semantic transition as the era of formation of Homo Singula­rity and beginning of their practical work with semantic infinity. The paper provides examples of such practical work in the fields of art and science. Homo Singularity not only prevents their physical and mental nature from destruction but also keeps the po­werful artificial intelligence under control by counte­ring big data with their ability to integrate the multiple into the single (whole) and make a transition from the discrete level of information to a continuous (infinite) level of meanings.

Geometric properties of infinite graphs and the Hardy–Littlewood maximal operator

We study different geometric properties on infinite graphs, related to the weak-type boundedness of the Hardy–Littlewood maximal averaging operator. In particular, we analyze the connections between the doubling condition, having finite dilation and overlapping indices, uniformly bounded degree, the equidistant comparison property and the weak-type boundedness of the centered Hardy–Littlewood maximal operator. Several non-trivial examples of infinite graphs are given to illustrate the differences among these properties.

A 3D two-scale multiplane cohesive-zone model for mixed-mode fracture with finite dilation

A 3D multi-scale cohesive-zone model (CZM) combining friction and finite dilation by a multi-plane approach (M-CZM), based on the concept of Representative Multiplane Element (RME), is developed within the mechanics of generalized continua for the analysis of mixed-mode fracture.The proposed M-CZM formulation captures the increase of measured fracture energy in mode II as a natural effect of multi-scale coupling between cohesion, friction and interlocking, employing a reduced set of micromechanical parameters characterized by a well-defined micromechanical interpretation. This permits to devise clear calibration and identification procedures for 3D fracture problems.Upon assessing the retrieval, by a regular 5-plane RME, of a quasi-isotropic response for fracture resistance and for dilation, the M-CZM is employed in FEM simulations of Double-Cantilever Beam (DCB) tests to obtain predictions of mixed mode I–II and mixed mode I–III fracture resistance.The DCB analyses show the key role of the characteristic height of asperities in determining the macroscopic fracture resistance in both mixed mode I–II and I–III interactions. Numerical results also show the independence of the mode-I fracture resistance on the geometry of the beam section and a marked dependence of the measured mixed-mode fracture resistance on the section aspect ratio.

A mixed-mode cohesive-zone model accounting for finite dilation and asperity degradation

A cohesive zone model is formulated to describe the mechanics of initiation and propagation of cracks and the associated asperity degradation and nonlinear dilation along structural interfaces of quasi-brittle materials, such as concrete, rocks and masonry, subjected to monotonic or cyclic loading. Using a two-scale approach, a cohesive-law is determined at each point of a smooth macroscale interface by resolving a problem at the micro-scale for a representative interface area (RIA), where the geometry of the asperities is modelled using three differently inclined microplanes. On each microplane a cohesive-frictional cohesive law is then used. In this paper, the finite depth of the asperities is accounted for by considering the progressive reduction in contact area between each couple of interfacing microplanes for increasing opening (macro-scale) relative-displacement. Furthermore, the rupture of the asperities and associated flattening of the fracture surface is captured by a progressive reduction of the inclination angles of the microplanes in the RIA. Numerical examples are reported to assess the sensitivity of the shear-stress slip curves and of the nonlinear dilation upon the geometry of the asperities in the RIA. Numerical–experimental comparisons are then presented to illustrate the predictive capability of the model in simulating granite rock joints subjected to monotonic and cyclic shear loading and the concrete-bar interaction in a pull-out test.

On differentiability of mappings with finite dilation

Abstract We study mappings f : (a,b) → Y with finite dilation having Lebesgue integrable majorant, where Y is a real normed vector space. We construct Lipschitz mapping f : (a,b) → Y, dim Y = ∞ , which is nowhere differentiable but its graph has everywhere trivial contingent. We show that if the contingent of the graph of a mapping with finite dilation is a nontrivial space, then f is almost everywhere differentiable.

Direct Measurement of the Surface Tension of a Soft Elastic Hydrogel: Exploration of Elastocapillary Instability in Adhesion

An adhesively stressed thin film of a soft hydrogel confined between two rigid flat substrates autoroughens with its dominant wavelength (λ) exhibiting pronounced dependence on the film thickness (H). A linear stability analysis confirmed that this long wavelength instability (λ ~ 7H) is due to an elastocapillary effect, the implementation of which required direct measurements of the surface tension and the elasticity of the gel. The surface tension of the gel was estimated from the fundamental spherical harmonic of a hemispherical cap of the gel that was excited by an external noise. The shear modulus (μ) of the gel was determined from its resonant shear mode in a confined geometry. During the course of this study, it was found that a high density steel ball submerges itself inside the gel by balancing its excess weight with the accumulated strain induced elastic force that allows another estimation of its elastic modulus. The large ratio (1.8 mm) of the surface tension to its elasticity ascertains the role of elastocapillarity in the adhesion-induced pattern formation with such gels. Experimental results are in accord with a linear stability analysis that predicts that the rescaled wavelength λ(μH/γ)(0.27) is linear with H, which also modifies the conventional stress to pull a flat rigid object out of a very soft film by a multiplicative factor: (γ/μH)(1/4). The analysis also suggests some new results related to the role of the finite dilation of a material in interfacial pattern formation that may have nontrivial consequences in the adhesive delamination of very thin and/or soft elastic films via self-generated cracks.

Open discrete mappings having local ACL n inverses

We consider open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f , so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and . For this class of mappings (or even for larger classes of open, discrete mappings) we generalize the important modular inequality of Poleckii, also using the modular estimates of the spherical rings from Cristea (Local homeomorphism having local ACL n inverses, Complex Var. Elliptic Equ. 53(1) (2008), 77–99). We continue the work from the same paper by generalizing some basic facts from the theory of quasiregular mappings. We give equicontinuity results, Picard, Montel and Liouville type theorems, estimates of the modulus of continuity, analogues of Schwarz's lemma, eliminability results and boundary extensions theorems. Together with the multiple extensions of Zoric's theorem from the paper, we establish strong generalizations of one of the most important theorems from the theory of quasiregular mappings. We also extend similar results given in some recent classes of functions larger than the class of quasiregular functions, as the class of mappings of finite distortion and satisfying condition (𝒜), or the class of mappings of finite dilatation with dilatation in the bounded mean oscillation class.

On solvent diffusion in a solid with large dilatation

A theoretical framework is developed for solvent diffusion in solids with large dilatation due to solvent induced swelling. The resulting non-linear governing differential equations are not amenable to a closed form solution, and hence, a solution technique based on the finite element method is presented and implemented. Several demonstration problems are solved using an in-house finite element code (NOVA-3D) to highlight the importance of including the effect of finite dilatation in a diffusion model for hygroscopic materials. The effect of finite dilatation on the calculation of diffusivity is discussed, as well as the presence of large shearing stresses induced by the concentration gradient.

Absolutely Continuous Functions of Several Variables

A class of “absolutely continuous functions” ofnvariables is introduced. The absolute continuity implies continuity, weak differentiability with gradient inLn, differentiability almost everywhere, area, coarea and degree formulae. This gives a unified approach to some previously known conditions for the mentioned results based on monotonicity, finite dilatation or higher integrability of the gradient.

Axisymmetric three‐ and four‐node finite elements for large strain elastoplasticity

AbstractWithin the framework of the finite element method an application of the logarithmic strain space formulation of large strain elastoplasticity is illustrated for the examples of axisymmetric three‐node triangular and four‐node quadrilateral finite elements. The formulation of the large strain elastoplasticity is based on a strain space formulation in conjunction with logarithmic (or Hencky) strain tensors with respect to the reference configuration. It is therefore—from a material point of view—a full Lagrangian formulation. The use of logarithmic strains enables an additive split of finite dilatation and distortion, which are given by the logarithmic strain trace and deviator. As a consequence of the strain space formulation no stress tensors are involved in order to describe the plasticity. The stress which is work‐conjugate to the logarithmic strain follows from the stress‐strain relations and may be transformed to Cauchy stress. The desired finite element matrices are derived via the principle of virtual work applied to the Cauchy stress distribution of the current configuration. It should be noted that our considerations are not restricted to axisymmetry and that they remain valid for isoparametric, position‐ (displacement‐) based finite elements in general.

Weakly monotone functions

The definition of monotone function in the sense of Lebesgue is extended to the Sobolev spacesW 1,p ,p >n − 1. It is proven that such weakly monotone functions are continuous except in a singular set ofp-capacity zero that is empty in the casep =n. Applications to the regularity of mappings with finite dilatation appearing in nonlinear elasticity theory are given.

On mappings with integrable dilatation

A factorization of Stoilow’s type has been obtained for mappings in R 2 {\mathbb {R}^2} with integrable dilatation.

Certain conditions for holomorphicity in Hilbert spaces

New conditions for the C-differentiability of IR-differentiable mappings are obtained and a theorem on the holomorphicity of the mappings that have finite dilatation along cones is proved.