Basic Concepts of Strain and Tilt. Evelyn Roeloffs, USGS June 2008
|
|
- Leo Allison
- 6 years ago
- Views:
Transcription
1 Basic Concepts of Strain and Tilt Evelyn Roeloffs, USGS June
2 Coordinates Right-handed coordinate system, with positions along the three axes specified by x,y,z. x,y will usually be horizontal, and z will usually be up. We will often use the coordinate system x,y,z=east, (geographic) North, Up. Sometimes we will use 1,2,3 instead of x,y,z to allow equations to be written more compactly. The distinctions among the reference frames used for GPS are not important for the interpretation of strainmeter or tiltmeter data. Page 2 of 24
3 Displacements Displacement of a point is described by the three displacements in these respective directions denoted by u E, u N, and u UP. Page 3 of 24
4 Stress and Strain in One Dimension If L is the length of the rod and the force stretches it by dl, then strain, ε, is the dimensionless quantity ε = dl /L The stress, σ, is the force per unit area, in this case σ = F /πr 2 Since this is a linearly elastic rod, the stress is proportional to the strain: F /πr 2 = E(dL /L) or σ = Eε a) we consider ε positive when the rod gets longer; we consider σ to be positive when the rod is in tension. b) E is a material property called Young s modulus with dimensions of force per unit area, units hpa, MPa, GPa c) The stress σ is present throughout the length of the rod, not just at the end where the force is applied. d) Strain is uniform throughout the length of the rod; displacement increases linearly from the fixed to the free end. Page 4 of 24
5 Typical Values of Young s Modulus Elastic strains are small for many realistic situations. Material Young's modulus (GPa) Strain caused by a 0.1 MPa (apx 14.5 psi) load parallel to axis of a rod Rubber E-3 Spider thread 3.03E-3 Hair 10.01E-3 Brick 14 7E-6=7 microstrain Oak 14 7E-6=7 microstrain Concrete 27 4E-6=4 microstrain Marble 50 2E-6=2 microstrain Aluminum E-6=1.5 microstrain Granite E-6 = 1.4 microstrain Iron E-6=0.5 microstrain Table. Typical values of Young s modulus (from Page 5 of 24
6 Strain in Three Dimensions The strains are the spatial gradients of displacement. If displacement is uniform over a body (ie., every part of the body moves same distance in the same direction), then there is no strain. The body can be picked up, moved, or rotated and there will not necessarily be any strain. The equation for obtaining strain from displacements, in three dimensions, is ε ij = 1 u 2 i / x j + u j / x i [ ] or ε ij = 1 Δu 2 [ i /Δx j + Δu j /Δx i ] for i, j = 1,2,3 (or x, y,z, or East-North-Up). (1) Writing out a few of them: ε zz = 1 u 2 [ z / z + u z / z] = u z / z Δu z /Δz ε xy = 1 u 2 x / y + u y / x [ ] ε NN = 1 u 2 [ N / x N + u N / x N ] = u N / x N Δu N /Δx N ε NE = 1 u 2 [ N / x E + u E / x N ] = 1 u 2 E / x N + u N / x E [ ] = ε EN (a shear strain) Engineering shear strain: γ xy = 2ε xy = [ u x / y + u y / x] Page 6 of 24
7 3D Strain, continued The 3 displacements can each vary in any of 3 directions, so there are 9 i, j combinations. However, there are only 6 independent strains, because if i and j are different, then ε ij = ε ji Example: for the rod, strain in the direction parallel to the rod is the slope of the plot of displacement vs. distance along the rod: ε xx = u x / x = dl /L Page 7 of 24
8 Some basic assumptions: 1) Small region: ie., region in which it is OK to use the value of displacement at a point and its spatial gradients to estimate the displacement Size of region depends on the spatial distribution of displacement. For example, slip on the San Andreas fault produces a displacement field that has a jump across the fault. Can t use displacement and strain at a point SW of the fault to estimate displacement NE, even if points are only 100 m apart. For two points on the same side of the fault, the region might be about 100 m or more. For two sites several km from the fault, the displacement field is much smoother and observations one or more km apart are sufficient. 2) Small deformation: Generally we will be speaking of strains in range (0.1 nanostrain) to 10-4 (100 microstrain). This ranges goes from resolution of GTSM21 borehole strainmeter to approximate strain within 10 s of meters of a M7 fault rupture. 3) Only changes matter: For example, we will consider vertical stress changes caused by atmospheric pressure fluctuations, but we will not explicitly worry about the more or less constant overburden pressure. Page 8 of 24
9 Units of Strain Strain is dimensionless but is often referred to as if it had units A 1% change in volume is a volumetric strain of 0.01 = 10,000 microstrain = 10,000 parts-per-million (ppm) A 1-mm change in a 1-km baseline is a linear strain of 10-6 =1 microstrain 1 microstrain is sometimes written 1 µε 1 nanostrain=10-9 =0.001 microstrain, also called 1 part-per-billion (ppb) 1 nanostrain is sometimes written 1 nε We will mostly be considering strains that range from about 0.1 nanostrain (an approximate lower bound on resolution for borehole strainmeters) to 1000 microstrain. 10 microstrain is a ballpark value for the coseismic strain within 5 km of a M7 earthquake. Page 9 of 24
10 Sign Conventions for Strain In mathematical descriptions, increases of length or volume are always considered to be positive strains. There is also an unambiguous sign convention for shear strain based on its defining equation. ε xy = 1 u 2 [ x / y + u y / x] Page 10 of 24
11 Example: Strain Near Transition from Creeping to Locked on a Strike-Slip Fault The strain-displacement equations unambiguously define a sign convention for shear strain. However, an arbitrary sign convention is sometimes used in reporting data. It s always a good idea to check. If fault creep decreases from 30 mm/year to 0 over a 30 km reach of the fault, then ε yy increases by 1 microstrain/year. If the creeping zone is 100 m wide, then at the creeping end ε xy changes by -300 microstrain/year. Page 11 of 24
12 Matrix Notation It s convenient to represent strain using a matrix: ε 11 ε 12 ε 13 ε xx ε xy ε xz ε EE ε EN ε EZ ε 3 3 = ε 12 ε 22 ε 23 or ε 3 3 = ε xy ε yy ε yz or ε 3 3 = ε EN ε NN ε NZ ε 13 ε 23 ε 33 ε xz ε yz ε zz ε EZ ε NZ ε ZZ The term strain tensor is often used. This is because strain can be expressed in different coordinate systems ( transformed ) according to certain rules. We will often work in 2 horizontal dimensions, where ε 2 2 = ε xx ε xy ε xy ε yy The three independent strain components in 2D are usually written in vector form: ε xx ε yy ε xy 3 1 or ε xx + ε yy ε xx ε yy 2ε xy 3 1 Page 12 of 24
13 Form of the Horizontal Strain Tensor that We Will Use in this Workshop We will mostly use the form on the right and refer to its entries as follows: ε xx + ε yy Areal Strain ε a ε xx ε yy Differential Extension γ 1 2ε xy Engineering Shear Strain γ 2 Dashed line = initial shape Solid line = deformed shape All of the diagrams show positive strains. Page 13 of 24
14 Stress Stress has dimensions of force per unit area. Inside a deforming body, forces act on every small planar region. Force has magnitude and direction, and 3 components: one perpendicular to the surface, and two parallel to the surface, at right angles to each other. σ xx σ xy σ xz σ 3 3 = σ xy σ yy σ yz σ xz σ yz σ zz There can be 3 directions of force on each plane, and 3 orthogonal planes, each perpendicular to one of the coordinate axes, giving rise to 9 components of stress. We do not distinguish between σ ij and σ ji, because stresses arise from the forces that do not move (accelerate) the body. Mathematically, the requirement that σ ij = σ ji results from the equilibrium equations (see any elasticity text). Intuitively, the picture can be used to visualize what would happen if these shear stresses were not equal. Because σ ij = σ ji, there are only 6 independent stress components in 3 dimensions, similar to the situation for strain. Page 14 of 24
15 Types of Stress and Stress States The three stress components with two equal subscripts are referred to as normal stresses. They apply tension or compression along a specified coordinate axis. They act parallel to the normal to the face of the cube. Stress components with unequal subscripts are shear stresses. They apply equal and opposite forces on opposing faces of a cube of material, acting parallel to those faces. ( ) is called the mean stress, or sometimes the average The average of the three normal stresses 1 σ 3 xx + σ yy + σ zz stress. It is frequently written σ kk /3, using the repeated index to imply the sum. A simple and important stress state is that in which the three normal stresses are equal. This is referred to as hydrostatic or isotropic stress. Pressure in a liquid is equivalent to a hydrostatic stress state. See Engelder (1993) for discussions of stress states relevant to the lithosphere. Page 15 of 24
16 Stress-strain Relations The way in which stress and strain are coupled for a particular material is described by constitutive equations. For a linearly elastic medium, the constitutive relations basically say that strain is proportional to stress, in 3 dimensions. An isotropic material is one which has the same mechanical properties in all directions. The constitutive equations for normal stresses and strains, in an isotropic linearly elastic material, are: 2Gε xx = σ xx ν ( 1+ ν σ + σ + σ xx yy zz), 2Gε yy = σ yy ν ( 1+ ν σ + σ + σ xx yy zz), 2Gε zz = σ zz ν 1+ ν σ + σ + σ xx yy zz and the constitutive equations for shear stresses and strains are: 2Gε xy = σ xy, 2Gε xz = σ xz, 2Gε yz = σ yz (2d,e,f) All 6 constitutive equations can be written compactly as: 2Gε ij = σ ij ν 1+ ν σ kkδ ij (another way to write 2a-f) using the summation convention and δ ij =1 for i = j, and 0 for i j. ( ) (2a,b,c) Two material properties appear: G, the shear modulus (with units of force per unit area); and ν, the Poisson ratio (dimensionless). More elastic moduli are needed if the material is not isotropic, but that situation is beyond the scope of this short course. Page 16 of 24
17 Relationships Among Elastic Moduli Although only two material properties are needed to relate stress and strain in an isotropic elastic material, there are many ways to represent these properties. The inter-relationships useful in this short course are: Young s modulus, E Bulk modulus, K Shear modulus, G E 3(1 2ν)K 2G(1+ ν) K E 3(1 2ν) 2(1+ ν)g 3(1 2ν) G E 2(1+ ν) 3(1 2ν)K 2(1+ ν) Table. Relationships between elastic moduli We will use K and G frequently. They are examples of moduli, with dimensions of force/unit area. The Poisson ratio couples extension in one direction to contraction in the perpendicular directions. It is always >0 and <0.5, taking on the upper limit of 0.5 for liquids. The Poisson ratio is dimensionless and is not a modulus. To see the role of the bulk modulus, add up the equations relating the three normal stresses and strains: ( ε 11 + ε 22 + ε 33 ) = 3(1 2ν) σ 11 + σ 22 + σ 33 = 1 σ 11 + σ 22 + σ 33 2G(1+ ν) 3 K 3 The bulk modulus is the coefficient of proportionality between volumetric strain (ie., expansion or contraction), and mean stress change (for example, pressure). Page 17 of 24
18 Typical Values of Elastic Moduli and Poisson ratio Material Bulk Modulus (GPa) Shear modulus (GPa) Poisson ratio Air (0 C, x MPa) Water (25 C) Polycrystalline Ice Plagioclase Quartz Crystal Calcite Olivine Berea Sandstone Westerly Granite Tennessee Marble Table. Typical values of bulk modulus, shear modulus, and Poisson ratio Page 18 of 24
19 Sizes of relevant and/or familiar stresses and strains: Material immediately adjacent to a fault that experiences a seismic stress drop of 10 MPa undergoes strains order of 1000 microstrain. Barometric pressure drop during a typical storm is 20 millibars = 0.02 bars = MPa = 2 kpa =20 hpa Pressure underwater increases at rate of 2.3 psi/foot = 1 bar/10 meters. = 1MPa/100 meters; pressure difference from bottom to top of 10-foot-deep pool is 23 psi = apx 0.3 bar = 0.03 MPa. Page 19 of 24
20 Rotating Coordinates We ll need to switch back and forth between coordinate systems that have different horizontal orientations (the vertical direction remains the same in these notes; See any text on elasticity for the 3D formulas). ε x'x' ε y'y' ε x'y' 3 1 = 1 1+ cos2θ 1 cos2θ 2sin2θ 1 cos2θ 1+ cos2θ 2sin2θ 2 2sin2θ 2sin2θ 2cos2θ ε xx ε yy 3 3 ε xy 3 1 ε x'x' + ε y'y' ε x'x' ε y'y' 2ε x'y' = 0 cos2φ sin2φ 0 sin2φ cos2φ 3 3 ε xx + ε yy ' ε xx ε yy 2ε xy 3 1 (3a,b) The method for converting stresses to a rotated coordinate system is exactly the same as that for converting strains. Page 20 of 24
21 Invariants of Strain and Stress ε x'x' + ε y'y' ε x'x' ε y'y' 2ε x'y' = 0 cos2φ sin2φ 0 sin2φ cos2φ 3 3 ε xx + ε yy ' ε xx ε yy 2ε xy 3 1 (3b again) Inspecting equation (3b) shows that : ε x'x' + ε y'y' = ε xx + ε yy for any value of φ (4) This is also true of the volumetric strain in 3D, and of mean stresses. In other words, the areal and volumetric strain, and the mean stress, are the same (invariant) no matter what Cartesian coordinates they are expressed in. 2 In 2D there is one other invariant of strain, which is ε xy ε xx ε yy Principal Strains and Principal Stresses By setting ε x'y' =0 in equation (3b), φ can be found such that the shear stresses (and strains) vanish. In this coordinate system, the e stress and strain tensors have only diagonal elements, called principle stresses and strains., related by: ε I 0 0 1/ E ν / E ν / E σ I ε II 0 = ν / E 1/ E ν / E 0 σ II ε III ν / E ν / E 1/ E 0 0 σ III (5) Near the Earth s surface, the vertical stress is usually a principle stress, so the other two principle stresses are in the horizontal plane. Page 21 of 24
22 Reducing to Two Dimensions using the Plane Stress Assumption Plane stress: one of the principle stresses is zero, so that only the stress components in one plane are nonzero. This is the situation when the vertical stress is unchanging (for example, it equals the overburden) and can therefore be assumed to be zero. Shear strains in vertical planes also vanish, so the only nonzero strains are ε xx,ε yy, ε zz, andε xy. However, we can express ε zz in terms of ε xx and ε yy. To see this, write out the constitutive equations for the normal strains: 2Gε zz = ν 1+ ν (σ xx + σ yy ) 2G(ε xx + ε yy ) = 1 ν 1+ ν (σ xx + σ yy ) Comparing these two equations shows that ε zz = ν 1 ν (ε xx + ε yy ) (6) ε zz is of opposite sign from ε xx + ε yy ; when there is contraction in the horizontal plane, the vertical strain is extensional. Consequently, for plane stress the volumetric strain is smaller than the areal strain: 1 2ν ε xx + ε yy + ε zz = 1 ν (ε + ε xx yy ) for plane stress (7) Page 22 of 24
23 Tilt Tilt is defined simply as the change in inclination with respect to either the horizontal or the vertical. Tilt is conceptually different from strain - it does not necessarily entail deformation. That is, displacements can be distributed spatially such that there is no strain, but there are nonzero tilts. Tilt can be measured using vertical sensors in boreholes or horizontal sensors on the earth s surface. As for borehole strainmeters, tiltmeters in boreholes are usually considered to be at the surface for mathematical purposes. Two tiltmeter components are needed to characterize the change in inclination of a plane. Tilt is a vector. Figure. (A) Tilt without strain. (B) Tilt with strain. If one component is oriented East-West and the other North-South, tilts are related to displacements by: tilt EW = u x / z = u z / x and tilt NS = u y / z = u z / y Page 23 of 24
24 Borehole Strainmeters as Elastic Inclusions Page 24 of 24
Introduction to Strain and Borehole Strainmeter Data
Introduction to Strain and Borehole Strainmeter Data Evelyn Roeloffs U. S. Geological Survey Earthquake Science Center March 28, 2016 Evelyn Roeloffs, USGS ESC Strainmeters: Introduction March 28, 2016
More information3.2 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 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 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 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 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 information1 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 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 information[5] Stress and Strain
[5] Stress and Strain Page 1 of 34 [5] Stress and Strain [5.1] Internal Stress of Solids [5.2] Design of Simple Connections (will not be covered in class) [5.3] Deformation and Strain [5.4] Hooke s Law
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 informationUNIVERSITY 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 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 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 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 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 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 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 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 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 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 informationMechanics 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 informationSurface force on a volume element.
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
More informationMechanics 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 informationGeodynamics Lecture 5 Basics of elasticity
Geodynamics Lecture 5 Basics of elasticity Lecturer: David Whipp david.whipp@helsinki.fi! 16.9.2014 Geodynamics www.helsinki.fi/yliopisto 1 Goals of this lecture Introduce linear elasticity! Look at the
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 and Planar Constitutive Relations
3D and Planar Constitutive Relations A School on Mechanics of Fibre Reinforced Polymer Composites Knowledge Incubation for TEQIP Indian Institute of Technology Kanpur PM Mohite Department of Aerospace
More informationCONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS
Chapter 9 CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS Figure 9.1: Hooke memorial window, St. Helen s, Bishopsgate, City of London 211 212 CHAPTR 9. CONSTITUTIV RLATIONS FOR LINAR LASTIC SOLIDS 9.1 Mechanical
More 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 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 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 informationModule-4. Mechanical Properties of Metals
Module-4 Mechanical Properties of Metals Contents ) Elastic deformation and Plastic deformation ) Interpretation of tensile stress-strain curves 3) Yielding under multi-axial stress, Yield criteria, Macroscopic
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 informationIntroduction 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 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 informationEMA 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 informationStrength of Material. Shear Strain. Dr. Attaullah Shah
Strength of Material Shear Strain Dr. Attaullah Shah Shear Strain TRIAXIAL DEFORMATION Poisson's Ratio Relationship Between E, G, and ν BIAXIAL DEFORMATION Bulk Modulus of Elasticity or Modulus of Volume
More 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 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 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 5 Plane-Stress Stress-Strain Relations in a Global Coordinate System
Chapter 5 Plane-Stress Stress-Strain Relations in a Global Coordinate System One of the most important characteristics of structures made of fiber-reinforced materials, and one which dictates the manner
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 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 information16.20 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 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 informationStress-Strain Behavior
Stress-Strain Behavior 6.3 A specimen of aluminum having a rectangular cross section 10 mm 1.7 mm (0.4 in. 0.5 in.) is pulled in tension with 35,500 N (8000 lb f ) force, producing only elastic deformation.
More information1. A pure shear deformation is shown. The volume is unchanged. What is the strain tensor.
Elasticity Homework Problems 2014 Section 1. The Strain Tensor. 1. A pure shear deformation is shown. The volume is unchanged. What is the strain tensor. 2. Given a steel bar compressed with a deformation
More informationChapter 12. Static Equilibrium and Elasticity
Chapter 12 Static Equilibrium and Elasticity Static Equilibrium Equilibrium implies that the object moves with both constant velocity and constant angular velocity relative to an observer in an inertial
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 informationME 243. Mechanics of Solids
ME 243 Mechanics of Solids Lecture 2: Stress and Strain 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 informationGG611 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 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 informationMaterials and Methods The deformation within the process zone of a propagating fault can be modeled using an elastic approximation.
Materials and Methods The deformation within the process zone of a propagating fault can be modeled using an elastic approximation. In the process zone, stress amplitudes are poorly determined and much
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 informationChapter 2. Rubber Elasticity:
Chapter. Rubber Elasticity: The mechanical behavior of a rubber band, at first glance, might appear to be Hookean in that strain is close to 100% recoverable. However, the stress strain curve for a rubber
More informationElasticity 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 informationMohr's Circle and Earth Stress (The Elastic Earth)
Lect. 1 - Mohr s Circle and Earth Stress 6 Mohr's Circle and Earth Stress (The Elastic Earth) In the equations that we derived for Mohr s circle, we measured the angle, θ, as the angle between σ 1 and
More informationSTRESS, STRAIN AND DEFORMATION OF SOLIDS
VELAMMAL COLLEGE OF ENGINEERING AND TECHNOLOGY, MADURAI 625009 DEPARTMENT OF CIVIL ENGINEERING CE8301 STRENGTH OF MATERIALS I -------------------------------------------------------------------------------------------------------------------------------
More informationMECE 3321 MECHANICS OF SOLIDS CHAPTER 3
MECE 3321 MECHANICS OF SOLIDS CHAPTER 3 Samantha Ramirez TENSION AND COMPRESSION TESTS Tension and compression tests are used primarily to determine the relationship between σ avg and ε avg in any material.
More 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 Lectures & 3, 9/31 Aug 017 www.geosc.psu.edu/courses/geosc508 Discussion of Handin, JGR, 1969 and Chapter 1 Scholz, 00. Stress analysis and Mohr Circles Coulomb Failure
More information**********************************************************************
Department of Civil and Environmental Engineering School of Mining and Petroleum Engineering 3-33 Markin/CNRL Natural Resources Engineering Facility www.engineering.ualberta.ca/civil Tel: 780.492.4235
More 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 informationSamantha Ramirez, MSE. Stress. The intensity of the internal force acting on a specific plane (area) passing through a point. F 2
Samantha Ramirez, MSE Stress The intensity of the internal force acting on a specific plane (area) passing through a point. Δ ΔA Δ z Δ 1 2 ΔA Δ x Δ y ΔA is an infinitesimal size area with a uniform force
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 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 informationFig. 1. Circular fiber and interphase between the fiber and the matrix.
Finite element unit cell model based on ABAQUS for fiber reinforced composites Tian Tang Composites Manufacturing & Simulation Center, Purdue University West Lafayette, IN 47906 1. Problem Statement In
More informationNORMAL STRESS. The simplest form of stress is normal stress/direct stress, which is the stress perpendicular to the surface on which it acts.
NORMAL STRESS The simplest form of stress is normal stress/direct stress, which is the stress perpendicular to the surface on which it acts. σ = force/area = P/A where σ = the normal stress P = the centric
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 informationUnit I Stress and Strain
Unit I Stress and Strain Stress and strain at a point Tension, Compression, Shear Stress Hooke s Law Relationship among elastic constants Stress Strain Diagram for Mild Steel, TOR steel, Concrete Ultimate
More informationModule III - Macro-mechanics of Lamina. Lecture 23. Macro-Mechanics of Lamina
Module III - Macro-mechanics of Lamina Lecture 23 Macro-Mechanics of Lamina For better understanding of the macromechanics of lamina, the knowledge of the material properties in essential. Therefore, the
More informationCHAPTER 3 THE EFFECTS OF FORCES ON MATERIALS
CHAPTER THE EFFECTS OF FORCES ON MATERIALS EXERCISE 1, Page 50 1. A rectangular bar having a cross-sectional area of 80 mm has a tensile force of 0 kn applied to it. Determine the stress in the bar. Stress
More informationMechanics of Biomaterials
Mechanics of Biomaterials Lecture 7 Presented by Andrian Sue AMME498/998 Semester, 206 The University of Sydney Slide Mechanics Models The University of Sydney Slide 2 Last Week Using motion to find forces
More informationEART162: PLANETARY INTERIORS
EART162: PLANETARY INTERIORS Francis Nimmo Last Week Global gravity variations arise due to MoI difference (J 2 ) We can also determine C, the moment of inertia, either by observation (precession) or by
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 informationTopics. 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 informationUnderstand basic stress-strain response of engineering materials.
Module 3 Constitutive quations Learning Objectives Understand basic stress-strain response of engineering materials. Quantify the linear elastic stress-strain response in terms of tensorial quantities
More informationElasticity: Term Paper. Danielle Harper. University of Central Florida
Elasticity: Term Paper Danielle Harper University of Central Florida I. Abstract This research was conducted in order to experimentally test certain components of the theory of elasticity. The theory was
More information2/28/2006 Statics ( F.Robilliard) 1
2/28/2006 Statics (.Robilliard) 1 Extended Bodies: In our discussion so far, we have considered essentially only point masses, under the action of forces. We now broaden our considerations to extended
More informationChapter 2: Elasticity
OHP 1 Mechanical Properties of Materials Chapter 2: lasticity Prof. Wenjea J. Tseng ( 曾文甲 ) Department of Materials ngineering National Chung Hsing University wenjea@dragon.nchu.edu.tw Reference: W.F.
More informationChapter II: Reversible process and work
Chapter II: Reversible process and work 1- Process Defined by change in a system, a thermodynamic process is a passage of a thermodynamic system from an initial to a final state of thermodynamic equilibrium.
More informationANALYSIS 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(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 informationLoading σ Stress. Strain
hapter 2 Material Non-linearity In this chapter an overview of material non-linearity with regard to solid mechanics is presented. Initially, a general description of the constitutive relationships associated
More informationStress equilibrium in southern California from Maxwell stress function models fit to both earthquake data and a quasi-static dynamic simulation
Stress equilibrium in southern California from Maxwell stress function models fit to both earthquake data and a quasi-static dynamic simulation Peter Bird Dept. of Earth, Planetary, and Space Sciences
More informationGeology 229 Engineering Geology. Lecture 5. Engineering Properties of Rocks (West, Ch. 6)
Geology 229 Engineering Geology Lecture 5 Engineering Properties of Rocks (West, Ch. 6) Common mechanic properties: Density; Elastic properties: - elastic modulii Outline of this Lecture 1. Uniaxial rock
More informationSummary so far. Geological structures Earthquakes and their mechanisms Continuous versus block-like behavior Link with dynamics?
Summary so far Geodetic measurements velocities velocity gradient tensor (spatial derivatives of velocity) Velocity gradient tensor = strain rate (sym.) + rotation rate (antisym.) Strain rate tensor can
More informationSTANDARD SAMPLE. Reduced section " Diameter. Diameter. 2" Gauge length. Radius
MATERIAL PROPERTIES TENSILE MEASUREMENT F l l 0 A 0 F STANDARD SAMPLE Reduced section 2 " 1 4 0.505" Diameter 3 4 " Diameter 2" Gauge length 3 8 " Radius TYPICAL APPARATUS Load cell Extensometer Specimen
More informationCHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS
CHAPTER 6 MECHANICAL PROPERTIES OF METALS PROBLEM SOLUTIONS Concepts of Stress and Strain 6.1 Using mechanics of materials principles (i.e., equations of mechanical equilibrium applied to a free-body diagram),
More informationComposites 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 informationExercise: 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 informationChapter 13 ELASTIC PROPERTIES OF MATERIALS
Physics Including Human Applications 280 Chapter 13 ELASTIC PROPERTIES OF MATERIALS GOALS When you have mastered the contents of this chapter, you will be able to achieve the following goals: Definitions
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 informationMECHANICS OF MATERIALS
Third E CHAPTER 2 Stress MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University and Strain Axial Loading Contents Stress & Strain:
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 informationThe 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 informationσ = F/A ε = L/L σ ε a σ = Eε
Material and Property Information This chapter includes material from the book Practical Finite Element Analysis. It also has been reviewed and has additional material added by Sascha Beuermann. Hooke
More informationMAE 323: Chapter 4. Plane Stress and Plane Strain. The Stress Equilibrium Equation
The Stress Equilibrium Equation As we mentioned in Chapter 2, using the Galerkin formulation and a choice of shape functions, we can derive a discretized form of most differential equations. In Structural
More informationMechanical properties 1 Elastic behaviour of materials
MME131: Lecture 13 Mechanical properties 1 Elastic behaviour of materials A. K. M. B. Rashid Professor, Department of MME BUET, Dhaka Today s Topics Deformation of material under the action of a mechanical
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 informationUNIVERSITY PHYSICS I. Professor Meade Brooks, Collin College. Chapter 12: STATIC EQUILIBRIUM AND ELASTICITY
UNIVERSITY PHYSICS I Professor Meade Brooks, Collin College Chapter 12: STATIC EQUILIBRIUM AND ELASTICITY Two stilt walkers in standing position. All forces acting on each stilt walker balance out; neither
More information