Continuum Mechanics and the Finite Element Method

Size: px
Start display at page:

Download "Continuum Mechanics and the Finite Element Method"

Transcription

1 Continuum Mechanics and the Finite Element Method 1

2 Assignment 2 Due on March 2 midnight 2

3 Suppose you want to simulate this

4 The familiar mass-spring system l 0 l y i X y i x Spring length before/after l = x - y i l 0 = X - y i Deformation Measure Elastic Energy Forces f int (x) =

5 Mass Spring Systems Can be used to model arbitrary elastic/plastic objects, but Behavior depends on tessellation Find good spring layout Find good spring constants Different types of springs interfere No direct map to measurable material properties

6 Alternative Start from continuum mechanics Discretize with Finite Elements Decompose model into simple elements Setup & solve system of algebraic equations Advantages Accurate and controllable material behavior Largely independent of mesh structure

7 Mass Spring vs Continuum Mechanics Mass spring systems require: 1. Measure of Deformation 2. Material Model 3. Deformation Energy 4. Internal Forces We need to derive the same types of concepts using continuum mechanics principles

8 Continuum Mechanics: 3D Deformations For a deformable body, identify: 3 undeformed state R described by positions 3 deformed state ' R described by positions Displacement field u describes ' u : ' x X u(x ) in terms of X x z u(x) ' X x u(x ) y x

9 Continuum Mechanics: 3D Deformations Consider material points X 1 and X 2 such that d is infinitesimal, where d X Now consider deformed vector x z X 1 d X 2 d' x y 2 x 1 ( I u)d x 1 d' ' x 2 d' 2 X 1 Deformation gradient F= x X u xu xv xw u y y y v w u z v z w z

10 So Displacement field transforms points x X u(x ) Jacobian of displacement field (deformation gradient) transforms differentials (infinitesimal vectors) from undeformed to deformed d' ( I u) d Fd x F= X

11 Displacement Field and Deformation Gradient In general, displacement field is not explicitly described. Nevertheless, toy examples:

12 Displacement Field and Deformation Gradient In general, displacement field is not explicitly described. Nevertheless, toy examples:

13 Displacement Field and Deformation Gradient In general, displacement field is not explicitly described. Nevertheless, toy examples:

14 Displacement Field and Deformation Gradient In general, displacement field is not explicitly described. Nevertheless, toy examples:

15 Displacement Field and Deformation Gradient In general, displacement field is not explicitly described. Nevertheless, toy examples:

16 Measure of deformations Displacement field transforms points x X u(x ) Jacobian of displacement field (deformation gradient) transforms vectors d' ( I u) d Fd x F= X How can we describe deformations?

17 Back to spring deformation Deformation measure (strain): Similar to F Undeformed spring: = 1 Undeformed (infinitesimal) continuum volume: = I?

18 Strain (description of deformation in terms of relative displacement) Deformation measure (strain): Desirable property: if spring is undeformed, strain is 0 (no change in shape) Can we find a similar measure that would work for infinitesimal volumes?

19 3D Nonlinear Strain Idea: to quantify change in shape, measure change in squared length for any arbitrary vector d I F F d d d d d d d ) ( ' ' ' 2 2 T T T T Green strain ) ( 2 1 I F F E T

20 Strain (description of deformation in terms of relative displacement) Deformation measure (strain): Desirable property: if spring is undeformed, strain is 0 (no change in shape) Can we find a similar measure that would work for infinitesimal volumes? T Green strain E ( F F I) 1 2

21 3D Linear Strain Green strain is quadratic in displacements 2 1 w v w u w w v v u v w u v u u z z y z x y z y y x x z x y x ) ( ) ( u u u u I F F E T T T Neglecting quadratic term (small deformation assumption) leads to the linear Cauchy strain (small strain) Written out: Notation ε = 1 2 u + ut = 1 2 F + Ft I

22 3D Linear Strain Linear Cauchy strain Geometric interpretation y,v x,u i : normal strains i : shear strains 2 1 w v w u w w v v u v w u v u u z z y z x y z y y x x z x y x x y xy

23 Cauchy vs. Green strain Nonlinear Green strain is rotation-invariant Apply incremental rotation R to given deformation F to obtain F = RF Then E' 1 2 ( F' T F' I) E Linear Cauchy strain is not rotation-invariant ε = 1 2 F + F t ε artifacts for larger rotations

24 Mass Spring vs Continuum Mechanics Mass spring systems: 1. Measure of Deformation 2. Material Model 3. Deformation Energy 4. Internal Forces Continuum Mechanics: 1. Measure of Deformation: Green or Cauchy strain 2. Material Model

25 Material Model: linear isotropic material Material model links strain to energy (and stress) Linear isotropic material (generalized Hooke s law) Energy density Ψ = 1 2 λtr(ε)2 +μtr(ε 2 ) Lame parameters λ and μ are material constants related to Poisson Ratio and Young s modulus Interpretation tr ε 2 = ε F 2 penalizes all strain components equally tr ε 2 penalizes dilatations, i.e., volume changes

26 Volumetric Strain (dilatation, hydrostatic strain) Consider a cube with side length a For a given deformation ε, the volumetric strain is ΔV/V0 = (a 1 + ε 11 a 1 + ε 22 a 1 + ε 33 a 3 )/a 3 = ε 11 + ε 22 + ε 33 + O ε 2 tr ε a

27 Linear isotropic material Energy density: Ψ = 1 2 λtr(ε)2 +μtr(ε 2 ) Problem: Cauchy strain is not invariant under rotations artifacts for rotations Solutions: Corotational elasticity Nonlinear elasticity

28 Material Model: non-linear isotropic model Replace Cauchy strain with Green strain St. Venant-Kirchhoff material (StVK) Energy density: Ψ StVK = 1 2 λtr(e)2 +μtr(e 2 ) Rotation invariant!

29 Problems with StVK StVK softens under compression Ψ StVK = 1 2 λtr(e)2 +μtr E 2 Energy Deformation

30 Advanced nonlinear materials Green Strain E = 1 2 Ft F I = 1 (C I) 2 Split into deviatoric (i.e. shape changing/distortion) and volumetric (dilation, volume changing) deformations Volumetric: J = det F Deviatoric: C = det F 2/3 C Neo-Hookean material: Ψ NH = μ tr C I μln(j) + λ ln J Energy

31 Different Models St. Venant-Kirchoff Neo-Hookean Linear

32 Mass Spring vs Continuum Mechanics Mass spring systems: 1. Measure of Deformation 2. Material Model 3. Deformation Energy 4. Internal Forces Continuum Mechanics: 1. Measure of Deformation: Green or Cauchy strain 2. Material Model: linear, StVK, Neo-Hookean, etc 3. From Energy Density to Deformation Energy: Finite Element Discretization

33 Finite Element Discretization Divide domain into discrete elements, e.g., tetrahedra Explicitly store displacement values at nodes (x i ). Displacement field everywhere else obtained through interpolation: x X = N i X x i Deformation Gradient: F = x X = N X i x i i X t

34 ҧ ҧ Basis Functions Basis functions (aka shape functions) N i X j : R 3 R Satisfy delta-property: N i X j = δ ij Simplest choice: linear basis functions N i x, ҧ തy, z ҧ = a i x + b i തy + c i z + d i Compute N i (and N i X ) through X 4 ഥV e X 3 X 1 Ω e X2

35 Constant Strain Elements (Linear Basis Functions) Displacement field is continuous in space Deformation Gradient, strain, stress are not Constant Strain per element Deformation Gradient can be computed as F = ee -1 where e and E are matrices whose columns are edge vectors in undeformed and deformed configurations

36 Constant Strain Elements: From energy density to deformation energy Integrate energy density over the entire element: W e = න Ω e Ψ F If basis functions are linear: F is linear in x i F is constant throughout element: W e = න Ω e Ψ F = Ψ F തV e

37 Mass Spring vs Finite Element Method Mass spring systems: 1. Measure of Deformation 2. Material Model 3. Deformation Energy 4. Internal Forces Continuum Mechanics: 1. Measure of Deformation: Green or Cauchy strain 2. Material Model: linear, StVK, Neo-Hookean, etc 3. Deformation Energy: integrate over elements 4. Internal Forces:

38 FEM recipe Discretize into elements (triangles/tetraderons, etc) For each element Compute deformation gradient F = ee -1 Use material model to define energy density Ψ(F) Integrate over elements to compute energy: W Compute nodal forces as:

39 FEM recipe Area/volume of element f = W Ψ F = V x F x First Piola-Kirchhoff stress tensor P

40 FEM recipe Area/volume of element f = W Ψ F = V x F x First Piola-Kirchhoff stress tensor P

41 FEM recipe For a tetrahedron, this works out to: f 1 f 2 f 3 = VPE T ; f 4 = f 1 f 2 f 3 Additional reading:

42 Material Parameters Lame parameters λ and μ are material constants related to the fundamental physical parameters: Poisson s Ratio and Young s modulus (

43 Young s Modulus and Poisson Ratio Lame parameters λ and μ are material constants related to the fundamental physical parameters: Poisson s Ratio and Young s modulus ( 10 Kg 10 Kg Young s modulus (E), measure of stiffness Poisson s ratio (ν), relative transverse to axial deformation

44 What Do These Parameters Mean Stiffness is pretty intuitive 10 Kg OBJECT E= GPa Pascals = force/area 10 Kg OBJECT E=60 KPa

45 What Do These Parameters Mean Poisson s Ratio controls volume preservation OBJECT 1 1 OBJECT

46 What Do These Parameters Mean Poisson s Ratio controls volume preservation OBJECT 2 OBJECT 1/2

47 What Do These Parameters Mean Poisson s Ratio controls volume preservation OBJECT 2 Poisson s Ratio ( ) =? OBJECT 1/2

48 What Do These Parameters Mean Poisson s Ratio controls volume preservation OBJECT PR = 0.5 Poisson s Ratio ( ) = OBJECT PR = 0.5 1/2

49 What Do These Parameters Mean Poisson s ratio is between -1 and 0.5 OBJECT PR = 0.5 Poisson s Ratio ( ) = OBJECT PR =

50 Negative Poisson s Ratio

51 Measurement Where do material parameters come from?

52 Simple Measurement: Stiffness OBJECT

53 Simple Measurement What s the Force (Stress)? What s the Deformation (Strain)? 1kg OBJECT

54 Simple Measurement 1kg OBJECT

55 Simple Measurement 2kg OBJECT

56 Simple Measurement 3kg OBJECT

57 Simple Measurement 7kg OBJECT

58 Simple Measurement How do we get the stiffness? 7kg OBJECT

59 Simple Measurement How do we get the stiffness? 7kg OBJECT

60 Simple Measurement How do we get the stiffness? 7kg Stiffness OBJECT

61 Simple Measurement: Poisson s Ratio OBJECT

62 Simple Measurement: Poisson s Ratio OBJECT

63 Simple Measurement: Poisson s Ratio Compute changes in width and height OBJECT

64 Simple Measurement: Poisson s Ratio Poisson s Ratio OBJECT

65 Measurement Devices

66 Simulating Elastic Materials with CM+FEM You now have all the mathematical tools you need

67 Suppose you want to simulate this

68 Plastic and Elastic Materials Elastic Materials Objects return to their original shape in the absence of other forces Plastic Deformations: Object does not always return to its original shape

69 Example: Crushing a Coke Can Old Reference State New Reference State

70 Example: Crushing a van

71 New Reference State A Simple Model For Plasticity Recall our model for strain: Let s consider how to encode a change of reference shape into this metric Changing undeformed mesh is not easy! We want to exchange F with, a deformation gradient that takes into account the new shape of our object

72 Continuum Mechanics: Deformation deformation gradient maps undeformed vectors (local) to deformed (world) vectors Reference Space World Space

73 Continuum Mechanics: Deformation deformation gradient maps undeformed vectors (reference) to deformed (world) vectors Reference Space World Space

74 Continuum Mechanics: Deformation F transforms a vector from Reference space to World Space Reference Space World Space

75 Continuum Mechanics: Deformation Introduce a new space Reference Space Plastic Space World Space

76 Continuum Mechanics: Deformation Our goal is to use access to but we only have Reference Space Plastic Space World Space

77 Continuum Mechanics: Deformation Our goal is to use access to but we only have Reference Space Plastic Space World Space

78 Continuum Mechanics: Deformation Our goal is to use access to but we only have Keep an estimate of built incrementally per element,

79 How to Compute the Plastic Deformation Gradient We compute the strain/stress for each element during simulation When it gets above a certain threshold store F as

80 How to Compute the Plastic Deformation Gradient We compute the strain/stress for each element during simulation When it gets above a certain threshold store F as

81 How to Compute the Plastic Deformation Gradient Each subsequent simulation step uses

82 So now you too can simulate this

83 Questions?

84 93

Inverse 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 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 information

Measurement of deformation. Measurement of elastic force. Constitutive law. Finite element method

Measurement 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 information

Constitutive models. Constitutive model: determines P in terms of deformation

Constitutive models. Constitutive model: determines P in terms of deformation Constitutive models Constitutive model: determines P in terms of deformation Elastic material: P depends only on current F Hyperelastic material: work is independent of path strain energy density function

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

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost

Game Physics. Game and Media Technology Master Program - Utrecht University. Dr. Nicolas Pronost Game and Media Technology Master Program - Utrecht University Dr. Nicolas Pronost Soft body physics Soft bodies In reality, objects are not purely rigid for some it is a good approximation but if you hit

More information

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

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

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

Fundamentals of Linear Elasticity

Fundamentals 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 information

Surface force on a volume element.

Surface 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 information

Soft Bodies. Good approximation for hard ones. approximation breaks when objects break, or deform. Generalization: soft (deformable) bodies

Soft 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 information

Continuum mechanics V. Constitutive equations. 1. Constitutive equation: definition and basic axioms

Continuum 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 information

Constitutive Equations

Constitutive 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 information

Performance Evaluation of Various Smoothed Finite Element Methods with Tetrahedral Elements in Large Deformation Dynamic Analysis

Performance Evaluation of Various Smoothed Finite Element Methods with Tetrahedral Elements in Large Deformation Dynamic Analysis Performance Evaluation of Various Smoothed Finite Element Methods with Tetrahedral Elements in Large Deformation Dynamic Analysis Ryoya IIDA, Yuki ONISHI, Kenji AMAYA Tokyo Institute of Technology, Japan

More information

MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4

MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4 MITOCW MITRES2_002S10nonlinear_lec15_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources

More information

A Numerical Study of Finite Element Calculations for Incompressible Materials under Applied Boundary Displacements

A Numerical Study of Finite Element Calculations for Incompressible Materials under Applied Boundary Displacements A Numerical Study of Finite Element Calculations for Incompressible Materials under Applied Boundary Displacements A Thesis Submitted to the College of Graduate Studies and Research in Partial Fulfillment

More information

Nonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess

Nonlinear Theory of Elasticity. Dr.-Ing. Martin Ruess Nonlinear Theory of Elasticity Dr.-Ing. Martin Ruess geometry description Cartesian global coordinate system with base vectors of the Euclidian space orthonormal basis origin O point P domain of a deformable

More information

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February.

Engineering Sciences 241 Advanced Elasticity, Spring Distributed Thursday 8 February. Engineering Sciences 241 Advanced Elasticity, Spring 2001 J. R. Rice Homework Problems / Class Notes Mechanics of finite deformation (list of references at end) Distributed Thursday 8 February. Problems

More information

SEMM Mechanics PhD Preliminary Exam Spring Consider a two-dimensional rigid motion, whose displacement field is given by

SEMM 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 information

Mathematical Background

Mathematical 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 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

CIVL4332 L1 Introduction to Finite Element Method

CIVL4332 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 information

1 Nonlinear deformation

1 Nonlinear deformation NONLINEAR TRUSS 1 Nonlinear deformation When deformation and/or rotation of the truss are large, various strains and stresses can be defined and related by material laws. The material behavior can be expected

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

1.050 Engineering Mechanics. Lecture 22: Isotropic elasticity

1.050 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 information

Computational models of diamond anvil cell compression

Computational models of diamond anvil cell compression UDC 519.6 Computational models of diamond anvil cell compression A. I. Kondrat yev Independent Researcher, 5944 St. Alban Road, Pensacola, Florida 32503, USA Abstract. Diamond anvil cells (DAC) are extensively

More information

Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity

Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity Lecture #6: 3D Rate-independent Plasticity (cont.) Pressure-dependent plasticity by Borja Erice and Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling

More information

4 NON-LINEAR ANALYSIS

4 NON-LINEAR ANALYSIS 4 NON-INEAR ANAYSIS arge displacement elasticity theory, principle of virtual work arge displacement FEA with solid, thin slab, and bar models Virtual work density of internal forces revisited 4-1 SOURCES

More information

Linear Elasticity ( ) Objectives. Equipment. Introduction. ε is then

Linear Elasticity ( ) Objectives. Equipment. Introduction. ε is then Linear Elasticity Objectives In this lab you will measure the Young s Modulus of a steel wire. In the process, you will gain an understanding of the concepts of stress and strain. Equipment Young s Modulus

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

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

The Finite Element Method II

The Finite Element Method II [ 1 The Finite Element Method II Non-Linear finite element Use of Constitutive Relations Xinghong LIU Phd student 02.11.2007 [ 2 Finite element equilibrium equations: kinematic variables Displacement Strain-displacement

More information

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading

EMA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science (Mechanics of Materials) Chapter 2 Stress & Strain - Axial Loading MA 3702 Mechanics & Materials Science Zhe Cheng (2018) 2 Stress & Strain - Axial Loading Statics

More information

Mechanics of Biomaterials

Mechanics 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 information

Law of behavior very-rubber band: almost incompressible material

Law of behavior very-rubber band: almost incompressible material Titre : Loi de comportement hyperélastique : matériau pres[...] Date : 25/09/2013 Page : 1/9 Law of behavior very-rubber band: almost incompressible material Summary: One describes here the formulation

More information

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document

Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis. Supplementary Document Classification of Prostate Cancer Grades and T-Stages based on Tissue Elasticity Using Medical Image Analysis Supplementary Document Shan Yang, Vladimir Jojic, Jun Lian, Ronald Chen, Hongtu Zhu, Ming C.

More information

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

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

More information

INVERSE ANALYSIS METHODS OF IDENTIFYING CRUSTAL CHARACTERISTICS USING GPS ARRYA DATA

INVERSE ANALYSIS METHODS OF IDENTIFYING CRUSTAL CHARACTERISTICS USING GPS ARRYA DATA Problems in Solid Mechanics A Symposium in Honor of H.D. Bui Symi, Greece, July 3-8, 6 INVERSE ANALYSIS METHODS OF IDENTIFYING CRUSTAL CHARACTERISTICS USING GPS ARRYA DATA M. HORI (Earthquake Research

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

NONLINEAR CONTINUUM FORMULATIONS CONTENTS

NONLINEAR CONTINUUM FORMULATIONS CONTENTS NONLINEAR CONTINUUM FORMULATIONS CONTENTS Introduction to nonlinear continuum mechanics Descriptions of motion Measures of stresses and strains Updated and Total Lagrangian formulations Continuum shell

More information

UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES

UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the

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

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

CRITERIA FOR SELECTION OF FEM MODELS.

CRITERIA FOR SELECTION OF FEM MODELS. CRITERIA FOR SELECTION OF FEM MODELS. Prof. P. C.Vasani,Applied Mechanics Department, L. D. College of Engineering,Ahmedabad- 380015 Ph.(079) 7486320 [R] E-mail:pcv-im@eth.net 1. Criteria for Convergence.

More information

Lecture contents. Stress and strain Deformation potential. NNSE 618 Lecture #23

Lecture contents. Stress and strain Deformation potential. NNSE 618 Lecture #23 1 Lecture contents Stress and strain Deformation potential Few concepts from linear elasticity theory : Stress and Strain 6 independent components 2 Stress = force/area ( 3x3 symmetric tensor! ) ij ji

More information

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam MODELING OF CONCRETE MATERIALS AND STRUCTURES Class Meeting #1: Fundamentals Kaspar Willam University of Colorado at Boulder Notation: Direct and indicial tensor formulations Fundamentals: Stress and Strain

More information

MITOCW MITRES2_002S10nonlinear_lec05_300k-mp4

MITOCW MITRES2_002S10nonlinear_lec05_300k-mp4 MITOCW MITRES2_002S10nonlinear_lec05_300k-mp4 The following content is provided under a Creative Commons license. Your support will help MIT OpenCourseWare continue to offer high quality educational resources

More information

Elements of Rock Mechanics

Elements of Rock Mechanics Elements of Rock Mechanics Stress and strain Creep Constitutive equation Hooke's law Empirical relations Effects of porosity and fluids Anelasticity and viscoelasticity Reading: Shearer, 3 Stress Consider

More information

MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS

MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS 1 MECHANICS OF MATERIALS. EQUATIONS AND THEOREMS Version 2011-01-14 Stress tensor Definition of traction vector (1) Cauchy theorem (2) Equilibrium (3) Invariants (4) (5) (6) or, written in terms of principal

More information

International Journal of Pure and Applied Mathematics Volume 58 No ,

International Journal of Pure and Applied Mathematics Volume 58 No , International Journal of Pure and Applied Mathematics Volume 58 No. 2 2010, 195-208 A NOTE ON THE LINEARIZED FINITE THEORY OF ELASTICITY Maria Luisa Tonon Department of Mathematics University of Turin

More information

BAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS

BAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS Journal of Computational and Applied Mechanics, Vol.., No. 1., (2005), pp. 83 94 BAR ELEMENT WITH VARIATION OF CROSS-SECTION FOR GEOMETRIC NON-LINEAR ANALYSIS Vladimír Kutiš and Justín Murín Department

More information

Comparison of Models for Finite Plasticity

Comparison of Models for Finite Plasticity Comparison of Models for Finite Plasticity A numerical study Patrizio Neff and Christian Wieners California Institute of Technology (Universität Darmstadt) Universität Augsburg (Universität Heidelberg)

More information

Conservation of mass. Continuum Mechanics. Conservation of Momentum. Cauchy s Fundamental Postulate. # f body

Conservation of mass. Continuum Mechanics. Conservation of Momentum. Cauchy s Fundamental Postulate. # f body Continuum Mechanics We ll stick with the Lagrangian viewpoint for now Let s look at a deformable object World space: points x in the object as we see it Object space (or rest pose): points p in some reference

More information

Toward a novel approach for damage identification and health monitoring of bridge structures

Toward a novel approach for damage identification and health monitoring of bridge structures Toward a novel approach for damage identification and health monitoring of bridge structures Paolo Martino Calvi 1, Paolo Venini 1 1 Department of Structural Mechanics, University of Pavia, Italy E-mail:

More information

Chapter 3: Stress and Equilibrium of Deformable Bodies

Chapter 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 information

Other state variables include the temperature, θ, and the entropy, S, which are defined below.

Other state variables include the temperature, θ, and the entropy, S, which are defined below. Chapter 3 Thermodynamics In order to complete the formulation we need to express the stress tensor T and the heat-flux vector q in terms of other variables. These expressions are known as constitutive

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

Stress analysis of a stepped bar

Stress analysis of a stepped bar Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.

More information

Constitutive Relations

Constitutive Relations Constitutive Relations Dr. Andri Andriyana Centre de Mise en Forme des Matériaux, CEMEF UMR CNRS 7635 École des Mines de Paris, 06904 Sophia Antipolis, France Spring, 2008 Outline Outline 1 Review of field

More information

Deformable Materials 2 Adrien Treuille. source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes.

Deformable Materials 2 Adrien Treuille. source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes. Deformable Materials 2 Adrien Treuille source: Müller, Stam, James, Thürey. Real-Time Physics Class Notes. Goal Overview Strain (Recap) Stress From Strain to Stress Discretization Simulation Overview Strain

More information

Analytical Mechanics: Elastic Deformation

Analytical Mechanics: Elastic Deformation Analytical Mechanics: Elastic Deformation Shinichi Hirai Dept. Robotics, Ritsumeikan Univ. Shinichi Hirai (Dept. Robotics, Ritsumeikan Univ.) Analytical Mechanics: Elastic Deformation 1 / 60 Agenda Agenda

More information

Glossary. Glossary of Symbols. Glossary of Roman Symbols Glossary of Greek Symbols. Contents:

Glossary. Glossary of Symbols. Glossary of Roman Symbols Glossary of Greek Symbols. Contents: Glossary Glossary of Symbols Contents: Glossary of Roman Symbols Glossary of Greek Symbols Glossary G-l Glossary of Roman Symbols The Euclidean norm or "two-norm." For a vector a The Mooney-Rivlin material

More information

1 Useful Definitions or Concepts

1 Useful Definitions or Concepts 1 Useful Definitions or Concepts 1.1 Elastic constitutive laws One general type of elastic material model is the one called Cauchy elastic material, which depend on only the current local deformation of

More information

University of Sheffield The development of finite elements for 3D structural analysis in fire

University of Sheffield The development of finite elements for 3D structural analysis in fire The development of finite elements for 3D structural analysis in fire Chaoming Yu, I. W. Burgess, Z. Huang, R. J. Plank Department of Civil and Structural Engineering StiFF 05/09/2006 3D composite structures

More information

A new closed-form model for isotropic elastic sphere including new solutions for the free vibrations problem

A new closed-form model for isotropic elastic sphere including new solutions for the free vibrations problem A new closed-form model for isotropic elastic sphere including new solutions for the free vibrations problem E Hanukah Faculty of Mechanical Engineering, Technion Israel Institute of Technology, Haifa

More information

You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.

You 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 information

Fig. 1. Circular fiber and interphase between the fiber and the matrix.

Fig. 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 information

Understand basic stress-strain response of engineering materials.

Understand 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 information

Linear Cosserat elasticity, conformal curvature and bounded stiffness

Linear Cosserat elasticity, conformal curvature and bounded stiffness 1 Linear Cosserat elasticity, conformal curvature and bounded stiffness Patrizio Neff, Jena Jeong Chair of Nonlinear Analysis & Modelling, Uni Dui.-Essen Ecole Speciale des Travaux Publics, Cachan, Paris

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

A short review of continuum mechanics

A 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 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

Lectures on. Constitutive Modelling of Arteries. Ray Ogden

Lectures on. Constitutive Modelling of Arteries. Ray Ogden Lectures on Constitutive Modelling of Arteries Ray Ogden University of Aberdeen Xi an Jiaotong University April 2011 Overview of the Ingredients of Continuum Mechanics needed in Soft Tissue Biomechanics

More information

Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials

Concept Question Comment on the general features of the stress-strain response under this loading condition for both types of materials Module 5 Material failure Learning Objectives review the basic characteristics of the uni-axial stress-strain curves of ductile and brittle materials understand the need to develop failure criteria for

More information

FEM for elastic-plastic problems

FEM for elastic-plastic problems FEM for elastic-plastic problems Jerzy Pamin e-mail: JPamin@L5.pk.edu.pl With thanks to: P. Mika, A. Winnicki, A. Wosatko TNO DIANA http://www.tnodiana.com FEAP http://www.ce.berkeley.edu/feap Lecture

More information

2.1 Strain energy functions for incompressible materials

2.1 Strain energy functions for incompressible materials Chapter 2 Strain energy functions The aims of constitutive theories are to develop mathematical models for representing the real behavior of matter, to determine the material response and in general, to

More information

1. Background. is usually significantly lower than it is in uniaxial tension

1. Background. is usually significantly lower than it is in uniaxial tension NOTES ON QUANTIFYING MODES OF A SECOND- ORDER TENSOR. The mechanical behavior of rocks and rock-like materials (concrete, ceramics, etc.) strongly depends on the loading mode, defined by the values and

More information

CONSTITUTIVE RELATIONS FOR LINEAR ELASTIC SOLIDS

CONSTITUTIVE 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 information

DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS

DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS DEVELOPMENT OF A CONTINUUM PLASTICITY MODEL FOR THE COMMERCIAL FINITE ELEMENT CODE ABAQUS Mohsen Safaei, Wim De Waele Ghent University, Laboratory Soete, Belgium Abstract The present work relates to the

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

Hervé DELINGETTE. INRIA Sophia-Antipolis

Hervé DELINGETTE. INRIA Sophia-Antipolis Hervé DELINGETTE INRIA Sophia-Antipolis French National Research Institute in Computer Science and Automatic Control Asclepios : 3D Segmentation, Simulation Platform, Soft Tissue Modeling http://www.inria.fr/asclepios

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

COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS

COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS COMPUTATIONAL MODELING OF SHAPE MEMORY MATERIALS Jan Valdman Institute of Information Theory and Automation, Czech Academy of Sciences (Prague) based on joint works with Martin Kružík and Miroslav Frost

More information

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations

MODELING OF CONCRETE MATERIALS AND STRUCTURES. Kaspar Willam. Isotropic Elastic Models: Invariant vs Principal Formulations MODELING OF CONCRETE MATERIALS AND STRUCTURES Kaspar Willam University of Colorado at Boulder Class Meeting #2: Nonlinear Elastic Models Isotropic Elastic Models: Invariant vs Principal Formulations Elastic

More information

16.21 Techniques of Structural Analysis and Design Spring 2003 Unit #5 - Constitutive Equations

16.21 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 information

Discrete Analysis for Plate Bending Problems by Using Hybrid-type Penalty Method

Discrete Analysis for Plate Bending Problems by Using Hybrid-type Penalty Method 131 Bulletin of Research Center for Computing and Multimedia Studies, Hosei University, 21 (2008) Published online (http://hdl.handle.net/10114/1532) Discrete Analysis for Plate Bending Problems by Using

More information

3D Stress Tensors. 3D Stress Tensors, Eigenvalues and Rotations

3D 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 information

NORMAL 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. 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 information

. D CR Nomenclature D 1

. D CR Nomenclature D 1 . D CR Nomenclature D 1 Appendix D: CR NOMENCLATURE D 2 The notation used by different investigators working in CR formulations has not coalesced, since the topic is in flux. This Appendix identifies the

More information

Two problems in finite elasticity

Two problems in finite elasticity University of Wollongong Research Online University of Wollongong Thesis Collection 1954-2016 University of Wollongong Thesis Collections 2009 Two problems in finite elasticity Himanshuki Nilmini Padukka

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

Transactions on Engineering Sciences vol 6, 1994 WIT Press, ISSN

Transactions on Engineering Sciences vol 6, 1994 WIT Press,   ISSN Large strain FE-analyses of localized failure in snow C.H. Liu, G. Meschke, H.A. Mang Institute for Strength of Materials, Technical University of Vienna, A-1040 Karlsplatz 13/202, Vienna, Austria ABSTRACT

More information

Chapter 2. Rubber Elasticity:

Chapter 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 information

Analysis of thin plate structures using the absolute nodal coordinate formulation

Analysis of thin plate structures using the absolute nodal coordinate formulation 345 Analysis of thin plate structures using the absolute nodal coordinate formulation K Dufva 1 and A A Shabana 2 1 Department of Mechanical Engineering, Lappeenranta University of echnology, Lappeenranta,

More information

Bake, shake or break - and other applications for the FEM. 5: Do real-life experimentation using your FEM code

Bake, shake or break - and other applications for the FEM. 5: Do real-life experimentation using your FEM code Bake, shake or break - and other applications for the FEM Programming project in TMA4220 - part 2 by Kjetil André Johannessen TMA4220 - Numerical solution of partial differential equations using the finite

More information

What we should know about mechanics of materials

What we should know about mechanics of materials What we should know about mechanics of materials 0 John Maloney Van Vliet Group / Laboratory for Material Chemomechanics Department of Materials Science and Engineering Massachusetts Institute of Technology

More information

Load Cell Design Using COMSOL Multiphysics

Load 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 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

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

Simulation of elasticity, biomechanics and virtual surgery. Joseph M. Teran

Simulation of elasticity, biomechanics and virtual surgery. Joseph M. Teran Contents Simulation of elasticity, biomechanics and virtual surgery Joseph M. Teran 1 Simulation of elasticity, biomechanics and virtual surgery 3 Introduction 3 Real-time computing 4 Continuum mechanics

More information