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Lecture 1Introduction to Theory of elasticityand plasticityRules of the gamePrint version Lecture on Theory of Elasticity and Plasticity of

Dr. D. Dinev, Department of Structural Mechanics, UACEG

1.1

Contents

1 Introduction 11.1 Elasticity and plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Overview of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Course organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Mathematical preliminaries 62.1 Scalars, vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Kronecker delta and alternating symbol . . . . . . . . . . . . . . . . . . . . . . 82.4 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.5 Cartesian tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.6 Principal values and directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.7 Vector and tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.8 Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2

1 Introduction

1.1 Elasticity and plasticity

Introduction

Elasticity and plasticity What is the Theory of elasticity (TE)?

Branch of physics which deals with calculation of the deformation of solid bodies inequilibrium of applied forces

Theory of elasticity treats explicitly a linear or nonlinear response of structure toloading

What do we mean by a solid body? A solid body can sustain shear Body is and remains continuous during the deformation- neglecting its atomic struc-

ture, the body consists of continuous material points (we can infinitely zoom-inand still see numerous material points)

What does the modern TE deal with? Lab experiments- strain measurements, photoelasticity, fatigue, material description Theory- continuum mechanics, micromechanics, constitutive modeling Computation- finite elements, boundary elements, molecular mechanics

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1

Introduction

Elasticity and plasticity Which problems does the TE study?

All problems considering 2- or 3-dimensional formulation1.4

Introduction

Elasticity and plasticity Shell structures

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Introduction

Elasticity and plasticity Plate structures

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2

Introduction

Elasticity and plasticity

Disc structures (walls)1.7

Introduction

Mechanics of Materials (MoM)

Makes plausible but unsubstantial assumptions Most of the assumptions have a physical nature Deals mostly with ordinary differential equations Solve the complicated problems by coefficients from tables (i.e. stress concentration fac-

tors)

Elasticity and plasticity

More precise treatment Makes mathematical assumptions to help solve the equations Deals mostly with partial differential equations Allows us to assess the quality of the MoM-assumptions Uses more advanced mathematical tools- tensors, PDE, numerical solutions

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1.2 Overview of the course

Introduction

Overview of the course

Topics in this class Stress and relation with the internal forces Deformation and strain Equilibrium and compatibility Material behavior Elasticity problem formulation Energy principles 2-D formulation Finite element method Plate analysis

3

Shell theory Plasticity

Note A lot of mathematics Few videos and pictures

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Introduction

Overview of the course Textbooks

Elasticity theory, applications, and numerics, Martin H. Sadd, 2nd edition, Elsevier2009

Energy principles and variational methods in applied mechanics, J. N. Reddy, JohnWiley & Sons 2002

Fundamental finite element analysis and applications, M. Asghar Bhatti, John Wiley& Sons 2005

Theories and applications of plate analysis, Rudolph Szilard, John Wiley & Sons2004

Thin plates and shells, E. Ventsel and T. Krauthammer, Marcel Dekker 20011.10

Introduction

Overview of the course Other references

Elasticity in engineering mechanics, A. Boresi, K. Chong and J. Lee, John Wiley &Sons, 2011

Elasticity, J. R. Barber, 2nd edition, Kluwer academic publishers, 2004 Engineering elasticity, R. T. Fenner, Ellis Horwood Ltd, 1986 Advanced strength and applied elasticity, A. Ugural and S. Fenster, Prentice hall,

2003

Introduction to finite element method, C.A. Felippa, lecture notes, University of Col-orado at Boulder

Lecture handouts from different universities around the world1.11

1.3 Course organization

Introduction

Course organization Lecture notes- posted on a web-site: http://uacg.bg/?p=178&l=2&id=151&f=2&dp=23 Instructor

Dr. D. Dinev- Room 514, E-mail: [email protected] Teaching assistant

M. Ivanova Office hours

Instructor: . . . . . . . . . . . . TA: . . . . . . . . . . . .

Note For other time by appointment

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Introduction

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40 50 60 70 80 90 100

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Points

Course organization

Grading1.13

Introduction

Course organization

Grading is based on Homework- 15% Two mid-term exams- 50% Final exam- 35%

Participation Class will be taught with a mixture of lecture and student participation Class participation and attendance are expected of all students In-class discussions will be more valuable to you if you read the relevant sections

of the textbook before the class time1.14

Introduction

Course organization

Homeworks Homework is due at the beginning of the Thursday lectures The assigned problems for the HWs will be announced via web-site

Late homework policy Late homework will not be accepted and graded

Team work You are encouraged to discuss HW and class material with the instructor, the TAs

and your classmates

However, the submitted individual HW solutions and exams must involve only youreffort

Otherwise youll have terrible performance on the exam since you did not learn tothink for yourself

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2 Mathematical preliminaries

2.1 Scalars, vectors and tensors

Mathematical preliminaries

Scalars, vectors and tensor definitions Scalar quantities- represent a single magnitude at each point in space

Mass density- Temperature- T

Vector quantities- represent variables which are expressible in terms of components in a2-D or 3-D coordinate system

Displacement- u = ue1 + ve2 +we3where e1, e2 and e3 are unit basis vectors in the coordinate system Matrix quantities- represent variables which require more than three components to quan-

tify

Stress matrix

=

xx xy xzyx yy yzzx zy zz

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2.2 Index notation


Index notation Index notation is a shorthand scheme where a set of numbers is represented by a single

symbol with subscripts

ai =

a1a2a3

, ai j = a11 a12 a13a21 a22 a23

a31 a32 a33

a1 j first row ai1 first column

Addition and subtraction

aibi = a1b1a2b2

a3b3

ai jbi j =

a11b11 a12b12 a13b13a21b21 a22b22 a23b23a31b31 a32b32 a33b33

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Index notation Scalar multiplication

ai =

a1a2a3

, ai j = a11 a12 a13a21 a22 a23

a31 a32 a33

Outer multiplication (product)

aib j =

a1b1 a1b2 a1b3a2b1 a2b2 a2b3a3b1 a3b2 a3b3

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Index notation

Commutative, associative and distributive laws

ai +bi = bi +aiai jbk = bkai jai +(bi + ci) = (ai +bi)+ ciai(b jkc`) = (aib jk)c`ai j(bk + ck) = ai jbk +ai jck

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Index notation

Summation convention (Einsteins convention)- if a subscript appears twice in the sameterm, then summation over that subscript from one to three is implied

aii =3

i=1

aii = a11 +a22 +a33

ai jb j =3

j=1

ai jb j = ai1b1 +ai2b2 +ai3b3

j- dummy index subscript which is repeated into the notation (one side of theequation)

i- free index subscript which is not repeated into the notation1.20


Index notation- example

The matrix ai j and vector bi are

ai j =

1 2 00 4 32 1 2

, bi = 24

0

Determine the following quantities

aii = . . . ai jai j = . . . ai jb j = . . . ai ja jk = . . . ai jbib j = . . . bibi = . . . bib j = . . . Unsymmetric matrix decomposition

ai j =12(ai j +a ji) symmetric

+12(ai ja ji)

antisymmetric

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2.3 Kronecker delta and alternating symbol


Kronecker delta and alternating symbol Kronecker delta is defined as

i j ={

1 if i = j0 if i 6= j =

1 0 00 1 00 0 1

Properties of i j

i j = jiii = 3

i ja j =

11a1 +12a2 +13a3 = a1. . .. . .

= ai

i ja jk = aiki jai j = aiii ji j = 3

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Kronecker delta and alternating symbol Alternating (permutation) symbol is defined as

i jk =

+1 if i jk is an even permutation of 1,2,31 if i jk is an odd permutation of 1,2,30 otherwise Therefore

123 = 231 = 312 = 1321 = 132 = 213 =1112 = 131 = 222 = . . .= 0

Matrix determinant

det(ai j) = |ai j|=

a11 a12 a13a21 a22 a23a31 a32 a33

= i jka1ia2 ja3k = i jkai1a j2ak31.23

2.4 Coordinate transformations


Coordinate transformations

8

Consider two Cartesian coordinate systems with different orientation and basis vectors1.24



The basis vectors for the old (unprimed) and the new (primed) coordinate systems are

ei =

e1e2e3

, ei = e1e2

e3

Let Ni j denotes the cosine of the angle between xi-axis and x j-axis

Ni j = ei e j = cos(xi,x j)

The primed base vectors can be expressed in terms of those in the unprimed by relations

e1 = N11e1 +N12e2 +N13e3e2 = N21e1 +N22e2 +N23e3e3 = N31e1 +N32e2 +N33e3

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In matrix form

ei = Ni je jei = N jiej

An arbitrary vector can be written as

v = v1e1 + v2e2 + v3e3 = viei= v1e

1 + v

2e2 + v

3e3 = v

iei

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Or

v = viN jiej

Because v = vjej thus

vj = N jivi

Similarly

vi = Ni jvj

These relations constitute the transformation law for the Cartesian components of a vectorunder a change of orthogonal Cartesian coordinate system

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2.5 Cartesian tensors


Cartesian tensors General index notation scheme

a = a, zero order (scalar)ai = Nipap, first order (vector)ai j = NipN jqapq, second order (matrix)

ai jk = NipN jqNkrapqr, third order

. . .

A tensor is a generalization of the above mentioned quantities

Example The notation vi = Ni jv j is a relationship between two vectors which are transformed to

each other by a tensor (coordinate transformation). The multiplication of a vector by atensor results another vector (linear mapping).

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Cartesian tensors All second order tensors can be presented in matrix form

Ni j =

N11 N12 N13N21 N22 N23N31 N32 N33

Since Ni j can be presented as a matrix, all matrix operation for 33-matrix are valid The difference between a matrix and a tensor

We can multiply the three components of a vector vi by any 33-matrix The resulting three numbers (v1,v

2v3) may or may not represent the vector compo-

nents If they are the vector components, then the matrix represents the components of a

tensor Ni j If not, then the matrix is just an ordinary old matrix

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Cartesian tensors The second order tensor can be created by a dyadic (tensor or outer) product of the two

vectors v and v

N = vv = v1v1 v1v2 v1v3v2v1 v2v2 v2v3

v3v1 v3v2 v

3v3

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Transformation example The components of a first and a second order tensor in a particular coordinate frame are

given by

bi =

142

, ai j = 1 0 20 2 2

3 2 4

Determine the components of each tensor in a new coordinates found through a rotation of

60 about the x3-axis1.31

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Transformation example The rotation matrix is

Ni j = cos(xi,x j) = . . .

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Transformation example The transformation of the vector bi is

bi = Ni jb j = Mb = . . .

The second order tensor transformation isai j = NipN jpapq = NaN

T = . . .

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2.6 Principal values and directions


Principal values and directions for symmetric tensor The tensor transformation shows that there is a coordinate system in which the components

of the tensor take on maximum or minimum values If we choose a particular coordinate system that has been rotated so that the x3-axis lies

along the vector, then vector will have components

v =

00|v|

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Principal values and directions for symmetric tensor

Every tensor can be regarded as a transformation of one vector into another vector It is of interest to inquire there are certain vectors n which are transformed by a given

tensor A into multiples of themselves but scaled with some factors If such vectors exist they must satisfy the equation

A n = n, Ai jn j = ni Such vectors n are called eigenvectors of A The parameter is called eigenvalue and characterizes the change in length of the eigen-

vector n The above equation can be written as

(A I) n = 0, (Ai ji j)n j = 01.35



Because this is a homogeneous set of equations for n, a nontrivial solution will not existunless the determinant of the matrix (. . .) vanishes

det(A I) = 0, det(Ai ji j) = 0

Expanding the determinant produces a characteristic equation in terms of

3 + IA 2 IIA + IIIA = 01.36



The IA, IIA and IIIA are called the fundamental invariants of the tensor

IA = tr(A) = Aii = A11 +A22 +A33

IIA =12(tr(A)2 tr(A2))= 1

2(AiiA j jAi jAi j)

=

A11 A12A21 A22+ A22 A23A32 A33

+ A11 A13A31 A33

IIIA = det(A) = det(Ai j)

The roots of the characteristic equation determine the values for and each of these maybe back-substituted into (A I) n = 0 to solve for the associated principle directions n.

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Example

Determine the invariants and principal values and directions of the following tensor:

A =

3 1 11 0 21 2 0

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2.7 Vector and tensor algebra


Vector and tensor algebra Scalar product (dot product, inner product)

a b = |a||b|cos Magnitude of a vector

|a|= (a a)1/2

Vector product (cross-product)

ab = det e1 e2 e3a1 a2 a3

b1 b2 b3

Vector-matrix products

Aa = Ai ja j = a jAi jaT A = aiAi j = Ai jai

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Vector and tensor algebra Matrix-matrix products

AB = Ai jB jkABT = Ai jBk jAT B = A jiB jktr(AB) = Ai jB jitr(ABT ) = tr(AT B) = Ai jBi j

where ATi j = A ji and tr(A) = Aii = A11 +A22 +A331.40

2.8 Tensor calculus


Tensor calculus Common tensors used in field equations

a = a(x,y,z) = a(xi) = a(x) scalarai = ai(x,y,z) = ai(xi) = ai(x)vectorai j = ai j(x,y,z) = ai j(xi) = ai j(x) tensor

Comma notations for partial differentiation

a,i =xi

a

ai, j =x j

ai

ai j,k =xk

ai j

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Tensor calculus

Directional derivative Consider a scalar function . Find the derivative of the with respect of direction s

dds

=x

dxds

+y

dyds

+ z

dzds

The above expression can be presented as a dot product between two vectors

dds

=[

dxds

dyds

dzds

]xy z

= n The first vector represents the unit vector in the direction of s

n =dxds

e1 +dyds

e2 +dzds

e3

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Tensor calculus

Directional derivative The second vector is called the gradient of the scalar function and is defined by

= e1x

+ e2y

+ e3 z

The symbolic operator is called del operator (nabla operator) and is defined as

= e1x

+ e2y

+ e3 z

The operator 2 is called Laplacian operator and is defined as

2 = 2

x2+

2

y2+

2

z2

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Tensor calculus Common differential operations and similarities with multiplications

Name Operation Similarities OrderGradient of a scalar u vector Gradient of a vector u = ui, jeie j uv tensor

Divergence of a vector u = ui, j u v dot Curl of a vector u = i jkuk, jei uv cross

Laplacian of a vector 2u = u = ui,kkei

NoteThe -operator is a vector quantity

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-1.0

-0.5

0.0

0.5

1.0 -1.0

-0.5

0.0

0.5

1.0

-1.0

-0.5

0.0

0.5

1.0

Tensor calculus- example

A scalar field is presented as = x2 y21.45


-1

0

1

-1

0

1

-1

0

1


A vector field is u = 2xe1 +3yze2 + xye31.46


15

-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0

Tensor calculus- example Gradient of the scalar field is

= . . .1.47


Tensor calculus- example Laplacian of a scalar

2 = = . . . Divergence of a vector

u = . . . Gradient of a vector

u = . . .1.48


-1.0 -0.5 0.0 0.5 1.0

-1.0

-0.5

0.0

0.5

1.0


16

Curl of a vector

u = det e1 e2 e3

xy

z

2x 3yz xy

= . . .1.49


Tensor calculus

Gradient theorem S

n dS =

V dV

Divergence (Gauss) theorem S

u ndS =

V udV

Curl theorem S

undS =

VudV

where n is the outward normal vector to the surface S and V is the volume of the considereddomain

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The End

Welcome and good luck Any questions, opinions, discussions?

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IntroductionElasticity and plasticityOverview of the courseCourse organization

Mathematical preliminariesScalars, vectors and tensorsIndex notationKronecker delta and alternating symbolCoordinate transformationsCartesian tensorsPrincipal values and directionsVector and tensor algebraTensor calculus

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