# Theory of Elasticity & Plasticity.pdf

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Lecture 1 Introduction The rules of the game Print version Lecture on Theory of Elasticity and Plasticity of Dr. D. Dinev, Department of Structural Mechanics, UACEG 1.1

Contents 1

Introduction 1.1 Elasticity and plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Overview of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Course organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Mathematical preliminaries 2.1 Scalars, vectors and tensors . . . . . . . 2.2 Index notation . . . . . . . . . . . . . . 2.3 Kronecker delta and alternating symbol 2.4 Coordinate transformations . . . . . . . 2.5 Cartesian tensors . . . . . . . . . . . . 2.6 Principal values and directions . . . . . 2.7 Vector and tensor algebra . . . . . . . . 2.8 Tensor calculus . . . . . . . . . . . . .

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1.2

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 in equilibrium of applied forces – Theory of elasticity treats explicitly a linear or nonlinear response of structure to loading • 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 structure, the body consists of continuous material points (we can infinitely ”zoom-in” and 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 1.3

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Introduction

Elasticity and plasticity • Which problems does the TE study? – All problems considering 2- or 3-dimensional formulation 1.4

Introduction

Elasticity and plasticity • Shell structures 1.5

Introduction

Elasticity and plasticity • Plate structures 1.6

Introduction

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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 factors)

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 1.8

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 – Shell theory – Plasticity Note • A lot of mathematics • Few videos and pictures 1.9

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Introduction Overview of the course • Textbooks – Elasticity theory, applications, and numerics, Martin H. Sadd, 2nd edition, Elsevier 2009 – Energy principles and variational methods in applied mechanics, J. N. Reddy, John Wiley & 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 & Sons 2004 – Thin plates and shells, E. Ventsel and T. Krauthammer, Marcel Dekker 2001 1.10

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 Colorado at Boulder – Lecture handouts from different universities around the world 1.11

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 – Dr. A. Taushanov- Room 437 • Office hours – Instructor: Tues: 13-14; Thurs: 16-17 – TA: . . . . . . . . . . . . Note • For other time → by appointment 1.12

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Introduction 7

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Points

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 time 1.14

Introduction Course organization • Homeworks – Homework is due at the beginning of the Thursday lectures – The assigned problems for the HW’s 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 TA’s and your classmates – However, the submitted individual HW solutions and exams must involve only your effort – Otherwise you’ll have terrible performance on the exam since you did not learn to think for yourself 1.15

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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- ρ – Temperature- T • Vector quantities- represent variables which are expressible in terms of components in a 2-D or 3-D coordinate system – Displacement- u = ue1 + ve2 + we3 where e1 , e2 and e3 are unit basis vectors in the coordinate system • Matrix quantities- represent variables which require more than three components to quantify – Stress matrix 

σxx σ =  σyx σzx

 σxz σyz  σzz

σxy σyy σzy

1.16

2.2

Index notation

Mathematical preliminaries Index notation • Index notation is a shorthand scheme where a set of numbers is represented by a single symbol with subscripts     a1 a11 a12 a13 ai =  a2  , ai j =  a21 a22 a23  a3 a31 a32 a33 – a1 j – ai1

→ →

first row first column

 a1 ± b1 ai ± bi =  a2 ± b2  a3 ± b3  a11 ± b11 ai j ± bi j =  a21 ± b21 a31 ± b31

a12 ± b12 a22 ± b22 a32 ± b32

 a13 ± b13 a23 ± b23  a33 ± b33 1.17

Mathematical preliminaries Index notation • Scalar multiplication 

 λ a1 λ ai =  λ a2  , λ a3

λ a11 λ ai j =  λ a21 λ a31

λ a12 λ a22 λ a32

 λ a13 λ a23  λ a33

• Outer multiplication (product) 

a1 b1 ai b j =  a2 b1 a3 b1

a1 b2 a2 b2 a3 b2

 a1 b3 a2 b3  a3 b3 1.18

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Mathematical preliminaries Index notation • Commutative, associative and distributive laws ai + bi = bi + ai ai j bk = bk ai j ai + (bi + ci ) = (ai + bi ) + ci ai (b jk c` ) = (ai b jk )c` ai j (bk + ck ) = ai j bk + ai j ck 1.19

Mathematical preliminaries Index notation • Summation convention (Einstein’s convention)- if a subscript appears twice in the same term, then summation over that subscript from one to three is implied 3

aii = ∑ aii = a11 + a22 + a33 i=1

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ai j b j =

∑ ai j b j = ai1 b1 + ai2 b2 + ai3 b3

j=1

– j- dummy index– subscript which is repeated into the notation (one side of the equation) – i- free index– subscript which is not repeated into the notation 1.20

Mathematical preliminaries Index notation- example • The matrix ai j and vector bi are 

1 ai j =  0 2

 0 3 , 2

2 4 1

 2 bi =  4  0

• Determine the following quantities – aii = a11 + a22 + . . . = . . . (scalar)- no free index – ai j ai j = a11 a11 + a12 a12 + a13 a13 + . . . = 1 × 1 + 2 × 2 + . . . = . . . (scalar)- no free index – ai j b j = ai1 b1 + ai2 b2 + ai3 b3     a11 b1 + a12 b1 + a13 b3 ...  =  ...  ... = ... ... (vector)- one free index 1.21

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Mathematical preliminaries Index notation- example • Determine the following quantities ai j a jk = ai1 a1k + ai2 a2k + ai3 a3k  i = 1 a11 a1k + a12 a2k + a13 a3k =  i = 2 a21 a1k + . . . i = 3 a31 a1k + . . . The first expression gives the components of the 1-st row a11 a1k + a12 a2k + a13 a3k =  k = 1 a11 a11 + a12 a21 + a13 a31 = . . .  k = 2 a11 a12 + a12 a22 + a13 a32 = . . . k = 3 a11 a13 + a12 a23 + a13 a33 = . . . Finally 

1 ai j a jk =  6 6

10 19 10

 6 18  7

(matrix)- two free indexes

• ai j bi b j = a11 b1 b1 + a12 b1 b2 + a13 b1 b3 + . . . = . . . (scalar)- no free index 1.22

Mathematical preliminaries Index notation- example • Determine the following quantities – bi bi = b1 b1 + b2 b2 + . . . = . . . (scalar)- no free index   b1 b1 b1 b j bi b j =  b2 b j  =  . . . ... b3 b j 

b1 b2

b1 b3

  = ...

(matrix)- two free indexes 1.23

Mathematical preliminaries Index notation- example • Determine the following quantities – Unsymmetric matrix decomposition 1 1 ai j = (ai j + a ji ) + (ai j − a ji ) |2 {z } |2 {z } symmetric

antisymmetric

– Symmetric part 1 (ai j + a ji ) = . . . 2 – Antisymmetric part 1 (ai j − a ji ) = . . . 2 1.24

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2.3

Kronecker delta and alternating symbol

Mathematical preliminaries Kronecker delta and alternating symbol • Kronecker delta is defined as   1 1 if i = j = 0 δi j = 0 if i = 6 j 0

0 1 0

 0 0  1

• Properties of δi j δi j = δ ji δii = 3 

δ11 a1 + δ12 a2 + δ13 a3 = a1 δi j a j =  . . . = ai ... δi j a jk = aik δi j ai j = aii δi j δi j = 3 1.25

Mathematical preliminaries Kronecker delta and alternating symbol • Alternating (permutation) symbol is defined as   +1 if i jk is an even permutation of 1,2,3 −1 if i jk is an odd permutation of 1,2,3 εi jk =  0 otherwise • Therefore ε123 = ε231 = ε312 = 1 ε321 = ε132 = ε213 = −1 ε112 = ε131 = ε222 = . . . = 0 • Matrix determinant a11 det(ai j ) = |ai j | = a21 a31

a12 a22 a32

a13 a23 a33

= εi jk a1i a2 j a3k = εi jk ai1 a j2 ak3 1.26

2.4

Coordinate transformations

Mathematical preliminaries

Coordinate transformations 9

• Consider two Cartesian coordinate systems with different orientation and basis vectors 1.27

Mathematical preliminaries Coordinate transformations • The basis vectors for the old (unprimed) and the new (primed) coordinate systems are  0    e1 e1 ei =  e2  , e0i =  e02  e3 e03 • Let Ni j denotes the cosine of the angle between xi0 -axis and x j -axis Ni j = e0i · e j = cos(xi0 , x j ) • The primed base vectors can be expressed in terms of those in the unprimed by relations e01 = N11 e1 + N12 e2 + N13 e3 e02 = N21 e1 + N22 e2 + N23 e3 e03 = N31 e1 + N32 e2 + N33 e3 1.28

Mathematical preliminaries Coordinate transformations • In matrix form e0i = Ni j e j ei = N ji e0j • An arbitrary vector can be written as v = v1 e1 + v2 e2 + v3 e3 = vi ei = v01 e01 + v02 e02 + v03 e03 = v0i e0i 1.29

Mathematical preliminaries Coordinate transformations • Or v = vi N ji e0j • Because v = v0j e0j thus v0j = N ji vi • Similarly vi = Ni j v0j • These relations constitute the transformation law for the Cartesian components of a vector under a change of orthogonal Cartesian coordinate system 1.30

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2.5

Cartesian tensors

Mathematical preliminaries Cartesian tensors • General index notation scheme a0 = a,

zero order (scalar)

a0i = Nip a p ,

first order (vector)

a0i j = Nip N jq a pq , second order (matrix) a0i jk = Nip N jq Nkr a pqr , third order ... • A tensor is a generalization of the above mentioned quantities Example • The notation v0i = Ni j v j is a relationship between two vectors which are transformed to each other by a tensor (coordinate transformation). The multiplication of a vector by a tensor results another vector (linear mapping). 1.31

Mathematical preliminaries Cartesian tensors • All second order tensors can be presented in matrix form   N11 N12 N13 Ni j =  N21 N22 N23  N31 N32 N33 • Since Ni j can be presented as a matrix, all matrix operation for 3 × 3-matrix are valid • The difference between a matrix and a tensor – We can multiply the three components of a vector vi by any 3 × 3-matrix – The resulting three numbers (v01 , v02 v03 ) may or may not represent the vector components – 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 1.32

Mathematical preliminaries Cartesian tensors • The second order tensor can be created by a dyadic (tensor or outer) product of the two vectors v0 and v  0  v1 v1 v01 v2 v01 v3 N = v0 ⊗ v =  v02 v1 v02 v2 v02 v3  v03 v1 v03 v2 v03 v3 1.33

Mathematical preliminaries Transformation example • The components of a first and a second order tensor given by    1 1 bi =  4  , ai j =  0 3 2

in a particular coordinate frame are 0 2 2

 3 2  4

• Determine the components of each tensor in a new coordinates found through a rotation of 60◦ about the x3 -axis 1.34

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Mathematical preliminaries

Transformation example • The rotation matrix is cos 300◦ 0  Ni j = cos(xi , x j ) = cos 210◦ cos 90◦ 

cos 30◦ cos 300◦ cos 90◦

  1 cos 90◦  2√ cos 90◦  =  − 3 2 cos 0◦ 0

√ 3 2 1 2

0

 0  0  1 1.35

Mathematical preliminaries Transformation example • The transformation of the vector bi is   b0i = Ni j b j = 

1 2√ − 23

√ 3 2 1 2

0

0

• The second order tensor transformation is   √ 3 1 0 1 2  2√  1 0 a0i j = Nip N j p a pq =  − 3  0 2 2 3 0 0 1

  1 0   4 = ...  0 2 1

0 2 2

 1 3  2√ 2  − 3 2 4 0

√ 3 2 1 2

0

T 0  0  = ... 1 1.36

2.6

Principal values and directions

Mathematical preliminaries

Principal values and directions for symmetric tensor 12

• 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 x30 -axis lies along the vector, then vector will have components   0 v= 0  |v| 1.37

Mathematical preliminaries Principal values and directions for symmetric tensor • It is of interest to inquire whether there are certain vectors n that have only their lengths and not their orientation changed when operated upon by a given tensor A • That is, to seek vectors that are transformed into multiples of themselves • If such vectors exist they must satisfy the equation A · n = λ n,

Ai j n j = λ ni

• Such vectors n are called eigenvectors of A • The parameter λ is called eigenvalue and characterizes the change in length of the eigenvector n • The above equation can be written as (A − λ I) · n = 0,

(Ai j − λ δi j )n j = 0 1.38

Mathematical preliminaries Principal values and directions for symmetric tensor • Because this is a homogeneous set of equations for n, a nontrivial solution will not exist unless the determinant of the matrix (. . .) vanishes det(A − λ I) = 0,

det(Ai j − λ δi j ) = 0

• Expanding the determinant produces a characteristic equation in terms of λ −λ 3 + IA λ 2 − IIA λ + IIIA = 0 1.39

Mathematical preliminaries Principal values and directions for symmetric tensor • The IA , IIA and IIIA are called the fundamental invariants of the tensor IA = tr(A) = Aii = A11 + A22 + A33  1 1 IIA = tr(A)2 − tr(A2 ) = (Aii A j j − Ai j Ai j ) 2 2 A11 A12 A22 A23 A11 A13 + = + A21 A22 A32 A33 A31 A33

IIIA = det(A) = det(Ai j ) • The roots of the characteristic equation determine the values for λ and each of these may be back-substituted into (A − λ I) · n = 0 to solve for the associated principle directions n. 1.40

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Mathematical preliminaries Example • Determine the invariants and principal values and directions of the following tensor:   3 1 1 A= 1 0 2  1 2 0 • The invariants are IA = . . . ,

IIA = . . .

IIIA = . . .

• The characteristic equation is −λ 3 + 3λ 2 + 6λ − 8 = 0 • The roots are λ1 = −2, λ2 = 1 and λ3 = 4 1.41

Mathematical preliminaries Example • For λ1 = −2 we have (A − λ1 I) · n1 = 0      0 5 1 1 n11  1 2 2   n21  =  0  0 1 2 2 n31 • The homogeneous set of equations have linear dependent equations and the solution represents only the ratio between the solution set • Applying n31 = 1 and solving the first end second equations we get n1 = . . . • Similarly for λ2 = 1 and λ3 = 4 1.42

2.7

Vector and tensor algebra

Mathematical preliminaries Vector and tensor algebra • Scalar product (dot product, inner product) a · b = |a||b| cos θ • Magnitude of a vector |a| = (a · a)1/2 • Vector product (cross-product) 

e1 a × b = det  a1 b1

e2 a2 b2

 e1 3 a3  b3

• Vector-matrix products Aa = Ai j a j = a j Ai j aT A = ai Ai j = Ai j ai 1.43

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Mathematical preliminaries Vector and tensor algebra • Matrix-matrix products AB = Ai j B jk ABT = Ai j Bk j AT B = A ji B jk tr(AB) = Ai j B ji tr(ABT ) = tr(AT B) = Ai j Bi j where ATij = A ji and tr(A) = Aii = A11 + A22 + A33

2.8

1.44

Tensor calculus

Mathematical preliminaries Tensor calculus • Common tensors used in field equations a = a(x, y, z) = a(xi ) = a(x) − scalar ai = ai (x, y, z) = ai (xi ) = ai (x) − vector ai j = ai j (x, y, z) = ai j (xi ) = ai j (x) − tensor • Comma notations for partial differentiation ∂ a ∂ xi ∂ ai ai, j = ∂xj ∂ ai j,k = ai j ∂ xk

a,i =

1.45

Mathematical preliminaries Tensor calculus- example • Vector differentiation  ai, j =

∂ ai  = ∂xj

∂ a1 ∂x ∂ a2 ∂x ∂ a3 ∂x

∂ a1 ∂y ∂ a2 ∂y ∂ a3 ∂y

∂ a1 ∂z ∂ a2 ∂z ∂ a3 ∂z

   1.46

Mathematical preliminaries Tensor calculus • Directional derivative – Consider a scalar function φ . Find the derivative of the φ with respect of direction s ∂ φ dx ∂ φ dy ∂ φ dz dφ = + + ds ∂ x ds ∂ y ds ∂ z ds – The unit vector in the direction of s is dx dy dz n = e1 + e2 + e3 ds ds ds – The directional derivative can be expressed as a scalar product dφ = n · ∇φ ds 1.47

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Mathematical preliminaries Tensor calculus • Directional derivative – ∇φ is called the gradient of the scalar function φ and is defined by ∇φ = e1

∂φ ∂φ ∂φ + e2 + e3 ∂x ∂y ∂z

– The symbolic operator ∇ is called del operator (nabla operator) and is defined as ∇ = e1

∂ ∂ ∂ + e2 + e3 ∂x ∂y ∂z

– The operator ∇2 is called Laplacian operator and is defined as ∇2 =

∂2 ∂2 ∂2 + + ∂ x2 ∂ y2 ∂ z2 1.48

Mathematical preliminaries Tensor calculus • Common differential operations and similarities with multiplications Name Gradient of a scalar Gradient of a vector Divergence of a vector Curl of a vector Laplacian of a vector

Operation ∇φ ∇u = ui, j ei e j ∇ · u = ui, j ∇ × u = εi jk uk, j ei ∇2 u = ∇ · ∇u = ui,kk ei

Similarities ≈ λu ≈ u⊗v ≈ u·v ≈ u×v

Order vector ↑ tensor ↑ dot ↓ cross →

Note The ∇-operator is a vector quantity 1.49

Mathematical preliminaries Tensor calculus- example • Scalar and vector functions are φ = x2 − y2 and u = 2xe1 + 3yze2 + xye3 . Calculate the following expressions • Gradient of a scalar ∇φ = . . . • Laplacian of a scalar ∇2 φ = ∇ · ∇φ = . . . • Divergence of a vector ∇·u = ... • Gradient of a vector ∇u = . . . 1.50

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Mathematical preliminaries Tensor calculus- example • Curl of a vector 

e1

e2

e3

∇ × u = det 

∂ ∂x

∂ ∂y

∂ ∂z

 = ...

2x

3yz

xy 1.51

Mathematical preliminaries Tensor calculus • Divergence (Gauss) theorem Z

u · ndS =

S

Z

∇ · udV

V

where n is the outward normal vector to the surface S 1.52

Mathematical preliminaries

The End • Welcome and good luck • Any questions, opinions, discussions? 1.53

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