Mathematical Methods for Physics and Engineering : A Comprehensive Guide

26.12 PHYSICAL APPLICATIONS OF TENSORS

We can extend the idea of a second-order tensor that relates two vectors to a

situation where two physical second-order tensors are related by a fourth-order

tensor. The most common occurrence of such relationships is in the theory of

elasticity. This is not the place to give a detailed account of elasticity theory,

but suffice it to say that the local deformation of an elastic body at any interior

pointPcan be described by a second-order symmetric tensoreijcalled thestrain

tensor. It is given by

eij=

1

2

(

∂ui

∂xj

+

∂uj

∂xi

)

,

whereuis the displacement vector describing the strain of a small volume element

whose unstrained position relative to the origin isx. Similarly we can describe

the stress in the body atPby the second-order symmetricstress tensorpij;the

quantitypijis thexj-component of the stress vector acting across a plane through

Pwhose normal lies in thexi-direction. A generalisation of Hooke’s law then

relates the stress and strain tensors by

pij=cijklekl (26.46)

wherecijklis a fourth-order Cartesian tensor.

Assuming that the most general fourth-order isotropic tensor is

cijkl=λδijδkl+ηδikδjl+νδilδjk, (26.47)

find the form of (26.46) for an isotropic medium having Young’s modulusEand Poisson’s

ratioσ.

For an isotropic medium we must have an isotropic tensor forcijkl, and so we assume the
form (26.47). Substituting this into (26.46) yields

pij=λδijekk+ηeij+νeji.

Buteijis symmetric, and if we writeη+ν=2μ, then this takes the form

pij=λekkδij+2μeij,

in whichλandμare known asLam ́e constants. It will be noted that ifeij=0fori=j
then the same is true ofpij, i.e. the principal axes of the stress and strain tensors coincide.
Now consider a simple tension in thex 1 -direction, i.e.p 11 =Sbut all otherpij=0.
Then denotingekk(summed overk)byθwe have, in addition toeij=0fori=j,thethree
equations

S=λθ+2μe 11 ,

0=λθ+2μe 22 ,

0=λθ+2μe 33.

Adding them gives

S=θ(3λ+2μ).

Substituting forθfrom this into the first of the three, and recalling that Young’s modulus
is defined byS=Ee 11 ,givesEas

E=

μ(3λ+2μ)

λ+μ

. (26.48)