Aspects of topology of condensates and knotted solitons in condensed matter systems

The knotted solitons introduced by Faddeev and Niemi is presently a subject of great interest in particle and mathematical physics. In this paper we give a condensed matter interpretation of the recent results of Faddeev and Niemi.Comment: v2: Added a reference to the paper E. Babaev, L.D. Faddeev and A.J. Niemi cond-mat/0106152 where an exact equivalence was shown between the two-condensate Ginzburg-Landau model and a version of Faddeev model. Miscelaneous links related to knotted solitons are available at the author homepage at http://www.teorfys.uu.se/PEOPLE/egor/ . Animations of knotted solitons by Hietarinta and Salo are available at http://users.utu.fi/h/hietarin/knots/c45_p2.mp

Elastodynamics of radially inhomogeneous spherically anisotropic elastic materials in the Stroh formalism

A method is presented for solving elastodynamic problems in radially inhomogeneous elastic materials with spherical anisotropy, i.e.\ materials such that $c_{ijkl}= c_{ijkl}(r)$ in a spherical coordinate system ${r,\theta,\phi}$. The time harmonic displacement field $\mathbf{u}(r,\theta ,\phi)$ is expanded in a separation of variables form with dependence on $\theta,\phi$ described by vector spherical harmonics with $r$-dependent amplitudes. It is proved that such separation of variables solution is generally possible only if the spherical anisotropy is restricted to transverse isotropy with the principal axis in the radial direction, in which case the amplitudes are determined by a first-order ordinary differential system. Restricted forms of the displacement field, such as $\mathbf{u}(r,\theta)$, admit this type of separation of variables solutions for certain lower material symmetries. These results extend the Stroh formalism of elastodynamics in rectangular and cylindrical systems to spherical coordinates.Comment: 15 page

Neutrally reinforced holes in symmetrically laminated plates

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/76616/1/AIAA-46360-396.pd

Transformation cloaking and radial approximations for flexural waves in elastic plates

It is known that design of elastic cloaks is much more challenging than that of acoustic cloaks, cloaks of electromagnetic waves or scalar problems of antiplane shear. In this paper, we address fully the fourth-order problem and develop a model of a broadband invisibility cloak for channelling flexural waves in thin plates around finite inclusions. We also discuss an option to employ efficiently an elastic pre-stress and body forces to achieve such a result. An asymptotic derivation provides a rigorous link between the model in question and elastic wave propagation in thin solids. This is discussed in detail to show connection with non-symmetric formulations in vector elasticity studied in earlier work

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

Mineralized collagen fibrils have been usually analyzed like a two phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that, when Halpin-Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations and a new finite lement model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.The authors acknowledge the Ministerio de Economia y Competitividad the financial support given through the project DPI2010-20990 and the Generalitat Valenciana through the Programme Prometeo 2012/023. The authors thank Ms. Carla Gonzalez Carrillo by her help in the development of some of the numerical models.Vercher Martínez, A.; Giner Maravilla, E.; Arango Villegas, C.; Tarancón Caro, JE.; Fuenmayor Fernández, FJ. (2014). Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models. 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Nature 195:513Cusack S, Miller A (1979) Determination of the elastic constants of collagen by brillouin light scattering. J Mol Biol 135:39–51Doty S, Robinson RA, Schofield B (1976) Morphology of bone and histochemical staining characteristics of bone cells. In: Aurbach GD (ed) Handbook of physiology. American Physiology Soc, Washington, pp 3–23Erts D, Gathercole LJ, Atkins EDT (1994) Scanning probe microscopy of crystallites in calcified collagen. J Mater Sci Mater Med 5:200–206Faingold A, Sidney RC, Wagner HD (2012) Nanoindentation of osteonal bone lamellae. J Mech Biomech Materials 9:198–206Franzoso G, Zysset PK (2009) Elastic anisotropy of human cortical bone secondary osteons measured by nanoindentation. J Biomech Eng 131:021001Gebhardt W (1906) Über funktionell wichtige Anordnungsweisen der eineren und grösseren Bauelemente des Wirbeltierknochens. II. Spezieller Teil. Der Bau der Haversschen Lamellensysteme und seine funktionelle Bedeutung. Arch Entwickl Mech Org 20:187–322Gibson RF (1994) Principles of composite material mechanics. McGraw-Hill, New YorkGiraud-Guille M (1988) Twisted plywood architecture of collagen fibrils in human compact bone osteons. Calcif Tissue Int 42:167–180Gurtin ME (1972) The linear theory of elasticity. Handbuch der Physik VIa/ 2:1–296Halpin JC (1992) Primer on composite materials: analysis, 2nd edn. CRC Press, Taylor & Francis, Boca Raton, FloridaHassenkam T, Fantner GE, Cutroni JA, Weaver JC, Morse DE, Hanma PK (2004) High-resolution AFM imaging of intact and fractured trabecular bone. Bone 35:4–10Hohe J (2003) A direct homogenization approach for determination of the stiffness matrix for microheterogeneous plates with application to sandwich panels. Composites Part B 34:615–626Hulmes DJS, Wess TJ, Prockop DJ, Fratzl P (1995) Radial packing, order, and disorder in collagen fibrils. Biophys J 68:1661–1670Jäger I, Fratzl P (2000) Mineralized collagen fibrils: a mechanical model with a staggered arrangement of mineral particles. Biophys J 79:1737–1746Ji B, Gao H (2004) Mechanical properties of nanostructure of biological materials. J Mech Phy Sol 52:1963–1990Landis WJ, Hodgens KJ, Aerna J, Song MJ, McEwen BF (1996) Structural relations between collagen and mineral in bone as determined by high voltage electron microscopic tomography. Microsc Res Tech 33:192–202Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, San FranciscoLempriere BM (1968) Poisson’s ratio in orthotropic materials. Am Inst Aeronaut Astronaut J J6:2226–2227Lowenstam HA, Weiner S (1989) On biomineralization. Oxford University, New YorkLusis J, Woodhams RT, Xhantos M (1973) The effect of flake aspect ratio on flexural properties of mica reinforced plastics. Polym Eng Sci 13:139–145Martínez-Reina J, Domínguez J, García-Aznar JM (2011) Effect of porosity and mineral content on the elastic constants of cortical bone: a multiscale approach. Biomech Model Mechanobiol 10:309–322Orgel JPRO, Miller A, Irving TC, Fischetti RF, Hammersley AP, Wess TJ (2001) The in situ supermolecular structure of type I collagen. Structure 9:1061–1069Padawer GE, Beecher N (1970) On the strength and stiffness of planar reinforced plastic resins. Polym Eng Sci 10:185–192Pahr DH, Rammerstofer FG (2006) Buckling of honeycomb sandwiches: periodic finite element considerations. Comput Model Eng Sci 12:229–242Reisinger AG, Pahr DH, Zysset PK (2010) Sensitivity analysis and parametric study of elastic properties of an unidirectional mineralized bone fibril-array using mean field methods. Biomech Model Mechanobiol 9:499–510Reisinger AG, Pahr DH, Zysset PK (2011) Elastic anisotropy of bone lamellae as a function of fibril orientation pattern. Biomech Model Mechanobiol 10:67–77Rezkinov N, Almany-Magal R, Shahar R, Weiner S (2013) Three-dimensional imaging of collagen fibril organization in rat circumferential lamellar bone using a dual beam electron microscope reveals ordered and disordered sub-lamellar structures. Bone 52(2):676–683Rho JY, Kuhn-Spearing L, Zioupos P (1998) Mechanical properties and the hierarchical structure of bone. Med Eng Phys 20:92–102Rubin MA, Jasiuk I, Taylor J, Rubin J, Ganey T, Apkarian RP (2003) TEM analysis of the nanostructure of normal and osteoporotic human trabecular bone. Bone 33:270–282Suquet P (1987) Lecture notes in physics-homogenization techniques for composite media. Chapter IV. Springer, BerlinWagermaier W, Gupta HS, Gourrier A, Burghammer M, Roschger P, Fratzl P (2006) Spiral twisting of fiber orientation inside bone lamellae. Biointerphases 1:1–5Wagner HD, Weiner S (1992) On the relationship between the microstructure of bone and its mechanical stiffness. J Biomech 25:1311–1320Weiner S, Wagner HD (1998) The material bone: structure-mechanical function relations. Annu Rev Mater Sci 28:271–298Weiner S, Traub W, Wagner H (1999) Lamellar bone: structure-function relations. J Struct Biol 126:241–255Yao H, Ouyang L, Ching W (2007) Ab initio calculation of elastic constants of ceramic crystals. J Am Ceram 90:3194–3204Yoon YJ, Cowin SC (2008b) The estimated elastic constants for a single bone osteonal lamella. Biomech Model Mechanobiol 7:1–11Yuan F, Stock SR, Haeffner DR, Almer JD, Dunand DC, Brinson LC (2011) A new model to simulate the elastic properties of mineralized collagen fibril. Biomech Model Mechanobiol 10:147–160Zhang Z, Zhang YWF, Gao H (2010) On optimal hierarchy of load-bearing biological materials. Proc R Soc B 278:519–525Zuo S, Wei Y (2007) Effective elastic modulus of bone-like hierarchical materials. Acta Mechanica Solida Sinica 20:198–20

Crack kinking at the tip of a mode I crack in an orthotropic solid

The competition between crack penetration and crack kinking is addressed for a mode I macroscopic crack in an orthotropic elastic solid. Cohesive zones of finite peak strength and finite toughness are placed directly ahead of and orthogonal to the plane of the parent crack. The cohesive zone ahead of the crack tip is tensile in nature and leads to crack penetration, whereas the inclined zones slide without opening under a combined shear and normal traction, and give crack kinking. Thereby, the competition between continued crack growth by penetration ahead of the crack tip versus kinking is determined as a function of the relative strength and relative toughness of the cohesive zones. This competition is plotted in the form of a failure mechanism map, with the role of material orthotropy emphasized. Synergistic toughening is observed, whereby the parent crack tip is shielded by the activation of both the tensile and shear (kinking) cohesive zones, and the macroscopic toughness is elevated. The study is used to assess the degree to which various classes of composite have the tendency to undergo kinking

Eigenvalue problems associated with Korn's inequalities

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46189/1/205_2004_Article_BF00251798.pd

On Saint-Venant's principle in plane anisotropic elasticity

Methods involving energy-decay inequalities are applied in investigating Saint-Venant's principle for the planproblem of linear elastostatics for a wide class of anisotropic media. A lower bound (in terms of the elastic constants) is obtained for the rate of exponential decay of stresses and this is compared with the known result for the isotropic case. Par une méthode applicable à un très grand nombre de milieu anisotropique, l'auteur utilise les inégalités concernant la décroissance de l'énergie dans le cas d'un problème plan et dans l'hypothèse de conditions élastostatiques linéaires, l'auteur précise la validité du principe de Saint-Venant dans le cadre de ses applications. Une limite inférieure (en fonction des constantes élastiques) de la décroissance exponentielle des contraintes est mise en évidence et comparée aux résultats obtenus en milieux isotropes.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42676/1/10659_2004_Article_BF00125525.pd