Nonlinear Elasticity

W szczegółowy sposób omówione zostały założenia ogólnej nieliniowej teorii sprężystości i wyprowadzenia podstawowych elementów mechaniki ośrodka ciągłego, a także przytoczone zostały dowody fundamentalnych twierdzeń rozważanej teorii w ramach kinematyki, statyki i dynamiki ciał odkształcalnych. Omówiono kwestie formułowania związków konstytutywnych dla ciał izotropowych i anizotropowych zarówno o liniowej, jak i nieliniowej charakterystyce, ze szczególnym uwzględnieniem zagadnienia energetycznego sprzężenia miar naprężenia i odkształcenia oraz zastosowania rozkładu widmowego tensorów sprężystości. Osobno omówiono liniowy wariant teorii z różnymi jego sformułowaniami oraz metodami analitycznymi, m.in. zastosowaniem funkcji naprężeń, potencjałów, metod półodwrotnych oraz analizy zespolonej. Rozważania te poszerzono o opis teorii ogólniejszych – mechanikę ośrodka mikropolarnego Cosseratów oraz mechanikę ośrodka mikromorficznego Mindlina. W ramach teorii liniowej przedstawiono matematyczny opis zjawisk elastooptyki, propagacji płaskich i sferycznych fal sprężystych, a także propagacji sprężystych fal powierzchniowych. Przytoczono i udowodniono twierdzenia wariacyjne teorii liniowej oraz omówiono ich zastosowanie w ramach numerycznych metod rozwiązywania zagadnień tej teorii. Przeanalizowano rozwiązania trzynastu klasycznych zagadnień liniowej teorii sprężystości oraz zagadnień pokrewnych, ze szczególnym uwzględnieniem zagadnień dynamicznych. Całość uzupełniona została rozbudowanym dodatkiem poświęconym rachunkowi tensorowemu i absolutnemu rachunkowi różniczkowemu.

We present rigorous local and global bifurcation results for a concrete example from 3-dimensional nonlinear elastostatics — the problem of barrelling of compressed cylindrical columns. We use standard tools of bifurcation theory for the local analysis, already producing results that are rare in our field. For the global part we employ the generalized degree designed by Healey and Simpson to overcome the specific difficulties of 3-dimensional nonlinear elasticity. Ours are the first global bifurcation results for a problem from 3-dimensional nonlinear elastostatics not governed by ordinary differential equations. Moreover, our approach to the barrelling problem provides a paradigm for the solution of a large class of problems in nonlinear elastostatics concerning bifurcation from a homogeneously deformed state.

This paper represents a first attempt to derive one-dimensional models with non-convex strain energy starting from “genuine” three-dimensional, nonlinear, compressible, elasticity theory. Following the usual method of obtaining beam theories, we show here for a constrained kinematics appropriate for long cylinders governed by a polyconvex, objective, stored energy function, that the bar model originally proposed by Ericksen [3] is obtainable but enriched by an additional term in the strain gradient. This term, characteristic of nonsimple grade-2 materials, penalizes interfacial energies and makes single-interface two-phase solutions preferred. The resulting model has been proposed by a number of authors to describe the phenomenon of necking and cold drawing in polymeric fibers and, here, we discuss its suitability to interpret also the elastic-plastic behavior of metallic tensile bars under monotone loading.

Carbon-bonded refractories are widely used in steel industry. They are well known for their outstanding thermal shock behavior. The knowledge of the elastic properties at high temperatures will make simulations for thermal shock more effective or contribute to understand the material behavior based on its micro structural components as well as the contribution of each one. In this work measurements of the Young’s modulus of elasticity at high temperatures of carbon-bonded alumina refractories with two different amounts of carbon obtained by impulse excitation technique have been investigated. The measurements showed a significant increase of the Young's modulus of elasticity up to 1450°C independently of the carbon amount.  Furthermore, a clear hysteresis during cooling was observed.

► A kind of Elastic Composite, Reinforced Lightweight Concrete (ECRL.C) with the mentioned specifics is a type of "Resilient Composite Systems (RCS)" in which, contrary to the basic geometrical assumption of the flexure theory in Solid Mechanics, "the strain changes in the beam height during bending" is typically "Nonlinear". (The RCS could be counted as "The Methodically Reinforced Nonlinear Porous Materials", also having the high specific modulus of resilience in flexure.)

♦ Through employing this integrated structure, with significantly high strain capability and modulus of resilience in bending, we can achieve the high bearing capacities in beams with the secure fracture pattern, in less weight.

♦ Due to the system particulars and its behavior in bending, the usual calculation of the necessary equilibrium steel amount to attain the low-steel bending sections with the secure fracture pattern in the beams and its related limitations do not become propounded. Thereby, the strategic deadlock of the high possibility of the brittle fracture pattern in the bending elements made of the usual reinforced lightweight concretes, especially about the low-thickness bending elements as slabs, is unlocked.

♦ This simple, applied technology and the related components and systems can have several applications in the road and building industries. (It can also be used in making the resilient pieces and constructions "with appropriate behavior and high resistance against severe blasts and shocks.)

♦ Regarding the "strategic importance of Lightweight and Integrated Construction in the practical increase of the resistance and safety against earthquake" and considering the appropriate behavior of this "resilient", durable structure against the dynamic loads, shakes, impacts, blasts, and shocks and the possibility of making some lightweight and insulating, non-brittle, reinforced sandwich panels and pieces, this resilient system and its components can especially be useful in the "seismic areas".

♦ This system can also be employed in constructing the vibration and impact absorber bearing pieces & slabs, which can be used in the "Railroad & Subway Structures" too.

♦ Here, the "Resilient Composite Systems (RCS)" and particularly, the ECRLC as a type of the RCS have been concisely presented. [By the way, an instance of the said new structure and its components and the results of some performed experiments (as bending loading and compressive loading of the slabs made of this structure, similar to ASTM E 72 Standard) have been shown in the related pictures & figures.]
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/ ● Key Words: Strength of materials (solid mechanics), Civil (construction), Materials, Earthquake (resistance and safety), Resilient concrete (flexible concrete, bendable concrete, elastic concrete), Composite concrete, Lightweight concrete, Reinforced concrete, Fibered concrete, Lightweight and integrated construction, Rail (railroad, railway), Subway, Road, Bridge, Resilience, Energy absorption, Fracture pattern, Non-linear, Strain changes, Beam, Ductility, Toughness, Insulating (insulation), Thin, Slab, Roof, Ceiling, Wall (partition), Building, Tower, Plan of mixture, Insulating reinforced lightweight pieces, 3d, Sandwich panel, Dry mix, Plaster, Foam, Expanded polystyrene (EPS), Polypropylene, Pozzolan, Porous matrix (Pored matrix), Mesh (lattice), Cement, RCS, ECRLC
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/ ► Contents:
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* ABSTRACT
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* I. INTRODUCTION
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* II. WHAT ARE THE RESILIENT COMPOSITE SYSTEMS
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- A. General Review
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- B. Components
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. - 1) Mesh or Lattice
. - 2) Fibers or Strands
. - 3) "Matrix" With the Suitable Hollow "Pores (Voids)" and/or "Lightweight Aggregates" in its Context
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- C. More Explanations About the RCS
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- D. Why Are These Systems Called "Composite"?
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- E. The General Structural Particulars and Functional Criteria as the Necessary Specifications of the Compound Materials Generally Called "Resilient Composite Systems"
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. - 1) General Structural Criteria
. - 2) Functional Criteria (Required Specifications)
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* III. "ELASTIC COMPOSITE REINFORCED LIGHTWEIGHT CONCRETE (ECRLC)" AS A TYPE OF THE RESILIENT COMPOSITE SYSTEMS (RCS)
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- An Instance of the Lightweight Concrete That Could be Used in Making the ECRLC
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* IV. REVIEW OF SOME EXPERIMENTS, AND MORE DESCRIPTION ABOUT ECRLC
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* V. SUPPLEMENTARY ELEMENTS
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* VI. APPLICATIONS
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* VII. FINAL REVIEW
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* ACKNOWLEDGEMENTS
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* REFERENCES 
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***
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/ ● General Review:
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• A kind of "Elastic Composite, Reinforced Lightweight Concrete" with the said specifics is a type of the "Resilient Composite Systems (R.C.S.)" in which, contrary to the basic geometrical assumption of flexure theory in the Solid Mechanics, the strain changes in the beam height during bending is typically "Non-linear".
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• Indeed, the RCS, as the Elastic Composite, Reinforced Lightweight Concrete (ECRLC), do not behave as most of the solid materials in bending.
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• In the "Resilient Composite Systems", distributed pores and/or appropriate lightweight aggregates or beads, accompanied by the reticular structure of the strengthened conjoined matrix, bring about the expedient internal shape changes during bending and continuing the elasticity in bending with the said nonlinearly pattern. This means better distribution of the stresses and strains and better utilizing the potential capacities of the employed reinforcements in bending and tension; whereas, in the usual lightweight concretes for instance, distributed hollow pores (such as the gas bulbs in the cellular concretes) or lightweight aggregates (such as Plastic, Rubber or polystyrene beads or any other kind of lightweight aggregates such as Perlite and Vermiculite) decrease the modulus of resilience in bending and could increase "the possibility of beam fracture of brittle and primary compressive type" in bending (compared to the concrete with higher density) according to the case.
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• In this way, by using the mentioned method to make the said particular composite systems, we could considerably increase the modulus of resilience and bearing capacity in bending "together with" significant decrease of the weight and also the possibility of beam fracture of primary compressive type. Through making these particular integrated functioning systems, for the first time, the said (paradoxical) properties have been concomitantly fulfilled in "one functioning unit" altogether.
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• Respect to the special pattern of the strain changes during bending in the particular Resilient Composite System termed ECRLC, this system as an integrated functioning unit with the reticular arrangement and texture has more strain capability (particularly within the elastic limit), energy absorption and load bearing capacities in bending compared to the usual reinforced concrete beams.
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• Thereby, through employing this applied structure, solving some of the main problems in lightweight concretes application, especially the deadlock of brittle and insecure being of fracture pattern in many of the usual reinforced lightweight concrete structures, is provided; reaching to the high bearing capacities in bending elements (even with low dimensions & weights) is to hand, and getting access to a simple and practical opportunity for "qualitative development of possibilities of using lightweight concretes" (especially with oven-dry densities of < 1350-1400kg/m3 and compressive strengths of <14-17mpa, and even with oven-dry densities of < 800kg/m3) is conceivable.
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• Naturally, by more studies in the field, these structures and their applications in various fields could be developed more.
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/ ► FULL TEXT (Open Access):
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- http://arxiv.org/abs/1510.03933 
( https://arxiv.org/ftp/arxiv/papers/1510/1510.03933.pdf ) ;
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. Kamyar Esmaeili: "Elastic Composite Reinforced Lightweight Concrete as a Type of Resilient Composite Systems";
The Internet Journal of Innovative Technology and Creative Engineering (IJITCE); 2012; 2(8): 1-22.
- http://ia800305.us.archive.org/34/items/IJITCE/vol2no801.pdf [Also archived at: http://www.webcitation.org/6B2pFPpBh ] ;
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- https://www.scribd.com/doc/11530978/Elastic-Composite-Reinforced-Lightweight-Concrete-ECRLC-as-a-type-of-Resilient-Composite-Systems-RCS-http-arxiv-org-abs-1510-03933 ; 
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- https://sites.google.com/site/NEWSTRUCTURE1 ;
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- https://sites.google.com/site/newstructure1/publications-page/ELASTIC%20COMPOSITE%2C%20REINFORCED%20LIGHTWEIGHT%20CONCRETE%20AS%20A%20TYPE%20OF%20RESILIENT%20COMPOSITE%20SYSTEMS.pdf ;
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- http://www.pdf-archive.com/2015/09/22/elasticcomposite-reinforcedlightweightconcreteasatypeofrcs/elasticcomposite-reinforcedlightweightconcreteasatypeofrcs.pdf ;
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- The link to the document in PDF format on this site:
file:///C:/Users/pc5/Downloads/ELASTIC_COMPOSITE_REINFORCED_LIGHTWEIGHT.pdf ;
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- ...

Plastic hinge effects are defined as the moment at which the entire cross section has reached its yield stress. This is theoretically the maximum bending moment that a cross section can resist. When this point is reached a plastic hinge is developed and the structure becomes an unstable mechanism.

Change of type in the governing equations of equilibrium is examined in the context of a base neo-Hookean response augmented with unidirectional reinforcing that is characterized by a single additional constitutive parameter for strength of reinforcement. Plane deformations interpreted in terms of both local and global plane strain are considered. Loss of ordinary ellipticity is found to occur for sufficiently large strength of reinforcement under sufficiently severe deformation which necessarily involves contraction in the reinforcing direction. Loss of ellipticity in local plane strain is easily characterized, and its incipient breakdown is associated with the possible emergence of surfaces of weak discontinuity with orientation normals in the reinforcing direction. Loss of ellipticity in global plane strain is given a two-dimensional manifold characterization in a space involving 2 deformation parameters and the strength of reinforcing parameter. Orientation normals for the assoc...

The minimum principle of complementary energy is established for cable net- works involving only stress components as variables in geometrically nonlinear elasticity theory. The stress-strain relation is assumed to be piecewise linear. It is rather amazing that the complementary energy always attains minimum value at the equilibrium state irrespective of the stability of cable networks, contrary to the fact that

We define a class of deformations in $W^{1,p}(\Omega,\mathbb{R}^n)$, $p>n-1$, with positive Jacobian that do not exhibit cavitation.
We characterize that class in terms of the non-negativity of the topological degree and the equality between the distributional determinant and the pointwise determinant of the gradient.
Maps in this class are shown to satisfy a property of weak monotonicity, and, as a consequence, they enjoy an extra degree of regularity.
We also prove that these deformations are locally invertible; moreover, the neighbourhood of invertibility is stable along a weak convergent sequence in $W^{1,p}$, and the sequence of local inverses converges to the local inverse.
We use those features to show weak lower semicontinuity of functionals defined in the deformed configuration and functionals involving composition of maps.
We apply those results to prove existence of minimizers in some models for nematic elastomers and magnetoelasticity.

In order to reduce the costs related to corrosion damage in aircraft structures, it is vital to develop new robust, accurate and reliable damage detection methods. A possible answer to this problem is offered by newly developed nonlinear ultrasonic techniques, which monitors the nonlinear elastic wave propagation behaviour introduced by damage, to detect its presence and location.In this paper, a new nonlinear time reversal technique is presented for the detection and localization of a scattered zone (damage) in a multi-material medium. In particular, numerical findings on a friction stir-welded aluminium plate-like structure are reported. Damage was introduced in the heat affected zone and modelled using a multi-scale material constitutive model (Preisach–Mayergoyz space).Studies were conducted for two different transducer configurations. Particular attention was devoted to find the optimum time-reversed window to be re-emitted in the structures. The methodology was compared with traditional time-reversal acoustics (TRA), showing significant improvements. While the traditional TRA was not able to clearly localise the damage, the developed technique identified in a clear manner the faulted zone, showing its robustness to locate and characterize nonlinear sources, in presence of a multi-material medium.

"Se presenta el análisis no lineal dinámico en tres direcciones de 6 edificiosde concreto reforzado de 5, 12 y 20 pisos, diseñados con la NSR-10 yla zonificación de respuesta sísmica de Bogotá del año 2010. Estos 6 edificios,con planta típica idéntica, fueron sometidos a 78 señales sísmicas del sismo deQuetame (año 2008) registradas por 26 estaciones de la red de acelerógrafosde Bogotá (RAB). Para el diseño de los edificios se utilizaron los espectros derespuesta de aceleración absoluta para las zonas: «Piedemonte-B» y «Lacustre-500» de la microzonificación sísmica de Bogotá. Posteriormente, se calcularonlas rótulas plásticas para cada elemento estructural de los 6 edificios, y en elcaso de las columnas, también sus diagramas de interacción. Luego, las edificacion estridimensionales fueron sometidas simultáneamente a los registros contra el tiempo norte-sur, este-oeste y vertical. Para cada edificio sometidoa las señales de cada estación, se obtuvieron las demandas de deriva, los desplazamientosde la cubierta, el cortante en la base y los niveles de daño segúnestimadores internacionalmente aceptados. Finalmente, se generaron mapasde demandas de deriva y aceleración por edificio en la ciudad de Bogotá D.C.Los resultados sugieren que los edificios diseñados con la amenaza sísmica dela nueva microzonificación de Bogotá y con las estipulaciones de la NSR-10,presentarán daños excesivos asociados a estados límites como son los de «seguridadde vidas» para un sismo de baja magnitud como fue el de Quetame,que equivalió a solo una fracción de aquel que corresponde al comportamientoreal bajo el escenario de diseño"

Elastomers are being increasingly used for sealing and other applications in the oil and gas industry. Specifically developed elastomers possess durable properties and have the ability to withstand detrimental effects of heat, chemicals, and harsh environments. For successful modeling and simulation of various downhole processes, it is very important to determine the behavior of elastomer materials under realistic well conditions. Of special interest is the class known as swelling elastomers. This article reports some results from experiments conducted on mechanical testing and characterization of an inert (nonswelling) and a water-swelling elastomer (both belonging to the EPDM family) used for sealing purposes by a local petroleum development firm. Experiments were designed and conducted in accordance with standard ASTM test methods. Apart from regularly available testing equipment, some simple test rigs and fixtures were designed and fabricated. Elastomer behavior was tested for h...

Hysteretic damping is often modeled by means of linear viscoelastic approaches such as "nearly constant Attenuation (NCQ)" models. These models do not take into account nonlinear effects either on the stiffness or on the damping, which are well known features of soil dynamic behavior. The aim of this paper is to propose a mechanical model involving nonlinear viscoelastic behavior for isotropic materials. This model simultaneously takes into account nonlinear elasticity and nonlinear damping. On the one hand, the shear modulus is a function of the excitation level; on the other, the description of viscosity is based on a generalized Maxwell body involving non-linearity. This formulation is implemented into a 1D finite element approach for a dry soil. The validation of the model shows its ability to retrieve low amplitude ground motion response. For larger excitation levels, the analysis of seismic wave propagation in a nonlinear soil layer over an elastic bedrock leads to results which are physically satisfactory (lower amplitudes, larger time delays, higher frequency content).

The computer program VABS (Variational Asymptotic Beam Section Analysis) uses the vari- ational asymptotic method to split a three-dimensional nonlinear elasticity problem into a two- dimensional linear cross-sectional analysis and a one-dimensional, nonlinear beam problem. This is accomplished by taking advantage of certain small parameters inherent to beam-like structures. VABS is able to calculate the one-dimensional cross-sectional stiffness constants, with

We show that a model proposed by Rubin, Rosenau, and Gottlieb [J. Appl. Phys. 77 (1995) 4054], for dispersion caused by an inherent material characteristic length, belongs to the class of simple materials. Therefore, it is possible to generalize the idea of Rubin, Rosenau, and Gottlieb to include a wide range of material models, from nonlinear elasticity to turbulence. Using this insight, we are able to fine-tune nonlinear and dispersive effects in the theory of nonlinear elasticity in order to generate pulse solitary waves and also bulk travelling waves with compact support.

... Printed in the Netherlands. Simulation of ductile crack growth in thin aluminum panels using 3-D surface cohesive elements Y. ARUN ROY and ROBERT H. DODDS JR. ∗ Department of Civil Engineering, University of Illinois, Urbana, IL 61801, USA ...