Introduction to Theory of elasticity and plasticity Att 6521

Lecture 1
Introduction to Theory of elasticity
and plasticity
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
1.1 Elasticity and plasticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Overview of the course . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3 Course organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 Mathematical preliminaries 6
2.1 Scalars, vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Index notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Kronecker delta and alternating symbol . . . . . . . . . . . . . . . . . . . . . . 8
2.4 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 Cartesian tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.6 Principal values and directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.7 Vector and tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.8 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 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 struc-
ture, 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
1

Introduction
• Which problems does the TE study?
– All problems considering 2- or 3-dimensional formulation
1.4
Introduction
• Shell structures
1.5
Introduction
• Plate structures
1.6
2

Introduction
• 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)
• 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
3

– Shell theory
– Plasticity
Note
• A lot of mathematics
• Few videos and pictures
1.9
Introduction
• 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
• 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 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: ddinev_fce@uacg.bg
• Teaching assistant
– M. Ivanova
• Office hours
– Instructor: ............
– TA: ............
Note
• For other time → by appointment
1.12
4

Introduction
1
2
3
4
5
6
7
40 50 60 70 80 90 100
Grade
Points
Course organization
• Grading
1.13
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 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
5

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 quan-
tify
– Stress matrix
σ =


σxx σxy σxz
σyx σyy σyz
σzx σzy σzz


1.16
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 =


a1
a2
a3

, aij =


a11 a12 a13
a21 a22 a23
a31 a32 a33


– a1 j → first row
– ai1 → first column
• Addition and subtraction
ai ±bi =


a1 ±b1
a2 ±b2
a3 ±b3


aij ±bij =


a11 ±b11 a12 ±b12 a13 ±b13
a21 ±b21 a22 ±b22 a23 ±b23
a31 ±b31 a32 ±b32 a33 ±b33


1.17
Index notation
• Scalar multiplication
λai =


λa1
λa2
λa3

, λaij =


λa11 λa12 λa13
λa21 λa22 λa23
λa31 λa32 λa33


• Outer multiplication (product)
aibj =


a1b1 a1b2 a1b3
a2b1 a2b2 a2b3
a3b1 a3b2 a3b3


1.18
6

Index notation
• Commutative, associative and distributive laws
ai +bi = bi +ai
aijbk = bkaij
ai +(bi +ci) = (ai +bi)+ci
ai(bjkc ) = (aibjk)c
aij(bk +ck) = aijbk +aijck
1.19
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
aii =
3
∑
i=1
aii = a11 +a22 +a33
aijbj =
3
∑
j=1
aijbj = ai1b1 +ai2b2 +ai3b3
– 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
Index notation- example
• The matrix aij and vector bi are
aij =


1 2 0
0 4 3
2 1 2

, bi =


2
4
0


• Determine the following quantities
– aii = ...
– aijaij = ...
– aijbj = ...
– aijajk = ...
– aijbibj = ...
– bibi = ...
– bibj = ...
– Unsymmetric matrix decomposition
aij =
1
2
(aij +aji)
symmetric
+
1
2
(aij −aji)
antisymmetric
1.21
7

2.3 Kronecker delta and alternating symbol
Kronecker delta and alternating symbol
• Kronecker delta is deﬁned as
δij =
1 if i = j
0 if i = j
=


1 0 0
0 1 0
0 0 1


• Properties of δij
δij = δji
δii = 3
δijaj =


δ11a1 +δ12a2 +δ13a3 = a1
...
...
= ai
δijajk = aik
δijaij = aii
δijδij = 3
1.22
Kronecker delta and alternating symbol
• Alternating (permutation) symbol is deﬁned as
εijk =



+1 if ijk is an even permutation of 1,2,3
−1 if ijk is an odd permutation of 1,2,3
0 otherwise
• Therefore
ε123 = ε231 = ε312 = 1
ε321 = ε132 = ε213 = −1
ε112 = ε131 = ε222 = ... = 0
• Matrix determinant
det(aij) = |aij| =
a11 a12 a13
a21 a22 a23
a31 a32 a33
= εijka1ia2 ja3k = εijkai1aj2ak3
1.23
2.4 Coordinate transformations
Coordinate transformations
8

• Consider two Cartesian coordinate systems with different orientation and basis vectors
1.24
• The basis vectors for the old (unprimed) and the new (primed) coordinate systems are
ei =


e1
e2
e3

, ei =


e1
e2
e3


• Let Nij denotes the cosine of the angle between xi-axis and xj-axis
Nij = ei ·ej = cos(xi,xj)
• The primed base vectors can be expressed in terms of those in the unprimed by relations
e1 = N11e1 +N12e2 +N13e3
e2 = N21e1 +N22e2 +N23e3
e3 = N31e1 +N32e2 +N33e3
1.25
• In matrix form
ei = Nijej
ei = Njiej
• An arbitrary vector can be written as
v = v1e1 +v2e2 +v3e3 = viei
= v1e1 +v2e2 +v3e3 = viei
1.26
• Or
v = viNjiej
• Because v = vjej thus
vj = Njivi
• Similarly
vi = Nijvj
• These relations constitute the transformation law for the Cartesian components of a vector
under a change of orthogonal Cartesian coordinate system
1.27
9

2.5 Cartesian tensors
Cartesian tensors
• General index notation scheme
a = a, zero order (scalar)
ai = Nipap, ﬁrst order (vector)
aij = NipNjqapq, second order (matrix)
aijk = NipNjqNkrapqr, third order
...
• A tensor is a generalization of the above mentioned quantities
Example
• The notation vi = Nijvj 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).
1.28
Cartesian tensors
• All second order tensors can be presented in matrix form
Nij =


N11 N12 N13
N21 N22 N23
N31 N32 N33


• Since Nij 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 (v1,v2v3) may or may not represent the vector compo-
nents
– If they are the vector components, then the matrix represents the components of a
tensor Nij
– If not, then the matrix is just an ordinary old matrix
1.29
Cartesian tensors
• The second order tensor can be created by a dyadic (tensor or outer) product of the two
vectors v and v
N = v ⊗v =


v1v1 v1v2 v1v3
v2v1 v2v2 v2v3
v3v1 v3v2 v3v3


1.30
Transformation example
• The components of a ﬁrst and a second order tensor in a particular coordinate frame are
given by
bi =


1
4
2

, aij =


1 0 2
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
1.31
10

• The rotation matrix is
Nij = cos(xi,xj) = ...
1.32
• The transformation of the vector bi is
bi = Nijbj = Mb = ...
• The second order tensor transformation is
aij = NipNjpapq = NaNT
= ...
1.33
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 =


0
0
|v|


1.34
11

• 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 = λn, Aijnj = λni
• Such vectors n are called eigenvectors of A
• The parameter λ is called eigenvalue and characterizes the change in length of the eigen-
vector n
• The above equation can be written as
(A−λI)·n = 0, (Aij −λδij)nj = 0
1.35
• 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−λI) = 0, det(Aij −λδij) = 0
• Expanding the determinant produces a characteristic equation in terms of λ
−λ3
+IAλ2
−IIAλ +IIIA = 0
1.36
• The IA, IIA and IIIA are called the fundamental invariants of the tensor
IA = tr(A) = Aii = A11 +A22 +A33
IIA =
1
2
tr(A)2
−tr(A2
) =
1
2
(AiiAj j −AijAij)
=
A11 A12
A21 A22
+
A22 A23
A32 A33
+
A11 A13
A31 A33
IIIA = det(A) = det(Aij)
• The roots of the characteristic equation determine the values for λ and each of these may
be back-substituted into (A−λI)·n = 0 to solve for the associated principle directions n.
1.37
Example
• Determine the invariants and principal values and directions of the following tensor:
A =


3 1 1
1 0 2
1 2 0


1.38
12

2.7 Vector and tensor algebra
Vector and tensor algebra
• Scalar product (dot product, inner product)
a·b = |a||b|cosθ
• Magnitude of a vector
|a| = (a·a)1/2
• Vector product (cross-product)
a×b = det


e1 e2 e3
a1 a2 a3
b1 b2 b3


• Vector-matrix products
Aa = Aijaj = ajAij
aT
A = aiAij = Aijai
1.39
Vector and tensor algebra
• Matrix-matrix products
AB = AijBjk
ABT
= AijBk j
AT
B = AjiBjk
tr(AB) = AijBji
tr(ABT
) = tr(AT
B) = AijBij
where AT
ij = Aji and tr(A) = Aii = A11 +A22 +A33
1.40
2.8 Tensor calculus
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
aij = aij(x,y,z) = aij(xi) = aij(x)−tensor
• Comma notations for partial differentiation
a,i =
∂
∂xi
a
ai,j =
∂
∂xj
ai
aij,k =
∂
∂xk
aij
1.41
13

Tensor calculus
• Directional derivative
– Consider a scalar function φ. Find the derivative of the φ with respect of direction s
dφ
ds
=
∂φ
∂x
dx
ds
+
∂φ
∂y
dy
ds
+
∂φ
∂z
dz
ds
– The above expression can be presented as a dot product between two vectors
dφ
ds
= dx
ds
dy
ds
dz
ds



∂φ
∂x
∂φ
∂y
∂φ
∂z


 = n·∇φ
– The first vector represents the unit vector in the direction of s
n =
dx
ds
e1 +
dy
ds
e2 +
dz
ds
e3
1.42
Tensor calculus
• Directional derivative
– The second vector is called the gradient of the scalar function φ and is defined by
∇φ = e1
∂φ
∂x
+e2
∂φ
∂y
+e3
∂φ
∂z
– The symbolic operator ∇ is called del operator (nabla operator) and is defined as
∇ = e1
∂
∂x
+e2
∂
∂y
+e3
∂
∂z
– The operator ∇2 is called Laplacian operator and is defined as
∇2
=
∂2
∂x2
+
∂2
∂y2
+
∂2
∂z2
1.43
Tensor calculus
• Common differential operations and similarities with multiplications
Name Operation Similarities Order
Gradient of a scalar ∇φ ≈ λu vector ↑
Gradient of a vector ∇u = ui,jeiej ≈ u⊗v tensor ↑
Divergence of a vector ∇·u = ui,j ≈ u·v dot ↓
Curl of a vector ∇×u = εijkuk,jei ≈ u×v cross →
Laplacian of a vector ∇2u = ∇·∇u = ui,kkei
Note
The ∇-operator is a vector quantity
1.44
14

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 φ = x2 −y2
1.45
1
0
1
1
0
1
1
0
1
• A vector ﬁeld is u = 2xe1 +3yze2 +xye3
1.46
15

1.0 0.5 0.0 0.5 1.0
1.0
0.5
0.0
0.5
1.0
• Gradient of the scalar ﬁeld is
∇φ = ...
1.47
• Laplacian of a scalar
∇2
φ = ∇·∇φ = ...
• 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
∂
∂x
∂
∂y
∂
∂z
2x 3yz xy

 = ...
1.49
Tensor calculus
• Gradient theorem
S
nφ dS =
V
∇φ dV
• Divergence (Gauss) theorem
S
u·ndS =
V
∇·udV
• Curl theorem
S
u×ndS =
V
∇×udV
where n is the outward normal vector to the surface S and V is the volume of the considered
domain
1.50
The End
• Welcome and good luck
• Any questions, opinions, discussions?
1.51
17

Introduction to Theory of elasticity and plasticity Att 6521

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