geometric derivation of the infinitesimal strain tensor

7/21/2019 Geometric Derivation of the Infinitesimal Strain Tensor

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Infinitesimal strain theory - Wikipedia, the free ry

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Geometric derivation of the infinitesimal strain tensor

!onsider a t"o-dimensional deformation of an infinitesimal rectan#$lar materialelement "ith dimensions %y &'i#$re 1(, "hich after deformation, takes

the form of a rhom%$s) 'rom the #eometry of 'i#$re 1 "e ha*e

'or *ery small displacement #radients, i)e)

, , "e ha*e

+he normal strain in the -direction of the rectan#$lar element is defined

%y

and kno"in# that , "e ha*e

'i#$re 1) +"o-dimensional #eometric deformation of an

infinitesimal material element)
http://en.wikipedia.org/wiki/Infinitesimal_strain_theoryhttp://en.wikipedia.org/wiki/Infinitesimal_strain_theory


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Similarly, the normal strain in the -direction, and -direction, %ecomes

+he en#ineerin# shear strain, or the chan#e in an#le %et"een t"o ori#inally ortho#onal material lines, in this case line and , is defined as

'rom the #eometry of 'i#$re 1 "e ha*e

'or small rotations, i)e) and are "e ha*e

and, a#ain, for small displacement #radients, "e ha*e

th$s

y interchan#in# and and and , it can %e sho"n that

Similarly, for the - and - planes, "e ha*e


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It can %e seen that the tensorial shear strain components of the infinitesimal strain tensor can then %e epressed $sin# the en#ineerin# strain definition,

, as

Physical interpretation of the infinitesimal strain tensor

'rom finite strain theory "e ha*e

'or infinitesimal strains then "e ha*e

.i*idin# %y "e ha*e

'or small deformations "e ass$me that , th$s the second term of the left hand side %ecomes:

) +hen "e ha*e

"here , is the $nit *ector in the direction of , and the left-hand-side epression is the normal strain in the direction of ) 'or

the partic$lar case of in the direction, i)e) , "e ha*e


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Similarly, for and "e can find the normal strains and , respecti*ely) +herefore, the dia#onal elements of the

infinitesimal strain tensor are the normal strains in the coordinate directions)

Strain transformation rules

If "e choose an orthonormal coordinate system & ( "e can "rite the tensor in terms of components "ith respect to those %ase *ectors as

In matri form,

We can easily choose to $se another orthonormal coordinate system & ( instead) In that case the components of the tensor are different, say

+he components of the strain in the t"o coordinate systems are related %y

"here the /instein s$mmation con*ention for repeated indices has %een $sed and ) In matri form

or


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Strain invariants

!ertain operations on the strain tensor #i*e the same res$lt "itho$t re#ard to "hich orthonormal coordinate system is $sed to represent

the components of strain) +he res$lts of these operations are called strain invariants) +he most commonly $sed strain in*ariants are

In terms of components

Principal strains

It can %e sho"n that it is possi%le to find a coordinate system & ( in "hich the components of the strain tensor are

+he components of the strain tensor in the & ( coordinate system are called the principal strains and the directions are called the

directions of principal strain) Since there are no shear strain components in this coordinate system, the principal strains represent the maim$m

and minim$m stretches of an elemental *ol$me)

If "e are #i*en the components of the strain tensor in an ar%itrary orthonormal coordinate system, "e can find the principal strains $sin# an

ei#en*al$e decomposition determined %y sol*in# the system of e0$ations


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+his system of e0$ations is e0$i*alent to findin# the *ector alon# "hich the stress tensor %ecomes a p$re stretch "ith no shear component)

Volumetric strain

+he dilatation &the relati*e *ariation of the *ol$me( is the trace of the tensor:

ct$ally, if "e consider a c$%e "ith an ed#e len#th a, it is a 0$asi-c$%e after the deformation &the *ariations of the an#les do not chan#e the *ol$me(

"ith the dimensions and V a3, th$s

as "e consider small deformations,

therefore the form$la)


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eal *ariation of *ol$me &top( and the approimated one &%ottom(: the #reen dra"in# sho"s the estimated *ol$me and the oran#e dra"in# the ne#lected *ol$me

In case of p$re shear, "e can see that there is no chan#e of the *ol$me)

Strain deviator tensor

+he infinitesimal strain tensor , similarly to the !a$chy stress tensor, can %e epressed as the s$m of t"o other tensors:

1) a mean strain tensor or volumetric strain tensor or spherical strain tensor, , related to dilation or *ol$me chan#e and

2) a de*iatoric component called the strain deviator tensor, , related to distortion)

"here is the mean strain #i*en %y


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+he de*iatoric strain tensor can %e o%tained %y s$%tractin# the mean strain tensor from the infinitesimal strain tensor:

Octahedral strains

6et & ( %e the directions of the three principal strains) n octahedral plane is "hose normal makes e0$al an#les "ith the three

principal directions) +he en#ineerin# shear strain on an octahedral plane is called the octahedral shear strain and is #i*en %y

"here are the principal strains)

+he normal strain on an octahedral plane is #i*en %y

Equivalent strain

scalar 0$antity called the equivalent strain, or the *on ises e0$i*alent strain, is often $sed to descri%e the state of strain in solids)

Se*eral definitions of e0$i*alent strain can %e fo$nd in the literat$re) definition that is commonly $sed in the literat$re on plasticity is


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Special cases

Plane strain

In real en#ineerin# components, stress &and strain( are 3-.

tensors %$t in prismatic str$ct$res s$ch as a lon# metal %illet, the

len#th of the str$ct$re is m$ch #reater than the other t"o

dimensions) +he strains associated "ith len#th, i)e), the normalstrain and the shear strains and &if the len#th is the 3-

direction( are constrained %y near%y material and are small

compared to the cross-sectional strains) Plane strain is then an

accepta%le approimation) +he strain tensor for plane strain is

"ritten as:

in "hich the do$%le $nderline indicates a second order tensor)

+his strain state is calledplane strain) +he correspondin# stress

tensor is:

Plane strain state in a contin$$m)

in "hich the non-ero is needed to maintain the constraint ) +his stress term can %e temporarily remo*ed from the analysis to lea*eonly the in-plane terms, effecti*ely red$cin# the 3-. pro%lem to a m$ch simpler 2-. pro%lem)


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Antiplane strain

ntiplane strain is another special state of strain that can occ$r in a %ody, for instance in a re#ion close to a scre" dislocation) +he strain tensor

for antiplane strain is #i*en %y

Infinitesimal rotation tensor

+he infinitesimal strain tensor is defined as

+herefore the displacement #radient can %e epressed as

"here

+he 0$antity is the infinitesimal rotation tensor) +his tensor is ske" symmetric) 'or infinitesimal deformations the scalar components ofsatisfy the condition ) ;ote that the displacement #radient is small only if both the strain tensor and the rotation tensor are infinitesimal)

he a!ial vector

ske" symmetric second-order tensor has three independent scalar components) +hese three components are $sed to define an a!ial vector, ,

as follo"s

"here is the perm$tation sym%ol) In matri form


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+he aial *ector is also called the infinitesimal rotation vector) +he rotation *ector is related to the displacement #radient %y the relation

In inde notation

If and then the material $nder#oes an approimate ri#id %ody rotation of ma#nit$de aro$nd the *ector )

"elation bet#een the strain tensor and the rotation vector


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+he components of the strain tensor in a spherical coordinate system are #i*en %y>1?

geometric derivation of the infinitesimal strain tensor

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