DOCSLIB.ORG

Dispersion of Surface Waves in a Transversely Isotropic Half-Space Underlying a Liquid Layer

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Dispersion of Surface Waves in a Transversely Isotropic Half-Space Underlying a Liquid Layer Scientia Iranica A (2016) 23(6), 2469{2477 Sharif University of Technology Scientia Iranica Transactions A: Civil Engineering www.scientiairanica.com Dispersion of surface waves in a transversely isotropic half-space underlying a liquid layer A. Bagheria, A. Khojastehb, M. Rahimiana; and R. Attarnejada a. School of Civil Engineering, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran. b. School of Engineering Science, College of Engineering, University of Tehran, P.O. Box 11155-4563, Tehran, Iran. Received 23 November 2014; received in revised form 23 August 2015; accepted 28 November 2015 KEYWORDS Abstract. Surface waves dispersion is studied in a two-layer half-space consisting of a nite liquid layer overlying a transversely isotropic solid half-space. A couple of complete Surface wave; potential functions are utilized to uncouple the equation of motion of the transversely Dispersion; isotropic solid along with a displacement potential for the liquid. The frequency equation Liquid; and velocity dispersion curves are developed. Several solid materials are considered for Transversely isotropic; the bed, and both phase and group velocity curves are calculated. Higher modes are also Rayleigh wave; discussed and the respective curves are plotted. Various special cases are considered by Stoneley wave; letting the liquid layer depth take zero value, to form a solid half-space, or very large values Scholte wave; to form a full-space bi-material. Moreover, an isotropic bed material can be obtained as a Frequency equation; special case by appropriately setting the respective elastic constants. Reduced frequency Anisotropy. equations and numerical results are derived for each case to con rm the results with existing ones. © 2016 Sharif University of Technology. All rights reserved. 1. Introduction isotropy of the geologic materials [4]. This anisotropy can considerably a ect the dispersive properties of Propagation of elastic surface waves in various media surface waves. Studying wave propagation in trans- has been of importance and interest in di erent elds of versely isotropic solids is the research subject of many science such as seismology, earthquake engineering, ma- investigators due to its importance in di erent elds of terial engineering, geophysical studies, etc. The media science and technology. Stoneley [5] was the rst to consisting of a liquid layer lying over an isotropic solid point that transverse isotropy signi cantly in uences half-space has been studied by scientists interested in the wave propagation in a medium in comparison with elds such as ocean engineering, etc. [1-3]. isotropic one. Later, elastodynamics of transversely However, in a wide range of cases, there are isotropic media was subsequently studied by Synge [6], observations revealing the existence of anisotropy in Buchwald [7], and Payton [8]. Earth's crust. It is believed that the fact of following Rahimian et al. [9] presented an ecient ana- a preferred pattern instead of random orientations by lytical formulation to obtain the response of a three- materials depositing in the water and also existence of dimensional transversely isotropic half-space to a time- vertically aligned microcracks result in the transverse harmonic surface loading. Khojasteh et al. [10] extended the previous work to present explicit ex- pressions for the three-dimensional dynamic Green's *. Corresponding author. Tel.: +98 21 61112256; functions in a transversely isotropic half-space and Fax: +98 21 88078263 E-mail addresses: [email protected] (A. Bagheri); investigated the wave propagation. They continued the [email protected] (A. Khojasteh); [email protected] (M. work for a two-layer half-space [11] as well. Also, Kho- Rahimian); [email protected] (R. Attarnejad) jasteh et al. subsequently studied wave propagation in 2470 A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 @2u 1 @u u 1 @2u 1 @u transversely isotropic bi-materials, tri-materials, and a c r + r r +c r 2c multilayered half-space [12-14]. 11 @r2 r @r r2 66 r2 @2 11 r2 @ Surface waves dispersion in a model of a liquid layer overlying a transversely isotropic half-space has @2u (c + c ) 1 @2u 1 @u + c r + 11 12 r + been investigated in some papers. Abubakar and 44 @z2 2 r @r@ r2 @ Hudson [15] studied the dispersive properties of sur- face waves in such a system in the two-dimensional @2u + (c + c ) z + !2u = 0; case. Sharma et al. [16,17] obtained the frequency 13 44 @r@z s r equation of a three-layer model consisting of a liquid 2 2 2 layer lying over poroelastic and anisotropic layers. @ u 1 @u u 1 @ u @ u c66 + + c11 + c44 Sharma [18] also tackled the dispersion of Stoneley @r2 r @r r2 r2 @2 @z2 waves in an oceanic crust model. In addition to 2 Kumar and Miglani [19], Kumar and Kumar [20] 1 @ur (c11 + c12) 1 @ ur 1 @ur + 2c11 + solved the problem for the proposed oceanic crust r2 @ 2 r @r@ r2 @ models. In this paper, the dispersive characteristics 2 of a three-dimensional earth crust model, consisting 1 @ uz 2 + (c13 + c44) + s! u = 0; of a homogeneous transversely isotropic elastic me- r @@z dia overlaid by a homogeneous compressible liquid @2u 1 @u 1 @2u @2u layer, are investigated utilizing the potential method c z + z + z + c z 44 2 2 2 33 2 presented by Rahimian et al. [9] and extended by @r r @r r @ @z Khojasteh et al. [10]. The velocity dispersion curves of 2 2 @ ur 1 @ur 1 @ u surface waves are plotted. Arrival times of the waves + (c13 + c44) + + are distinctly dependent on the elastic properties of @r@z r @z r @@z the bed. Analyzing this phenomenon to realize the 2 earth crust's characteristics by reverse calculation is + s! uz = 0; (1) of signi cant importance in marine seismology and where the time factor exp(i!t) is suppressed. In order underground geophysical explorations. The formula- to uncouple these equations, a couple of potential func- tion can also be utilized to derive Green's functions of tions introduced by Eskandari-Ghadi [21] are applied. stress and displacement elds for various load types These two potential functions, F and , are related to in such a system which is applicable to earthquake displacement components u , u , and u as: engineering. r z @2F (r; ; z) 1 @(r; ; z) ur(r; ; z) = 3 ; 2. Statement of the problem and governing @r@z r @ equations 1 @2F (r; ; z) @(r; ; z) u(r; ; z) = 3 + ; The geometry of the system and the coordinate axes r @@z @r selected for the problem are shown in Figure 1. For @2 !2 the homogeneous transversely isotropic half-space, the u (r; ; z)= (1+ )r2 + 2 + s F (r; ; z); z 1 r @z2 c time-harmonic equations of motion are expressed as [9]: 66 (2) where: @2 1 @ 1 @2 r2 = + + ; (3) r @r2 r @r r2 @2 and also: c12 + c66 c44 c13 + c44 1 = ; 2 = ; 3 = : (4) c66 c66 c66 Rewriting the equation of motion (Eq. (1)) in terms of the potential functions yields to the two separate partial di erential equations as: @2 r2r2 + !2 F = 0; 1 2 @z2 2 Figure 1. Geometry of the system. r0 = 0; (5) A. Bagheri et al./Scientia Iranica, Transactions A: Civil Engineering 23 (2016) 2469{2477 2471 d2 de ning: r~ 2 r~ 2 + !2 F~m(z) = 0; 1m 2m dz2 m 1 @2 1 !2 r2 = r2 + + s ; i = 0; 1; 2; (6) i r 2 2 s @z i c66 ~ 2 m i r0m~m = 0; (13) and: in which: = 1 0 !2 1 d2 r~ 2 = s 2 + ; i = 0; 1; 2: (14) c c im 2 2 44 11 ic66 s dz 1 = 2 = ; 2 = 1 + 1 = ; i c66 c66 The general solutions for equations in Eq. (13) are: delta 1 1 1 c = + 1 + 33 ; (7) 2 2 F~m(; z) =A ()e1z + B ()e1z + C ()e2z s c44s2 c11s1 c11 c44 m m m m where: 2z + Dm()e ; 1 s = p : (8) 0 m 3z 3z 2 ~m(; z) = Em()e + Fm()e ; (15) Note that s1 and s2 are not zero or pure imaginary introducing: numbers with respect to the positive de niteness of the r strain energy function [22]: 1p 2 4 2 1 = a + b + c + d + e; 4 2 2 2 c33c44s + c13 + 2c13c44 c11c33 s + c11c44 = 0: (9) r 1p = a2 + b c4 + d2 + e; The two real roots of this equation can be easily 2 2 obtained since no unknowns of odd order exist in s Eq. (9). Hence s1 and s2 are obtained by Eq. (10) as !2 2 s shown in Box I. With respect to the angular direction, 3 = s0 : (16) c66 Fourier series expansion is now applied as follows: X1 In the above terms: im [F (r; ; z); (r; ; z)] = [Fm(r; z); m(r; z)] e : 1 2 2 1 (11) a = (s + s ); 2 1 2 Also, with respect to the radial direction, mth order 1 2 1 1 Hankel transform may be expressed as: b = s! + ; h i 2 c33 c44 ~m m F (; z); ~ (; z) 2 2 2 c = (s2 s1) ; Z 1 = [F (r; z); (r; z)] rJ (r)dr; 1 1 m d = 2 !2 + (s2 + s2) 0 s 1 2 c33 c44 [F (r; z); (r; z)] c11 1 1 Z 2 + ; 1 h i c33 c11 c44 ~m m = F (; z); ~ (; z) Jm(r)d: (12) 0 2 2 4 1 1 Applying Hankel transform and Fourier series expan- e = s! : (17) c33 c44 sion, one may arrive at the two following ordinary di erential equations for transformed components of The unknown functions Am, Bm, Cm, Dm, Em, and and F and as [9]: Fm will be determined from the boundary conditions.
Load more

Recommended publications
  • Transversely Isotropic Plasticity with Application to Fiber-Reinforced Plastics
    6. LS-DYNA Anwenderforum, Frankenthal 2007 Material II Transversely Isotropic Plasticity with Application to Fiber-Reinforced Plastics M. Vogler1, S. Kolling2, R. Rolfes1 1 Leibniz University Hannover, Institute for Structural Analysis, 30167 Hannover 2 DaimlerChrysler AG, EP/SPB, HPC X271, 71059 Sindelfingen Abstract: In this article a constitutive formulation for transversely isotropic materials is presented taking large plastic deformation at small elastic strains into account. A scalar damage model is used for the approximation of the unloading behavior. Furthermore, a failure surface is assumed taking the influence of triaxiality on fracture into account. Regular- ization is considered by the introduction of an internal length for the computation of the fracture energy. The present formulation is applied to the simulation of tensile tests for a molded glass fibre reinforced polyurethane material. Keywords: Anisotropy, Transversely Isotropy, Structural Tensors, Plasticity, Anisotropic Damage, Failure, Fiber Reinforced Plastics © 2007 Copyright by DYNAmore GmbH D - II - 55 Material II 6. LS-DYNA Anwenderforum, Frankenthal 2007 1 Introduction Most of the materials that are used in the automotive industry are anisotropic to some degree. This material behavior can be observed in metals as well as in non-metallic ma- terials like molded components of fibre reinforced plastics among others. In metals, the anisotropy is induced during the manufacturing process, e.g. during sheet metal forming and due to direction of rolling. In fibre-reinforced plastics, the anisotropy is determined by the direction of the fibres. The state-of-the-art in the numerical simulation of structural parts made from fibre re- inforced plastics represents Glaser’s model [1].
    [Show full text]
  • Constitutive Relations: Transverse Isotropy and Isotropy
    Objectives_template Module 3: 3D Constitutive Equations Lecture 11: Constitutive Relations: Transverse Isotropy and Isotropy The Lecture Contains: Transverse Isotropy Isotropic Bodies Homework References file:///D|/Web%20Course%20(Ganesh%20Rana)/Dr.%20Mohite/CompositeMaterials/lecture11/11_1.htm[8/18/2014 12:10:09 PM] Objectives_template Module 3: 3D Constitutive Equations Lecture 11: Constitutive relations: Transverse isotropy and isotropy Transverse Isotropy: Introduction: In this lecture, we are going to see some more simplifications of constitutive equation and develop the relation for isotropic materials. First we will see the development of transverse isotropy and then we will reduce from it to isotropy. First Approach: Invariance Approach This is obtained from an orthotropic material. Here, we develop the constitutive relation for a material with transverse isotropy in x2-x3 plane (this is used in lamina/laminae/laminate modeling). This is obtained with the following form of the change of axes. (3.30) Now, we have Figure 3.6: State of stress (a) in x1, x2, x3 system (b) with x1-x2 and x1-x3 planes of symmetry From this, the strains in transformed coordinate system are given as: file:///D|/Web%20Course%20(Ganesh%20Rana)/Dr.%20Mohite/CompositeMaterials/lecture11/11_2.htm[8/18/2014 12:10:09 PM] Objectives_template (3.31) Here, it is to be noted that the shear strains are the tensorial shear strain terms. For any angle α, (3.32) and therefore, W must reduce to the form (3.33) Then, for W to be invariant we must have Now, let us write the left hand side of the above equation using the matrix as given in Equation (3.26) and engineering shear strains.
    [Show full text]
  • Eulerian Formulation and Level Set Models for Incompressible Fluid-Structure Interaction
    ESAIM: M2AN 42 (2008) 471–492 ESAIM: Mathematical Modelling and Numerical Analysis DOI: 10.1051/m2an:2008013 www.esaim-m2an.org EULERIAN FORMULATION AND LEVEL SET MODELS FOR INCOMPRESSIBLE FLUID-STRUCTURE INTERACTION Georges-Henri Cottet1, Emmanuel Maitre1 and Thomas Milcent1 Abstract. This paper is devoted to Eulerian models for incompressible fluid-structure systems. These models are primarily derived for computational purposes as they allow to simulate in a rather straight- forward way complex 3D systems. We first analyze the level set model of immersed membranes proposed in [Cottet and Maitre, Math. Models Methods Appl. Sci. 16 (2006) 415–438]. We in particular show that this model can be interpreted as a generalization of so-called Korteweg fluids. We then extend this model to more generic fluid-structure systems. In this framework, assuming anisotropy, the membrane model appears as a formal limit system when the elastic body width vanishes. We finally provide some numerical experiments which illustrate this claim. Mathematics Subject Classification. 76D05, 74B20, 74F10. Received July 25, 2007. Revised December 20, 2007. Published online April 1st, 2008. 1. Introduction Fluid-structure interaction problems remain among the most challenging problems in computational mechan- ics. The difficulties in the simulation of these problems stem form the fact that they rely on the coupling of models of different nature: Lagrangian in the solid and Eulerian in the fluid. The so-called ALE (for Arbitrary Lagrangian Eulerian) methods cope with this difficulty by adapting the fluid solver along the deformations of the solid medium in the direction normal to the interface. These methods allow to accurately account for continuity of stresses and velocities at the interface but they are difficult to implement and time-consuming in particular for 3D systems undergoing large deformations.
    [Show full text]
  • On Transversely Isotropic Media with Ellipsoidal Slowness Surfaces
    On Transversely Isotropic Elastic Media with Ellipsoidal Slowness Surfaces a; ;1 b;2 Anna L. Mazzucato ∗ , Lizabeth V. Rachele aDepartment of Mathematics, Pennsylvania State University, University Park, PA 16802, USA bInverse Problems Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA Abstract We consider inhomogeneous elastic media with ellipsoidal slowness surfaces. We describe all classes of transversely isotropic media for which the sheets associated to each wave mode are ellipsoids. These media have the property that elastic waves in each mode propagate along geodesic segments of certain Riemannian metrics. In particular, we study the intersection of the sheets of the slowness surface. In view of applications to the analysis of propagation of singularities along rays, we give pointwise conditions that guarantee that the sheet of the slowness surface corresponding to a given wave mode is disjoint from the others. We also investigate the smoothness of the associated polarization vectors as functions of position and direction. We employ coordinate and frame-independent methods, suitable to the study of the dynamic inverse boundary problem in elasticity. Key words: Elastodynamics, transverse isotropy, anisotropy, ellipsoidal slowness surface 2000 MSC: 74Bxx, 35Lxx 1 Introduction and Main Results In this paper we consider inhomogeneous, transversely isotropic elastic media for which the associated slowness surface has ellipsoidal sheets. Transverse isotropy is characterized by the property that at each point the elastic response of the medium is isotropic in the plane orthogonal to the so-called fiber direction. We consider media in which the fiber direction may vary smoothly from point to point. Examples of transversely isotropic elastic media include hexagonal crystals and biological tissue, such as muscle.
    [Show full text]
  • Invariant Formulation of Hyperelastic Transverse Isotropy Based on Polyconvex Free Energy Functions Joorg€ Schrooder€ A,*, Patrizio Neff B
    International Journal of Solids and Structures 40 (2003) 401–445 www.elsevier.com/locate/ijsolstr Invariant formulation of hyperelastic transverse isotropy based on polyconvex free energy functions Joorg€ Schrooder€ a,*, Patrizio Neff b a Institut f€ur Mechanik, Fachbereich 10, Universit€at Essen, 45117 Essen, Universit€atsstr. 15, Germany b Fachbereich Mathematik, Technische Universita€t Darmstadt, 64289 Darmstadt, Schloßgartenstr. 7, Germany Received 7November 2001; received in revised form 2 August 2002 Abstract In this paper we propose a formulation of polyconvex anisotropic hyperelasticity at finite strains. The main goal is the representation of the governing constitutive equations within the framework of the invariant theory which auto- matically fulfill the polyconvexity condition in the sense of Ball in order to guarantee the existence of minimizers. Based on the introduction of additional argument tensors, the so-called structural tensors, the free energies and the anisotropic stress response functions are represented by scalar-valued and tensor-valued isotropic tensor functions, respectively. In order to obtain various free energies to model specific problems which permit the matching of data stemming from experiments, we assume an additive structure. A variety of isotropic and anisotropic functions for transversely isotropic material behaviour are derived, where each individual term fulfills a priori the polyconvexity condition. The tensor generators for the stresses and moduli are evaluated in detail and some representative numerical examples are pre- sented. Furthermore, we propose an extension to orthotropic symmetry. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear elasticity; Anisotropy; Polyconvexity; Existence of minimizers 1. Introduction Anisotropic materials have a wide range of applications, e.g.
    [Show full text]
  • Development of P-Y Criterion for Anisotropic Rock And
    DEVELOPMENT OF P-Y CRITERION FOR ANISOTROPIC ROCK AND COHESIVE INTERMEDIATE GEOMATERIALS A Dissertation Presented to The Graduate Faculty of the University of Akron In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Ehab Salem Shatnawi August, 2008 DEVELOPMENT OF P-Y CRITERION FOR ANISOTROPIC ROCK AND COHESIVE INTERMEDIATE GEOMATERIALS Ehab Salem Shatnawi Dissertation Approved: Accepted: ______________________________ ______________________________ Advisor Department Chair Dr. Robert Liang Dr. Wieslaw Binienda ______________________________ ______________________________ Committee Member Dean of the College Dr. Craig Menzemer Dr. George K. Haritos ______________________________ ______________________________ Committee Member Dean of the Graduate School Dr. Daren Zywicki Dr. George R. Newkome ______________________________ ______________________________ Committee Member Date Dr. Xiaosheng Gao ______________________________ Committee Member Dr. Kevin Kreider ii ABSTRACT Rock-socketed drilled shaft foundations are commonly used to resist large axial and lateral loads applied to structures or as a means to stabilize an unstable slope with either marginal factor of safety or experiencing continuing slope movements. One of the widely used approaches for analyzing the response of drilled shafts under lateral loads is the p-y approach. Although there are past and ongoing research efforts to develop pertinent p-y criterion for the laterally loaded rock-socketed drilled shafts, most of these p-y curves were derived from basic assumptions that the rock mass behaves as an isotropic continuum. The assumption of isotropy may not be applicable to the rock mass with intrinsic anisotropy or the rock formation with distinguishing joints and bedding planes. Therefore, there is a need to develop a p-y curve criterion that can take into account the effects of rock anisotropy on the p-y curve of laterally loaded drilled shafts.
    [Show full text]
  • ON the ELASTIC MODULI and COMPLIANCES of TRANSVERSELY ISOTROPIC and ORTHOTROPIC MATERIALS Vlado A
    Journal of Mechanics of Materials and Structures ON THE ELASTIC MODULI AND COMPLIANCES OF TRANSVERSELY ISOTROPIC AND ORTHOTROPIC MATERIALS Vlado A. Lubarda and Michelle C. Chen Volume 3, Nº 1 January 2008 mathematical sciences publishers JOURNAL OF MECHANICS OF MATERIALS AND STRUCTURES Vol. 3, No. 1, 2008 ON THE ELASTIC MODULI AND COMPLIANCES OF TRANSVERSELY ISOTROPIC AND ORTHOTROPIC MATERIALS VLADO A. LUBARDA AND MICHELLE C. CHEN The relationships between the elastic moduli and compliances of transversely isotropic and orthotropic materials, which correspond to different appealing sets of linearly independent fourth-order base tensors used to cast the elastic moduli and compliances tensors, are derived by performing explicit inversions of the involved fourth-order tensors. The deduced sets of elastic constants are related to each other and to common engineering constants expressed in the Voigt notation with respect to the coordinate axes aligned along the directions orthogonal to the planes of material symmetry. The results are applied to a transversely isotropic monocrystalline zinc and an orthotropic human femural bone. 1. Introduction There has been a significant amount of research devoted to tensorial representation of elastic constants of anisotropic materials, which are the components of the fourth-order tensors in three-dimensional geometrical space, or the second-order tensors in six-dimensional stress or strain space. This was par- ticularly important for the spectral decompositions of the fourth-order stiffness and compliance tensors and the determination of the corresponding eigenvalues and eigentensors for materials with different types of elastic anisotropy. The spectral decomposition, for example, allows an additive decomposition of the elastic strain energy into a sum of six or fewer uncoupled energy modes, which are given by trace products of the corresponding pairs of stress and strain eigentensors.
    [Show full text]
  • Behaviour of Anisotropic Rocks Authors: M
    Behaviour of anisotropic rocks Authors: M. Sc. Mohamed Ismael, M. Sc. Lifu Chang & Prof. Dr.-Ing. habil. Heinz Konietzky (Geotechnical Institute, TU Bergakademie Freiberg) 1 Introduction ............................................................................................................. 2 2 Features causing rock anisotropy ........................................................................... 2 2.1 Primary structures ............................................................................................ 2 2.2 Secondary structures ........................................................................................ 4 2.3 Discontinuity frequency .................................................................................... 4 3 Observation and measurement of rock anisotropy ................................................. 5 3.1 Strength anisotropy .......................................................................................... 5 3.2 Stiffness anisotropy ........................................................................................ 11 3.3 Permeability anisotropy .................................................................................. 11 3.4 Seismic anisotropy ......................................................................................... 12 4 Classification of rock anisotropy ........................................................................... 13 4.1 Point load index .............................................................................................. 13 4.2
    [Show full text]
  • Structure and Frictional Properties of the Leg Joint of the Beetle Pachnoda Marginata (Scarabaeidae, Cetoniinae) As an Inspiration for Technical Joints
    biomimetics Article Structure and Frictional Properties of the Leg Joint of the Beetle Pachnoda marginata (Scarabaeidae, Cetoniinae) as an Inspiration for Technical Joints Steffen Vagts 1,* , Josef Schlattmann 1, Alexander Kovalev 2 and Stanislav N. Gorb 2 1 Department of System Technologies and Engineering Design Methodology, Hamburg University of Technology, Denickestr. 22, D-21079 Hamburg, Germany; [email protected] 2 Department of Functional Morphology and Biomechanics, Kiel University, Am Botanischen Garten 9, D-24118 Kiel, Germany; [email protected] (A.K.); [email protected] (S.N.G.) * Correspondence: steff[email protected]; Tel.:+49-40-428-784-422 Received: 21 February 2020; Accepted: 15 April 2020; Published: 20 April 2020 Abstract: The efficient locomotion of insects is not only inspiring for control algorithms but also promises innovations for the reduction of friction in joints. After previous analysis of the leg kinematics and the topological characterization of the contacting joint surfaces in the beetle Pachnoda marginata, in the present paper, we report on the measurement of the coefficient of friction within the leg joints exhibiting an anisotropic frictional behavior in different sliding directions. In addition, the simulation of the mechanical behavior of a single microstructural element helped us to understand the interactions between the contact parts of this tribological system. These findings were partly transferred to a technical contact pair which is typical for such an application as joint connectors in the automotive field. This innovation helped to reduce the coefficient of friction under dry sliding conditions up to 17%. Keywords: locomotion; walking; leg; joints; insects; Arthropoda; friction; coefficient of friction; biotribology; biomimetics 1.
    [Show full text]
  • Universal Relations for Transversely Isotropic Elastic Materials
    UniversalRelations for 'ftansverselyIsotropic Elastic Materials R. C. BATRA Department of Engineering Science and Mechanics, M/C 0219, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA (Received 8 March 2002; Final version 13 March 2002) Dedicated to Millard E Beatty, respected colleague and friend. Abstract: We adoptHayes and Knops'sapproach and deriveuniversal relations for finite deformationsof a transverselyisotropic elastic material. Explicit universalrelations are obtained for homogeneousdeformations correspondingto triaxial stretches,simple shear,and simultaneousshear and extension.Universal relations are also derivedfor five families of nonhomogeneousdeformations. 1. INTRODUcnON Truesdell and Noll [1, section 54] and Beatty [2] have lucidly stated the importance of universal relations for finite deformations of isotropic elastic materials; the same remarks apply to transverselyisotropic elastic materials. Beatty [2, 3] has derived a class of universal relations for constrainedand unconstrainedisotropic elastic materials. Pucci and Saccomandi [4], have given a general approach for finding universal relations in continuum mechanics. Saccomandi and Vianello [5] have given a variational characterization of the universal relations for hemitropic materials. They have proved that the classof transverselyhemitropic hyperelasticbodies is characterizedby the condition that the inner product betweenSC - CS and W N vanishes for all deformations of the body; S is the second Piola-Kirchhoff stress tensor, C the right
    [Show full text]
  • PDF (Chapter 15. Anisotropy)
    Theory of the Earth Don L. Anderson Chapter 15. Anisotropy Boston: Blackwell Scientific Publications, c1989 Copyright transferred to the author September 2, 1998. You are granted permission for individual, educational, research and noncommercial reproduction, distribution, display and performance of this work in any format. Recommended citation: Anderson, Don L. Theory of the Earth. Boston: Blackwell Scientific Publications, 1989. http://resolver.caltech.edu/CaltechBOOK:1989.001 A scanned image of the entire book may be found at the following persistent URL: http://resolver.caltech.edu/CaltechBook:1989.001 Abstract: Anisotropy is responsible for the largest variations in seismic velocities; changes in the preferred orientation of mantle minerals, or in the direction of seismic waves, cause larger changes in velocity than can be accounted for by changes in temperature, composition or mineralogy. Therefore, discussions of velocity gradients, both radial and lateral, and chemistry and mineralogy of the mantle must allow for the presence of anisotropy. Anisotropy is not a second-order effect. Seismic data that are interpreted in terms of isotropic theory can lead to models that are not even approximately correct. On the other hand a wealth of new information regarding mantle mineralogy and flow will be available as the anisotropy of the mantle becomes better understood. Anisotropy And perpendicular now and now transverse, Pierce the dark soil and as they pierce and pass Make bare the secrets of the Earth's deep heart. nisotropy is responsible for the largest variations in tropic solid with these same elastic constants can be mod- A seismic velocities; changes in the preferred orienta- eled exactly as a stack of isotropic layers.
    [Show full text]
  • The Elastic and Electric Fields for Three-Dimensional Contact For
    International Journal of Solids and Structures 37 (2000) 3201±3229 www.elsevier.com/locate/ijsolstr The elastic and electric ®elds for three-dimensional contact for transversely isotropic piezoelectric materials Hao-Jiang Ding*, Peng-Fei Hou, Feng-Lin Guo Department of Civil Engineering, Zhejiang University, Hangzhou, 310027, People's Republic of China Received 30 May 1998; in revised form 14 January 1999 Abstract Firstly, the extended Boussinesq and Cerruti solutions for point forces and point charge acting on the surface of a transversely isotropic piezoelectric half-space are derived. Secondly, aiming at a series of common three-dimensional contact including spherical contact, a conical indentor and an upright circular ¯at punch on a transversely isotropic piezoelectric half-space, we solve for their elastic and electric ®elds in smooth and frictional cases by ®rst evaluating the displacement functions and then dierentiating. The displacement functions can be obtained by integrating the extended Boussinesq or Cerruti solutions in the contact region. Then, when only normal pressure is loaded, the stresses in the half-space of PZT-4 piezoelectric ceramic are compared in the ®gures with those of the transversely isotropic material which are assumed to have the same elastic constants as those of PZT-4. Meanwhile, the electric components in the half-space of PZT-4 piezoelectric ceramic are also shown in the same ®gures. # 2000 Elsevier Science Ltd. All rights reserved. 1. Introduction Since Hertz (1882) published his classic article `On the contact of elastic solids', the research on the contact of elastic materials has been conducted for more than one hundred years, and a lot of scientists including mathematicians, physicists and civil engineers all contributed to this area.
    [Show full text]