Chapter 3 LAMINATED MODEL DERIVATION

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Chapter 3 LAMINATED MODEL DERIVATION"

Transcription

1 17 Chapter 3 LAMINATED MODEL DERIVATION 3.1 Fundamental Poisson Equation The simplest version of the frictionless laminated model was originally introduced in 1961 by Salamon, and more recently explored in 1991 (Salamon, 1962; Salamon, 1991b; Yang, 1992). In this model, the media consists of a stack of strata laminations where the interfaces between beds, including the ground surface, are all horizontal, and free of shear stresses and cohesion (see Figure 3.1). In the general version of this model, the elastic modulus, Poisson s Ratio, and thickness of the j-th bed are E j, v j, and t j, respectively. The theory of thin plates is used as the basis of the model (Ugural and Fenster, 1975). From this theory, the relationship between the vertical deflection (w) of Figure 3.1 Schematic of laminated overburden.

2 18 the middle plane of a horizontal plate and the resultant transverse pressure (p) acting on the plate is defined by: (3.1) where D is the flexural rigidity of a plate: (3.2) and 4 denotes the bi-harmonic operator in the xy plane, specifically: (3.3) Throughout this thesis, positive normal stresses and strains signify compression, and the positive z-axis points vertically downward. This sign convention necessitates that vertical displacement in the upward direction is taken to be positive. As shown in Figure 3.1, the resultant transverse pressure on the (j+1)-th plate can be written as: (3.4) where b and t denote the induced vertical stress at the bottom and the top of the (j+1)-th plate, respectively. It should be noted that in the first approximation, the thickness (t) of the plate will not change provided that no traction acts on the face of the bent plate and that the stretching of the middle plane is neglected (Ugural and Fenster, 1975). Thus, the change in thickness of an individual plate can be attributed solely to the effect of the vertical stress.

3 19 If we assume that the compressive stress acting across the top half of layer j+1 and the bottom half of layer j is t, then the change in distance between the middle planes of the two layers is: (3.5) Solving for the stress on the top of the (j+1)-th layer: (3.6) Similarly, the stress on the bottom of the (j+1)-th layer is: (3.7) Substituting equations 3.6 and 3.7 into equation 3.4 gives: (3.8) Factoring out common terms: (3.9)

4 20 Adding a few additional terms in preparation for a finite-difference representation: (3.10) and rearranging: (3.11) If we use to represent the finite-difference operator such that: (3.12) Then, the previous equation 3.11 can be written as: (3.13) This equation controls the strata pressure for a frictionless laminated model containing distinct layers. If we consider the simplification where the overburden is composed of

5 21 homogeneous stratifications such that the thickness (t), elastic modulus (E) and Poisson s Ratio (v) are identical for each layer, then the above equation (3.13) simplifies greatly to: (3.14) Substituting back into equation 3.1 we get: (3.15) Now, if the thickness of the layers are small in respect to the areal extent of the problem, the finite-difference approximation in equation 3.15 can be represented as a differential operator: (3.16) Substituting this differential operator into 3.15, simplifying and rearranging gives: (3.17) Making a substitution of: (3.18)

6 22 we get: (3.19) This is a fourth-order, partial-differential equation, which mathematically transforms the homogeneous, layered medium into a quasi-continuum that maintains the flexibility of the laminated rock mass. In order to eliminate the bi-harmonic operator ( 4 ) and solve equation 3.19, a double Fourier integral transform is used. The Fourier transform of the vertical displacement, w(x,y,z), with respect to x and y is defined as (Weinberger, 1965): (3.20) where w is the transformed vertical displacement function and 1 and 2 are the transformation variables. In conjunction with the Fourier transform, the inverse Fourier transform is defined as: (3.21) The feature of the Fourier integral transform that makes it useful for solving higher order differential equations and for reducing partial differential equations to ordinary differential equations is that the transform of the derivative of the original function is simply the transform of the original equation multiplied by -i. In our case, this feature can be expressed as: (3.22)

7 23 If the double Fourier transform is applied to equation 3.19, the result is the transformed equation: (3.23) where: (3.24) The general solution for equation 3.23 takes the form: (3.25) where C 1 and C 2 are the constants of integration to be determined from boundary conditions. To solve this equation (3.25), an infinite laminated medium is assumed, and the origin of the z-axis is set to be at the seam level with the positive z-axis pointing downward. Also, compression on the seam is specified as positive and upward movement of the overburden is specified as positive. Next, the domain of the vertical displacement is broken into two parts; above the seam where z is negative (w (-) ) and below the seam where z is positive (w (+) ). Then, applying the boundary condition that the vertical displacement must go to 0 (w 0) as the distance from the seam goes to infinity (z ± ), the solution for the displacements in the roof and floor takes the form: (3.26) where symmetry between the roof and floor displacements dictates that the constant of integration, C, is the same for both equations. Also, knowing that the vertical stress,, is related to the change in displacement by the elastic modulus, E;

8 24 (3.27) the equations for the vertical stress can be written as: (3.28) By examination of equation 3.26, it can be seen that: (3.29) Rearranging these equations we get: (3.30) Now, using the critical identity that the inverse Fourier transform of: (3.31) where 2 is the Laplace operator in the x-y plane: (3.32)

9 25 the inverse transform can be taken of equations 3.30: (3.33) (3.34) Then, since at the seam level the stresses on the roof and floor are equal: and the convergence, s, can be represented as: (3.35) the equations 3.33 can be rearranged as: (3.36) Now expanding the Laplace operator and realizing that the seam level stresses are induced stresses, the fundamental equation for a laminated overburden with homogeneous stratifications can be written: (3.37) This second-order, partial-differential equation relates the convergence in the seam to the induced stress at the seam level in a layered media.

10 Displacement and Stress Influence Functions Equation 3.37 can be used to solve for the displacements at the seam level once the lamination properties are determined. However, in order to solve for displacements and/or stresses remote from the seam, an influence or kernel function which relates the seam convergence to the remote displacement needs to be derived. Following Yang s (1992) derivation, the boundary conditions for a concentrated unit convergence,, applied at (0, 0, 0) are: (3.38) Then the Fourier transform of the first boundary condition is: (3.39) Taking equations 3.26 at z = 0, and substituting these into equation 3.39, the value of C can be found as: (3.40) Substituting this result back into equations 3.26 and then taking the inverse Fourier transform results in the influence function for vertical displacement (W) from a unit point seam convergence: (3.41)

11 27 Using the identity in equation 3.27, the above equation can be used to determine the influence function for vertical stress from a unit point seam convergence: (3.42) 3.3 Numerical Solution of Fundamental Equation The fundamental differential equation which relates the convergence (s) in the seam to the induced stress ( i ) at the seam level in a layered media was derived in equation 3.37 and is repeated here: (3.43) This is a classic second-order, elliptical, partial-differential equation, and many numerical techniques have been developed for solving this type of equation. Examining the right side of this equation, it is found that the induced stress is the only variable which is not a material constant. In the simplest case of an opening in the seam, the induced stress ( i ) is equal to the negative of the primitive ( q ), or overburden stress. However, when there is material in the seam supporting the roof, the induced stress in the surrounding laminations is reduced by the support of the coal or other seam material ( c ). In general, the amount of support provided by the seam material is a function of the seam convergence, c (s), and in the case of failed material or gob, this support would typically be a non-linear function of the seam convergence. The calculation of surface-effect stresses ( s ) are more complicated. For calculating the effects of a traction-free plane at the ground surface, the technique of a mirror-image seam is used (Yang, 1992; Salamon 1991b). Initially, the seam is considered to

12 28 be in an infinite medium and the appropriate seam displacements are determined. Then, a fictitious mirror-image seam is placed above the ground surface at a distance equal to the depth (see Figure 3.2). This fictitious seam is also assumed to be in an infinite medium; however, the calculated convergence in the actual seam is exactly mirrored as divergence in the mirror-image seam. Thus, the distributions of convergence and divergence are identical in magnitude but opposite in sign. Consequently, the sum of the propagated displacements and stresses (after equations 3.41 and 3.42) from the two seams is zero at a plane midway between the two seams, at the ground surface. Thus, the union of the two infinite media solutions corresponds to the effect of the actual seam at finite depth. Figure 3.2 Schematic of mirror-image and multiple-seam stress calculation.

13 29 However, at the level of the actual seam, the propagated stresses from the mirror-image seam contribute to the total induced stress on the actual seam elements. In fact, every element in the mirror-image seam propagates a small incremental stress to every element in the actual seam based on equation So, the surface effect stress ( s ) on any given seam element is equal to the numerical integration of the incremental stresses propagated from the mirror-image seam. (The details of the surface-effect stress calculation are explained further in section ) The calculation of multiple-seam stresses ( m ) is very similar to the calculation of surface-effect stresses. However, instead of a mirror-image seam with mirrored divergence, for the multiple-seam stress calculation, the second seam has an independent mine plan and therefore, an independent displacement distribution (see Figure 3.2). Once again, every element in the second seam propagates a small incremental stress to every element in the actual seam (and vice versa). Therefore, the multiple-seam stress ( m ) on any given seam element in the actual seam is the numerical sum of the incremental stresses propagated from every element of the second seam. (The details of the multipleseam stress calculation are explained further in section ). From the proceeding paragraphs, it is seen that the surface-effect and multiple-seam components of the induced stress are extremely complicated functions of in-seam and off-seam convergence. Thus, in general, the total induced stress can be the sum of many factors, some of which may be non-linear functions of the seam convergence. (3.44) Once it is determined that the induced stress can be a non-linear function of the seam convergence, the choice of solution techniques is generally limited to iterative procedures. Also, from a practical perspective, a robust solution algorithm with accelerated convergence is desirable, and to stay compatible with MULSIM it is desired to solve the seam convergence distribution on an even grid. Considering these factors, a

14 30 central-difference approximation using a Gauss-Siedel iterative scheme with Successive Over-Relaxation (SOR) was chosen as the solution technique (Ames, 1992). With this numerical technique, equation 3.43 can be solved over a gridded area using successive iterations of the kernel equation: (3.45) where: O is the over-relaxation factor (between 1 and 2); x is the grid dimension; the superscript r refers to the iteration number; and the subscripts j and k refer to the horizontal and vertical grid locations, respectively, on the finite-difference grid such that s 2,2 is the convergence value at the grid intersection 2 over and 2 up from the origin. This Gauss-Siedel solution scheme with SOR provides a number of numerical and computational advantages for the practical solution of equation Adjustment of the over-relaxation factor (O) allows the number of iterations for the convergence of the finite-difference solution to be minimized. Using the convergence values that were recently updated during the present iteration improves the speed of convergence and allows the convergence values to be stored in a single array that is updated as the solution progresses through the grid. Finally, the iterative solution technique allows the calculation of the non-linear induced stress to be smoothly incorporated into the normal iteration cycle.

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

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method

Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Module 4 : Deflection of Structures Lecture 4 : Strain Energy Method Objectives In this course you will learn the following Deflection by strain energy method. Evaluation of strain energy in member under

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

Chapter 2 CONTINUUM MECHANICS PROBLEMS

Chapter 2 CONTINUUM MECHANICS PROBLEMS Chapter 2 CONTINUUM MECHANICS PROBLEMS The concept of treating solids and fluids as though they are continuous media, rather thancomposedofdiscretemolecules, is one that is widely used in most branches

More information

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD

ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY THE SUPERPOSITION METHOD Journal of Sound and Vibration (1999) 219(2), 265 277 Article No. jsvi.1998.1874, available online at http://www.idealibrary.com.on ACCURATE FREE VIBRATION ANALYSIS OF POINT SUPPORTED MINDLIN PLATES BY

More information

Chapter 12 Partial Differential Equations

Chapter 12 Partial Differential Equations Chapter 12 Partial Differential Equations Advanced Engineering Mathematics Wei-Ta Chu National Chung Cheng University wtchu@cs.ccu.edu.tw 1 2 12.1 Basic Concepts of PDEs Partial Differential Equation A

More information

Methods of Interpreting Ground Stress Based on Underground Stress Measurements and Numerical Modelling

Methods of Interpreting Ground Stress Based on Underground Stress Measurements and Numerical Modelling University of Wollongong Research Online Coal Operators' Conference Faculty of Engineering and Information Sciences 2006 Methods of Interpreting Ground Stress Based on Underground Stress Measurements and

More information

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur

Sound Propagation through Media. Nachiketa Tiwari Indian Institute of Technology Kanpur Sound Propagation through Media Nachiketa Tiwari Indian Institute of Technology Kanpur LECTURE-13 WAVE PROPAGATION IN SOLIDS Longitudinal Vibrations In Thin Plates Unlike 3-D solids, thin plates have surfaces

More information

Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS. Introduction

Lecture 4: PRELIMINARY CONCEPTS OF STRUCTURAL ANALYSIS. Introduction Introduction In this class we will focus on the structural analysis of framed structures. We will learn about the flexibility method first, and then learn how to use the primary analytical tools associated

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

Final Exam Solution Dynamics :45 12:15. Problem 1 Bateau

Final Exam Solution Dynamics :45 12:15. Problem 1 Bateau Final Exam Solution Dynamics 2 191157140 31-01-2013 8:45 12:15 Problem 1 Bateau Bateau is a trapeze act by Cirque du Soleil in which artists perform aerial maneuvers on a boat shaped structure. The boat

More information

Practice Test SHM with Answers

Practice Test SHM with Answers Practice Test SHM with Answers MPC 1) If we double the frequency of a system undergoing simple harmonic motion, which of the following statements about that system are true? (There could be more than one

More information

Classification of partial differential equations and their solution characteristics

Classification of partial differential equations and their solution characteristics 9 TH INDO GERMAN WINTER ACADEMY 2010 Classification of partial differential equations and their solution characteristics By Ankita Bhutani IIT Roorkee Tutors: Prof. V. Buwa Prof. S. V. R. Rao Prof. U.

More information

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.

Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.

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

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

If the number of unknown reaction components are equal to the number of equations, the structure is known as statically determinate.

If the number of unknown reaction components are equal to the number of equations, the structure is known as statically determinate. 1 of 6 EQUILIBRIUM OF A RIGID BODY AND ANALYSIS OF ETRUCTURAS II 9.1 reactions in supports and joints of a two-dimensional structure and statically indeterminate reactions: Statically indeterminate structures

More information

A *69>H>N6 #DJGC6A DG C<>C::G>C<,8>:C8:H /DA 'D 2:6G - ( - ) +"' ( + -"( (' (& -+" % '('%"' +"-2 ( -!"',- % )% -.C>K:GH>IN D; AF69>HH>6,-+

A *69>H>N6 #DJGC6A DG C<>C::G>C<,8>:C8:H /DA 'D 2:6G - ( - ) +' ( + -( (' (& -+ % '('%' +-2 ( -!',- % )% -.C>K:GH>IN D; AF69>HH>6,-+ The primary objective is to determine whether the structural efficiency of plates can be improved with variable thickness The large displacement analysis of steel plate with variable thickness at direction

More information

Use a highlighter to mark the most important parts, or the parts. you want to remember in the background information.

Use a highlighter to mark the most important parts, or the parts. you want to remember in the background information. P a g e 1 Name A Fault Model Purpose: To explore the types of faults and how they affect the geosphere Background Information: A fault is an area of stress in the earth where broken rocks slide past each

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

Unit 4 Lesson 3 Mountain Building. Copyright Houghton Mifflin Harcourt Publishing Company

Unit 4 Lesson 3 Mountain Building. Copyright Houghton Mifflin Harcourt Publishing Company Stressed Out How can tectonic plate motion cause deformation? The movement of tectonic plates causes stress on rock structures. Stress is the amount of force per unit area that is placed on an object.

More information

Unit M1.5 Statically Indeterminate Systems

Unit M1.5 Statically Indeterminate Systems Unit M1.5 Statically Indeterminate Systems Readings: CDL 2.1, 2.3, 2.4, 2.7 16.001/002 -- Unified Engineering Department of Aeronautics and Astronautics Massachusetts Institute of Technology LEARNING OBJECTIVES

More information

SETTLEMENT TROUGH DUE TO TUNNELING IN COHESIVE GROUND

SETTLEMENT TROUGH DUE TO TUNNELING IN COHESIVE GROUND Indian Geotechnical Journal, 41(), 11, 64-75 SETTLEMENT TROUGH DUE TO TUNNELING IN COHESIVE GROUND Mohammed Y. Fattah 1, Kais T. Shlash and Nahla M. Salim 3 Key words Tunnel, clay, finite elements, settlement,

More information

Composites Design and Analysis. Stress Strain Relationship

Composites Design and Analysis. Stress Strain Relationship Composites Design and Analysis Stress Strain Relationship Composite design and analysis Laminate Theory Manufacturing Methods Materials Composite Materials Design / Analysis Engineer Design Guidelines

More information

Dynamics Manual. Version 7

Dynamics Manual. Version 7 Dynamics Manual Version 7 DYNAMICS MANUAL TABLE OF CONTENTS 1 Introduction...1-1 1.1 About this manual...1-1 2 Tutorial...2-1 2.1 Dynamic analysis of a generator on an elastic foundation...2-1 2.1.1 Input...2-1

More information

DYNAMIC ANALYSIS OF PILES IN SAND BASED ON SOIL-PILE INTERACTION

DYNAMIC ANALYSIS OF PILES IN SAND BASED ON SOIL-PILE INTERACTION October 1-17,, Beijing, China DYNAMIC ANALYSIS OF PILES IN SAND BASED ON SOIL-PILE INTERACTION Mohammad M. Ahmadi 1 and Mahdi Ehsani 1 Assistant Professor, Dept. of Civil Engineering, Geotechnical Group,

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

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

IGJ PROOFS SETTLEMENT TROUGH DUE TO TUNNELING IN COHESIVE GROUND. Surface Settlement. Introduction. Indian Geotechnical Journal, 41(2), 2011, 64-75

IGJ PROOFS SETTLEMENT TROUGH DUE TO TUNNELING IN COHESIVE GROUND. Surface Settlement. Introduction. Indian Geotechnical Journal, 41(2), 2011, 64-75 Indian Geotechnical Journal, 41(), 11, 64-75 SETTLEMENT TROUGH DUE TO TUNNELING IN COHESIVE GROUND Key words Tunnel, clay, finite elements, settlement, complex variable Introduction The construction of

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

PLAT DAN CANGKANG (TKS 4219)

PLAT DAN CANGKANG (TKS 4219) PLAT DAN CANGKANG (TKS 4219) SESI I: PLATES Dr.Eng. Achfas Zacoeb Dept. of Civil Engineering Brawijaya University INTRODUCTION Plates are straight, plane, two-dimensional structural components of which

More information

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method

Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method Module 2 Analysis of Statically Indeterminate Structures by the Matrix Force Method Lesson 8 The Force Method of Analysis: Beams Instructional Objectives After reading this chapter the student will be

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

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

FINITE-VOLUME SOLUTION OF DIFFUSION EQUATION AND APPLICATION TO MODEL PROBLEMS

FINITE-VOLUME SOLUTION OF DIFFUSION EQUATION AND APPLICATION TO MODEL PROBLEMS IJRET: International Journal of Research in Engineering and Technology eissn: 39-63 pissn: 3-738 FINITE-VOLUME SOLUTION OF DIFFUSION EQUATION AND APPLICATION TO MODEL PROBLEMS Asish Mitra Reviewer: Heat

More information

Instabilities and Dynamic Rupture in a Frictional Interface

Instabilities and Dynamic Rupture in a Frictional Interface Instabilities and Dynamic Rupture in a Frictional Interface Laurent BAILLET LGIT (Laboratoire de Géophysique Interne et Tectonophysique) Grenoble France laurent.baillet@ujf-grenoble.fr http://www-lgit.obs.ujf-grenoble.fr/users/lbaillet/

More information

DETERMINING THE STRESS PATTERN IN THE HH RAILROAD TIES DUE TO DYNAMIC LOADS 1

DETERMINING THE STRESS PATTERN IN THE HH RAILROAD TIES DUE TO DYNAMIC LOADS 1 PERIODICA POLYTECHNICA SER. CIV. ENG. VOL. 46, NO. 1, PP. 125 148 (2002) DETERMINING THE STRESS PATTERN IN THE HH RAILROAD TIES DUE TO DYNAMIC LOADS 1 Nándor LIEGNER Department of Highway and Railway Engineering

More information

An accelerated predictor-corrector scheme for 3D crack growth simulations

An accelerated predictor-corrector scheme for 3D crack growth simulations An accelerated predictor-corrector scheme for 3D crack growth simulations W. Weber 1 and G. Kuhn 2 1,2 1 Institute of Applied Mechanics, University of Erlangen-Nuremberg Egerlandstraße 5, 91058 Erlangen,

More information

Numerical Modeling for Different Types of Fractures

Numerical Modeling for Different Types of Fractures umerical Modeling for Different Types of Fractures Xiaoqin Cui* CREWES Department of Geoscience University of Calgary Canada xicui@ucalgary.ca and Laurence R. Lines Edward S. Krebes Department of Geoscience

More information

CVEEN 7330 Modeling Exercise 2c

CVEEN 7330 Modeling Exercise 2c CVEEN 7330 Modeling Exercise 2c Table of Contents Table of Contents... 1 Objectives:... 2 FLAC Input:... 2 DEEPSOIL INPUTS:... 5 Required Outputs from FLAC:... 6 Required Output from DEEPSOIL:... 6 Additional

More information

Micro-meso draping modelling of non-crimp fabrics

Micro-meso draping modelling of non-crimp fabrics Micro-meso draping modelling of non-crimp fabrics Oleksandr Vorobiov 1, Dr. Th. Bischoff 1, Dr. A. Tulke 1 1 FTA Forschungsgesellschaft für Textiltechnik mbh 1 Introduction Non-crimp fabrics (NCFs) are

More information

Analyzing effect of fluid flow on surface subsidence

Analyzing effect of fluid flow on surface subsidence Analyzing effect of fluid flow on surface subsidence in mining area Y. Abousleiman", M. Bai\ H. Zhang', T. Liu" and J.-C. Roegiers* a. School of Engineering and Architecture, The Lebanese American University,

More information

FIXED BEAMS IN BENDING

FIXED BEAMS IN BENDING FIXED BEAMS IN BENDING INTRODUCTION Fixed or built-in beams are commonly used in building construction because they possess high rigidity in comparison to simply supported beams. When a simply supported

More information

Chapter 5: Ball Grid Array (BGA)

Chapter 5: Ball Grid Array (BGA) Chapter 5: Ball Grid Array (BGA) 5.1 Development of the Models The following sequence of pictures explains schematically how the FE-model of the Ball Grid Array (BGA) was developed. Initially a single

More information

Main Means of Rock Stress Measurement

Main Means of Rock Stress Measurement Real Stress Distributions through Sedimentary Strata and Implications for Reservoir Development and Potential Gas and Coal Development Strategies Ian Gray Sigra Pty Ltd 93 Colebard St West, Acacia Ridge,

More information

Equivalent electrostatic capacitance Computation using FreeFEM++

Equivalent electrostatic capacitance Computation using FreeFEM++ Equivalent electrostatic capacitance Computation using FreeFEM++ P. Ventura*, F. Hecht** *PV R&D Consulting, Nice, France **Laboratoire Jacques Louis Lions, Université Pierre et Marie Curie, Paris, France

More information

Mining. Slope stability analysis at highway BR-153 using numerical models. Mineração. Abstract. 1. Introduction

Mining. Slope stability analysis at highway BR-153 using numerical models. Mineração. Abstract. 1. Introduction Mining Mineração http://dx.doi.org/10.1590/0370-44672015690040 Ricardo Hundelshaussen Rubio Engenheiro Industrial / Doutorando Universidade Federal do Rio Grande do Sul - UFRS Departamento de Engenharia

More information

A study of the critical condition of a battened column and a frame by classical methods

A study of the critical condition of a battened column and a frame by classical methods University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 003 A study of the critical condition of a battened column and a frame by classical methods Jamal A.H Bekdache

More information

Chapter 6: Cross-Sectional Properties of Structural Members

Chapter 6: Cross-Sectional Properties of Structural Members Chapter 6: Cross-Sectional Properties of Structural Members Introduction Beam design requires the knowledge of the following. Material strengths (allowable stresses) Critical shear and moment values Cross

More information

EE C245 ME C218 Introduction to MEMS Design

EE C245 ME C218 Introduction to MEMS Design EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 16: Energy

More information

Hydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition

Hydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition Fluid Structure Interaction and Moving Boundary Problems IV 63 Hydroelastic vibration of a rectangular perforated plate with a simply supported boundary condition K.-H. Jeong, G.-M. Lee, T.-W. Kim & J.-I.

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

Brittle Deformation. Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm

Brittle Deformation. Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm Lecture 6 Brittle Deformation Earth Structure (2 nd Edition), 2004 W.W. Norton & Co, New York Slide show by Ben van der Pluijm WW Norton, unless noted otherwise Brittle deformation EarthStructure (2 nd

More information

Flow and Transport. c(s, t)s ds,

Flow and Transport. c(s, t)s ds, Flow and Transport 1. The Transport Equation We shall describe the transport of a dissolved chemical by water that is traveling with uniform velocity ν through a long thin tube G with uniform cross section

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

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

Forces in Earth s Crust

Forces in Earth s Crust Forces in Earth s Crust (pages 180 186) Types of Stress (page 181) Key Concept: Tension, compression, and shearing work over millions of years to change the shape and volume of rock. When Earth s plates

More information

VORONOI APPLIED ELEMENT METHOD FOR STRUCTURAL ANALYSIS: THEORY AND APPLICATION FOR LINEAR AND NON-LINEAR MATERIALS

VORONOI APPLIED ELEMENT METHOD FOR STRUCTURAL ANALYSIS: THEORY AND APPLICATION FOR LINEAR AND NON-LINEAR MATERIALS The 4 th World Conference on Earthquake Engineering October -7, 008, Beijing, China VORONOI APPLIED ELEMENT METHOD FOR STRUCTURAL ANALYSIS: THEORY AND APPLICATION FOR LINEAR AND NON-LINEAR MATERIALS K.

More information

Chapter 15 Structures

Chapter 15 Structures Chapter 15 Structures Plummer/McGeary/Carlson (c) The McGraw-Hill Companies, Inc. TECTONIC FORCES AT WORK Stress & Strain Stress Strain Compressive stress Shortening strain Tensional stress stretching

More information

Optimum Size of Coal Pillar Dimensions Serving Mechanised Caving Longwall Face in a Thick Seam

Optimum Size of Coal Pillar Dimensions Serving Mechanised Caving Longwall Face in a Thick Seam University of Wollongong Research Online Coal Operators' Conference Faculty of Engineering and Information Sciences 2016 Optimum Size of Coal Pillar Dimensions Serving Mechanised Caving Longwall Face in

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

HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLÜGGE. BAND VIa/2 FESTKÖRPERMECHANIK II BANDHERAUSGEBER C.TRUESDELL MIT 25 FIGUREN

HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLÜGGE. BAND VIa/2 FESTKÖRPERMECHANIK II BANDHERAUSGEBER C.TRUESDELL MIT 25 FIGUREN HANDBUCH DER PHYSIK HERAUSGEGEBEN VON S. FLÜGGE BAND VIa/2 FESTKÖRPERMECHANIK II BANDHERAUSGEBER C.TRUESDELL MIT 25 FIGUREN SPRINGER-VERLAG BERLIN HEIDELBERG NEWYORK 1972 Contents. The Linear Theory of

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

Plane Strain Test for Metal Sheet Characterization

Plane Strain Test for Metal Sheet Characterization Plane Strain Test for Metal Sheet Characterization Paulo Flores 1, Felix Bonnet 2 and Anne-Marie Habraken 3 1 DIM, University of Concepción, Edmundo Larenas 270, Concepción, Chile 2 ENS - Cachan, Avenue

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

EQUILIBRIUM and ELASTICITY

EQUILIBRIUM and ELASTICITY PH 221-1D Spring 2013 EQUILIBRIUM and ELASTICITY Lectures 30-32 Chapter 12 (Halliday/Resnick/Walker, Fundamentals of Physics 9 th edition) 1 Chapter 12 Equilibrium and Elasticity In this chapter we will

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

Calculation of periodic roof weighting interval in longwall mining using finite element method

Calculation of periodic roof weighting interval in longwall mining using finite element method Calculation of periodic roof weighting interval in longwall mining using finite element method Navid Hosseini 1, Kamran Goshtasbi 2, Behdeen Oraee-Mirzamani 3 Abstract The state of periodic loading and

More information

L e c t u r e. D r. S a s s a n M o h a s s e b

L e c t u r e. D r. S a s s a n M o h a s s e b The Scaled smteam@gmx.ch Boundary Finite www.erdbebenschutz.ch Element Method Lecture A L e c t u r e A1 D r. S a s s a n M o h a s s e b V i s i t i n g P r o f e s s o r M. I. T. C a m b r i d g e December

More information

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11

Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 11 Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras Module - 01 Lecture - 11 Last class, what we did is, we looked at a method called superposition

More information

7.5 Elastic Buckling Columns and Buckling

7.5 Elastic Buckling Columns and Buckling 7.5 Elastic Buckling The initial theory of the buckling of columns was worked out by Euler in 1757, a nice example of a theory preceding the application, the application mainly being for the later invented

More information

Equilibrium & Elasticity

Equilibrium & Elasticity PHYS 101 Previous Exam Problems CHAPTER 12 Equilibrium & Elasticity Static equilibrium Elasticity 1. A uniform steel bar of length 3.0 m and weight 20 N rests on two supports (A and B) at its ends. A block

More information

Finite element modelling of infinitely wide Angle-ply FRP. laminates

Finite element modelling of infinitely wide Angle-ply FRP. laminates www.ijaser.com 2012 by the authors Licensee IJASER- Under Creative Commons License 3.0 editorial@ijaser.com Research article ISSN 2277 9442 Finite element modelling of infinitely wide Angle-ply FRP laminates

More information

Chapter 2: Deflections of Structures

Chapter 2: Deflections of Structures Chapter 2: Deflections of Structures Fig. 4.1. (Fig. 2.1.) ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 1 (2.1) (4.1) (2.2) Fig.4.2 Fig.2.2 ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 2

More information

Oscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Mass-spring system Energy in SHM Pendulums

Oscillations. PHYS 101 Previous Exam Problems CHAPTER. Simple harmonic motion Mass-spring system Energy in SHM Pendulums PHYS 101 Previous Exam Problems CHAPTER 15 Oscillations Simple harmonic motion Mass-spring system Energy in SHM Pendulums 1. The displacement of a particle oscillating along the x axis is given as a function

More information

Piezoelectric Bimorph Response with Imperfect Bonding Conditions

Piezoelectric Bimorph Response with Imperfect Bonding Conditions Copyright c 28 ICCES ICCES, vol.6, no.3, pp.5-56 Piezoelectric Bimorph Response with Imperfect Bonding Conditions Milazzo A., Alaimo A. and Benedetti I. Summary The effect of the finite stiffness bonding

More information

2D Embankment and Slope Analysis (Numerical)

2D Embankment and Slope Analysis (Numerical) 2D Embankment and Slope Analysis (Numerical) Page 1 2D Embankment and Slope Analysis (Numerical) Sunday, August 14, 2011 Reading Assignment Lecture Notes Other Materials FLAC Manual 1. 2. Homework Assignment

More information

(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e

(MPa) compute (a) The traction vector acting on an internal material plane with normal n ( e1 e EN10: Continuum Mechanics Homework : Kinetics Due 1:00 noon Friday February 4th School of Engineering Brown University 1. For the Cauchy stress tensor with components 100 5 50 0 00 (MPa) compute (a) The

More information

14. *14.8 CASTIGLIANO S THEOREM

14. *14.8 CASTIGLIANO S THEOREM *14.8 CASTIGLIANO S THEOREM Consider a body of arbitrary shape subjected to a series of n forces P 1, P 2, P n. Since external work done by forces is equal to internal strain energy stored in body, by

More information

FRICTION INDUCED IRREVERSIBLE STRETCHING OF SUBSTRATE FILMS BY RECOR- DING WITH A CATAMARAN GLIDER ON THIN FILM FLOPPY DISKS

FRICTION INDUCED IRREVERSIBLE STRETCHING OF SUBSTRATE FILMS BY RECOR- DING WITH A CATAMARAN GLIDER ON THIN FILM FLOPPY DISKS Journal of the Magnetics Society of Japan Vol. 15 Supplement, No. S2 (1991) 1991 by The Magnetics Society of Japan FRICTION INDUCED IRREVERSIBLE STRETCHING OF SUBSTRATE FILMS BY RECOR- DING WITH A CATAMARAN

More information

Chapter 2 Basis for Indeterminate Structures

Chapter 2 Basis for Indeterminate Structures Chapter - Basis for the Analysis of Indeterminate Structures.1 Introduction... 3.1.1 Background... 3.1. Basis of Structural Analysis... 4. Small Displacements... 6..1 Introduction... 6.. Derivation...

More information

VIBRATION PROBLEMS IN ENGINEERING

VIBRATION PROBLEMS IN ENGINEERING VIBRATION PROBLEMS IN ENGINEERING FIFTH EDITION W. WEAVER, JR. Professor Emeritus of Structural Engineering The Late S. P. TIMOSHENKO Professor Emeritus of Engineering Mechanics The Late D. H. YOUNG Professor

More information

Waves propagation in an arbitrary direction in heat conducting orthotropic elastic composites

Waves propagation in an arbitrary direction in heat conducting orthotropic elastic composites Rakenteiden Mekaniikka (Journal of Structural Mechanics) Vol. 46 No 03 pp. 4-5 Waves propagation in an arbitrary direction in heat conducting orthotropic elastic composites K. L. Verma Summary. Dispersion

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

THE INFLUENCE OF THERMAL ACTIONS AND COMPLEX SUPPORT CONDITIONS ON THE MECHANICAL STATE OF SANDWICH STRUCTURE

THE INFLUENCE OF THERMAL ACTIONS AND COMPLEX SUPPORT CONDITIONS ON THE MECHANICAL STATE OF SANDWICH STRUCTURE Journal of Applied Mathematics and Computational Mechanics 013, 1(4), 13-1 THE INFLUENCE OF THERMAL ACTIONS AND COMPLEX SUPPORT CONDITIONS ON THE MECHANICAL STATE OF SANDWICH STRUCTURE Jolanta Błaszczuk

More information

Static & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering

Static & Dynamic. Analysis of Structures. Edward L.Wilson. University of California, Berkeley. Fourth Edition. Professor Emeritus of Civil Engineering Static & Dynamic Analysis of Structures A Physical Approach With Emphasis on Earthquake Engineering Edward LWilson Professor Emeritus of Civil Engineering University of California, Berkeley Fourth Edition

More information

Introduction to Finite Element Method

Introduction to Finite Element Method Introduction to Finite Element Method Dr. Rakesh K Kapania Aerospace and Ocean Engineering Department Virginia Polytechnic Institute and State University, Blacksburg, VA AOE 524, Vehicle Structures Summer,

More information

SIMULATION OF PLANE STRAIN FIBER COMPOSITE PLATES IN BENDING THROUGH A BEM/ACA/HM FORMULATION

SIMULATION OF PLANE STRAIN FIBER COMPOSITE PLATES IN BENDING THROUGH A BEM/ACA/HM FORMULATION 8 th GRACM International Congress on Computational Mechanics Volos, 12 July 15 July 2015 SIMULATION OF PLANE STRAIN FIBER COMPOSITE PLATES IN BENDING THROUGH A BEM/ACA/HM FORMULATION Theodore V. Gortsas

More information

U.S. South America Workshop. Mechanics and Advanced Materials Research and Education. Rio de Janeiro, Brazil. August 2 6, Steven L.

U.S. South America Workshop. Mechanics and Advanced Materials Research and Education. Rio de Janeiro, Brazil. August 2 6, Steven L. Computational Modeling of Composite and Functionally Graded Materials U.S. South America Workshop Mechanics and Advanced Materials Research and Education Rio de Janeiro, Brazil August 2 6, 2002 Steven

More information

ADVANCES IN THE PROGRESSIVE DAMAGE ANALYSIS OF COMPOSITES

ADVANCES IN THE PROGRESSIVE DAMAGE ANALYSIS OF COMPOSITES NAFEMS WORLD CONGRESS 13, SALZBURG, AUSTRIA ADVANCES IN THE PROGRESSIVE DAMAGE ANALYSIS OF M. Bruyneel, J.P. Delsemme, P. Jetteur (LMS Samtech, Belgium); A.C. Goupil (ISMANS, France). Dr. Ir. M. Bruyneel,

More information

Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon.

Materials: engineering, science, processing and design, 2nd edition Copyright (c)2010 Michael Ashby, Hugh Shercliff, David Cebon. Modes of Loading (1) tension (a) (2) compression (b) (3) bending (c) (4) torsion (d) and combinations of them (e) Figure 4.2 1 Standard Solution to Elastic Problems Three common modes of loading: (a) tie

More information

Shape Earth. Plate Boundaries. Building. Building

Shape Earth. Plate Boundaries. Building. Building Chapter Introduction Lesson 1 Lesson 2 Lesson 3 Lesson 4 Chapter Wrap-Up Forces That Shape Earth Landforms at Plate Boundaries Mountain Building Continent Building How is Earth s surface shaped by plate

More information

Application of a non-local failure criterion to a crack in heterogeneous media S. Bavaglia*, S.E. Mikhailov*

Application of a non-local failure criterion to a crack in heterogeneous media S. Bavaglia*, S.E. Mikhailov* Application of a non-local failure criterion to a crack in heterogeneous media S. Bavaglia*, S.E. Mikhailov* University of Perugia, Italy Email: mic@unipg.it ^Wessex Institute of Technology, Ashurst Lodge,

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

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

An alternative multi-region BEM technique for layered soil problems

An alternative multi-region BEM technique for layered soil problems An alternative multi-region BM technique for layered soil problems D.B. Ribeiro & J.B. Paiva Structural ngineering Department, São Carlos ngineering School, University of São Paulo, Brazil. Abstract Different

More information

Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate

Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate Buckling Behavior of 3D Randomly Oriented CNT Reinforced Nanocomposite Plate Outline Introduction Representative Volume Element (RVE) Periodic Boundary Conditions on RVE Homogenization Method Analytical

More information

NORCO COLLEGE SLO to PLO MATRIX

NORCO COLLEGE SLO to PLO MATRIX SLO to PLO MATRI CERTIFICATE/PROGRAM: Math ADT COURSE: MAT-1A Calculus I Calculate the limit of a function. SLO 2 Determine the continuity of a function. Find the derivatives of algebraic and transcendental

More information

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design

Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Seismic Pushover Analysis Using AASHTO Guide Specifications for LRFD Seismic Bridge Design Elmer E. Marx, Alaska Department of Transportation and Public Facilities Michael Keever, California Department

More information

UNIT I SOLID STATE PHYSICS

UNIT I SOLID STATE PHYSICS UNIT I SOLID STATE PHYSICS CHAPTER 1 CRYSTAL STRUCTURE 1.1 INTRODUCTION When two atoms are brought together, two kinds of forces: attraction and repulsion come into play. The force of attraction increases

More information