Continuum Mechanics for Hillslopes: Part III

Slides:

Advertisements

Similar presentations

III. Strain and Stress Strain Stress Rheology Reading Suppe, Chapter 3 Twiss&Moores, chapter 15.

Advertisements

1cs533d-term Notes  No lecture Thursday (apologies)

1 MAE 5130: VISCOUS FLOWS Lecture 3: Kinematic Properties August 24, 2010 Mechanical and Aerospace Engineering Department Florida Institute of Technology.

APPLIED MECHANICS Lecture 10 Slovak University of Technology

M. A. Farjoo.  The stiffness can be defined by appropriate stress – strain relations.  The components of any engineering constant can be expressed in.

ECIV 520 A Structural Analysis II

An Introduction to Stress and Strain

ENGR 225 Section 2.1. Review ~ Force per Area Normal Stress –The average normal stress can be determined from σ = P / A, where P is the internal axial.

MECH 221 FLUID MECHANICS (Fall 06/07) Chapter 4: FLUID KINETMATICS

Expectations after today’s lecture

A Generalized Frame work Viscous Fluid Flow… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.

Fluids. Eulerian View  In a Lagrangian view each body is described at each point in space. Difficult for a fluid with many particles.  In an Eulerian.

Forces. Normal Stress A stress measures the surface force per unit area.  Elastic for small changes A normal stress acts normal to a surface.  Compression.

Conservation Laws for Continua

CHAPTER OBJECTIVES Define concept of normal strain

Review (2 nd order tensors): Tensor – Linear mapping of a vector onto another vector Tensor components in a Cartesian basis (3x3 matrix): Basis change.

College of Physics Science & Technology YANGZHOU UNIVERSITYCHINA Chapter 11ROTATION 11.1 The Motion of Rigid Bodies Rigid bodies A rigid body is.

Theories of Stress and Strain

Chapter 9: Differential Analysis of Fluid Flow SCHOOL OF BIOPROCESS ENGINEERING, UNIVERSITI MALAYSIA PERLIS.

Chapter 10 Rotation of a Rigid Object about a Fixed Axis.

CHAPTER 4 DEFORMATION AND STRAIN. Deformation describes the complete transformation from the initial to the final geometry (shape, position and orientation)

STRAIN RATE, ROTATION RATE AND ISOTROPY

KINEMATICS Kinematics describes fluid flow without analyzing the forces responsibly for flow generation. Thereby it doesn’t matter what kind of liquid.

III. Strain and Stress Strain Stress Rheology Reading Suppe, Chapter 3

1 MAE 5130: VISCOUS FLOWS Momentum Equation: The Navier-Stokes Equations, Part 2 September 9, 2010 Mechanical and Aerospace Engineering Department Florida.

Fluid Flows due to Pure Mechanical Forces… P M V Subbarao Professor Mechanical Engineering Department I I T Delhi Construction of Navier-Stokes Equations.

Dr. Wang Xingbo Fall ， 2005 Mathematical & Mechanical Method in Mechanical Engineering.

Lecture 3 Kinematics Part I

Chapter 10 Rotational Motion.

Finite strain I Rigid body deformations Strain measurements The strain tensor Faith of material lines.

GG313 Lecture 14 Coordinate Transformations Strain.

Rotational Kinetic Energy An object rotating about some axis with an angular speed, , has rotational kinetic energy even though it may not have.

Chapter 4 Rock Mechanics Strain

Elasticity I Ali K. Abdel-Fattah. Elasticity In physics, elasticity is a physical property of materials which return to their original shape after they.

Abj 4.2.2: Pressure, Pressure Force, and Fluid Motion Without Flow [Q2 and Q3] Area as A Vector Component of Area Vector – Projected Area Net Area.

Strength of Materials Malayer University Department of Civil Engineering Taught by: Dr. Ali Reza Bagherieh In The Name of God.

 2005 Pearson Education South Asia Pte Ltd 2. Strain 1 CHAPTER OBJECTIVES Define concept of normal strain Define concept of shear strain Determine normal.

Today: Back to stress stress in the earth Strain Measuring Strain Strain Examples Today: Back to stress stress in the earth Strain Measuring Strain Strain.

Structural Geology Deformation and Strain – Homogeneous Strain, Strain Ellipsoid, Strain Path, Coaxial and Noncoaxial Strain Lecture 7 – Spring 2016 Deformation.

 2005 Pearson Education South Asia Pte Ltd 2. Strain 1 CHAPTER OBJECTIVES Define concept of normal strain Define concept of shear strain Determine normal.

 2005 Pearson Education South Asia Pte Ltd 2. Strain 1 CHAPTER OBJECTIVES To define the concept of normal strain To define the concept of shear strain.

INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 1)

6. Strain Assoc.Prof.Dr. Ahmet Zafer Şenalp Mechanical Engineering Department Gebze Technical University.

Chapter 10 Lecture 18: Rotation of a Rigid Object about a Fixed Axis: II.

Introduction to strain and borehole strainmeter data

Geology 5640/6640 Introduction to Seismology 28 Jan 2015 © A.R. Lowry 2015 Read for Fri 30 Jan: S&W (§ ) Last time: The Strain Tensor Stress.

CHAPTER OBJECTIVES Define concept of normal strain Define concept of shear strain Determine normal and shear strain in engineering applications 1.

 2005 Pearson Education South Asia Pte Ltd 2. Strain 1 CHAPTER OBJECTIVES Define concept of normal strain Define concept of shear strain Determine normal.

ME 7980 Cardiovascular Biofluid Mechanics

PHYS 1443 – Section 001 Lecture #19

Introduction to Seismology

Chapter 9: Differential Analysis of Fluid Flow

PHYS 1443 – Section 003 Lecture #18

COORDINATE TRANSFORMATION

EXAMPLE 2.1 Rod below is subjected to temperature increase along its axis, creating a normal strain of z = 40(10−3)z1/2, where z is given in meters.

PHYS 1441 – Section 002 Lecture #22

PHYS 1443 – Section 003 Lecture #15

Continuum Mechanics for Hillslopes: Part IV

Continuum Mechanics for Hillslopes: Part V

1st part of paper due 3/10/06 1” margins, 12 pt type, double spaced, ~ 1-2 pages Geologic Setting Location (generally provide geologic location map) Basic.

Concepts of stress and strain

Chapter 2 Strain.

CHAPTER OBJECTIVES Define concept of normal strain

CHAPTER OBJECTIVES Define concept of normal strain

Lab 10: Strain Analysis.

Concepts of stress and strain

CHAPTER OBJECTIVES Define concept of normal strain

PHYS 1443 – Section 003 Lecture #15

Presentation transcript:

Continuum Mechanics for Hillslopes: Part III Today we will focus on Deformation and Strain. Conservation Laws and Constitutive Relations on Thursday. 12/4/2018

Deformation Driven by both body forces and stresses Style and rate of deformation differs based on material properties (liquids, solids, etc.) Deformation described by a ‘displacement field’ Vectors connect positions before and after deformation Rigid-body translation Rigid-body rotation Distortion (strain) 12/4/2018

Normal Strain Elongation of contraction of a displacement vector. 12/4/2018

Normal Strain Displacement of point b can be described as: Displacement of point a PLUS Product of the gradient of displacement and the original line length PLUS An expansion series of higher order terms (using Taylor’s Theorem) 12/4/2018

Normal Strain 12/4/2018

Normal Strain (by definition: the normal component of strain is a change in line length) (note: strain is a dimensionless quantity) 12/4/2018

Normal Strain For infinitesimal strains, can assume only linear relationships matter. Assumption good for strains as large as 0.1% or even 1%. Works for large strains, if considered over short periods of time. 12/4/2018

Normal Strain By definition: positive in elongation. Relates infinitesimal normal strain to the gradient of displacement, along a coordinate direction. Note subscripts: If related: normal If unequal: shear 12/4/2018

Normal Strain (Area) Fractional change in area 12/4/2018

Normal Strain (Area) Calculating the area Of the final region, A1 Substituting the expression last into: 12/4/2018

Normal Strain (Area) and because are <<1, their product is very small. Thus, and in 3 dimensions, dilation is: 12/4/2018

Shear Strain The change in angle between lines that were originally perpendicular. Rotation α1 is positive in ccw direction because produces a displacement in the + y direction. Same for α2. When α1 = α2, this is pure shear 12/4/2018

Shear Strain By the small angle approximation where: 12/4/2018

Shear Strain DEFINING: The average angular change from the original right angle of the elemental area (average shear strain): Plugging in from above: Or: Finding components as symmetric: 12/4/2018

Shear Strain Same derivations can be done for: Many engineering applications use the total shear strain (the sum of the angular changes, α1 + α2), But most geological analyses use the average shear strain. 12/4/2018

Combined normal strain and average shear strain give a strain tensor: Total shear strain would remove the ½’s from the off-diagonal terms. 12/4/2018

Application: Debris Flows 12/4/2018

Rotation 12/4/2018

Rotation 12/4/2018

Rotation 12/4/2018

Strain in Alternate Coordinate Systems 12/4/2018

Strain in Alternate Coordinate Systems 12/4/2018

Rate of Deformation 12/4/2018