theory of elasticity & plasticity.pdf
TRANSCRIPT
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Lecture 1IntroductionThe rules 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 . . . . . . . . . . . . . . . . . . . . . . 92.4 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5 Cartesian tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.6 Principal values and directions . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.7 Vector and tensor algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 Tensor calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 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 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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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 structures1.5
Introduction
Elasticity and plasticity
Plate structures1.6
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 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
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
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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, 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
Dr. A. Taushanov- Room 437 Office hours
Instructor: Tues: 13-14; Thurs: 16-17 TA: . . . . . . . . . . . .
Note
For other time by appointment1.12
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Introduction
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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
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
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
1.17
Mathematical preliminaries
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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Mathematical preliminaries
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
1.19
Mathematical preliminaries
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
Mathematical preliminaries
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 = a11 +a22 + . . .= . . . (scalar)- no free index ai jai j = a11a11 + a12a12 + a13a13 + . . . = 1 1+ 2 2+ . . . = . . . (scalar)- no free
index
ai jb j = ai1b1 +ai2b2 +ai3b3
=
a11b1 +a12b1 +a13b3. . .. . .
= . . .. . .. . .
(vector)- one free index
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Mathematical preliminaries
Index notation- example
Determine the following quantities
ai ja jk = ai1a1k +ai2a2k +ai3a3k
=
i = 1 a11a1k +a12a2k +a13a3ki = 2 a21a1k + . . .i = 3 a31a1k + . . .
The first expression gives the components of the 1-st row
a11a1k +a12a2k +a13a3k = k = 1 a11a11 +a12a21 +a13a31 = . . .k = 2 a11a12 +a12a22 +a13a32 = . . .k = 3 a11a13 +a12a23 +a13a33 = . . .
Finally
ai ja jk =
1 10 66 19 186 10 7
(matrix)- two free indexes ai jbib j = a11b1b1 +a12b1b2 +a13b1b3 + . . .= . . . (scalar)- no free index
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Mathematical preliminaries
Index notation- example
Determine the following quantities bibi = b1b1 +b2b2 + . . .= . . . (scalar)- no free index
bib j =
b1b jb2b jb3b j
= b1b1 b1b2 b1b3. . .
. . .
= . . .(matrix)- two free indexes
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Mathematical preliminaries
Index notation- example
Determine the following quantities Unsymmetric matrix decomposition
ai j =12(ai j +a ji) symmetric
+12(ai ja ji)
antisymmetric
Symmetric part
12(ai j +a ji) = . . .
Antisymmetric part
12(ai ja ji) = . . .
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2.3 Kronecker delta and alternating symbol
Mathematical preliminaries
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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Mathematical preliminaries
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 j2ak31.26
2.4 Coordinate transformations
Mathematical preliminaries
Coordinate transformations
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Consider two Cartesian coordinate systems with different orientation and basis vectors1.27
Mathematical preliminaries
Coordinate transformations
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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Mathematical preliminaries
Coordinate transformations
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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Mathematical preliminaries
Coordinate transformations
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
Mathematical preliminaries
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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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 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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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 = vv = v1v1 v1v2 v1v3v2v1 v2v2 v2v3
v3v1 v3v2 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 30 2 2
3 2 4
Determine the components of each tensor in a new coordinates found through a rotation of
60 about the x3-axis1.34
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Mathematical preliminaries
Transformation example
The rotation matrix is
Ni j = cos(xi,x j) =
cos300 cos30 cos90cos210 cos300 cos90cos90 cos90 cos0
= 12
3
2 0
32
12 0
0 0 1
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Mathematical preliminaries
Transformation example
The transformation of the vector bi is
bi = Ni jb j =
12
32 0
32
12 0
0 0 1
14
2
= . . . The second order tensor transformation is
ai j = NipN jpapq =
12
32 0
32
12 0
0 0 1
1 0 30 2 2
3 2 4
12
3
2 0
32
12 0
0 0 1
T
= . . .
1.36
2.6 Principal values and directions
Mathematical preliminaries
Principal values and directions for symmetric tensor
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The tensor transformation shows that there is a coordinate system in which the componentsof 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|
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 lengthsand 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 = 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 = 01.38
Mathematical preliminaries
Principal values and directions for symmetric tensor
Because this is a homogeneous set of equations for n, a nontrivial solution will not 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 = 01.39
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
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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Mathematical preliminaries
Example
Determine the invariants and principal values and directions of the following tensor:
A =
3 1 11 0 21 2 0
The invariants are
IA = . . . , IIA = . . . IIIA = . . .
The characteristic equation is
3 +3 2 +6 8 = 0
The roots are 1 =2, 2 = 1 and 3 = 41.41
Mathematical preliminaries
Example
For 1 =2 we have (A1I) n1 = 0 5 1 11 2 21 2 2
n11n21n31
= 00
0
The homogeneous set of equations have linear dependent equations and the solution rep-
resents only the ratio between the solution set Applying n31 = 1 and solving the first end second equations we get
n1 = . . .
Similarly for 2 = 1 and 3 = 41.42
2.7 Vector and tensor algebra
Mathematical preliminaries
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 e13a1 a2 a3
b1 b2 b3
Vector-matrix products
Aa = Ai ja j = a jAi jaT A = aiAi j = Ai jai
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Mathematical preliminaries
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 +A331.44
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 =xi
a
ai, j =x j
ai
ai j,k =xk
ai j
1.45
Mathematical preliminaries
Tensor calculus- example Vector differentiation
ai, j =aix j
=
a1x
a1y
a1 z
a2x
a2y
a2 z
a3x
a3y
a3 z
1.46
Mathematical preliminaries
Tensor calculus Directional derivative
Consider a scalar function . Find the derivative of the with respect of direction sdds
=x
dxds
+y
dyds
+ z
dzds
The unit vector in the direction of s is
n =dxds
e1 +dyds
e2 +dzds
e3
The directional derivative can be expressed as a scalar productdds
= n 1.47
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Mathematical preliminaries
Tensor calculus
Directional derivative 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
1.48
Mathematical preliminaries
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
1.49
Mathematical preliminaries
Tensor calculus- example
Scalar and vector functions are = x2 y2 and u = 2xe1 + 3yze2 + xye3. Calculate thefollowing expressions Gradient of a scalar
= . . .
Laplacian of a scalar
2 = = . . .
Divergence of a vector
u = . . .
Gradient of a vector
u = . . .
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Mathematical preliminaries
Tensor calculus- example
Curl of a vector
u = det e1 e2 e3
xy
z
2x 3yz xy
= . . .1.51
Mathematical preliminaries
Tensor calculus
Divergence (Gauss) theorem S
u ndS =
V udV
where n is the outward normal vector to the surface S1.52
Mathematical preliminaries
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