Common Variable Types in Elasticity Elasticity theory is a mathematical model of material deformation. Using principles of continuum mechanics, it is formula ted in terms of many diere nt types of eld variables specied at spatial points in t he body under study. Some examples include: Scalars - Single magnitude mass density ρ, temperature T, modulus of elasticity E, . . . Vectors –Three componen ts in three dimensions displacement vector Matrices – ine components in three dimensions stress matrix Other – !ariables "ith more than nine components Chapter 1 Mathematical Preliminaries ,e # , e $ , e % are unit basis vectors ElasticityTheory, Applications and Numerics M.H. Sadd , University of Rhode Island 3 2 1 e e e u w v u + + = σ τ τ τ σ τ τ τ σ = σ zzy zx yzy yx xzxy x ] [
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Elasticity theory is a mathematical model of material deformation.Using principles of continuum mechanics, it is formulated in terms ofmany dierent types of eld variables specied at spatial points in thebody under study. Some examples include:
Scalars - Single magnitude
mass density ρ, temperature T , modulus of elasticity E, . . .
Vectors – Three components in three dimensions displacement vector
IndexTensor !otation&ith the "ide variety of variables, elasticity formulation ma'esuse of a tensor formalism using index notation. This enables
e(cient representation of all variables and governing e)uationsusing a single standardi*ed method.+ndex notation is a shorthand scheme"hereby a "hole set of numbers orcomponents can be represented by asingle symbol "ith subscripts+n general a symbol aij…k "ith N distinct indices represents %N
distinct numbersddition, subtraction, multiplication and e)uality of indexsymbols are dened in the normal fashion- e.g.
dierent coordinate systems re)uiresdevelopment of transformation rulesfor scalar, vector, matrix and higherorder variables 2 a concept connected"ith basic denitions of tensorvariables. The t"o 3artesian frames4 x #,x $,x %5 and dier only
by orientationUsing 6otation7atrix
transformationla"s for3artesian vectorcomponents
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Cartesian Tensorseneral Trans+ormation a.sScalars, vectors, matrices, and higher order )uantities can berepresented by an index notational scheme, and thus all)uantities may then be referred to as tensors of dierent orders. The transformation properties of a vector can be used toestablish the general transformation properties of these tensors.
6estricting the transformations to those only bet"een 3artesiancoordinate systems, the general set of transformation relationsfor various orders are:
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Example 1-2 Transformation ExamplesThe components of a first and second order tensor in a particular coordinate frame are given by
8etermine the components of each tensor in a ne"coordinate system found through a rotation of 91o 4π9radians5 about the x %0axis. 3hoose a countercloc'"ise
rotation "hen vie"ing do"n the negative x %0axis, see
;igure #0$. The original and primed coordinate systems are sho"n in;igure #0$. The solution starts by determining therotation matrix for this case
The transformation for the vector )uantity follo"s from e)uation4#.<.#5$
and the second order tensor 4matrix5 transformsaccording to 4#.<.#5%
x 3
x #
x $
x ′
#
x ′
$
x ′
3
91o
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
Second Order Tensors The direction determined by unit vector n is said to be a principaldirection or eigenvector of the symmetric second order tensor aij
if there exists a parameter λ 4 principal value or eigenvalue5 suchthat
6elation is a homogeneous system of three linear algebraic
e)uations in the un'no"ns n#, n$, n%. The system possessesnontrivial solution if and only if determinant of coe(cient matrixvanishes
scalars Ia, IIa and IIIa are called the fundamental invariants of
the tensor aij
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
+t is al"ays possible to identify a right0handed 3artesian
coordinate system such that each axes lie along principaldirections of any given symmetric second order tensor. Suchaxes are called the principal axes of the tensor, and the basisvectors are the principal directions =n4#5, n4$5 , n4%5>
x 3
x 1
x /
Principal )xesOri*inal i&en )xes
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
8etermine the invariants, and principal values anddirections of
;irst determine the principal invariants
The characteristic e)uation then becomes
Thus for this case all principal values aredistinct;or the λ# / < root, e)uation 4#.9.#5 gives the
system
"hich gives a normali*ed solution
+n similar fashion the other t"o principal directions arefound to be+t is easily veried that these directions are mutually orthogonal.ote for this case, the transformation matrix Qij dened by 4#.?.#5
becomes
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island
+f dierentiation index is distinct, order of the tensor "ill beincreased by one- e.g. derivative operation on a vector producesa second order tensor or matrix
Elasticity Theory, Applications and NumericsM.H. Sadd , University of Rhode Island