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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
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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-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
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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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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
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– 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: ddinev_fce@uacg.bg
• Teaching assistant
– M. Ivanova• Office hours
– Instructor: . . . . . . . . . . . .
– TA: . . . . . . . . . . . .
Note• For other time→ by appointment
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Introduction
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GradePoints
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 sectionsof the textbook before the class time
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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’sand your classmates
– However, the submitted individual HW solutions and exams must involve only youreffort
– Otherwise you’ll 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- ρ
– 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 +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 σxzσyx σyy σyzσzx σzy σzz
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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
ai =
a1a2a3
, ai j =
a11 a12 a13a21 a22 a23a31 a32 a33
– a1 j → first row– ai1 → first column
• Addition and subtraction
ai±bi =
a1±b1a2±b2a3±b3
ai j±bi j =
a11±b11 a12±b12 a13±b13a21±b21 a22±b22 a23±b23a31±b31 a32±b32 a33±b33
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Mathematical preliminaries
Index notation• Scalar multiplication
λai =
λa1λa2λa3
, λai j =
λa11 λa12 λa13λa21 λa22 λa23λa31 λa32 λa33
• Outer multiplication (product)
aib j =
a1b1 a1b2 a1b3a2b1 a2b2 a2b3a3b1 a3b2 a3b3
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Mathematical preliminaries
Index notation
• Commutative, associative and distributive laws
ai +bi = bi +ai
ai jbk = bkai j
ai +(bi + ci) = (ai +bi)+ ci
ai(b jkc`) = (aib jk)c`ai j(bk + ck) = ai jbk +ai jck
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Mathematical preliminaries
Index notation
• Summation convention (Einstein’s 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 notation
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Mathematical preliminaries
Index notation- example
• The matrix ai j and vector bi are
ai j =
1 2 00 4 32 1 2
, bi =
240
• 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 j−a ji)︸ ︷︷ ︸
antisymmetric
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2.3 Kronecker delta and alternating symbol
Mathematical preliminaries
Kronecker delta and alternating symbol• Kronecker delta is defined as
δi j =
{1 if i = j0 if i 6= j =
1 0 00 1 00 0 1
• Properties of δi j
δi j = δ ji
δii = 3
δi ja j =
δ11a1 +δ12a2 +δ13a3 = a1. . .. . .
= ai
δi ja jk = aik
δi jai j = aii
δi jδi j = 31.22
Mathematical preliminaries
Kronecker delta and alternating symbol• Alternating (permutation) symbol is defined as
εi jk =
+1 if i jk is an even permutation of 1,2,3−1 if i jk is an odd permutation of 1,2,30 otherwise
• Therefore
ε123 = ε231 = ε312 = 1ε321 = ε132 = ε213 =−1ε112 = ε131 = ε222 = . . .= 0
• Matrix determinant
det(ai j) = |ai j|=
∣∣∣∣∣∣a11 a12 a13a21 a22 a23a31 a32 a33
∣∣∣∣∣∣= εi jka1ia2 ja3k = εi jkai1a j2ak3
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2.4 Coordinate transformations
Mathematical preliminaries
Coordinate transformations
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• Consider two Cartesian coordinate systems with different orientation and basis vectors1.24
Mathematical preliminaries
Coordinate transformations
• The basis vectors for the old (unprimed) and the new (primed) coordinate systems are
ei =
e1e2e3
, e′i =
e′1e′2e′3
• Let Ni j denotes the cosine of the angle between x′i-axis and x j-axis
Ni j = e′i · e j = cos(x′i,x j)
• The primed base vectors can be expressed in terms of those in the unprimed by relations
e′1 = N11e1 +N12e2 +N13e3
e′2 = N21e1 +N22e2 +N23e3
e′3 = N31e1 +N32e2 +N33e3
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Mathematical preliminaries
Coordinate transformations
• In matrix form
e′i = Ni je j
ei = N jie′j
• An arbitrary vector can be written as
v = v1e1 + v2e2 + v3e3 = viei
= v′1e′1 + v′2e′2 + v′3e′3 = v′ie′i
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Mathematical preliminaries
Coordinate transformations
• Or
v = viN jie′j
• Because v = v′je′j thus
v′j = N jivi
• Similarly
vi = Ni jv′j
• 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
Mathematical preliminaries
Cartesian tensors• General index notation scheme
a′ = a, zero order (scalar)a′i = Nipap, first order (vector)a′i j = NipN jqapq, second order (matrix)
a′i jk = NipN jqNkrapqr, third order
. . .
• A tensor is a generalization of the above mentioned quantities
Example• The notation v′i = 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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Mathematical preliminaries
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 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 (v′1,v
′2v′3) 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 matrix1.29
Mathematical preliminaries
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 =
v′1v1 v′1v2 v′1v3v′2v1 v′2v2 v′2v3v′3v1 v′3v2 v′3v3
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Mathematical preliminaries
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 23 2 4
• Determine the components of each tensor in a new coordinates found through a rotation of
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Mathematical preliminaries
Transformation example• The rotation matrix is
Ni j = cos(x′i,x j) = . . .
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Mathematical preliminaries
Transformation example• The transformation of the vector bi is
b′i = Ni jb j = Mb = . . .
• The second order tensor transformation is
a′i j = NipN jpapq = NaNT = . . .
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2.6 Principal values and directions
Mathematical preliminaries
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 x′3-axis lies
along the vector, then vector will have components
v =
00|v|
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Mathematical preliminaries
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 = λn, Ai jn j = 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−λ I) ·n = 0, (Ai j−λδi j)n j = 0
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Mathematical preliminaries
Principal values and directions for symmetric tensor
• Because this is a homogeneous set of equations for n, a nontrivial solution will not existunless the determinant of the matrix (. . .) vanishes
det(A−λ I) = 0, det(Ai j−λδi j) = 0
• Expanding the determinant produces a characteristic equation in terms of λ
−λ3 + IAλ
2− IIAλ + IIIA = 0
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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
IIA =12(tr(A)2− tr(A2)
)=
12(AiiA j j−Ai 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 λ and each of these maybe back-substituted into (A−λ I) ·n = 0 to solve for the associated principle directions n.
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Mathematical preliminaries
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
Mathematical preliminaries
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 e3a1 a2 a3b1 b2 b3
• Vector-matrix products
Aa = Ai ja j = a jAi j
aT A = aiAi j = Ai jai
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Mathematical preliminaries
Vector and tensor algebra• Matrix-matrix products
AB = Ai jB jk
ABT = Ai jBk j
AT B = A jiB jk
tr(AB) = Ai jB ji
tr(ABT ) = tr(AT B) = Ai jBi j
where ATi j = A ji and tr(A) = Aii = A11 +A22 +A33
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2.8 Tensor calculus
Mathematical preliminaries
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 =∂
∂xia
ai, j =∂
∂x jai
ai j,k =∂
∂xkai j
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Mathematical preliminaries
Tensor calculus
• Directional derivative
– Consider a scalar function φ . Find the derivative of the φ with respect of direction s
dφ
ds=
∂φ
∂xdxds
+∂φ
∂ydyds
+∂φ
∂ zdzds
– The above expression can be presented as a dot product between two vectors
dφ
ds=[
dxds
dyds
dzds
]∂φ
∂x∂φ
∂y∂φ
∂ z
= n ·∇φ
– The first vector represents the unit vector in the direction of s
n =dxds
e1 +dyds
e2 +dzds
e3
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Mathematical preliminaries
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
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Mathematical preliminaries
Tensor calculus• Common differential operations and similarities with multiplications
Name Operation Similarities OrderGradient of a scalar ∇φ ≈ λu vector ↑Gradient of a vector ∇u = ui, jeie j ≈ u⊗v tensor ↑
Divergence of a vector ∇ ·u = ui, j ≈ u ·v dot ↓Curl of a vector ∇×u = εi jkuk, jei ≈ u×v cross→
Laplacian of a vector ∇2u = ∇ ·∇u = ui,kkei
NoteThe ∇-operator is a vector quantity
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Mathematical preliminaries
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Tensor calculus- example
• A scalar field is presented as φ = x2− y2
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Mathematical preliminaries
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• A vector field is u = 2xe1 +3yze2 + xye3
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Mathematical preliminaries
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Tensor calculus- example• Gradient of the scalar field is
∇φ = . . .
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Mathematical preliminaries
Tensor calculus- example• Laplacian of a scalar
∇2φ = ∇ ·∇φ = . . .
• Divergence of a vector
∇ ·u = . . .
• Gradient of a vector
∇u = . . .
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Mathematical preliminaries
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• Curl of a vector
∇×u = det
e1 e2 e3∂
∂x∂
∂y∂
∂ z2x 3yz xy
= . . .
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Mathematical preliminaries
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 considereddomain
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Mathematical preliminaries
The End
• Welcome and good luck• Any questions, opinions, discussions?
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