Surface force on a volume element.
|
|
- Maximillian Fox
- 5 years ago
- Views:
Transcription
1 STRESS and STRAIN Reading: Section. of Stein and Wysession. In this section, we will see how Newton s second law and Generalized Hooke s law can be used to characterize the response of continuous medium to applied forces. The stress tensor describes the forces acting on internal surfaces of a deformable continuous medium. The strain tensor describes the distortion of (or the variation in displacement within) the body. Stress alone will not cause wave motion, but the equation of motion describes how spatial variations in stress produce acceleration. Stress () is related to strain (ε) by: Stress In D: = Eε where E is Young s modulus (NBE has dimension of stress). We consider two types of force that can act on a body:. Body force (f i ) this force acts everywhere in the body and the net force is proportional to the volume of the body. E.g., gravity, electro-magnetics.. Surface force (T i ) this force acts on the surface of a body and the net force is proportional to the surface area of the body. Consider the forces acting on a small volume, V, with surface S, within a large continuous medium. F n V Surface force on a volume element. (Figure, Adapted from Stein & Wyssession, 00) Stress = F, or force per unit area (N/m ), N/m = Pa (Pascal) A
2 atm= 000mbar = bar = 0 5 Pa (force equivalent to the apple sauce from one apple spread over m ) Examples: 40km ~ 4 GPa 660km ~ GPa CMB ~ 5 GPa Center of the earth ~ 60 GPa The surface force F acts on each element of surface which has a unit normal vector, n. The forces acting on the surfaces of a volume element can be described by three traction vectors. By definition the traction vector, T, is the limit of the surface force per unit area at any point as the area becomes infinitesimal. F Stress vector = Traction = T = lim = ( T, T,T ) δs 0 δs Each traction vector acts on a surface perpendicular to a coordinate axis: T (i) is the traction vector working on the surface with its unit normal n in direction I (Fig. ). T () T () T () T () T () X X T () X Traction vectors on the faces of a volume element. Fig. (Adapted from Stein & Wyssession, 00) REMEMBER: we have made the assumption/approximation that the medium is a continuum. For a more detailed approach refer to L. E. Malvern, 969, Introduction to the Mechanics of a Continuous Medium, Englewood Cliffs, N. J., Prentice-Hall.
3 Equation of motion Using Newton s second low we can write the equation of motion: F i = ma i F i = body forces + surface forces = f i + T i = ma i = m δ u i δt or, see Fig. : F i = T + f dv = ρ u i i i i i i V. t As the internal surfaces are, in general, not known we need to get an expression independent of or i. This can be done by realizing that i is the orthogonal projection of surface along axis (i): i =cosφ i =n i d, with φ i the angle between the normal vector n and T i (see Fig. ). If we assume that the system is in equilibrium (i.e., a i =0) and that dv/ goes to zero, then the equation above is equal to zero and division by gives: T i = i n + i n + i n = ji n j ji ( j ) = T i n T Stress components on the faces of a tetrahedron. Fig. (Adapted from Stein & Wyssession, 00)
4 .50 Introduction to Seismology /9/05 In the absence of body forces, the stress tensor is symmetric ( ij = ji ), therefore there are only 6 independent elements. The diagonal elements represent the normal stress and the off- diagonal elements the shear stress. A symmetric tensor can also be diagonalized; in this case that means that the body can be rotated such that the tractions become parallel to the normals to the surfaces being investigated. In other words, T i = ji n j = ij n j = λ n i This is an eigenvalue/eigenvector problem: ( ij λδ ij ) n j = 0 I n = 0 ( λ ) Where λ is the eigenvalue or principle stresses and n is the eigenvector or principle axis. Taking the determinant of the stress tensor gives a cubic equation for λ and three solutions which can be plugged back in to give three eigenvectors, the principal stress axes. Digitalization makes all the shear components disappear and the remaining diagonal components are the principle stresses. 0 0 = Where the magnitude of the components are: Types of stress:. Uni-axial stress e.g., =0, =0, 0. Plane stress e.g. 0, =0, 0. Pure shear this is an example of plane stress where two of the stress components are equal in magnitude but opposite in direction: e.g., = 4. Isotropic, lithostatic, hydrostatic stress same stress everywhere, where pressure, or mean stress, M: M = ( + + ) 4
5 5. Deviatoric stress defined as the remaining stress state after the effect of the mean stress has been removed. The deviatoric stress gives rise to motion. This is the most important stress in seismology. Strain When stress is applied to a non-rigid body deformation occurs. This deformation can be described by the strain tensor. Strain is a relative measurement and is therefore dimensionless. Undeformed Deformed x+ δx x δx u+ δu u δu Change in relative displacement during deformation. (Adapted from Stein & Wyssession, 00) u ( x + δx ) u ( x ) δ u u = u In D the strain is given by: ε xx = = +, where we used δ x δ x x x the linearization: u ( x + δx ) u ( x ) +δ u (x). This is justified as long as the change in displacement is smooth over a distance δx: u δu δx (infinitesimal strain theory). x x u u ( x ) =u ( x 0 )+ d x u u u x x x d u ( x ) =u ( x 0 ) + d =u ( x0 ) +Jd etc d 5
6 Where J is the Jacobian transformation tensor. J = ε + Ω ε is the symmetric matrix strain tensor ε ij Ω is the antisymmetric matrix rotation tensor Ω ij u u u u 0 u x x x u Ω ij = u 0 () x x () () 0 In seismology we are interested only in the distortion of the material (strain tensor) and not the rigid body rotation (rotation tensor). The trace (tr) of the strain tensor is u u = u i tr(ε) = u + + =.u, which is also known as the cubic dilatation (Θ). x x x i = x i Divergence of the displacement field relates to the relative change in volume. The trace of the rotation vector is zero, i.e. a rigid body rotation does not involve a volume change. References Stein, S. & M. Wysession, 00, An Introduction to Seismology, Earthquakes, and Earth Structure, Malden, M.A., Blackwell Publishing Ltd. 6
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 informationPEAT 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 informationMacroscopic 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 informationElements 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 informationContinuum 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 information3D 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 informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING CAMBRIDGE, MASSACHUSETTS 02139
MASSACHUSETTS NSTTUTE OF TECHNOLOGY DEPARTMENT OF MATERALS SCENCE AND ENGNEERNG CAMBRDGE, MASSACHUSETTS 39 3. MECHANCAL PROPERTES OF MATERALS PROBLEM SET SOLUTONS Reading Ashby, M.F., 98, Tensors: Notes
More informationChapter 3: Stress and Equilibrium of Deformable Bodies
Ch3-Stress-Equilibrium Page 1 Chapter 3: Stress and Equilibrium of Deformable Bodies When structures / deformable bodies are acted upon by loads, they build up internal forces (stresses) within them to
More informationBasic 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 informationMechanics 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 informationFundamentals of Linear Elasticity
Fundamentals of Linear Elasticity Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research of the Polish Academy
More informationA short review of continuum mechanics
A short review of continuum mechanics Professor Anette M. Karlsson, Department of Mechanical ngineering, UD September, 006 This is a short and arbitrary review of continuum mechanics. Most of this material
More informationElements 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 informationElastic Wave Theory. LeRoy Dorman Room 436 Ritter Hall Tel: Based on notes by C. R. Bentley. Version 1.
Elastic Wave Theory LeRoy Dorman Room 436 Ritter Hall Tel: 4-2406 email: ldorman@ucsd.edu Based on notes by C. R. Bentley. Version 1.1, 20090323 1 Chapter 1 Tensors Elastic wave theory is concerned with
More informationVYSOKÁ 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 information3 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 informationMathematical Background
CHAPTER ONE Mathematical Background This book assumes a background in the fundamentals of solid mechanics and the mechanical behavior of materials, including elasticity, plasticity, and friction. A previous
More informationChapter 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 informationContinuum Mechanics and the Finite Element Method
Continuum Mechanics and the Finite Element Method 1 Assignment 2 Due on March 2 nd @ midnight 2 Suppose you want to simulate this The familiar mass-spring system l 0 l y i X y i x Spring length before/after
More informationStress, 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 information3D Stress Tensors. 3D Stress Tensors, Eigenvalues and Rotations
3D Stress Tensors 3D Stress Tensors, Eigenvalues and Rotations Recall that we can think of an n x n matrix Mij as a transformation matrix that transforms a vector xi to give a new vector yj (first index
More informationMechanics 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 informationK. F. Graff, "Wave Motion in Elastic Solids," Dover, NY 1975 (inexpensive!)
Phys 598 EW Elastic Waves Fall 2015 Standard texts and monographs, all with a view towards solving classical problems in elastic wave propagation: K. F. Graff, "Wave Motion in Elastic Solids," Dover, NY
More informationSEISMOLOGY 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 informationExample 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 informationNumerical 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 informationCH.4. STRESS. Continuum Mechanics Course (MMC)
CH.4. STRESS Continuum Mechanics Course (MMC) Overview Forces Acting on a Continuum Body Cauchy s Postulates Stress Tensor Stress Tensor Components Scientific Notation Engineering Notation Sign Criterion
More informationSoft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies
Soft-Body Physics Soft Bodies Realistic objects are not purely rigid. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies Deformed
More informationBasic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008
Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June 2008 1 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and
More informationInverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros
Inverse Design (and a lightweight introduction to the Finite Element Method) Stelian Coros Computational Design Forward design: direct manipulation of design parameters Level of abstraction Exploration
More informationFundamentals of Fluid Dynamics: Elementary Viscous Flow
Fundamentals of Fluid Dynamics: Elementary Viscous Flow Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pan.pl/ tzielins/ Institute of Fundamental Technological Research
More informationSymmetry and Properties of Crystals (MSE638) Stress and Strain Tensor
Symmetry and Properties of Crystals (MSE638) Stress and Strain Tensor Somnath Bhowmick Materials Science and Engineering, IIT Kanpur April 6, 2018 Tensile test and Hooke s Law Upto certain strain (0.75),
More informationLecture Notes 3
12.005 Lecture Notes 3 Tensors Most physical quantities that are important in continuum mechanics like temperature, force, and stress can be represented by a tensor. Temperature can be specified by stating
More informationLecture 8: Tissue Mechanics
Computational Biology Group (CoBi), D-BSSE, ETHZ Lecture 8: Tissue Mechanics Prof Dagmar Iber, PhD DPhil MSc Computational Biology 2015/16 7. Mai 2016 2 / 57 Contents 1 Introduction to Elastic Materials
More informationLecture 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 information1.050 Engineering Mechanics. Lecture 22: Isotropic elasticity
1.050 Engineering Mechanics Lecture 22: Isotropic elasticity 1.050 Content overview I. Dimensional analysis 1. On monsters, mice and mushrooms 2. Similarity relations: Important engineering tools II. Stresses
More informationStrain analysis.
Strain analysis ecalais@purdue.edu Plates vs. continuum Gordon and Stein, 1991 Most plates are rigid at the until know we have studied a purely discontinuous approach where plates are
More informationComputational Astrophysics
Computational Astrophysics Lecture 1: Introduction to numerical methods Lecture 2:The SPH formulation Lecture 3: Construction of SPH smoothing functions Lecture 4: SPH for general dynamic flow Lecture
More informationMAE4700/5700 Finite Element Analysis for Mechanical and Aerospace Design
MAE4700/5700 Finite Element Analsis for Mechanical and Aerospace Design Cornell Universit, Fall 2009 Nicholas Zabaras Materials Process Design and Control Laborator Sible School of Mechanical and Aerospace
More informationPhysics of Continuous media
Physics of Continuous media Sourendu Gupta TIFR, Mumbai, India Classical Mechanics 2012 October 26, 2012 Deformations of continuous media If a body is deformed, we say that the point which originally had
More informationBasic Theorems in Dynamic Elasticity
Basic Theorems in Dynamic Elasticity 1. Stress-Strain relationships 2. Equation of motion 3. Uniqueness and reciprocity theorems 4. Elastodynamic Green s function 5. Representation theorems Víctor M. CRUZ-ATIENZA
More informationDynamic analysis. 1. Force and stress
Dynamic analysis 1. Force and stress Dynamics is the part of structural geology that involves energy, force, stress, and strength. It's very important to distinguish dynamic concepts from kinematic ones.
More informationMechanics 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 informationGame 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 informationContinuum 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 informationFinite 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 informationMidterm Examination. Please initial the statement below to show that you have read it
EN75: Advanced Mechanics of Solids Midterm Examination School of Engineering Brown University NAME: General Instructions No collaboration of any kind is permitted on this examination. You may use two pages
More informationModern Seismology Lecture Outline
Modern Seismology Lecture Outline Seismic networks and data centres Mathematical background for time series analysis Seismic processing, applications Filtering Correlation Instrument correction, Transfer
More informationRock 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 informationChapter 6: Momentum Analysis
6-1 Introduction 6-2Newton s Law and Conservation of Momentum 6-3 Choosing a Control Volume 6-4 Forces Acting on a Control Volume 6-5Linear Momentum Equation 6-6 Angular Momentum 6-7 The Second Law of
More informationINTRODUCTION TO STRAIN
SIMPLE STRAIN INTRODUCTION TO STRAIN In general terms, Strain is a geometric quantity that measures the deformation of a body. There are two types of strain: normal strain: characterizes dimensional changes,
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationChapter 3. Forces, Momentum & Stress. 3.1 Newtonian mechanics: a very brief résumé
Chapter 3 Forces, Momentum & Stress 3.1 Newtonian mechanics: a very brief résumé In classical Newtonian particle mechanics, particles (lumps of matter) only experience acceleration when acted on by external
More informationMECH 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 informationCourse No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu
Course No: (1 st version: for graduate students) Course Name: Continuum Mechanics Offered by: Chyanbin Hwu 2011. 11. 25 Contents: 1. Introduction 1.1 Basic Concepts of Continuum Mechanics 1.2 The Need
More informationCONTINUUM MECHANICS. lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern
CONTINUUM MECHANICS lecture notes 2003 jp dr.-ing. habil. ellen kuhl technical university of kaiserslautern Contents Tensor calculus. Tensor algebra.................................... Vector algebra.................................
More informationSolutions for Fundamentals of Continuum Mechanics. John W. Rudnicki
Solutions for Fundamentals of Continuum Mechanics John W. Rudnicki December, 015 ii Contents I Mathematical Preliminaries 1 1 Vectors 3 Tensors 7 3 Cartesian Coordinates 9 4 Vector (Cross) Product 13 5
More informationcos(θ)sin(θ) Alternative Exercise Correct Correct θ = 0 skiladæmi 10 Part A Part B Part C Due: 11:59pm on Wednesday, November 11, 2015
skiladæmi 10 Due: 11:59pm on Wednesday, November 11, 015 You will receive no credit for items you complete after the assignment is due Grading Policy Alternative Exercise 1115 A bar with cross sectional
More informationPart 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA
Part 5 ACOUSTIC WAVE PROPAGATION IN ANISOTROPIC MEDIA Review of Fundamentals displacement-strain relation stress-strain relation balance of momentum (deformation) (constitutive equation) (Newton's Law)
More informationNavier-Stokes Equation: Principle of Conservation of Momentum
Navier-tokes Equation: Principle of Conservation of Momentum R. hankar ubramanian Department of Chemical and Biomolecular Engineering Clarkson University Newton formulated the principle of conservation
More informationAdvanced Structural Analysis EGF Section Properties and Bending
Advanced Structural Analysis EGF316 3. Section Properties and Bending 3.1 Loads in beams When we analyse beams, we need to consider various types of loads acting on them, for example, axial forces, shear
More informationChapter 6: Momentum Analysis of Flow Systems
Chapter 6: Momentum Analysis of Flow Systems Introduction Fluid flow problems can be analyzed using one of three basic approaches: differential, experimental, and integral (or control volume). In Chap.
More informationProfessor George C. Johnson. ME185 - Introduction to Continuum Mechanics. Midterm Exam II. ) (1) x
Spring, 997 ME85 - Introduction to Continuum Mechanics Midterm Exam II roblem. (+ points) (a) Let ρ be the mass density, v be the velocity vector, be the Cauchy stress tensor, and b be the body force per
More informationConstitutive Equations
Constitutive quations David Roylance Department of Materials Science and ngineering Massachusetts Institute of Technology Cambridge, MA 0239 October 4, 2000 Introduction The modules on kinematics (Module
More informationSEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by
SEMM Mechanics PhD Preliminary Exam Spring 2014 1. Consider a two-dimensional rigid motion, whose displacement field is given by u(x) = [cos(β)x 1 + sin(β)x 2 X 1 ]e 1 + [ sin(β)x 1 + cos(β)x 2 X 2 ]e
More informationMechanical 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 informationFoundations of Solid Mechanics. James R. Rice. Harvard University. January, 1998
Manuscript for publication as Chapter 2 of the book Mechanics and Materials: Fundamentals and Linkages (eds. M. A. Meyers, R. W. Armstrong, and H. Kirchner), Wiley, publication expected 1998. Foundations
More informationLecture notes Models of Mechanics
Lecture notes Models of Mechanics Anders Klarbring Division of Mechanics, Linköping University, Sweden Lecture 7: Small deformation theories Klarbring (Mechanics, LiU) Lecture notes Linköping 2012 1 /
More information3.22 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 informationChapter 16: Elastic Solids
Chapter 16: Elastic Solids Chapter 16: Elastic Solids... 366 16.1 Introduction... 367 16.2 The Elastic Strain... 368 16.2.1 The displacement vector... 368 16.2.2 The deformation gradient... 368 16.2.3
More informationThe science of elasticity
The science of elasticity In 1676 Hooke realized that 1.Every kind of solid changes shape when a mechanical force acts on it. 2.It is this change of shape which enables the solid to supply the reaction
More informationCIVL4332 L1 Introduction to Finite Element Method
CIVL L Introduction to Finite Element Method CIVL L Introduction to Finite Element Method by Joe Gattas, Faris Albermani Introduction The FEM is a numerical technique for solving physical problems such
More informationRotational motion of a rigid body spinning around a rotational axis ˆn;
Physics 106a, Caltech 15 November, 2018 Lecture 14: Rotations The motion of solid bodies So far, we have been studying the motion of point particles, which are essentially just translational. Bodies with
More informationUnit 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 informationTensor Transformations and the Maximum Shear Stress. (Draft 1, 1/28/07)
Tensor Transformations and the Maximum Shear Stress (Draft 1, 1/28/07) Introduction The order of a tensor is the number of subscripts it has. For each subscript it is multiplied by a direction cosine array
More informationMECH 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 information16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations
6.2 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive quations Constitutive quations For elastic materials: If the relation is linear: Û σ ij = σ ij (ɛ) = ρ () ɛ ij σ ij =
More informationContinuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms
Continuum mechanics office Math 0.107 ales.janka@unifr.ch http://perso.unifr.ch/ales.janka/mechanics Mars 16, 2011, Université de Fribourg 1. Constitutive equation: definition and basic axioms Constitutive
More information20. 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 informationTensor fields. Tensor fields: Outline. Chantal Oberson Ausoni
Tensor fields Chantal Oberson Ausoni 7.8.2014 ICS Summer school Roscoff - Visualization at the interfaces 28.7-8.8, 2014 1 Tensor fields: Outline 1. TENSOR FIELDS: DEFINITION 2. PROPERTIES OF SECOND-ORDER
More informationTheory at a Glance (for IES, GATE, PSU)
1. Stress and Strain Theory at a Glance (for IES, GATE, PSU) 1.1 Stress () When a material is subjected to an external force, a resisting force is set up within the component. The internal resistance force
More information202 Index. failure, 26 field equation, 122 force, 1
Index acceleration, 12, 161 admissible function, 155 admissible stress, 32 Airy's stress function, 122, 124 d'alembert's principle, 165, 167, 177 amplitude, 171 analogy, 76 anisotropic material, 20 aperiodic
More informationCVEN 5161 Advanced Mechanics of Materials I
CVEN 5161 Advanced Mechanics of Materials I Instructor: Kaspar J. Willam Revised Version of Class Notes Fall 2003 Chapter 1 Preliminaries The mathematical tools behind stress and strain are housed in Linear
More informationContinuum Mechanics Fundamentals
Continuum Mechanics Fundamentals James R. Rice, notes for ES 220, 12 November 2009; corrections 9 December 2009 Conserved Quantities Let a conseved quantity have amount F per unit volume. Examples are
More informationYou may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
MATHEMATICAL TRIPOS Part III Thursday 1 June 2006 1.30 to 4.30 PAPER 76 NONLINEAR CONTINUUM MECHANICS Attempt FOUR questions. There are SIX questions in total. The questions carry equal weight. STATIONERY
More informationExercise: 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 informationVariational 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 informationLoad Cell Design Using COMSOL Multiphysics
Load Cell Design Using COMSOL Multiphysics Andrei Marchidan, Tarah N. Sullivan and Joseph L. Palladino Department of Engineering, Trinity College, Hartford, CT 06106, USA joseph.palladino@trincoll.edu
More informationLinearized 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(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 informationChapter 2: Fluid Dynamics Review
7 Chapter 2: Fluid Dynamics Review This chapter serves as a short review of basic fluid mechanics. We derive the relevant transport equations (or conservation equations), state Newton s viscosity law leading
More informationNDT&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 informationDynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet
Dynamics of the Mantle and Lithosphere ETH Zürich Continuum Mechanics in Geodynamics: Equation cheat sheet or all equations you will probably ever need Definitions 1. Coordinate system. x,y,z or x 1,x
More informationModule 2 Stresses in machine elements
Module 2 Stresses in machine elements Lesson 3 Strain analysis Instructional Objectives At the end of this lesson, the student should learn Normal and shear strains. 3-D strain matri. Constitutive equation;
More informationDIAGONALIZATION OF THE STRESS TENSOR
DIAGONALIZATION OF THE STRESS TENSOR INTRODUCTION By the use of Cauchy s theorem we are able to reduce the number of stress components in the stress tensor to only nine values. An additional simplification
More informationLecture Notes 8
12.005 Lecture Notes 8 Assertion: most of the stress tensor in the Earth is close to "lithostatic," τ ij ~ -ρgd δ ij, where ρ is the average density of the overburden, g is gravitational acceleration,
More informationSeismology. Chapter Historical perspective
Chapter 4 Seismology 4.1 Historical perspective 1678 Hooke Hooke s Law F = c u (or σ = Eɛ) 1760 Mitchell Recognition that ground motion due to earthquakes is related to wave propagation 1821 Navier Equation
More informationLecture 14: Strain Examples. GEOS 655 Tectonic Geodesy Jeff Freymueller
Lecture 14: Strain Examples GEOS 655 Tectonic Geodesy Jeff Freymueller A Worked Example Consider this case of pure shear deformation, and two vectors dx 1 and dx 2. How do they rotate? We ll look at vector
More informationMeasurement of deformation. Measurement of elastic force. Constitutive law. Finite element method
Deformable Bodies Deformation x p(x) Given a rest shape x and its deformed configuration p(x), how large is the internal restoring force f(p)? To answer this question, we need a way to measure deformation
More informationChapter Two: Mechanical Properties of materials
Chapter Two: Mechanical Properties of materials Time : 16 Hours An important consideration in the choice of a material is the way it behave when subjected to force. The mechanical properties of a material
More information