SEISMOLOGY Lecture 9: Anisotropy, Attenuation and Anelasticity
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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]. -
Carbon – Science and Technology
© Applied Science Innovations Pvt. Ltd., India Carbon – Sci. Tech. 1 (2010) 139 - 143 Carbon – Science and Technology ASI ISSN 0974 – 0546 http://www.applied-science-innovations.com ARTICLE Received :29/03/2010, Accepted :02/09/2010 ----------------------------------------------------------------------------------------------------------------------------- Ablation morphologies of different types of carbon in carbon/carbon composites Jian Yin, Hongbo Zhang, Xiang Xiong, Jinlv Zuo State Key Laboratory of Powder Metallurgy, Central South University, Lushan South Road, Changsha, China. --------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction : Carbon/carbon (C/C) composites the ablation morphology and formation mechanism of combine good mechanical properties and designable each types of carbon. capabilities of composites and excellent ultrahigh temperature properties of carbon materials. They have In this study, ablation morphologies of resin-based low densities, high specific strength, good thermal carbon, carbon fibers, pyrolytic carbon with smooth stability, high thermal conductivity, low thermal laminar structure and rough laminar structure has been expansion coefficient and excellent ablation properties investigated in detail and their formation mechanisms [1]. As C/C composites show excellent characteristics were discussed. in both structural design and functional application, 2 Experimental : they have become one of the most competitive high temperature materials widely used in aviation and 2.1 Preparation of C/C composites : Bulk needled polyacrylonitrile (PAN) carbon fiber felts spacecraft industry [2 - 4]. In particular, they are were used as reinforcements. Three kinds of C/C considered to be the most suitable materials for solid composites, labeled as sample A, B and C, were rocket motor nozzles. prepared. Sample A is a C/C composite mainly with C/C composites are composed of carbon fibers and smooth laminar pyrolytic carbon, sample B is a C/C carbon matrices. -
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. -
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. -
Identifying the Elastic Isotropy of Architectured Materials Based On
Identifying the elastic isotropy of architectured materials based on deep learning method Anran Wei a, Jie Xiong b, Weidong Yang c, Fenglin Guo a, d, * a Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China b Department of Mechanical Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong, China c School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China d State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China * Corresponding author, E-mail: [email protected] Abstract: With the achievement on the additive manufacturing, the mechanical properties of architectured materials can be precisely designed by tailoring microstructures. As one of the primary design objectives, the elastic isotropy is of great significance for many engineering applications. However, the prevailing experimental and numerical methods are normally too costly and time-consuming to determine the elastic isotropy of architectured materials with tens of thousands of possible microstructures in design space. The quick mechanical characterization is thus desired for the advanced design of architectured materials. Here, a deep learning-based approach is developed as a portable and efficient tool to identify the elastic isotropy of architectured materials directly from the images of their representative microstructures with arbitrary component distributions. The measure of elastic isotropy for architectured materials is derived firstly in this paper to construct a database with associated images of microstructures. Then a convolutional neural network is trained with the database. It is found that the convolutional neural network shows good performance on the isotropy identification. -
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. -
Global Mantle Flow and the Development of Seismic Anisotropy: Differences Between the Oceanic and Continental Upper Mantle Clinton P
Global Mantle Flow and the Development of Seismic Anisotropy: Differences Between the Oceanic and Continental Upper Mantle Clinton P. Conrad Department of Earth and Planetary Sciences Johns Hopkins University Baltimore, MD 21218, USA Mark D. Behn Department of Geology and Geophysics Woods Hole Oceanographic Institution Woods Hole, MA 02543, USA Paul G. Silver Department of Terrestrial Magnetism Carnegie Institution of Washington Washington, DC 20015, USA Submitted to Journal of Geophysical Research : July 3, 2006 Viscous shear in the asthenosphere accommodates relative motion between the Earth’s surface plates and underlying mantle, generating lattice-preferred orientation (LPO) in olivine aggregates and a seismically anisotropic fabric. Because this fabric develops with the evolving mantle flow field, observations of seismic anisotropy directly constrain flow patterns if the contribution of lithospheric fossil anisotropy is small. We use global viscous mantle flow models to characterize the relationship between asthenospheric deformation and LPO, and constrain the asthenospheric contribution to anisotropy using a global compilation of observed shear-wave splitting measurements. For asthenosphere >500 km from plate boundaries, simple shear rotates the LPO toward the infinite strain axis (ISA, the LPO after infinite deformation) faster than the ISA changes along flow lines. Thus, we expect ISA to approximate LPO throughout most of the asthenosphere, greatly simplifying LPO predictions because strain integration along flow lines is unnecessary. Approximating LPO ~ISA and assuming A-type fabric (olivine a-axis parallel to ISA), we find that mantle flow driven by both plate motions and mantle density heterogeneity successfully predicts oceanic anisotropy (average misfit = 12º). Continental anisotropy is less well fit (average misfit = 39º), but lateral variations in lithospheric thickness improve the fit in some continental areas. -
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. -
Seismic Anisotropy: a Review of Studies by Japanese Researchers
J. Phys. Earth, 43, 301-319, 1995 Seismic Anisotropy: A Review of Studies by Japanese Researchers Satoshi Kaneshima Departmentof Earth and Planetary Physics, Faculty of Science, The Universityof Tokyo, Bunkyo-ku,Tokyo 113,Japan 1. Preface Studies on seismic anisotropy in Japan began in the mid 60's, nearly at the same time its importance for geodynamicswas revealed by Hess (1964). Work in the 60's and 70's consisted mainly of studies of the theoretical and/or observational aspects of seismic surface waves. The latter half of the 70' and the 80's, on the other hand, may be characterized as being rich in body wave observations. Elastic wave propagation in anisotropic media is currently under extensiveresearch in exploration seismologyin the U.S. and Europe. Numerous laboratory experiments in the fields of mineral physics and rock mechanics have been performed throughout these decades. This article gives a review of research concerning seismic anisotropy by Japanese authors. Studies by foreign authors are also referred to whenever they are considered to be essential in the context of this article. The review is divided into five sections which deal with: (1) laboratory measurements of elastic anisotropy of rocks or minerals, (2) papers on rock deformation and the origin of seismic anisotropy relevant to tectonics and geodynarnics,(3) theoretical studies on the Earth's free oscillations, (4) theory and observations for surface waves, and (5) body wave studies. Added at the end of this article is a brief outlook on some current topics on seismic anisotropy as well as future problems to be solved. -
F:\2014 Papers\2009 Papers\Relativity Without Light
Relativity without Light: A Further Suggestion Shan Gao Research Center for Philosophy of Science and Technology, Shanxi University, Taiyuan 030006, P. R. China Department of Philosophy, University of Chinese Academy of Sciences, Beijing 100049, P. R. China E-mail: [email protected] The role of the light postulate in special relativity is reexamined. The existing theory of relativity without light shows that one can deduce Lorentz-like transformations with an undetermined invariant speed based on homogeneity of space and time, isotropy of space and the principle of relativity. However, since the transformations can be Lorentzian or Galilean, depending on the finiteness of the invariant speed, a further postulate is needed to determine the speed in order to establish a real connection between the theory and special relativity. In this paper, I argue that a certain discreteness of space-time, whose existence is strongly suggested by the combination of quantum theory and general relativity, may result in the existence of a maximum and invariant speed when combing with the principle of relativity, and thus it can determine the finiteness of the speed in the theory of relativity without light. According to this analysis, the speed constant c in special relativity is not the actual speed of light, but the ratio between the minimum length and the shortest time of discrete space-time. This suggests a more complete theory of relativity, the theory of relativity in discrete space-time, which is based on the principle of relativity and the constancy of the minimum size of discrete space-time. 1. Introduction Special relativity was originally based on two postulates: the principle of relativity and the constancy of the speed of light. -
A Review of Nonparametric Hypothesis Tests of Isotropy Properties in Spatial Data Zachary D
Submitted to Statistical Science A Review of Nonparametric Hypothesis Tests of Isotropy Properties in Spatial Data Zachary D. Weller and Jennifer A. Hoeting Department of Statistics, Colorado State University Abstract. An important aspect of modeling spatially-referenced data is appropriately specifying the covariance function of the random field. A practitioner working with spatial data is presented a number of choices regarding the structure of the dependence between observations. One of these choices is determining whether or not an isotropic covariance func- tion is appropriate. Isotropy implies that spatial dependence does not de- pend on the direction of the spatial separation between sampling locations. Misspecification of isotropy properties (directional dependence) can lead to misleading inferences, e.g., inaccurate predictions and parameter es- timates. A researcher may use graphical diagnostics, such as directional sample variograms, to decide whether the assumption of isotropy is rea- sonable. These graphical techniques can be difficult to assess, open to sub- jective interpretations, and misleading. Hypothesis tests of the assumption of isotropy may be more desirable. To this end, a number of tests of direc- tional dependence have been developed using both the spatial and spectral representations of random fields. We provide an overview of nonparametric methods available to test the hypotheses of isotropy and symmetry in spa- tial data. We summarize test properties, discuss important considerations and recommendations in choosing and implementing a test, compare sev- eral of the methods via a simulation study, and propose a number of open research questions. Several of the reviewed methods can be implemented in R using our package spTest, available on CRAN. -
The Upper Mantle and Transition Zone
The Upper Mantle and Transition Zone Daniel J. Frost* DOI: 10.2113/GSELEMENTS.4.3.171 he upper mantle is the source of almost all magmas. It contains major body wave velocity structure, such as PREM (preliminary reference transitions in rheological and thermal behaviour that control the character Earth model) (e.g. Dziewonski and Tof plate tectonics and the style of mantle dynamics. Essential parameters Anderson 1981). in any model to describe these phenomena are the mantle’s compositional The transition zone, between 410 and thermal structure. Most samples of the mantle come from the lithosphere. and 660 km, is an excellent region Although the composition of the underlying asthenospheric mantle can be to perform such a comparison estimated, this is made difficult by the fact that this part of the mantle partially because it is free of the complex thermal and chemical structure melts and differentiates before samples ever reach the surface. The composition imparted on the shallow mantle by and conditions in the mantle at depths significantly below the lithosphere must the lithosphere and melting be interpreted from geophysical observations combined with experimental processes. It contains a number of seismic discontinuities—sharp jumps data on mineral and rock properties. Fortunately, the transition zone, which in seismic velocity, that are gener- extends from approximately 410 to 660 km, has a number of characteristic ally accepted to arise from mineral globally observed seismic properties that should ultimately place essential phase transformations (Agee 1998). These discontinuities have certain constraints on the compositional and thermal state of the mantle. features that correlate directly with characteristics of the mineral trans- KEYWORDS: seismic discontinuity, phase transformation, pyrolite, wadsleyite, ringwoodite formations, such as the proportions of the transforming minerals and the temperature at the discontinu- INTRODUCTION ity.