MATERIAL ELASTIC ANISOTROPIC command

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "MATERIAL ELASTIC ANISOTROPIC command"

Transcription

1 MATERIAL ELASTIC ANISOTROPIC command.. Synopsis The MATERIAL ELASTIC ANISOTROPIC command is used to specify the parameters associated with an anisotropic linear elastic material idealization. Syntax The following syntax is used to describe an anisotropic elastic material idealization: MATerial ELAstic ANIsotropic NUMber # (EScription string ) ( C11 #.# ) ( C12 #.# ) ( C13 #.#) ( C14 #.# ) ( C15 #.#) ( C16 #.#) ( C22 #.#) ( C23 #.#) ( C24 #.#) ( C25 #.#) ( C26 #.#) ( C33 #.#) ( C34 #.#) ( C35 #.#) ( C36 #.#) ( C44 #.#) ( C45 #.#) ( C46 #.#) ( C55 #.#) ( C56 #.#) ( C66 #.#).. 1 V. N. Kaliakin

2 Explanatory Notes The NUMBER keyword is used to specify the (global) number of the material associated with the incompressible isotropic elastic idealization. The default material number is one (1). The optional alphanumeric string associated with the ESCRIPTION keyword must be enclosed in double quotes ( ). It is used solely to describe the material being idealized to the analyst. The ESCRIPTION string is printed as part of the echo of the material. For a three-dimensional state of stress, this constitutive relation is written in the following form: σ 11 C 11 C 12 C 13 C 14 C 15 C 16 ε 11 σ 22 C 12 C 22 C 23 C 24 C 25 C 26 ε 22 σ 33 = C 13 C 23 C 33 C 34 C 35 C 36 ε 33 σ 12 C 14 C 24 C 34 C 44 C 45 C 46 γ 12 σ 13 C 15 C 25 C 35 C 45 C 55 C 56 γ 13 C 16 C 26 C 36 C 46 C 56 C 66 σ 23 For conditions of plane stress, plane strain and torsionless axisymmetry, the constitutive relation reduces to σ 11 σ 22 σ 33 σ 12 C 11 C 12 C 13 C 14 = C 12 C 22 C 23 C 24 C 13 C 23 C 33 C 34 C 14 C 24 C 34 C 44 where loading has been assumed to be applied only in the 1-2 plane. Note that in the above expressions, the engineering measure of shearing strains, which is twice the tensorial values, has been employed; i.e., γ 12 = 2ε 12, γ 13 = 2ε 13, and γ 23 = 2ε 23. Orthotropic Elastic Material Idealization: For a material through each point of which pass three mutually perpendicular planes of elastic symmetry. If similar planes are parallel at all points in the material, then taking (x 1, x 2, x 3 ) coordinate axes normal to these planes (i.e., along the principal directions) it follows that there should be no interaction between the various shear components or between the shear and normal components. The material is described by twelve elastic constants, only nine of which are independent. The entries in the three-dimensional constitutive matrix now become ε 11 ε 22 ε 33 γ 12 γ 23 (1) (2) C 11 = E 1(1 ν 23 ν 32 ) ; C 12 = C 21 = E 1(ν 21 + ν 31 ν 23 ) (3) C 13 = C 31 = E 1(ν 31 + ν 21 ν 32 ) ; C 22 = E 2(1 ν 13 ν 31 ) (4) 2 V. N. Kaliakin

3 C 23 = C 32 = E 2(ν 32 + ν 12 ν 31 ) ; C 33 = E 3(1 ν 12 ν 21 ) (5) C 44 = G 12 ; C 55 = G 13 ; C 66 = G 23 (6) where = 1 ν 12 ν 21 ν 13 ν 31 ν 23 ν 32 2ν 12 ν 31 ν 23. Examples of orthotropic materials include fiber reinforced composite beams, wood (as a first approximation), sheet metal and honeycomb. Transversely Isotropic Elastic Material Idealization: Such materials possess the following properties: through all points there pass parallel planes of elastic symmetry in which all directions are elastically equivalent (i.e., planes of isotropy). Thus at each point there exists one principal direction and an infinite number of principal directions in a plane normal to the first direction. Five parameter values characterize a transversely isotropic elastic material. Taking the x 3 -axis normal to the plane of isotropy, with the x 1 and x 2 axes directed arbitrarily in this plane, these are E 1 is the elastic modulus for compression or tension in the plane of isotropy. E 2 is the elastic modulus for compression or tension in a direction normal to the plane of isotropy. ν 11 is Poisson s ratio 1 characterizing transverse contraction in the plane of isotropy when tension is applied in the plane of isotropy. ν 21 is Poisson s ratio characterizing transverse contraction when tension is applied in the plane of isotropy. Similarly, ν 12 is Poisson s ratio characterizing contraction normal to the plane of isotropy when tension is applied in this plane. G 2 is the shear modulus associated with the plane of isotropy and any plane perpendicular to it. The entries in the three-dimensional constitutive matrix are thus C 11 = C 22 = C 12 = E 1 [E 2 E 1 (ν 21 ) 2 ] (1 + ν 11 ) [E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 ] E 1 [E 1 (ν 21 ) 2 + E 2 ν 11 ] (1 + ν 11 ) [E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 ] (7) (8) C 13 = C 23 = C 33 = 1 Named in honor of S.. Poisson ( ). E 1 E 2 ν 21 E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 (9) (E 2 ) 2 (1 ν 11 ) E 2 (1 ν 11 ) 2E 1 (ν 21 ) 2 (10) 3 V. N. Kaliakin

4 C 44 = E 1 2(1 + ν 11 ) ; C 55 = C 66 = G 2 (11) Isotropic Elastic Material Idealization: The simplest linearly elastic material, for which the elastic behavior is independent of the orientation of the coordinate axes, is called isotropic. For such a material the constitutive relations simplify in that only two material constants are required to completely describe the material behavior; e.g., the elastic (Young s) modulus E and Poisson s ratio ν, the Lamé constants λ and µ, or the bulk modulus K and the shear modulus G. The entries in the three-dimensional constitutive matrix now become C 11 = C 22 = C 33 = C 12 = C 13 = C 23 = C 44 = C 55 = C 66 = E(1 ν) (1 + ν)(1 2ν) = λ + 2µ = K G (12) Eν (1 + ν)(1 2ν) = λ = K 2 3 G (13) E 2(1 + ν) = µ = G (14) Further details pertaining to the finite element analysis of elastic bodies are given in Chapter 12 of [1]. 4 V. N. Kaliakin

5 Example of Command Usage Idealization of an Anisotropic Elastic Material The following commands are sufficient to describe the anisotropic elastic material idealization material elastic aniso num 2 desc hypothetical hybrid material c e+07 c e+07 c e+07 c & c e+07 c e+07 & c c e+07 c c e+07 & Idealization of Wood (an Orthotropic Elastic Material) In this example the linear anisotropic material idealization is used to describe an orthotropic material (ouglas Fir). Twelve parameters, nine of which are independent, characterize an orthotropic material. The three principal axes of material symmetry are oriented with respect to the grain directions and growth rings [2]. enoting the longitudinal, tangential and radial directions by the letters L, T, and R, respectively, the following values are used to characterize ouglas Fir at approximately 12% moisture content: [2] E T /E L = E R /E L = ν LR = ν RL = ν LT = ν T L = ν RT = ν T R = G LR /E L = G LT /E L = G RT /E L = where E L = 12, 600 MP a (an average value, obtained from bending tests). enoting the longitudinal, transverse and radial directions by 1, 2, and 3, respectively, it follows that and E 1 = 12, 600 MP a E 2 = 0.050(12, 600) = MP a E 3 = 0.068(12, 600) = MP a 5 V. N. Kaliakin

6 and G 12 = 0.078(12, 600) = MP a G 13 = 0.064(12, 600) = MP a G 23 = 0.007(12, 600) = 88.2 MP a Finally, since ν 12 = 0.449, ν 13 = 0.292, and ν 23 = 0.390, it follows that ν 21 = E 2ν 12 E 1 = (0.050)(0.449) = ν 31 = E 3ν 13 = 0.068(0.292) = E 1 ν 32 = E ( ) 3ν = (0.390) = E For an orthotropic material the entries in C are computed as follows: = 1 ν 12 ν 21 ν 13 ν 31 ν 23 ν 32 ν 12 ν 31 ν 23 ν 21 ν 13 ν 32 C 11 = E 1(1 ν 23 ν 32 ) C 21 = E 2(ν 12 + ν 13 ν 32 ) C 31 = E 3(ν 13 + ν 12 ν 23 ) Thus, for the present material idealization, C 12 = E 1(ν 21 + ν 31 ν 23 ), C 13 = E 1(ν 31 + ν 21 ν 32 ), C 22 = E 2(1 ν 13 ν 31 ), C 23 = E 2(ν 32 + ν 12 ν 31 ), C 32 = E 3(ν 23 + ν 21 ν 13 ), C 33 = E 3(1 ν 12 ν 21 ) C 44 = G 12 ; C 55 = G 13 ; C 66 = G 23 = 1 (0.449)( ) (0.292)( ) (0.390)(0.5304) 2(0.449)( )(0.390) = C 11 = MP a C 12 = MP a C 13 = MP a C 22 = MP a C 23 = MP a C 33 = MP a C 44 = G 12 = 0.078(12, 600) = MP a C 55 = G 13 = 0.064(12, 600) = MP a C 66 = G 23 = 0.007(12, 600) = 88.2 MP a The following commands are sufficient to describe the orthotropic elastic material idealization: 6 V. N. Kaliakin

7 material elastic aniso num 1 & desc orthotropic material entered using anisotropic command c c c & c c c & c c c & Idealization of an Isotropic Elastic Material In this example the linear anisotropic material idealization is used to describe an isotropic material. Two parameters characterize such a material; in this case an elastic modulus E = 20, 000 and a Poisson s ratio of ν = The entries in C for an isotropic material are computed as follows: E(1 ν) C 11 = C 22 = C 33 = = 24, 000 (1 + ν)(1 2ν) Eν C 12 = C 13 = C 23 = = 8, 000 (1 + ν)(1 2ν) E C 44 = C 55 = C 66 = = 8, 000 2(1 + ν) The following commands are sufficient to describe the isotropic elastic material idealization: material elastic aniso num 1 & desc isotropic material entered using anisotropic command c c c & c c c & c c c & 7 V. N. Kaliakin

8 Bibliography [1] Kaliakin, V. N., Approximate Solution Techniques, Numerical Modeling and Finite Element Methods. New York: Marcel ekker, Inc. (2001). [2] U. S. epartment of Agriculture, The Encyclopedia of Wood. New York, NY: Skyhorse Publishing (2007). 8

MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command.

MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command. MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command Synopsis The MATERIAL ELASTIC HERRMANN INCOMPRESSIBLE command is used to specify the parameters associated with an isotropic, linear elastic material idealization

More information

Chapter 3. Load and Stress Analysis

Chapter 3. Load and Stress Analysis Chapter 3 Load and Stress Analysis 2 Shear Force and Bending Moments in Beams Internal shear force V & bending moment M must ensure equilibrium Fig. 3 2 Sign Conventions for Bending and Shear Fig. 3 3

More information

Module 7: Micromechanics Lecture 29: Background of Concentric Cylinder Assemblage Model. Introduction. The Lecture Contains

Module 7: Micromechanics Lecture 29: Background of Concentric Cylinder Assemblage Model. Introduction. The Lecture Contains Introduction In this lecture we are going to introduce a new micromechanics model to determine the fibrous composite effective properties in terms of properties of its individual phases. In this model

More information

EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES

EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES EFFECT OF ELLIPTIC OR CIRCULAR HOLES ON THE STRESS DISTRIBUTION IN PLATES OF WOOD OR PLYWOOD CONSIDERED AS ORTHOTROPIC MATERIALS Information Revied and Reaffirmed March 1956 No. 1510 EFFECT OF ELLIPTIC

More information

3D and Planar Constitutive Relations

3D and Planar Constitutive Relations 3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace

More information

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2017 lecture five mechanics www.carttalk.com of materials Mechanics of Materials 1 Mechanics of Materials MECHANICS MATERIALS

More information

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture

five Mechanics of Materials 1 ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN DR. ANNE NICHOLS SUMMER 2014 lecture five mechanics www.carttalk.com of materials Mechanics of Materials 1 Mechanics of Materials MECHANICS MATERIALS

More information

Understand basic stress-strain response of engineering materials.

Understand basic stress-strain response of engineering materials. Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities

More information

2. Mechanics of Materials: Strain. 3. Hookes's Law

2. Mechanics of Materials: Strain. 3. Hookes's Law Mechanics of Materials Course: WB3413, Dredging Processes 1 Fundamental Theory Required for Sand, Clay and Rock Cutting 1. Mechanics of Materials: Stress 1. Introduction 2. Plane Stress and Coordinate

More information

6. Bending CHAPTER OBJECTIVES

6. Bending CHAPTER OBJECTIVES CHAPTER OBJECTIVES Determine stress in members caused by bending Discuss how to establish shear and moment diagrams for a beam or shaft Determine largest shear and moment in a member, and specify where

More information

TRESS - STRAIN RELATIONS

TRESS - STRAIN RELATIONS TRESS - STRAIN RELATIONS Stress Strain Relations: Hook's law, states that within the elastic limits the stress is proportional to t is impossible to describe the entire stress strain curve with simple

More information

ME 582 Advanced Materials Science. Chapter 2 Macromechanical Analysis of a Lamina (Part 2)

ME 582 Advanced Materials Science. Chapter 2 Macromechanical Analysis of a Lamina (Part 2) ME 582 Advanced Materials Science Chapter 2 Macromechanical Analysis of a Lamina (Part 2) Laboratory for Composite Materials Research Department of Mechanical Engineering University of South Alabama, Mobile,

More information

Continuum Mechanics. Continuum Mechanics and Constitutive Equations

Continuum Mechanics. Continuum Mechanics and Constitutive Equations Continuum Mechanics Continuum Mechanics and Constitutive Equations Continuum mechanics pertains to the description of mechanical behavior of materials under the assumption that the material is a uniform

More information

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical

More information

Example-3. Title. Description. Cylindrical Hole in an Infinite Mohr-Coulomb Medium

Example-3. Title. Description. Cylindrical Hole in an Infinite Mohr-Coulomb Medium Example-3 Title Cylindrical Hole in an Infinite Mohr-Coulomb Medium Description The problem concerns the determination of stresses and displacements for the case of a cylindrical hole in an infinite elasto-plastic

More information

THEORY OF PLATES AND SHELLS

THEORY OF PLATES AND SHELLS THEORY OF PLATES AND SHELLS S. TIMOSHENKO Professor Emeritus of Engineering Mechanics Stanford University S. WOINOWSKY-KRIEGER Professor of Engineering Mechanics Laval University SECOND EDITION MCGRAW-HILL

More information

Stresses and Strains in flexible Pavements

Stresses and Strains in flexible Pavements Stresses and Strains in flexible Pavements Multi Layered Elastic System Assumptions in Multi Layered Elastic Systems The material properties of each layer are homogeneous property at point A i is the same

More information

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE

Tuesday, February 11, Chapter 3. Load and Stress Analysis. Dr. Mohammad Suliman Abuhaiba, PE 1 Chapter 3 Load and Stress Analysis 2 Chapter Outline Equilibrium & Free-Body Diagrams Shear Force and Bending Moments in Beams Singularity Functions Stress Cartesian Stress Components Mohr s Circle for

More information

LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS

LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS XXII. LAMINATION THEORY FOR THE STRENGTH OF FIBER COMPOSITE MATERIALS Introduction The lamination theory for the elastic stiffness of fiber composite materials is the backbone of the entire field, it holds

More information

**********************************************************************

********************************************************************** Department of Civil and Environmental Engineering School of Mining and Petroleum Engineering 3-33 Markin/CNRL Natural Resources Engineering Facility www.engineering.ualberta.ca/civil Tel: 780.492.4235

More information

Finite Element Analysis of Wood and Composite Structured Hockey sticks

Finite Element Analysis of Wood and Composite Structured Hockey sticks Finite Element Analysis of Wood and Composite Structured Hockey sticks 605 Finite Element Analysis Professor Ian Grosse May 15, 2003 By: Michael O Brien Project Statement The direction of this project

More information

Chapter 2 - Macromechanical Analysis of a Lamina. Exercise Set. 2.1 The number of independent elastic constants in three dimensions are: 2.

Chapter 2 - Macromechanical Analysis of a Lamina. Exercise Set. 2.1 The number of independent elastic constants in three dimensions are: 2. Chapter - Macromechanical Analysis of a Lamina Exercise Set. The number of independent elastic constants in three dimensions are: Anisotropic Monoclinic 3 Orthotropic 9 Transversely Orthotropic 5 Isotropic.

More information

Modeling of Fiber-Reinforced Membrane Materials Daniel Balzani. (Acknowledgement: Anna Zahn) Tasks Week 2 Winter term 2014

Modeling of Fiber-Reinforced Membrane Materials Daniel Balzani. (Acknowledgement: Anna Zahn) Tasks Week 2 Winter term 2014 Institute of echanics and Shell Structures Faculty Civil Engineering Chair of echanics odeling of Fiber-Reinforced embrane aterials OOC@TU9 Daniel Balani (Acknowledgement: Anna Zahn Tasks Week 2 Winter

More information

ELASTICITY (MDM 10203)

ELASTICITY (MDM 10203) LASTICITY (MDM 10203) Lecture Module 5: 3D Constitutive Relations Dr. Waluyo Adi Siswanto University Tun Hussein Onn Malaysia Generalised Hooke's Law In one dimensional system: = (basic Hooke's law) Considering

More information

Bending of Simply Supported Isotropic and Composite Laminate Plates

Bending of Simply Supported Isotropic and Composite Laminate Plates Bending of Simply Supported Isotropic and Composite Laminate Plates Ernesto Gutierrez-Miravete 1 Isotropic Plates Consider simply a supported rectangular plate of isotropic material (length a, width b,

More information

Cellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996).

Cellular solid structures with unbounded thermal expansion. Roderic Lakes. Journal of Materials Science Letters, 15, (1996). 1 Cellular solid structures with unbounded thermal expansion Roderic Lakes Journal of Materials Science Letters, 15, 475-477 (1996). Abstract Material microstructures are presented which can exhibit coefficients

More information

DESIGN OF LAMINATES FOR IN-PLANE LOADING

DESIGN OF LAMINATES FOR IN-PLANE LOADING DESIGN OF LAMINATES FOR IN-PLANOADING G. VERCHERY ISMANS 44 avenue F.A. Bartholdi, 72000 Le Mans, France Georges.Verchery@m4x.org SUMMARY This work relates to the design of laminated structures primarily

More information

6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa ( psi) and

6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa ( psi) and 6.4 A cylindrical specimen of a titanium alloy having an elastic modulus of 107 GPa (15.5 10 6 psi) and an original diameter of 3.8 mm (0.15 in.) will experience only elastic deformation when a tensile

More information

Tensor Visualization. CSC 7443: Scientific Information Visualization

Tensor Visualization. CSC 7443: Scientific Information Visualization Tensor Visualization Tensor data A tensor is a multivariate quantity Scalar is a tensor of rank zero s = s(x,y,z) Vector is a tensor of rank one v = (v x,v y,v z ) For a symmetric tensor of rank 2, its

More information

Lectures on. Constitutive Modelling of Arteries. Ray Ogden

Lectures on. Constitutive Modelling of Arteries. Ray Ogden Lectures on Constitutive Modelling of Arteries Ray Ogden University of Aberdeen Xi an Jiaotong University April 2011 Overview of the Ingredients of Continuum Mechanics needed in Soft Tissue Biomechanics

More information

CHAPTER -6- BENDING Part -1-

CHAPTER -6- BENDING Part -1- Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and

More information

Plane and axisymmetric models in Mentat & MARC. Tutorial with some Background

Plane and axisymmetric models in Mentat & MARC. Tutorial with some Background Plane and axisymmetric models in Mentat & MARC Tutorial with some Background Eindhoven University of Technology Department of Mechanical Engineering Piet J.G. Schreurs Lambèrt C.A. van Breemen March 6,

More information

Strength of Material. Shear Strain. Dr. Attaullah Shah

Strength of Material. Shear Strain. Dr. Attaullah Shah Strength of Material Shear Strain Dr. Attaullah Shah Shear Strain TRIAXIAL DEFORMATION Poisson's Ratio Relationship Between E, G, and ν BIAXIAL DEFORMATION Bulk Modulus of Elasticity or Modulus of Volume

More information

SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS

SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS SANDWICH COMPOSITE BEAMS for STRUCTURAL APPLICATIONS de Aguiar, José M., josemaguiar@gmail.com Faculdade de Tecnologia de São Paulo, FATEC-SP Centro Estadual de Educação Tecnológica Paula Souza. CEETEPS

More information

Stress-Strain Behavior

Stress-Strain Behavior Stress-Strain Behavior 6.3 A specimen of aluminum having a rectangular cross section 10 mm 1.7 mm (0.4 in. 0.5 in.) is pulled in tension with 35,500 N (8000 lb f ) force, producing only elastic deformation.

More information

International Journal of Advanced Engineering Technology E-ISSN

International Journal of Advanced Engineering Technology E-ISSN Research Article INTEGRATED FORCE METHOD FOR FIBER REINFORCED COMPOSITE PLATE BENDING PROBLEMS Doiphode G. S., Patodi S. C.* Address for Correspondence Assistant Professor, Applied Mechanics Department,

More information

Geology 229 Engineering Geology. Lecture 5. Engineering Properties of Rocks (West, Ch. 6)

Geology 229 Engineering Geology. Lecture 5. Engineering Properties of Rocks (West, Ch. 6) Geology 229 Engineering Geology Lecture 5 Engineering Properties of Rocks (West, Ch. 6) Common mechanic properties: Density; Elastic properties: - elastic modulii Outline of this Lecture 1. Uniaxial rock

More information

Lecture 8 Viscoelasticity and Deformation

Lecture 8 Viscoelasticity and Deformation HW#5 Due 2/13 (Friday) Lab #1 Due 2/18 (Next Wednesday) For Friday Read: pg 130 168 (rest of Chpt. 4) 1 Poisson s Ratio, μ (pg. 115) Ratio of the strain in the direction perpendicular to the applied force

More information

7.4 The Elementary Beam Theory

7.4 The Elementary Beam Theory 7.4 The Elementary Beam Theory In this section, problems involving long and slender beams are addressed. s with pressure vessels, the geometry of the beam, and the specific type of loading which will be

More information

RESEARCH PROJECT NO. 26

RESEARCH PROJECT NO. 26 RESEARCH PROJECT NO. 26 DETERMINATION OF POISSON'S RATIO IN AD1 TESTING BY UNIVERSITY OF MICHIGAN REPORT PREPARED BY BELA V. KOVACS and JOHN R. KEOUGH \ MEMBER / / DUCTILE IRON \ SOCIETY Issued by the

More information

Basic concepts to start Mechanics of Materials

Basic concepts to start Mechanics of Materials Basic concepts to start Mechanics of Materials Georges Cailletaud Centre des Matériaux Ecole des Mines de Paris/CNRS Notations Notations (maths) (1/2) A vector v (element of a vectorial space) can be seen

More information

Mechanics of Solids notes

Mechanics of Solids notes Mechanics of Solids notes 1 UNIT II Pure Bending Loading restrictions: As we are aware of the fact internal reactions developed on any cross-section of a beam may consists of a resultant normal force,

More information

Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS

Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA UNESCO EOLSS MECHANICS OF MATERIALS Jeff Brown Hope College, Department of Engineering, 27 Graves Pl., Holland, Michigan, USA Keywords: Solid mechanics, stress, strain, yield strength Contents 1. Introduction 2. Stress

More information

Materials and Structures. Indian Institute of Technology Kanpur

Materials and Structures. Indian Institute of Technology Kanpur Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 15 Behavior of Unidirectional Composites Lecture Overview Mt Material ilaxes in unidirectional

More information

CORRELATING OFF-AXIS TENSION TESTS TO SHEAR MODULUS OF WOOD-BASED PANELS

CORRELATING OFF-AXIS TENSION TESTS TO SHEAR MODULUS OF WOOD-BASED PANELS CORRELATING OFF-AXIS TENSION TESTS TO SHEAR MODULUS OF WOOD-BASED PANELS By Edmond P. Saliklis 1 and Robert H. Falk ABSTRACT: The weakness of existing relationships correlating off-axis modulus of elasticity

More information

2.1 Strain energy functions for incompressible materials

2.1 Strain energy functions for incompressible materials Chapter 2 Strain energy functions The aims of constitutive theories are to develop mathematical models for representing the real behavior of matter, to determine the material response and in general, to

More information

3. BEAMS: STRAIN, STRESS, DEFLECTIONS

3. BEAMS: STRAIN, STRESS, DEFLECTIONS 3. BEAMS: STRAIN, STRESS, DEFLECTIONS The beam, or flexural member, is frequently encountered in structures and machines, and its elementary stress analysis constitutes one of the more interesting facets

More information

Generic Strategies to Implement Material Grading in Finite Element Methods for Isotropic and Anisotropic Materials

Generic Strategies to Implement Material Grading in Finite Element Methods for Isotropic and Anisotropic Materials University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Engineering Mechanics Dissertations & Theses Mechanical & Materials Engineering, Department of Winter 12-9-2011 Generic

More information

4.MECHANICAL PROPERTIES OF MATERIALS

4.MECHANICAL PROPERTIES OF MATERIALS 4.MECHANICAL PROPERTIES OF MATERIALS The diagram representing the relation between stress and strain in a given material is an important characteristic of the material. To obtain the stress-strain diagram

More information

Esben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer

Esben Byskov. Elementary Continuum. Mechanics for Everyone. With Applications to Structural Mechanics. Springer Esben Byskov Elementary Continuum Mechanics for Everyone With Applications to Structural Mechanics Springer Contents Preface v Contents ix Introduction What Is Continuum Mechanics? "I Need Continuum Mechanics

More information

Composite Laminate Modeling

Composite Laminate Modeling omposite 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 WHAT THIS WHITE PAPER

More information

MATERIAL MECHANICS, SE2126 COMPUTER LAB 4 MICRO MECHANICS. E E v E E E E E v E E + + = m f f. f f

MATERIAL MECHANICS, SE2126 COMPUTER LAB 4 MICRO MECHANICS. E E v E E E E E v E E + + = m f f. f f MATRIAL MCHANICS, S226 COMPUTR LAB 4 MICRO MCHANICS 2 2 2 f m f f m T m f m f f m v v + + = + PART A SPHRICAL PARTICL INCLUSION Consider a solid granular material, a so called particle composite, shown

More information

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #2: Nonlinear Elastic Models Isotropic Elastic Models: Invariant vs Principal Formulations Elastic

More information

Stress-induced transverse isotropy in rocks

Stress-induced transverse isotropy in rocks Stanford Exploration Project, Report 80, May 15, 2001, pages 1 318 Stress-induced transverse isotropy in rocks Lawrence M. Schwartz, 1 William F. Murphy, III, 1 and James G. Berryman 1 ABSTRACT The application

More information

7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses

7 TRANSVERSE SHEAR transverse shear stress longitudinal shear stresses 7 TRANSVERSE SHEAR Before we develop a relationship that describes the shear-stress distribution over the cross section of a beam, we will make some preliminary remarks regarding the way shear acts within

More information

A Suggested Analytical Solution for Vibration of Honeycombs Sandwich Combined Plate Structure

A Suggested Analytical Solution for Vibration of Honeycombs Sandwich Combined Plate Structure International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:16 No:04 9 A Suggested Analytical Solution for Vibration of Honeycombs Sandwich Combined Plate Structure Muhsin J. Jweeg College

More information

COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.)

COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.) Narvik University College (Høgskolen i Narvik) EXAMINATION TASK COURSE STE6289 Modern Materials and Computations (Moderne materialer og beregninger 7.5 stp.) CLASS: Master students in Engineering Design

More information

Enhancing Prediction Accuracy In Sift Theory

Enhancing Prediction Accuracy In Sift Theory 18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS Enhancing Prediction Accuracy In Sift Theory J. Wang 1 *, W. K. Chiu 1 Defence Science and Technology Organisation, Fishermans Bend, Australia, Department

More information

9 Strength Theories of Lamina

9 Strength Theories of Lamina 9 trength Theories of Lamina 9- TRENGTH O ORTHOTROPIC LAMINA or isotropic materials the simplest method to predict failure is to compare the applied stresses to the strengths or some other allowable stresses.

More information

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING )

Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) Chapter 5 CENTRIC TENSION OR COMPRESSION ( AXIAL LOADING ) 5.1 DEFINITION A construction member is subjected to centric (axial) tension or compression if in any cross section the single distinct stress

More information

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive

More information

THE BENDING STIFFNESSES OF CORRUGATED BOARD

THE BENDING STIFFNESSES OF CORRUGATED BOARD AMD-Vol. 145/MD-Vol. 36, Mechanics of Cellulosic Materials ASME 1992 THE BENDING STIFFNESSES OF CORRUGATED BOARD S. Luo and J. C. Suhling Department of Mechanical Engineering Auburn University Auburn,

More information

Validation of the Resonalyser method: an inverse method for material identification

Validation of the Resonalyser method: an inverse method for material identification Validation of the Resonalyser method: an inverse method for material identification T. Lauwagie, H. Sol,. Roebben 3, W. Heylen and Y. Shi Katholieke Universiteit Leuven (KUL) Department of Mechanical ngineering

More information

Fracture Mechanics of Composites with Residual Thermal Stresses

Fracture Mechanics of Composites with Residual Thermal Stresses J. A. Nairn Material Science & Engineering, University of Utah, Salt Lake City, Utah 84 Fracture Mechanics of Composites with Residual Thermal Stresses The problem of calculating the energy release rate

More information

MATERIAL MECHANICS, SE2126 COMPUTER LAB 3 VISCOELASTICITY. k a. N t

MATERIAL MECHANICS, SE2126 COMPUTER LAB 3 VISCOELASTICITY. k a. N t MATERIAL MECHANICS, SE2126 COMPUTER LAB 3 VISCOELASTICITY N t i Gt () G0 1 i ( 1 e τ = α ) i= 1 k a k b τ PART A RELAXING PLASTIC PAPERCLIP Consider an ordinary paperclip made of plastic, as they more

More information

S HEAR S TRENGTH IN P RINCIPAL P LANE OF W OOD

S HEAR S TRENGTH IN P RINCIPAL P LANE OF W OOD S HEAR S TRENGTH IN P RINCIPAL P LANE OF W OOD By Jen Y. Liu 1 and Lester H. Floeter 2 ABSTRACT: In this study, the writers used Tsai and Wu s tensor polynomial theory to rederive a formula originally

More information

Summary PHY101 ( 2 ) T / Hanadi Al Harbi

Summary PHY101 ( 2 ) T / Hanadi Al Harbi الكمية Physical Quantity القانون Low التعريف Definition الوحدة SI Unit Linear Momentum P = mθ be equal to the mass of an object times its velocity. Kg. m/s vector quantity Stress F \ A the external force

More information

Elasticity: Term Paper. Danielle Harper. University of Central Florida

Elasticity: Term Paper. Danielle Harper. University of Central Florida Elasticity: Term Paper Danielle Harper University of Central Florida I. Abstract This research was conducted in order to experimentally test certain components of the theory of elasticity. The theory was

More information

Outline. Tensile-Test Specimen and Machine. Stress-Strain Curve. Review of Mechanical Properties. Mechanical Behaviour

Outline. Tensile-Test Specimen and Machine. Stress-Strain Curve. Review of Mechanical Properties. Mechanical Behaviour Tensile-Test Specimen and Machine Review of Mechanical Properties Outline Tensile test True stress - true strain (flow curve) mechanical properties: - Resilience - Ductility - Toughness - Hardness A standard

More information

Properties of a chiral honeycomb with a Poisson's ratio -1

Properties of a chiral honeycomb with a Poisson's ratio -1 Properties of a chiral honeycomb with a Poisson's ratio -1 D. Prall, R. S. Lakes Int. J. of Mechanical Sciences, 39, 305-314, (1996). Abstract A theoretical and experimental investigation is conducted

More information

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit

More information

FEM Modeling of a 3D Printed Carbon Fiber Pylon

FEM Modeling of a 3D Printed Carbon Fiber Pylon FEM Modeling of a 3D Printed Carbon Fiber Pylon I. López G.*, B. Chiné, and J.L. León S. Costa Rica Institute of Technology, School of Materials Science and Engineering, Cartago, Costa Rica *Corresponding

More information

UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation.

UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. UNIT 1 STRESS STRAIN AND DEFORMATION OF SOLIDS, STATES OF STRESS 1. Define stress. When an external force acts on a body, it undergoes deformation. At the same time the body resists deformation. The magnitude

More information

Symmetric Bending of Beams

Symmetric Bending of Beams Symmetric Bending of Beams beam is any long structural member on which loads act perpendicular to the longitudinal axis. Learning objectives Understand the theory, its limitations and its applications

More information

PLAIN WEAVE REINFORCEMENT IN C/C COMPOSITES VISUALISED IN 3D FOR ELASTIC PARAMETRES

PLAIN WEAVE REINFORCEMENT IN C/C COMPOSITES VISUALISED IN 3D FOR ELASTIC PARAMETRES THE 19 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS PLAIN WEAVE REINFORCEMENT IN C/C COMPOSITES VISUALISED IN 3D FOR ELASTIC PARAMETRES P. Tesinova Technical University of Liberec, Faculty of Textile

More information

The Relationship between the Applied Torque and Stresses in Post-Tension Structures

The Relationship between the Applied Torque and Stresses in Post-Tension Structures ECNDT 6 - Poster 218 The Relationship between the Applied Torque and Stresses in Post-Tension Structures Fui Kiew LIEW, Sinin HAMDAN * and Mohd. Shahril OSMAN, Faculty of Engineering, Universiti Malaysia

More information

The Design of Polyurethane Parts: Using Closed Solutions and Finite Element Analysis to Obtain Optimal Results

The Design of Polyurethane Parts: Using Closed Solutions and Finite Element Analysis to Obtain Optimal Results The Design of Polyurethane Parts: Using Closed Solutions and Finite Element Analysis to Obtain Optimal Results By: Richard Palinkas George Nybakken Ian Laskowitz Chemtura Corporation Overview How does

More information

MECE 3321 MECHANICS OF SOLIDS CHAPTER 3

MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 Samantha Ramirez TENSION AND COMPRESSION TESTS Tension and compression tests are used primarily to determine the relationship between σ avg and ε avg in any material.

More information

UNIT I SIMPLE STRESSES AND STRAINS

UNIT I SIMPLE STRESSES AND STRAINS Subject with Code : SM-1(15A01303) Year & Sem: II-B.Tech & I-Sem SIDDHARTH GROUP OF INSTITUTIONS :: PUTTUR Siddharth Nagar, Narayanavanam Road 517583 QUESTION BANK (DESCRIPTIVE) UNIT I SIMPLE STRESSES

More information

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics

More information

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship

Chapter 5 Elastic Strain, Deflection, and Stability 1. Elastic Stress-Strain Relationship Chapter 5 Elastic Strain, Deflection, and Stability Elastic Stress-Strain Relationship A stress in the x-direction causes a strain in the x-direction by σ x also causes a strain in the y-direction & z-direction

More information

five mechanics of materials Mechanics of Materials Mechanics of Materials Knowledge Required MECHANICS MATERIALS

five mechanics of materials Mechanics of Materials Mechanics of Materials Knowledge Required MECHANICS MATERIALS RCHITECTUR STRUCTURES: FORM, BEHVIOR, ND DESIGN DR. NNE NICHOS SUMMER 2014 Mechanics o Materials MECHNICS MTERIS lecture ive mechanics o materials www.carttalk.com Mechanics o Materials 1 rchitectural

More information

Passive Damping Characteristics of Carbon Epoxy Composite Plates

Passive Damping Characteristics of Carbon Epoxy Composite Plates Journal of aterials Science and Engineering A 6 (1-2) (2016) 35-42 doi: 10.17265/2161-6213/2016.1-2.005 D DAVID PUBLISHIG Passive Damping Characteristics of Carbon Epoxy Composite Plates Dileep Kumar K

More information

Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008

Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008 Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and

More information

Anisotropic modeling of short fibers reinforced thermoplastics materials with LS-DYNA

Anisotropic modeling of short fibers reinforced thermoplastics materials with LS-DYNA Anisotropic modeling of short fibers reinforced thermoplastics materials with LS-DYNA Alexandre Hatt 1 1 Faurecia Automotive Seating, Simplified Limited Liability Company 1 Abstract / Summary Polymer thermoplastics

More information

BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG

BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE 2 ND YEAR STUDENTS OF THE UACEG BOOK OF COURSE WORKS ON STRENGTH OF MATERIALS FOR THE ND YEAR STUDENTS OF THE UACEG Assoc.Prof. Dr. Svetlana Lilkova-Markova, Chief. Assist. Prof. Dimitar Lolov Sofia, 011 STRENGTH OF MATERIALS GENERAL

More information

DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS

DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS Mohsen Safaei, Wim De Waele Ghent University, Laboratory Soete, Belgium Abstract The present work relates to the

More information

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction

COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS. Abstract. Introduction COMPRESSION AND BENDING STIFFNESS OF FIBER-REINFORCED ELASTOMERIC BEARINGS Hsiang-Chuan Tsai, National Taiwan University of Science and Technology, Taipei, Taiwan James M. Kelly, University of California,

More information

ULTRASONIC STUDIES OF STRESSES AND PLASTIC DEFORMATION IN STEEL DURING TENSION AND COMPRESSION. J. Frankel and W. Scholz*

ULTRASONIC STUDIES OF STRESSES AND PLASTIC DEFORMATION IN STEEL DURING TENSION AND COMPRESSION. J. Frankel and W. Scholz* ULTRASONC STUDES OF STRESSES AND PLASTC DEFORMATON N STEEL DURNG TENSON AND COMPRESSON J. Frankel and W. Scholz* US Army Armament Research, Development, & Engineering Ct~ Close Combat Armaments Center

More information

This Technical Note describes how the program checks column capacity or designs reinforced concrete columns when the ACI code is selected.

This Technical Note describes how the program checks column capacity or designs reinforced concrete columns when the ACI code is selected. COMPUTERS AND STRUCTURES, INC., BERKELEY, CALIFORNIA DECEMBER 2001 CONCRETE FRAME DESIGN ACI-318-99 Technical Note This Technical Note describes how the program checks column capacity or designs reinforced

More information

1.103 CIVIL ENGINEERING MATERIALS LABORATORY (1-2-3) Dr. J.T. Germaine Spring 2004 LABORATORY ASSIGNMENT NUMBER 6

1.103 CIVIL ENGINEERING MATERIALS LABORATORY (1-2-3) Dr. J.T. Germaine Spring 2004 LABORATORY ASSIGNMENT NUMBER 6 1.103 CIVIL ENGINEERING MATERIALS LABORATORY (1-2-3) Dr. J.T. Germaine MIT Spring 2004 LABORATORY ASSIGNMENT NUMBER 6 COMPRESSION TESTING AND ANISOTROPY OF WOOD Purpose: Reading: During this laboratory

More information

Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA

Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA Mechanics of Composite Materials, Second Edition Autar K Kaw University of South Florida, Tampa, USA What programs are in PROMAL? Master Menu The master menu screen with five separate applications from

More information

STRENGTH AND STIFFNESS REDUCTION OF LARGE NOTCHED BEAMS

STRENGTH AND STIFFNESS REDUCTION OF LARGE NOTCHED BEAMS STRENGTH AND STIFFNESS REDUCTION OF LARGE NOTCHED BEAMS By Joseph F. Murphy 1 ABSTRACT: Four large glulam beams with notches on the tension side were tested for strength and stiffness. Using either bending

More information

MAE 323: Chapter 4. Plane Stress and Plane Strain. The Stress Equilibrium Equation

MAE 323: Chapter 4. Plane Stress and Plane Strain. The Stress Equilibrium Equation The Stress Equilibrium Equation As we mentioned in Chapter 2, using the Galerkin formulation and a choice of shape functions, we can derive a discretized form of most differential equations. In Structural

More information

14. LS-DYNA Forum 2016

14. LS-DYNA Forum 2016 14. LS-DYNA Forum 2016 A Novel Approach to Model Laminated Glass R. Böhm, A. Haufe, A. Erhart DYNAmore GmbH Stuttgart 1 Content Introduction and Motivation Common approach to model laminated glass New

More information

INFLUENCE KINDS OF MATERIALS ON THE POISSON S RATIO OF WOVEN FABRICS

INFLUENCE KINDS OF MATERIALS ON THE POISSON S RATIO OF WOVEN FABRICS ISSN 1846-6168 (Print), ISSN 1848-5588 (Online) ID: TG-217816142553 Original scientific paper INFLUENCE KINDS OF MATERIALS ON THE POISSON S RATIO OF WOVEN FABRICS Željko PENAVA, Diana ŠIMIĆ PENAVA, Željko

More information

MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS

MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS 1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal

More information

Analysis of Planar Truss

Analysis of Planar Truss Analysis of Planar Truss Although the APES computer program is not a specific matrix structural code, it can none the less be used to analyze simple structures. In this example, the following statically

More information

Materials and Structures. Indian Institute of Technology Kanpur

Materials and Structures. Indian Institute of Technology Kanpur Introduction to Composite Materials and Structures Nachiketa Tiwari Indian Institute of Technology Kanpur Lecture 16 Behavior of Unidirectional Composites Lecture Overview Mt Material ilaxes in unidirectional

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information