DOCSLIB.ORG

Fractured Rock Masses As Equivalent Continua – a Numerical Study

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Fractured Rock Masses As Equivalent Continua – a Numerical Study Fractured Rock Masses as Equivalent Continua – A Numerical Study Ki-Bok Min April 2004 TRITA-LWR PHD 1011 ISSN 1650-8602 ISRN KTH/LWR/PHD 1011-SE ISBN 91-7283-757-8 To my father and mother in Korea and my lovely wife, Mi-Suk Lee Fractured Rock Masses as Equivalent Continua – A Numerical Study ACKNOWLEDGEMENT It is my great pleasure to be able to thank the people who have helped and supported me in various ways. First of all, I would like to express my deepest gratitude to my main supervisor, Dr. Lanru Jing of the Royal Institute of Technology (KTH) for his academic guidance throughout the duration of the study. I am greatly indebted to his advice, suggestion and all the support for my research work. I also would like to express my sincere gratitude to my co- supervisor, Prof. Ove Stephansson of GeoForschungsZentrum (GFZ), Germany for the trust on my work and endless encouragement, which led me to come to this stage with confidence. I also thank Dr. Jonny Rutqvist of Lawrence Berkeley National Laboratory (LBNL), USA as my third supervisor for his enlightening advice and discussion, through my two visits to Berkeley and numerous email and telephone correspondences. Main financial support for the study was given by European Commission and additional support was provided by the Swedish Nuclear Inspective (SKI) and the French National Radioactive Waste Management Agency (ANDRA). I would like to thank Dr. Henning von Maravic of EC, Dr. Fritz Kautsky of SKI and Dr. Kun Su of ANDRA for their en- gagements. I was very fortunate to be involved in the prestigious international projects, DECO- VALEX and BENCHPAR. I would like to thank the colleagues in the projects for their comments, fruitful discussion and cooperation during the course of workshops. In particu- lar, I would like thank Prof. John Hudson of Imperial College, UK for his insightful com- ments and especially encouragement. I would like to extend my sincere gratitude to Dr. Chin-Fu Tsang of LBNL for suggesting many ideas on my research, collaboration on the two papers and hospitality during my visits to Berkeley. I also would like to thank Philipp Blum of then Univ. of Birmingham, Johan Öhman of Uppsala Univ., Dr. Johan Anderson of JA Streamflow, Dr. Leslie Knight of Nirex, Dr. Auli Niemi of Uppsala Univ. for very fruitful interactions. I am grateful to the former and current colleagues at Engineering Geology and Geophysics (EGG) group at KTH for all the comments, discussion and cooperations. Ms. Ulla Eng- berg is specially thanked for having been supportive of many aspect of my life in Stock- holm, especially my early days. Diego Mas of Itasca Geomekanik is acknowledged for our collaborations and fruitful discussions. I greatly appreciate the expert review of the sum- mary of the thesis by Tomofumi Koyama and I wish him the best of luck in both profes- sional and private life in Stockholm. I would like to thank Britt Chow for effective and kind help with all administrative matters. I would like to thank my ex-supervisor, Prof. Chung-In Lee of Seoul National University, for giving me enormous moral support and advice for my PhD study through many email correspondences. Prof. Jae-Dong Kim of Kangwon National University is acknowledged for the help in finding this wonderful place and encouragement. I would like to convey my special thanks to a number of seniors in rock mechanics community in Korea, including Dr. Hee-Suk Lee of SK E & C, for their encouragement for the initiation and continuation of my PhD study. i Ki-Bok Min TRITA-LWR PHD 1011 Contribution by my wife, Mi-Suk Lee, is immeasurable if invisible in this thesis. Her ex- ceptional support and understanding during my thesis work have been essential and it made me to complete the thesis successfully. Also I would like to share much of this pleasure with Mi-Suk’s family in Korea, especially new-born nephew and nieces. I am deeply grateful to my parents for their exceptional moral support on my study and I would like to thank my sisters and brother for encouraging my study. I am excited and so happy to give this thesis to my parents who have been extremely enthusiastic for the higher education of their four children. Tack så mycket!! Ki-Bok Min, Stockholm, April 2004 ii Fractured Rock Masses as Equivalent Continua – A Numerical Study ABSTRACT In this thesis, fractured rock masses are treated as equivalent continua for large-scale analyses of rock engineering projects. Systematic developments are made for the determi- nation of equivalent mechanical and hydraulic properties of fractured rock masses using a hybrid discrete fracture network - distinct element method (DFN-DEM) approach. The determined equivalent properties are then used for a far-field finite element analysis of the thermo-mechanical impacts on the stress, deformation and permeability of fractured rocks surrounding a hypothetical geological repository of nuclear waste. The geological data were extracted from the results of an extensive site investigation programme at Sellafield, UK, conducted by Nirex UK Ltd. The scale dependencies of the hydraulic and mechanical properties were investigated by using multiple realizations of the fracture system geometry with increasing model sizes until properly defined hydraulic and mechanical representative elementary volumes (REVs) were reached. The validity of the second order permeability tensor and the fourth- order mechanical compliance tensor were tested for continuum analyses at larger scales. The REV was determined to be around 5 m for mechanical and hydraulic data in this study. Analysis of the stress-dependent mechanical and hydraulic properties shows that the effect of rock stresses is crucial. The elastic moduli increase significantly with the increase of stress and an empirical equation of stress-dependent elastic modulus is suggested based on results of numerical experiments. Calculations of the Poisson’s ratios suggest greater val- ues than are normally assumed in practice. Depending on the state of stress, permeability decreases or increases with increasing compressive stress. Stress-induced flow channeling effect is captured by numerical modeling for the first time and detailed mechanisms of shear dilation of fractures are provided. Based on the numerical experiments, a set of em- pirical equations was suggested for the stress-dependent permeability, considering both normal deformation and shear dilation of fractures. Thermo-mechanical impact on the performance of a hypothetical repository at a far-field scale (5 km by 1 km) was investigated with the stress-dependent equivalent properties de- termined at the REV scale. This analysis shows that mechanical responses vary signifi- cantly depending on how the mechanical properties were determined. The change of per- meability due to the thermal loading is, however, not significant in this particular case. The thesis provides a framework for systematic analysis of large-scale engineering appli- cations in fractured rock masses, such as geological repositories of nuclear wastes. Keyword: Fractured rock masses, Equivalent Continuum, Discrete Fracture Network (DFN), Distinct Element Method (DEM), Finite Element Method (FEM), Nuclear Waste Disposal, Coupled Thermo-Hydro-Mechanical Processes iii Ki-Bok Min TRITA-LWR PHD 1011 iv Fractured Rock Masses as Equivalent Continua – A Numerical Study SAMMANFATTNING I denna avhandling behandlas metoder för analys av spruckna bergmassor som kontinuer- liga material. En systematisk utveckling av metoder för ekvivalenta mekaniska och hyd- rauliska egenskaper hos spruckna bergmassor har skett med användninga av diskreta spricknätsmodeller och distinkt elementmetod (DFN-DEM). De ekvivalenta materialmo- dellerna har sedan tillämpats på storskaliga analyser av ett hypotetiskt slutförvar för kärn- avfall med hjälp av finit elementmetod. Spänningar, deformationer, flöden och termome- kanisk respons hos bergmassan har analyserats. Geologiska data för studien baseras på resultaten av platsundersökningarna vid Sellafield i Storbritannien, som genomfört av Ni- rex UK Ltd. Skalberoendet hos de hydrauliska och mekaniska egenskaperna har undersökts genom an- vändande av upprepade analyser av spricksystem med samma statistiska egenskaper men med ökande modellstorlek tills representativa hydrauliska och mekaniska elementarvoly- mer (REV) uppnåddes. Skalan för den fastställda elementarvolymen REV för den testade bergmassan är ungefär 5m för konstanta mekaniska och hydrauliska egenskaper. Studien av spänningsberoende mekaniska och hydrauliska egenskaper visar att effekten av bergspänningarna är kritisk. Elasticitetsmodulen ökar signifikant med ökad spänning och ett empiriskt samband för elasticitetsmodulens spänningsberoende presenteras. Det beräk- nade Poissonsförhållandet är mycket högre för bergmassorna än de typiska värden som används i praktiken. Permeabiliteten hos bergmassan minskar eller ökar med ökad spän- ning och är beroende av spänningstillståndet. Effekter av spänningsinducerad kanalström- ning i sprickorna fångas upp genom numerisk modellering och redovisas för första gången och mekanismerna bakom dilatation av sprickorna i samband med skjuvning redovisas. Baserat på numeriska experiment föreslås en uppsättning av empiriska ekvationer för spänningsberoende permeabilitet i samband med normalbelastning och skjuvning av sprickorna och bergmassan. Den termomekaniska responsen hos bergmassan från ett hypotetisk slutförvar för radioak- tivt avfall i skalan 5km x 1 km har studerats med den utvecklade DFN-DEM tekniken för ekvivalenta
Load more

Recommended publications
  • Static Analysis of Isotropic, Orthotropic and Functionally Graded Material Beams
    Journal of Multidisciplinary Engineering Science and Technology (JMEST) ISSN: 2458-9403 Vol. 3 Issue 5, May - 2016 Static analysis of isotropic, orthotropic and functionally graded material beams Waleed M. Soliman M. Adnan Elshafei M. A. Kamel Dep. of Aeronautical Engineering Dep. of Aeronautical Engineering Dep. of Aeronautical Engineering Military Technical College Military Technical College Military Technical College Cairo, Egypt Cairo, Egypt Cairo, Egypt [email protected] [email protected] [email protected] Abstract—This paper presents static analysis of degrees of freedom for each lamina, and it can be isotropic, orthotropic and Functionally Graded used for long and short beams, this laminated finite Materials (FGMs) beams using a Finite Element element model gives good results for both stresses and deflections when compared with other solutions. Method (FEM). Ansys Workbench15 has been used to build up several models to simulate In 1993 Lidstrom [2] have used the total potential different types of beams with different boundary energy formulation to analyze equilibrium for a conditions, all beams have been subjected to both moderate deflection 3-D beam element, the condensed two-node element reduced the size of the of uniformly distributed and transversal point problem, compared with the three-node element, but loads within the experience of Timoshenko Beam increased the computing time. The condensed two- Theory and First order Shear Deformation Theory. node system was less numerically stable than the The material properties are assumed to be three-node system. Because of this fact, it was not temperature-independent, and are graded in the possible to evaluate the third and fourth-order thickness direction according to a simple power differentials of the strain energy function, and thus not law distribution of the volume fractions of the possible to determine the types of criticality constituents.
    [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]
  • Orthotropic Material Homogenization of Composite Materials Including Damping
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Lirias Orthotropic material homogenization of composite materials including damping A. Nateghia,e, A. Rezaeia,e,, E. Deckersa,d, S. Jonckheerea,b,d, C. Claeysa,d, B. Pluymersa,d, W. Van Paepegemc, W. Desmeta,d a KU Leuven, Department of Mechanical Engineering, Celestijnenlaan 300B box 2420, 3001 Heverlee, Belgium. E-mail: [email protected]. b Siemens Industry Software NV, Digital Factory, Product Lifecycle Management – Simulation and Test Solutions, Interleuvenlaan 68, B-3001 Leuven, Belgium. c Ghent University, Department of Materials Science & Engineering, Technologiepark-Zwijnaarde 903, 9052 Zwijnaarde, Belgium. d Member of Flanders Make. e SIM vzw, Technologiepark Zwijnaarde 935, B-9052 Ghent, Belgium. Introduction In the multiscale analysis of composite materials obtaining the effective equivalent material properties of the composite from the properties of its constituents is an important bridge between different scales. Although, a variety of homogenization methods are already in use, there is still a need for further development of novel approaches which can deal with complexities such as frequency dependency of material properties; which is specially of importance when dealing with damped structures. Time-domain methods Frequency-domain method Material homogenization is performed in two separate steps: Wave dispersion based homogenization is performed to a static step and a dynamic one. provide homogenized damping and material
    [Show full text]
  • Analysis of Deformation
    Chapter 7: Constitutive Equations Definition: In the previous chapters we’ve learned about the definition and meaning of the concepts of stress and strain. One is an objective measure of load and the other is an objective measure of deformation. In fluids, one talks about the rate-of-deformation as opposed to simply strain (i.e. deformation alone or by itself). We all know though that deformation is caused by loads (i.e. there must be a relationship between stress and strain). A relationship between stress and strain (or rate-of-deformation tensor) is simply called a “constitutive equation”. Below we will describe how such equations are formulated. Constitutive equations between stress and strain are normally written based on phenomenological (i.e. experimental) observations and some assumption(s) on the physical behavior or response of a material to loading. Such equations can and should always be tested against experimental observations. Although there is almost an infinite amount of different materials, leading one to conclude that there is an equivalently infinite amount of constitutive equations or relations that describe such materials behavior, it turns out that there are really three major equations that cover the behavior of a wide range of materials of applied interest. One equation describes stress and small strain in solids and called “Hooke’s law”. The other two equations describe the behavior of fluidic materials. Hookean Elastic Solid: We will start explaining these equations by considering Hooke’s law first. Hooke’s law simply states that the stress tensor is assumed to be linearly related to the strain tensor.
    [Show full text]
  • Numerical Simulation of Excavations Within Jointed Rock of Infinite Extent
    NUMERICAL SIMULATION OF EXCAVATIONS WITHIN JOINTED ROCK OF INFINITE EXTENT by ALEXANDROS I . SOFIANOS (M.Sc.,D.I.C.) May 1984- - A thesis submitted for the degree of Doctor of Philosophy of the University of London Rock Mechanics Section, R.S.M.,Imperial College London SW7 2BP -2- A bstract The subject of the thesis is the development of a program to study i the behaviour of stratified and jointed rock masses around excavations. The rock mass is divided into two regions,one which is. supposed to * exhibit linear elastic behaviour,and the other which will include discontinuities that behave inelastically.The former has been simulated by a boundary integral plane strain orthotropic module,and the latter by quadratic joint,plane strain and membrane elements.The # two modules are coupled in one program.Sequences of loading include static point,pressure,body,and residual loads,construetion and excavation, and quasistatic earthquake load.The program is interactive with graphics. Problems of infinite or finite extent may be solved. Errors due to the coupling of the two numerical methods have been analysed. Through a survey of constitutive laws, idealizations of behaviour and test results for intact rock and discontinuities,appropriate models have been selected and parameter ♦ ranges identif i ed. The representation of the rock mass as an equivalent orthotropic elastic continuum has been investigated and programmed. Simplified theoretical solutions developed for the problem of a wedge on the roof of an opening have been compared with the computed results.A problem of open stoping is analysed. * ACKNOWLEDGEMENTS The author wishes to acknowledge the contribution of all members of the Rock Mechanics group at Imperial College to this work, and its full financial support by the State Scholarship Foundation of * G reece.
    [Show full text]
  • Composite Laminate Modeling
    Composite Laminate Modeling White Paper for Femap and NX Nastran Users Venkata Bheemreddy, Ph.D., Senior Staff Mechanical Engineer Adrian Jensen, PE, Senior Staff Mechanical Engineer Predictive Engineering Femap 11.1.2 White Paper 2014 WHAT THIS WHITE PAPER COVERS This note is intended for new engineers interested in modeling composites and experienced engineers who would like to get acquainted with the Femap interface. This note is intended to accompany a technical seminar and will provide you a starting background on composites. The following topics are covered: o A little background on the mechanics of composites and how micromechanics can be leveraged to obtain composite material properties o 2D composite laminate modeling Defining a material model, layup, property card and material angles Symmetric vs. unsymmetric laminate and why this is important Results post processing o 3D composite laminate modeling Defining a material model, layup, property card and ply/stack orientation When is a 3D model preferred over a 2D model o Modeling a sandwich composite Methods of modeling a sandwich composite 3D vs. 2D sandwich composite models and their pros and cons o Failure modeling of a 2D composite laminate Defining a laminate failure model Post processing laminate and lamina failure indices Predictive Engineering Document, Feel Free to Share With Your Colleagues Page 2 of 66 Predictive Engineering Femap 11.1.2 White Paper 2014 TABLE OF CONTENTS 1. INTRODUCTION ...........................................................................................................................................................
    [Show full text]
  • Dynamic Analysis of a Viscoelastic Orthotropic Cracked Body Using the Extended Finite Element Method
    Dynamic analysis of a viscoelastic orthotropic cracked body using the extended finite element method M. Toolabia, A. S. Fallah a,*, P. M. Baiz b, L.A. Louca a a Department of Civil and Environmental Engineering, Skempton Building, South Kensington Campus, Imperial College London SW7 2AZ b Department of Aeronautics, Roderic Hill Building, South Kensington Campus, Imperial College London SW7 2AZ Abstract The extended finite element method (XFEM) is found promising in approximating solutions to locally non-smooth features such as jumps, kinks, high gradients, inclusions, voids, shocks, boundary layers or cracks in solid or fluid mechanics problems. The XFEM uses the properties of the partition of unity finite element method (PUFEM) to represent the discontinuities without the corresponding finite element mesh requirements. In the present study numerical simulations of a dynamically loaded orthotropic viscoelastic cracked body are performed using XFEM and the J-integral and stress intensity factors (SIF’s) are calculated. This is achieved by fully (reproducing elements) or partially (blending elements) enriching the elements in the vicinity of the crack tip or body. The enrichment type is restricted to extrinsic mesh-based topological local enrichment in the current work. Thus two types of enrichment functions are adopted viz. the Heaviside step function replicating a jump across the crack and the asymptotic crack tip function particular to the element containing the crack tip or its immediately adjacent ones. A constitutive model for strain-rate dependent moduli and Poisson ratios (viscoelasticity) is formulated. A symmetric double cantilever beam (DCB) of a generic orthotropic material (mixed mode fracture) is studied using the developed XFEM code.
    [Show full text]
  • The Interaction Integral for Fracture of Orthotropic Functionally Graded Materials: Evaluation of Stress Intensity Factors Jeong-Ho Kim, Glaucio H
    International Journal of Solids and Structures 40 (2003) 3967–4001 www.elsevier.com/locate/ijsolstr The interaction integral for fracture of orthotropic functionally graded materials: evaluation of stress intensity factors Jeong-Ho Kim, Glaucio H. Paulino * Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Newmark Laboratory, 205North Mathews Avenue, Urbana, IL 61801, USA Received 26 July 2002; received in revised form 11 March 2003 Abstract The interaction integral is an accurate and robust scheme for evaluating mixed-mode stress intensity factors. This paper extends the concept to orthotropic functionally graded materials and addresses fracture mechanics problems with arbitrarily oriented straight and/or curved cracks. The gradation of orthotropic material properties are smooth func- tions of spatial coordinates, which are integrated into the element stiffness matrix using the so-called ‘‘generalized isoparametric formulation’’. The types of orthotropic material gradation considered include exponential, radial, and hyperbolic-tangent functions. Stress intensity factors for mode I and mixed-mode two-dimensional problems are evaluated by means of the interaction integral and the finite element method. Extensive computational experiments have been performed to validate the proposed formulation. The accuracy of numerical results is discussed by com- parison with available analytical, semi-analytical, or numerical solutions. Ó 2003 Elsevier Science Ltd. All rights reserved. Keywords: Functionally graded material (FGM); Fracture mechanics; Stress intensity factor (SIF); Interaction integral; Finite element method (FEM); Generalized isoparametric formulation (GIF) 1. Introduction Functionally graded materials (FGMs) are a new class of composites in which the volume fraction of constituent materials vary smoothly, giving a nonuniform microstructure with continuously graded macro- properties (Hirai, 1993; Paulino et al., 2002).
    [Show full text]
  • Plane Strain Theory for an Orthotropic Cylinder with Pure Bending
    Appendix A Plane Strain Theory for an Orthotropic Cylinder with Pure Bending Lekhnitskii (1963) derives a general theory for the state of stress in an elastic body bounded by a cylindrical surface and possessing cylindrical anisotropy where the stresses do not vary along the generator. In what follows, the analysis given in Archer (1976) for the transversely isotropic material model is extended to the orthotropic case (Archer 1985). The equations of equilibrium in cylindrical coordinates after dropping dependence on the axial coordinate, and the "torsion" equilibrium equation, and body force terms become ou, +! O't"re + u, -O"e =O or r oe r (A.1) O't"re +! OO"e + 2 't"re =O or r o(} r and the generalized Hooke's law for the orthotropic case e, [all [ Be l= a12 (A.2) Sz a13 where IX., IX9 , and IXz are prescribed strains as measured by strain relief experiments (Nicholson 1971, 1973a, Sasaki et al. 1978) and taken as independent ofz; u., ue, O"z, -r,e are stress components; e., Be, Bz, y9, are the strain components; r, (}, z the cylindrical coordinates; and all ... %6 the elastic constants. The shear stresses 't"ez and 't"rz have been uncoupled from the system (A.1-A.2) and are applicable to torsion problems which do not concern us here. The relevant strain-displacement relations are ou 1 ov u e,=or ' ee=~ oe+r (A.3) ow 1 ou ov v Bz= OZ ' l're=~ o(} +or-~ A modification of the Lekhnitskii (1963) derivation which includes the growth strains follows if we set (A.4) then since Sz and IXz are independent of z, we may write D +1Xz=a33 (Ar sin(}+ Brcos (}+C) , (A.5) 206 Plain Strain Theory for an Orthotropic Cylinder with Pure Bending where A and B correspond to "pure bending" stress terms and C to a "uniform axial" stress term.
    [Show full text]
  • Analysis of a Kinked Crack in an Anisotropic Material Under Antiplane Deformation† H
    Journal of Mechanical Science and Technology 26 (2) (2012) 411~419 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-1025-4 Analysis of a kinked crack in an anisotropic material under antiplane deformation† H. G. Beom*, J. W. Lee and C. B. Cui Department of Mechanical Engineering, Inha University, 253 Yonghyun-dong, Incheon, 402-751, Korea (Manuscript Received July 21, 2011; Revised September 23, 2011; Accepted September 23, 2011) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract This paper analyzes the asymptotic problem of a kinked crack in an anisotropic material under antiplane deformation. Using the linear transformation method proposed in this paper, a solution to the asymptotic problem of a kinked crack in an anisotropic material can be obtained from the solution of the corresponding isotropic kinked crack problem. The exact solution of the stress intensity factor for the kinked crack in the anisotropic material is obtained from the solution of the isotropic problem. The effect of the kink angle and two ani- sotropic parameters on the stress intensity factor is discussed for the inclined orthotropic material as well as the anisotropic material. In order to verify the exact solution of the stress intensity factor, numerical calculations are performed by using finite element analysis.
    [Show full text]
  • Orthotropic Materials
    Kapitel 2 Orthotropic Materials 2.1 Elastic Strain matrix Elastic strains are related to stresses by Hooke's law, as stated below. The stress- strain relationship is in each material point formulated in the local cartesian coor- dinate system. "e = Cσ (1) σ is the stress vector, which uses the same convention as the strain vector. The stress vector is given below. T σ = [σl σr σt ¿lr ¿lt ¿rt] (2) C is the material compliance matrix. The compliance matrix is given as 2 3 1 ¡ ºrl ¡ ºtl 0 0 0 El Er Et 6 ¡ ºlr 1 ¡ ºtr 0 0 0 7 6 El Er Et 7 6 º º 1 7 6 ¡ lt ¡ rt 0 0 0 7 C = 6 El Er Et 7 (3) 6 0 0 0 1 0 0 7 6 Glr 7 4 0 0 0 0 1 0 5 Glt 0 0 0 0 0 1 Grt E's are moduli of elasticity, G's are shear moduli and º's are Poissons ratios. The compliance matrix should be symmetric, which gives rise to the following restric- 4 KAPITEL 2. ORTHOTROPIC MATERIALS 2.2. COORDINATE SYSTEMS tions: Er Et Et ºrl = ºlr ; ºtl = ºlt ; ºtr = ºrt (4) El El Er The inverse of the compliance matrix is the material stiness matrix, D, which give the stresses produced by an elastic strain state. 2 3 El (1 ¡ ºrtºtr)=k El (ºrtºtl + ºrl)=k El (ºrlºtr + ºtl)=k 0 0 0 6 7 6 Er (ºltºtr + ºlr)=k Er (1 ¡ ºltºtl)=k Er (ºlrºtl + ºtr)=k 0 0 0 7 6 7 6 7 D = 6 Et (ºlrºrt + ºlt)=k Et (ºltºrl + ºrt)=k Et (1 ¡ ºlrºrl)=k 0 0 0 7 6 7 6 0 0 0 Glr 0 0 7 4 5 0 0 0 0 Glt 0 0 0 0 0 0 Grt (5) where k = 1 ¡ ºrtºtr ¡ ºlrºrl ¡ ºltºtl ¡ ºlrºrtºtl ¡ ºrlºtrºlt.
    [Show full text]
  • An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials
    NASA/TM–2011-217314 An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials Michael P. Nemeth Langley Research Center, Hampton, Virginia December 2011 NASA STI Program . in Profile Since its founding, NASA has been dedicated to CONFERENCE PUBLICATION. Collected the advancement of aeronautics and space science. papers from scientific and technical The NASA scientific and technical information (STI) conferences, symposia, seminars, or other program plays a key part in helping NASA maintain meetings sponsored or co-sponsored by NASA. this important role. SPECIAL PUBLICATION. Scientific, The NASA STI program operates under the technical, or historical information from NASA auspices of the Agency Chief Information Officer. It programs, projects, and missions, often collects, organizes, provides for archiving, and concerned with subjects having substantial disseminates NASA’s STI. The NASA STI program public interest. provides access to the NASA Aeronautics and Space Database and its public interface, the NASA Technical TECHNICAL TRANSLATION. English- Report Server, thus providing one of the largest language translations of foreign scientific and collections of aeronautical and space science STI in technical material pertinent to NASA’s mission. the world. Results are published in both non-NASA channels and by NASA in the NASA STI Report Specialized services also include creating custom Series, which includes the following report types: thesauri, building customized databases, and organizing and publishing research results. TECHNICAL PUBLICATION. Reports of completed research or a major significant phase For more information about the NASA STI of research that present the results of NASA program, see the following: programs and include extensive data or theoretical analysis.
    [Show full text]