Chapter 2 Governing Equations

Size: px
Start display at page:

Download "Chapter 2 Governing Equations"

Transcription

1 Chapter Governing Equations Abstract In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is assumed to be isotropic, continuous and infinite. Although this subject has been treated in various approaches in a number of references, it may be appropriate to develop a general procedure for solution of a number of essential soil dynamic problems based on the first principles of elasticity. The main scope of this chapter is to present a general systematic solution for addressing the problems considered in the next two chapters. This general solution is presented in two-dimensional Fourier domain in terms of elastic dilatation, and two elastic rotation components, which are used for decoupling of the involved partial differential equations. This chapter is also presenting the required fundamental of elasto-dynamics to achieve the set goal for the chapter. Keywords Stress strain Boundary displacements relation Elastic rotations Boundary stresses In this chapter fundamental governing equations for propagation of a harmonic disturbance on the surface of an elastic half-space is presented. The elastic media is assumed to be isotropic, continuous and infinite. Although this subject has been treated in various approaches in a number of references such as Lamb (904), Timoshenko and Goodier (95), Sneddon (95), Ewing and Jardetzky (957), and many others, it may be appropriate to develop a general procedure for solution of a number of essential soil dynamic problems based on the first principles of elasticity. The main scope of this chapter is to present a general systematic solution for addressing the problems considered in the next two chapters. This general solution is presented in two-dimensional Fourier domain in terms of elastic dilatation, and two elastic rotation components, which are used for decoupling of the involved partial differential equations. This chapter is also presenting the required fundamental of elasto-dynamics to achieve the set goal for the chapter. H.R. Hamidzadeh et al., Wave Propagation in Solid and Porous Half-Space Media, DOI 0.007/ , Springer ScienceCBusiness Media New York 04 5

2 6 Governing Equations Fig.. System of stresses on an infinitesimal element in the elastic medium z σ σ zz zz + z dz τ τ zy τ zx zx + dz τ zy + dz z z σ xx τ τ yz xy τ yz + dy τ σ yx y yy τ τ yz σ τ xz yy xz + dx σ yy + x τ yz τ yx y τ xy τ yx + dx τ xy + dx x σ σ xx x xx + dx x τ τ zx zy dy y σ zz x. Derivation of Equations of Motion The equations of motion for an elastic half-space medium can be obtained by deriving the equilibrium equations for an infinitesimal element in terms of the applied stresses. This element is assumed to be an elastic body with an applied orthogonal stresses as shown in Fig... The equations of motion for the element can be expressed by writing Newton s second law in three directions (see Shames and Cozzarelli 99). These equations yy yx zx C X (.) zy xy C Y zz xz yz C Z (.3) where is the mass density of the medium; ij and ij are direct and shear stresses on the plane perpendicular to j direction and along the direction of i; X, Y,and Z are internal forces per unit volume in the x, y, andz directions; u, v, andw are displacements of a point in the medium in the x, y, andz directions.. Stress Strain Relation For a homogeneous isotropic material, i.e. a material having the same properties in all directions, Hooke s Law can be written by the following equations:

3 .3 Strains in Terms of Displacements 7 " xx D E xx yy C zz (.4) " yy D yy. zz C xx / (.5) E " zz D zz xx C yy E (.6) where E and are elastic modulus and Poisson ratio of the medium. Adding Eqs. (.4) (.6) yields: E " xx C " yy C " zz D. / xx C yy C zz (.7) After substituting yy C zz from Eq. (.4) into Eq. (.7) and simplifying, this yields: E xx D. C /. / " C E. C / " xx (.8) " D " xx C " yy C " zz (.9) Introducing Lame constants ( and G) the above equation can be written as: Following the same procedure, yy and zz become xx D " C G" xx (.0) yy D " C G" yy (.) zz D " C G" zz (.) The familiar equations relating shear stresses to strains are: xy D yx D G xy (.3) yz D zy D G yz (.4) zx D xz D G zx (.5).3 Strains in Terms of Displacements Since strains generally vary from point to point, the mathematical definitions of strain must relate to an infinitesimal element. Consider an element in the.x; y/ plane (Fig..). During straining, point A experiences displacements u and v in the x and y directions. Displacements of other points are also shown in Fig...

4 8 Governing Equations Fig.. Elastic deformation of the cubic element in the.x;y/ plane v v+ dy y C y u u+ dy y C ' ϕ D D' dy A u dx v δz θ B u u+ dx x B' v v+ dx x x On this basis, the mathematical expressions for the linear strains as described by Ford and Alexander (963) and many others are " xx " yy (.6) (.7) " zz (.8) These three strains are direct strains, but shear strains due to small deformation can be presented in terms of rotation. On the other hand, angles of rotation of any side of the element are very small, therefore, by definition: Because dx xy D C ' D v C dx v and dy >>,Eq.(.9) becomes u C dy u (.9) xy (.0) Consequently, other shear strains are given by yz zx (.) (.)

5 .5 Equations of Motion 9.4 Elastic Rotations in Terms of Displacements BasedonFig.. the elastic rotation about the z-axis is defined to be:! z D. '/ (.3) Considering that both angles of and ' are very small then this component of elastic rotation can be expressed as! (.4) Similarly, other rotations are! (.5) (.6).5 Equations of Motion Now, having substituted Eqs. (.0) (.) and(.6) (.8) into Eqs. (.) (.3), the equations of motion for the element C G" xx/ G yx G zx C X @ " C G"yy C G yz G xy C Y C G" zz/ G xz G yz C Z (.9) Substituting for strains in Eqs. (.7) (.9), in terms of volumetric strain and displacements, the equations of motion become:. C C Gr u C X (.30). C C Gr v C Y (.3). C C Gr w C Z (.3)

6 0 Governing Equations where r is the Laplacian operator. If the body force is neglected and variations of the displacements are harmonic with circular frequency of, then Eqs. (.30) (.3) can be written in the following form:. C C Gr u D u (.33). C C Gr v D v (.34). C C Gr w D w (.35) The above equations can be represented in terms of dilatation and rotations by differentiating Eqs. (.33) (.35) with respect to x, y, andz, then adding them together:. C G/r " C Gr D 0 (.36) Taking into account the definition of volumetric strain, the above equation becomes r C G r " C " D 0 (.37) or r C " D 0 (.38) where D C C D C G (.39) (.40) C is the velocity of waves of dilatation in the media. Subtracting the derivative of (.34) with respect to x from the derivative of (.33) with respect to y gives. " C Gr. @u " Gr D After simplifying the above equation (.4) r C!z D 0 (.4)

7 .6 Displacements in Terms of Dilatation and Rotation Components where D C C D G (.43) (.44) C is velocity of waves of distortion. Finally, subtracting the derivative of Eq. (.34) with respect to z from the derivativeof Eq. (.35) with respect to y results: r C!x D 0 (.45).6 Displacements in Terms of Dilatation and Rotation Components Since the general solutions of equations of motion are given in terms of dilatation and rotation components, in order to determine displacements, they must be given in terms of dilatation and rotation components. Using Eqs. (.4) (.6), the Laplacian operator of the displacement in the x-direction in terms of dilatation and rotation components is r u u v w C! y xx yy z zz z (.46) Similarly, the corresponding expression for the Laplacian operator of v and w are r v x r w y (.47) (.48) By substituting these equations into Eqs. (.33) (.35), the displacements will be found in terms of dilatation and components of rotations:

8 Governing Equations. C G z. C G x. C C y D u (.49) D v (.50) D w (.5) After some rearrangement and reduction, the above equations become: u C v C w x (.5) (.53) (.54).7 StressesinTermsofDilatation and Rotation Components In order to get the stresses in terms of the dilatation and rotation components, Eq. (.) can be written as follows: yy D " C G" yy D. C G/ " yy C " xx C " zz (.55) and, in terms of displacement components, it is yy D. C (.56) Having substituted derivatives of Eqs. (.5) (.54) in the above relation, it yields: C ".C x yy D " " x (.57)

9 .7 Stresses in Terms of Dilatation and Rotation Components 3 simplification, the above equation yields C " x yy z " " Similarly, the other direct stresses are C " z xx y " " and C " y zz x " " (.58) (.59) (.60) Shear stresses in the dilatation strain and the elastic rotation components can be obtained as presented in the following. For instance, the shear stress, xy,interms of displacements, xy D G (.6) Substituting for u and v from Eqs. (.5) (.54), this shear stress be presented by the following equation xy D " z z Differentiating Eqs. (.4) (.6) with respect to x, y, x y z y (.6) (.63) (.64) (.65) And adding them all it x y z D 0 (.66)

10 4 Governing Equations The above equations demonstrate that the! y depends on the other two elastic rotations! x and! z. After y = from the above equation into Eq. (.6) and simplifying, the shear stress xy is obtained: xy D @! z x (.67) Following the same procedure, the shear stress yz will be presented by the following equation: yz D @! x z (.68).8 Fourier Transformation of Equations of Motion, Boundary Stresses, and Displacements In order to reduce the three partial differential equations of (.38), (.4), and (.45) into three ordinary differential equations in terms of y, complex Fourier transformation was used. The boundary conditions must then be treated in the same way, so that, instead of having relations in partial derivatives with respect to x, y, and z, relations are obtained in terms of the derivatives with respect to y only. By applying the two-dimensional Fourier transforms (see Appendix A) on the equations of motion, the reduced equations p C q N" D 0 p C q N!x D 0 p C q N!z D 0 (.7) where N", N! x,and N! z are the corresponding double complex Fourier transformations of ",! x,and! z. The transformed stresses at the surface of the half-space (.x; z/ plane) in terms of the transform of the dilatation and elastic rotation components are C N" N yy D C N! x N! z N" p C q (.7) For the shear stresses, ip d N" N xy D G dy C d d C p q N! z pq N! x (.73)

11 .9 General Solution of Transformed Equations of Motion 5 and iq d N" N yz D G dy d d p C q N! x C pq N! z (.74) The double complex Fourier transform of the displacements will be transforms of Eqs. (.5) (.54) Nu D i p N" C Nv D.ip N! z iq N! x Nw D i q N! z C i q N! y N! x N! y can be obtained by transforming equation (.6). (.76) i p N! y (.77) N! y D iq Nu C ip Nw (.78) Substituting N! y from the above equation into Eqs. (.75)and(.77), and solving for Nu and Nw results in: Nu D i Nv D Nw D i p N" C pq= q N! x C p = q N! z (.79) d N" iq dy N! x C ip N! z (.80) q = q N" pq= q N! z q N! x (.8).9 General Solution of Transformed Equations of Motion The general solution of Eqs. (.69) (.7) as a function of y is N" D A " exp. y/ (.8) N! x D A x exp. y/ (.83) N! z D A z exp. 3 y/ (.84)

12 6 Governing Equations where values of,,and 3 must be negative to satisfy the boundary conditions requirements for N", N! x,and N! z as y approaches to infinity. Substituting the above in Eqs. (.69) (.7) results in: From the last two equations, it is obvious that D p C q (.85) D p C q (.86) 3 D p C q (.87) D 3 (.88) Values of A ", A x,anda z are functions of p and q and depend on the boundary conditions of the problem. This means that they are dependent on the complex double Fourier transform of the stresses, which act as external excitations on the surface of the half-space medium. In order to evaluate these arbitrary functions, Eqs. (.7) (.74) must be satisfied by the boundary conditions that are expressed by three stress components yy, xy,and yz, which are applied on the surface of elastic half-space (x;z) wherey is zero. Substituting y D 0 in Eqs. (.8) (.84) and placing them into Eqs. (.7) (.74) the double complex Fourier transform of the stresses on the surface of half-space will be presented by the following equations: q p C i p i q A " 4pq A " C 4i q A " C q C A x 4i p A x C p A z D N yy.p; q/ G A z D N xy.p; q/ G A x C 4pq A z D N yz.p; q/ G (.89) (.90) (.9) The solutions to the above set of equations are given by the values of A ", A x,anda z. These values can be expressed as A " D D " D A x D D x D A z D D z D (.9) (.93) (.94)

13 .9 General Solution of Transformed Equations of Motion 7 where q p C i p D D 6 4 i q 4i q 4pq q 3 4i p p 7 4pq 5 (.95) or D D 4 ˆ.p;q/ (.96) and ˆ.p;q/D p C q 4 p C q (.97) This is the well-known function associated with Rayleigh surface waves, also N yy =G 4i q 4i p D " D N xy =G 4pq p ˇ N yz =G q 4pq ˇ q p C N yy=g 4i p D x D i p N xy =G p ˇ i q N yz =G 4pq ˇ q p C 4i q N yy =G D z D i p 4pq N xy =G ˇ i q q N yz =G ˇ 4 (.98) (.99) (.00) Equations (.8) through (.84) along with Eqs. (.89) (.9) present the general solution for the elastic dilatation N" and elastic rotations N! x and N! z for the surface of the medium due to applied stresses on the surface in the Fourier domain. These general equations will be utilized to determine the displacements on the surface of an elastic half-space for the two specific surface stress distributions caused by a vertical/horizontal harmonic point force that will be addressed in Chaps. 3 and 4, respectively.

14

Basic Equations of Elasticity

Basic Equations of Elasticity A Basic Equations of Elasticity A.1 STRESS The state of stress at any point in a loaded bo is defined completely in terms of the nine components of stress: σ xx,σ yy,σ zz,σ xy,σ yx,σ yz,σ zy,σ zx,andσ

More information

3D Elasticity Theory

3D Elasticity Theory 3D lasticity Theory Many structural analysis problems are analysed using the theory of elasticity in which Hooke s law is used to enforce proportionality between stress and strain at any deformation level.

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Standard Solids and Fracture Fluids: Mechanical, Chemical Effects Effective Stress Dilatancy Hardening and Stability Mead, 1925

More information

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16.

NDT&E Methods: UT. VJ Technologies CAVITY INSPECTION. Nondestructive Testing & Evaluation TPU Lecture Course 2015/16. CAVITY INSPECTION NDT&E Methods: UT VJ Technologies NDT&E Methods: UT 6. NDT&E: Introduction to Methods 6.1. Ultrasonic Testing: Basics of Elasto-Dynamics 6.2. Principles of Measurement 6.3. The Pulse-Echo

More information

Lecture 8. Stress Strain in Multi-dimension

Lecture 8. Stress Strain in Multi-dimension Lecture 8. Stress Strain in Multi-dimension Module. General Field Equations General Field Equations [] Equilibrium Equations in Elastic bodies xx x y z yx zx f x 0, etc [2] Kinematics xx u x x,etc. [3]

More information

PEAT SEISMOLOGY Lecture 2: Continuum mechanics

PEAT SEISMOLOGY Lecture 2: Continuum mechanics PEAT8002 - SEISMOLOGY Lecture 2: Continuum mechanics Nick Rawlinson Research School of Earth Sciences Australian National University Strain Strain is the formal description of the change in shape of a

More information

Macroscopic theory Rock as 'elastic continuum'

Macroscopic theory Rock as 'elastic continuum' Elasticity and Seismic Waves Macroscopic theory Rock as 'elastic continuum' Elastic body is deformed in response to stress Two types of deformation: Change in volume and shape Equations of motion Wave

More information

Unit IV State of stress in Three Dimensions

Unit IV State of stress in Three Dimensions Unit IV State of stress in Three Dimensions State of stress in Three Dimensions References Punmia B.C.,"Theory of Structures" (SMTS) Vol II, Laxmi Publishing Pvt Ltd, New Delhi 2004. Rattan.S.S., "Strength

More information

Wave and Elasticity Equations

Wave and Elasticity Equations 1 Wave and lasticity quations Now let us consider the vibrating string problem which is modeled by the one-dimensional wave equation. Suppose that a taut string is suspended by its extremes at the points

More information

12. Stresses and Strains

12. Stresses and Strains 12. Stresses and Strains Finite Element Method Differential Equation Weak Formulation Approximating Functions Weighted Residuals FEM - Formulation Classification of Problems Scalar Vector 1-D T(x) u(x)

More information

Strain Transformation equations

Strain Transformation equations Strain Transformation equations R. Chandramouli Associate Dean-Research SASTRA University, Thanjavur-613 401 Joint Initiative of IITs and IISc Funded by MHRD Page 1 of 8 Table of Contents 1. Stress transformation

More information

Chapter 5. The Differential Forms of the Fundamental Laws

Chapter 5. The Differential Forms of the Fundamental Laws Chapter 5 The Differential Forms of the Fundamental Laws 1 5.1 Introduction Two primary methods in deriving the differential forms of fundamental laws: Gauss s Theorem: Allows area integrals of the equations

More information

Introduction to Seismology Spring 2008

Introduction to Seismology Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 12.510 Introduction to Seismology Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Stress and Strain

More information

Finite Element Method in Geotechnical Engineering

Finite Element Method in Geotechnical Engineering Finite Element Method in Geotechnical Engineering Short Course on + Dynamics Boulder, Colorado January 5-8, 2004 Stein Sture Professor of Civil Engineering University of Colorado at Boulder Contents Steps

More information

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept.

AE/ME 339. Computational Fluid Dynamics (CFD) K. M. Isaac. Momentum equation. Computational Fluid Dynamics (AE/ME 339) MAEEM Dept. AE/ME 339 Computational Fluid Dynamics (CFD) 9//005 Topic7_NS_ F0 1 Momentum equation 9//005 Topic7_NS_ F0 1 Consider the moving fluid element model shown in Figure.b Basis is Newton s nd Law which says

More information

Mechanics PhD Preliminary Spring 2017

Mechanics PhD Preliminary Spring 2017 Mechanics PhD Preliminary Spring 2017 1. (10 points) Consider a body Ω that is assembled by gluing together two separate bodies along a flat interface. The normal vector to the interface is given by n

More information

Chapter 9: Differential Analysis of Fluid Flow

Chapter 9: Differential Analysis of Fluid Flow of Fluid Flow Objectives 1. Understand how the differential equations of mass and momentum conservation are derived. 2. Calculate the stream function and pressure field, and plot streamlines for a known

More information

Stress, Strain, Mohr s Circle

Stress, Strain, Mohr s Circle Stress, Strain, Mohr s Circle The fundamental quantities in solid mechanics are stresses and strains. In accordance with the continuum mechanics assumption, the molecular structure of materials is neglected

More information

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any

A DARK GREY P O N T, with a Switch Tail, and a small Star on the Forehead. Any Y Y Y X X «/ YY Y Y ««Y x ) & \ & & } # Y \#$& / Y Y X» \\ / X X X x & Y Y X «q «z \x» = q Y # % \ & [ & Z \ & { + % ) / / «q zy» / & / / / & x x X / % % ) Y x X Y $ Z % Y Y x x } / % «] «] # z» & Y X»

More information

Chapter 9: Differential Analysis

Chapter 9: Differential Analysis 9-1 Introduction 9-2 Conservation of Mass 9-3 The Stream Function 9-4 Conservation of Linear Momentum 9-5 Navier Stokes Equation 9-6 Differential Analysis Problems Recall 9-1 Introduction (1) Chap 5: Control

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

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Overview Milestones in continuum mechanics Concepts of modulus and stiffness. Stress-strain relations Elasticity Surface and body

More information

Mechanical Properties of Materials

Mechanical Properties of Materials Mechanical Properties of Materials Strains Material Model Stresses Learning objectives Understand the qualitative and quantitative description of mechanical properties of materials. Learn the logic of

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Preliminary Math Concept of Stress Stress Components Equilibrium

More information

Solving the torsion problem for isotropic matrial with a rectangular cross section using the FEM and FVM methods with triangular elements

Solving the torsion problem for isotropic matrial with a rectangular cross section using the FEM and FVM methods with triangular elements Solving the torsion problem for isotropic matrial with a rectangular cross section using the FEM and FVM methods with triangular elements Nasser M. Abbasi. June 0, 04 Contents Introduction. Problem setup...................................

More information

GG611 Structural Geology Sec1on Steve Martel POST 805

GG611 Structural Geology Sec1on Steve Martel POST 805 GG611 Structural Geology Sec1on Steve Martel POST 805 smartel@hawaii.edu Lecture 5 Mechanics Stress, Strain, and Rheology 11/6/16 GG611 1 Stresses Control How Rock Fractures hkp://hvo.wr.usgs.gov/kilauea/update/images.html

More information

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso

MECH 5312 Solid Mechanics II. Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso MECH 5312 Solid Mechanics II Dr. Calvin M. Stewart Department of Mechanical Engineering The University of Texas at El Paso Table of Contents Thermodynamics Derivation Hooke s Law: Anisotropic Elasticity

More information

M E 320 Professor John M. Cimbala Lecture 10

M E 320 Professor John M. Cimbala Lecture 10 M E 320 Professor John M. Cimbala Lecture 10 Today, we will: Finish our example problem rates of motion and deformation of fluid particles Discuss the Reynolds Transport Theorem (RTT) Show how the RTT

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

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth

Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth Rock Rheology GEOL 5700 Physics and Chemistry of the Solid Earth References: Turcotte and Schubert, Geodynamics, Sections 2.1,-2.4, 2.7, 3.1-3.8, 6.1, 6.2, 6.8, 7.1-7.4. Jaeger and Cook, Fundamentals of

More information

Useful Formulae ( )

Useful Formulae ( ) Appendix A Useful Formulae (985-989-993-) 34 Jeremić et al. A.. CHAPTER SUMMARY AND HIGHLIGHTS page: 35 of 536 A. Chapter Summary and Highlights A. Stress and Strain This section reviews small deformation

More information

DEFORMATION PATTERN IN ELASTIC CRUST

DEFORMATION PATTERN IN ELASTIC CRUST DEFORMATION PATTERN IN ELASTIC CRUST Stress and force in 2D Strain : normal and shear Elastic medium equations Vertical fault in elastic medium => arctangent General elastic dislocation (Okada s formulas)

More information

In this section, mathematical description of the motion of fluid elements moving in a flow field is

In this section, mathematical description of the motion of fluid elements moving in a flow field is Jun. 05, 015 Chapter 6. Differential Analysis of Fluid Flow 6.1 Fluid Element Kinematics In this section, mathematical description of the motion of fluid elements moving in a flow field is given. A small

More information

Crystal Relaxation, Elasticity, and Lattice Dynamics

Crystal Relaxation, Elasticity, and Lattice Dynamics http://exciting-code.org Crystal Relaxation, Elasticity, and Lattice Dynamics Pasquale Pavone Humboldt-Universität zu Berlin http://exciting-code.org PART I: Structure Optimization Pasquale Pavone Humboldt-Universität

More information

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t)

V (r,t) = i ˆ u( x, y,z,t) + ˆ j v( x, y,z,t) + k ˆ w( x, y, z,t) IV. DIFFERENTIAL RELATIONS FOR A FLUID PARTICLE This chapter presents the development and application of the basic differential equations of fluid motion. Simplifications in the general equations and common

More information

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004

Elements of Continuum Elasticity. David M. Parks Mechanics and Materials II February 25, 2004 Elements of Continuum Elasticity David M. Parks Mechanics and Materials II 2.002 February 25, 2004 Solid Mechanics in 3 Dimensions: stress/equilibrium, strain/displacement, and intro to linear elastic

More information

Example 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60.

Example 3.7 Consider the undeformed configuration of a solid as shown in Figure 3.60. 162 3. The linear 3-D elasticity mathematical model The 3-D elasticity model is of great importance, since it is our highest order hierarchical model assuming linear elastic behavior. Therefore, it provides

More information

Properties of the stress tensor

Properties of the stress tensor Appendix C Properties of the stress tensor Some of the basic properties of the stress tensor and traction vector are reviewed in the following. C.1 The traction vector Let us assume that the state of stress

More information

Introduction to Fluid Mechanics

Introduction to Fluid Mechanics Introduction to Fluid Mechanics Tien-Tsan Shieh April 16, 2009 What is a Fluid? The key distinction between a fluid and a solid lies in the mode of resistance to change of shape. The fluid, unlike the

More information

Continuum mechanism: Stress and strain

Continuum mechanism: Stress and strain Continuum mechanics deals with the relation between forces (stress, σ) and deformation (strain, ε), or deformation rate (strain rate, ε). Solid materials, rigid, usually deform elastically, that is the

More information

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA

VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA VYSOKÁ ŠKOLA BÁŇSKÁ TECHNICKÁ UNIVERZITA OSTRAVA FAKULTA METALURGIE A MATERIÁLOVÉHO INŽENÝRSTVÍ APPLIED MECHANICS Study Support Leo Václavek Ostrava 2015 Title:Applied Mechanics Code: Author: doc. Ing.

More information

1 Stress and Strain. Introduction

1 Stress and Strain. Introduction 1 Stress and Strain Introduction This book is concerned with the mechanical behavior of materials. The term mechanical behavior refers to the response of materials to forces. Under load, a material may

More information

EE C247B ME C218 Introduction to MEMS Design Spring 2017

EE C247B ME C218 Introduction to MEMS Design Spring 2017 247B/M 28: Introduction to MMS Design Lecture 0m2: Mechanics of Materials CTN 2/6/7 Outline C247B M C28 Introduction to MMS Design Spring 207 Prof. Clark T.- Reading: Senturia, Chpt. 8 Lecture Topics:

More information

ANALYSIS OF STRAINS CONCEPT OF STRAIN

ANALYSIS OF STRAINS CONCEPT OF STRAIN ANALYSIS OF STRAINS CONCEPT OF STRAIN Concept of strain : if a bar is subjected to a direct load, and hence a stress the bar will change in length. If the bar has an original length L and changes by an

More information

Variational principles in mechanics

Variational principles in mechanics CHAPTER Variational principles in mechanics.1 Linear Elasticity n D Figure.1: A domain and its boundary = D [. Consider a domain Ω R 3 with its boundary = D [ of normal n (see Figure.1). The problem of

More information

2 Introduction to mechanics

2 Introduction to mechanics 21 Motivation Thermodynamic bodies are being characterized by two competing opposite phenomena, energy and entropy which some researchers in thermodynamics would classify as cause and chance or determinancy

More information

Proceedings of Meetings on Acoustics

Proceedings of Meetings on Acoustics Proceedings of Meetings on Acoustics Volume 19, 13 http://acousticalsociety.org/ ICA 13 Montreal Montreal, Canada - 7 June 13 Structural Acoustics and Vibration Session 4aSA: Applications in Structural

More information

Mechanics of materials Lecture 4 Strain and deformation

Mechanics of materials Lecture 4 Strain and deformation Mechanics of materials Lecture 4 Strain and deformation Reijo Kouhia Tampere University of Technology Department of Mechanical Engineering and Industrial Design Wednesday 3 rd February, 206 of a continuum

More information

Mechanics of Earthquakes and Faulting

Mechanics of Earthquakes and Faulting Mechanics of Earthquakes and Faulting www.geosc.psu.edu/courses/geosc508 Surface and body forces Tensors, Mohr circles. Theoretical strength of materials Defects Stress concentrations Griffith failure

More information

HIGHER-ORDER THEORIES

HIGHER-ORDER THEORIES HIGHER-ORDER THEORIES THIRD-ORDER SHEAR DEFORMATION PLATE THEORY LAYERWISE LAMINATE THEORY J.N. Reddy 1 Third-Order Shear Deformation Plate Theory Assumed Displacement Field µ u(x y z t) u 0 (x y t) +

More information

Stress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning

Stress/Strain. Outline. Lecture 1. Stress. Strain. Plane Stress and Plane Strain. Materials. ME EN 372 Andrew Ning Stress/Strain Lecture 1 ME EN 372 Andrew Ning aning@byu.edu Outline Stress Strain Plane Stress and Plane Strain Materials otes and News [I had leftover time and so was also able to go through Section 3.1

More information

Exercise: concepts from chapter 5

Exercise: concepts from chapter 5 Reading: Fundamentals of Structural Geology, Ch 5 1) Study the oöids depicted in Figure 1a and 1b. Figure 1a Figure 1b Figure 1. Nearly undeformed (1a) and significantly deformed (1b) oöids with spherulitic

More information

The Kinematic Equations

The Kinematic Equations The Kinematic Equations David Roylance Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, MA 0139 September 19, 000 Introduction The kinematic or strain-displacement

More information

THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM

THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM Internat. J. Math. & Math. Sci. Vol., No. 8 () 59 56 S67 Hindawi Publishing Corp. THE REFLECTION PHENOMENA OF SV-WAVES IN A GENERALIZED THERMOELASTIC MEDIUM ABO-EL-NOUR N. ABD-ALLA and AMIRA A. S. AL-DAWY

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

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly

Consider an elastic spring as shown in the Fig.2.4. When the spring is slowly .3 Strain Energy Consider an elastic spring as shown in the Fig..4. When the spring is slowly pulled, it deflects by a small amount u 1. When the load is removed from the spring, it goes back to the original

More information

Linearized theory of elasticity

Linearized theory of elasticity Linearized theory of elasticity Arie Verhoeven averhoev@win.tue.nl CASA Seminar, May 24, 2006 Seminar: Continuum mechanics 1 Stress and stress principles Bart Nowak March 8 2 Strain and deformation Mark

More information

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1

3 2 6 Solve the initial value problem u ( t) 3. a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 Math Problem a- If A has eigenvalues λ =, λ = 1 and corresponding eigenvectors 1 3 6 Solve the initial value problem u ( t) = Au( t) with u (0) =. 3 1 u 1 =, u 1 3 = b- True or false and why 1. if A is

More information

Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory

Topics. GG612 Structural Geology Sec3on Steve Martel POST 805 Lecture 4 Mechanics: Stress and Elas3city Theory GG612 Structural Geology Sec3on Steve Martel POST 805 smartel@hawaii.edu Lecture 4 Mechanics: Stress and Elas3city Theory 11/6/15 GG611 1 Topics 1. Stress vectors (trac3ons) 2. Stress at a point 3. Cauchy

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

4. Mathematical models used in engineering structural analysis

4. Mathematical models used in engineering structural analysis 4. Mathematical models used in engineering structural analysis In this chapter we pursue a formidable task to present the most important mathematical models in structural mechanics. In order to best situate

More information

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship

3.2 Hooke s law anisotropic elasticity Robert Hooke ( ) Most general relationship 3.2 Hooke s law anisotropic elasticity Robert Hooke (1635-1703) Most general relationship σ = C ε + C ε + C ε + C γ + C γ + C γ 11 12 yy 13 zz 14 xy 15 xz 16 yz σ = C ε + C ε + C ε + C γ + C γ + C γ yy

More information

PEAT SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II

PEAT SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II PEAT8002 - SEISMOLOGY Lecture 12: Earthquake source mechanisms and radiation patterns II Nick Rawlinson Research School of Earth Sciences Australian National University Waveform modelling P-wave first-motions

More information

3. The linear 3-D elasticity mathematical model

3. The linear 3-D elasticity mathematical model 3. The linear 3-D elasticity mathematical model In Chapter we examined some fundamental conditions that should be satisfied in the modeling of all deformable solids and structures. The study of truss structures

More information

Elasticity in two dimensions 1

Elasticity in two dimensions 1 Elasticity in two dimensions 1 Elasticity in two dimensions Chapters 3 and 4 of Mechanics of the Cell, as well as its Appendix D, contain selected results for the elastic behavior of materials in two and

More information

LASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE

LASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE LASER GENERATED THERMOELASTIC WAVES IN AN ANISOTROPIC INFINITE PLATE H. M. Al-Qahtani and S. K. Datta University of Colorado Boulder CO 839-7 ABSTRACT. An analysis of the propagation of thermoelastic waves

More information

STRAIN. Normal Strain: The elongation or contractions of a line segment per unit length is referred to as normal strain denoted by Greek symbol.

STRAIN. Normal Strain: The elongation or contractions of a line segment per unit length is referred to as normal strain denoted by Greek symbol. STRAIN In engineering the deformation of a body is specified using the concept of normal strain and shear strain whenever a force is applied to a body, it will tend to change the body s shape and size.

More information

GG612 Lecture 3. Outline

GG612 Lecture 3. Outline GG61 Lecture 3 Strain and Stress Should complete infinitesimal strain by adding rota>on. Outline Matrix Opera+ons Strain 1 General concepts Homogeneous strain 3 Matrix representa>ons 4 Squares of line

More information

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich

UNIVERSITY OF SASKATCHEWAN ME MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich UNIVERSITY OF SASKATCHEWAN ME 313.3 MECHANICS OF MATERIALS I FINAL EXAM DECEMBER 13, 2008 Professor A. Dolovich A CLOSED BOOK EXAMINATION TIME: 3 HOURS For Marker s Use Only LAST NAME (printed): FIRST

More information

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI

Lecture 8 Analyzing the diffusion weighted signal. Room CSB 272 this week! Please install AFNI Lecture 8 Analyzing the diffusion weighted signal Room CSB 272 this week! Please install AFNI http://afni.nimh.nih.gov/afni/ Next lecture, DTI For this lecture, think in terms of a single voxel We re still

More information

Unit 13 Review of Simple Beam Theory

Unit 13 Review of Simple Beam Theory MIT - 16.0 Fall, 00 Unit 13 Review of Simple Beam Theory Readings: Review Unified Engineering notes on Beam Theory BMP 3.8, 3.9, 3.10 T & G 10-15 Paul A. Lagace, Ph.D. Professor of Aeronautics & Astronautics

More information

Receiver. Johana Brokešová Charles University in Prague

Receiver. Johana Brokešová Charles University in Prague Propagation of seismic waves - theoretical background Receiver Johana Brokešová Charles University in Prague Seismic waves = waves in elastic continuum a model of the medium through which the waves propagate

More information

19. Principal Stresses

19. Principal Stresses 19. Principal Stresses I Main Topics A Cauchy s formula B Principal stresses (eigenvectors and eigenvalues) C Example 10/24/18 GG303 1 19. Principal Stresses hkp://hvo.wr.usgs.gov/kilauea/update/images.html

More information

7.2.1 Seismic waves. Waves in a mass- spring system

7.2.1 Seismic waves. Waves in a mass- spring system 7..1 Seismic waves Waves in a mass- spring system Acoustic waves in a liquid or gas Seismic waves in a solid Surface waves Wavefronts, rays and geometrical attenuation Amplitude and energy Waves in a mass-

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

Chapter 1 Fluid Characteristics

Chapter 1 Fluid Characteristics Chapter 1 Fluid Characteristics 1.1 Introduction 1.1.1 Phases Solid increasing increasing spacing and intermolecular liquid latitude of cohesive Fluid gas (vapor) molecular force plasma motion 1.1.2 Fluidity

More information

Chapter 1 Fracture Physics

Chapter 1 Fracture Physics Chapter 1 Fracture Physics In this chapter we give a brief account of the theory of fracture, restricting ourselves to brittle fracture. The structure of this chapter is as follows. The first part describes

More information

Exercise: concepts from chapter 8

Exercise: concepts from chapter 8 Reading: Fundamentals of Structural Geology, Ch 8 1) The following exercises explore elementary concepts associated with a linear elastic material that is isotropic and homogeneous with respect to elastic

More information

LOWELL WEEKLY JOURNAL

LOWELL WEEKLY JOURNAL Y -» $ 5 Y 7 Y Y -Y- Q x Q» 75»»/ q } # ]»\ - - $ { Q» / X x»»- 3 q $ 9 ) Y q - 5 5 3 3 3 7 Q q - - Q _»»/Q Y - 9 - - - )- [ X 7» -» - )»? / /? Q Y»» # X Q» - -?» Q ) Q \ Q - - - 3? 7» -? #»»» 7 - / Q

More information

SEISMOLOGY I. Laurea Magistralis in Physics of the Earth and of the Environment. Elasticity. Fabio ROMANELLI

SEISMOLOGY I. Laurea Magistralis in Physics of the Earth and of the Environment. Elasticity. Fabio ROMANELLI SEISMOLOGY I Laurea Magistralis in Physics of the Earth and of the Environment Elasticity Fabio ROMANELLI Dept. Earth Sciences Università degli studi di Trieste romanel@dst.units.it 1 Elasticity and Seismic

More information

ME 243. Lecture 10: Combined stresses

ME 243. Lecture 10: Combined stresses ME 243 Mechanics of Solids Lecture 10: Combined stresses Ahmad Shahedi Shakil Lecturer, Dept. of Mechanical Engg, BUET E-mail: sshakil@me.buet.ac.bd, shakil6791@gmail.com Website: teacher.buet.ac.bd/sshakil

More information

DEFORMATION PATTERN IN ELASTIC CRUST

DEFORMATION PATTERN IN ELASTIC CRUST DEFORMATION PATTERN IN ELASTIC CRUST Stress and force in 2D Strain : normal and shear Elastic medium equations Vertical fault in elastic medium => arctangent General elastic dislocation (Okada s formulas)

More information

20. Rheology & Linear Elasticity

20. Rheology & Linear Elasticity I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava

More information

Applications of Eigenvalues & Eigenvectors

Applications of Eigenvalues & Eigenvectors Applications of Eigenvalues & Eigenvectors Louie L. Yaw Walla Walla University Engineering Department For Linear Algebra Class November 17, 214 Outline 1 The eigenvalue/eigenvector problem 2 Principal

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

Mechanical Behavior of Composite Tapered Lamina

Mechanical Behavior of Composite Tapered Lamina International Journal of Engineering Research and Development e-issn: 2278-067X, p-issn: 2278-800X, www.ijerd.com Volume 10, Issue 8 (August 2014), PP.19-27 Mechanical Behavior of Composite Tapered Lamina

More information

Add-on unidirectional elastic metamaterial plate cloak

Add-on unidirectional elastic metamaterial plate cloak Add-on unidirectional elastic metamaterial plate cloak Min Kyung Lee *a and Yoon Young Kim **a,b a Department of Mechanical and Aerospace Engineering, Seoul National University, Gwanak-ro, Gwanak-gu, Seoul,

More information

How are calculus, alchemy and forging coins related?

How are calculus, alchemy and forging coins related? BMOLE 452-689 Transport Chapter 8. Transport in Porous Media Text Book: Transport Phenomena in Biological Systems Authors: Truskey, Yuan, Katz Focus on what is presented in class and problems Dr. Corey

More information

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature

Chapter 1. Continuum mechanics review. 1.1 Definitions and nomenclature Chapter 1 Continuum mechanics review We will assume some familiarity with continuum mechanics as discussed in the context of an introductory geodynamics course; a good reference for such problems is Turcotte

More information

Stress transformation and Mohr s circle for stresses

Stress transformation and Mohr s circle for stresses Stress transformation and Mohr s circle for stresses 1.1 General State of stress Consider a certain body, subjected to external force. The force F is acting on the surface over an area da of the surface.

More information

16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity

16.20 HANDOUT #2 Fall, 2002 Review of General Elasticity 6.20 HANDOUT #2 Fall, 2002 Review of General Elasticity NOTATION REVIEW (e.g., for strain) Engineering Contracted Engineering Tensor Tensor ε x = ε = ε xx = ε ε y = ε 2 = ε yy = ε 22 ε z = ε 3 = ε zz =

More information

ELASTICITY (MDM 10203)

ELASTICITY (MDM 10203) ELASTICITY () Lecture Module 3: Fundamental Stress and Strain University Tun Hussein Onn Malaysia Normal Stress inconstant stress distribution σ= dp da P = da A dimensional Area of σ and A σ A 3 dimensional

More information

Equilibrium of Deformable Body

Equilibrium of Deformable Body Equilibrium of Deformable Body Review Static Equilibrium If a body is in static equilibrium under the action applied external forces, the Newton s Second Law provides us six scalar equations of equilibrium

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

3.22 Mechanical Properties of Materials Spring 2008

3.22 Mechanical Properties of Materials Spring 2008 MIT OpenCourseWare http://ocw.mit.edu 3.22 Mechanical Properties of Materials Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. Quiz #1 Example

More information

Aircraft Structures Kirchhoff-Love Plates

Aircraft Structures Kirchhoff-Love Plates University of Liège erospace & Mechanical Engineering ircraft Structures Kirchhoff-Love Plates Ludovic Noels Computational & Multiscale Mechanics of Materials CM3 http://www.ltas-cm3.ulg.ac.be/ Chemin

More information

On propagation of Love waves in an infinite transversely isotropic poroelastic layer

On propagation of Love waves in an infinite transversely isotropic poroelastic layer Journal of Physics: Conference Series PAPER OPEN ACCESS On propagation of Love waves in an infinite transversely isotropic poroelastic layer To cite this article: C Nageswara Nath et al 2015 J. Phys.:

More information

1 Hooke s law, stiffness, and compliance

1 Hooke s law, stiffness, and compliance Non-quilibrium Continuum Physics TA session #5 TA: Yohai Bar Sinai 3.04.206 Linear elasticity I This TA session is the first of three at least, maybe more) in which we ll dive deep deep into linear elasticity

More information

Numerical Modelling in Geosciences. Lecture 6 Deformation

Numerical Modelling in Geosciences. Lecture 6 Deformation Numerical Modelling in Geosciences Lecture 6 Deformation Tensor Second-rank tensor stress ), strain ), strain rate ) Invariants quantities independent of the coordinate system): - First invariant trace:!!

More information

Notes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n.

Notes: Most of the material presented in this chapter is taken from Jackson, Chap. 2, 3, and 4, and Di Bartolo, Chap. 2. 2π nx i a. ( ) = G n. Chapter. Electrostatic II Notes: Most of the material presented in this chapter is taken from Jackson, Chap.,, and 4, and Di Bartolo, Chap... Mathematical Considerations.. The Fourier series and the Fourier

More information