Stress, Strain and Deformation in Solids

7/23/2019 Stress, Strain and Deformation in Solids

http://slidepdf.com/reader/full/stress-strain-and-deformation-in-solids 1/38

©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids 1

Chapter 8 - Stress, Strain and Deformation in Solids

For a beam in bendin, !e are often interested in determinin thetrans"erse displa#ement alon the beam as sho!n belo!.

x

y

$ % y

u x

$ % displa#ement in & dire#tion yu x =

'o do that, !e !o(ld start !ith Conser"ation of )inear*oment(m. +nfort(natel&, C)* is in terms of stress, σ .Conse(entl&, !e m(st ha"e some !a& to relate stress todeformation. We !ill need additional e(ations as follo!s:

Constitutive relations - relate stress to strain

Kinematic relations - relate strain to displa#ement $radients%




n the st(d& of the motion of a solid or fl(id, !e !ill find itne#essar& to des#ribe the /inemati# beha"ior of a #ontin((m

bod& b& definin epressions #alled strains in terms of the gradients of displacement components. n the eample belo!, !e#onsider an elasti# bar of lenth ). f the bar is s(be#ted b& anaial for#e F, it !ill stret#h

an amo(nt δ as sho!n in

fi(re b%. 'he (antit& δ)

is a meas(re of the #hanein lenth relati"e to theoriinal lenth and is

defined to be the axial strain for the bar.

a% (ndeformed

b% stret#hed $deformed%

)

) 3 δ

F F




n fi(re d%, a shear load is applied thatis parallel to the top s(rfa#e as sho!n.

'he anle θ meas(res the amo(nt theoriinal riht anle in fi(re #% has#haned from a riht anle, and the

angle θ is related to the shear strain.

What #a(ses strain5

• *e#hani#al loads $for#es,

press(res, et#.%

• 'emperat(re #hane $thermalepansion%

• *oist(re absorption

66 Stress $relationship bet!een stress and strain is the#onstit(ti"e relationship%

#% (ndeformed

d% sheared $deformed%

θF

70o



©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids

n this #hapter !e !ill mathemati#all& formali9e these simpleideas to de"elop epressions for strains in terms of displa#ement

#omponents. We !ill #onsider t!o approa#hes:

1% mathemati#all& pre#ise approa#h and2% a simpler eometri#al approa#h.



©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids

Deformations in solids are #hara#teri9ed b& displa#ementsof points and b& elonations and rotations of line sementsin the solid.

.nitial State of ;od& Deformed State

&

9

<

=

<6

=6

r r*

dr dr*

u(r)

u(r+dr)

r >position "e#tor of point < $initial%

r* >position "e#tor of point <6 $deformed%

u(r) >displa#ement "e#tor of point <

dr > "e#tor line sement bet!een < and =



©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids ?

@ote the follo!in eometr& (antities:< mo"es to <6,= mo"es to =6,line sement <-= deforms $stret#hesrotates% to <6-=6:

r > position "e#tor of point < $initial state%

6r > position "e#tor of point < $in deformed state%$ %u r > displa#ement "e#tor of point < $from initial todeformed state%

dr > "e#tor line sement bet!een < and = $initial state%6

dr

r > "e#tor line sement bet!een < and = $deformed

state%$ %u r dr + > displa#ement "e#tor of point = $from initial to

deformed state%



©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids A

From eometr& of the "e#tors, !e #an !rite t!o e(ations:

displa#ement "e#tor of point <: 6$ %u r r r = −r r r r

and6$ % $ %dr u r dr u r dr + + = +r r r r r r r or 6 $ % $ %dr dr u r dr u r = + + −r r r r r r r

'he last t!o terms represent the #hane $radient% indispla#ement u !ith respe#t to position r , i.e.,

$ % $ % $ %u r dr u r du dr u+ − = = • ∇

⇐displacement gradientmatrix $44%. @ote: nots&mmetri#B

∇ =

u

u x

x u y

x u z

x

u x y u y y u z y

u x

z u y

z u z

z

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂




Definition of strain. Strain is a meas(re of the deformationand rotation of line sements. Consider t!o material

elements 1dr 2dr , !hi#h (ndero deformations that !ill

brin them into ne! lo#ations6

1dr

r

62

dr r

respe#ti"el&.

.nitial State of ;od& Deformed State

&

9

r r*

u(r)

dr* 1

dr 1

dr 2

dr* 2




<re"io(sl&, !e obtained for a line sement 6dr r :

6 $ % $ % $ %dr dr u r dr u r dr du dr dr u= + + − = + = + • ∇r r r r r r r r r r r r

We !ill obtain identi#al epressions for the t!o line

sements6

1

dr r

and6

2

dr r

defined in the fi(re abo"e:

6 $ %1 1 1

dr dr dr u= + • ∇r r r r and 6 $ %

2 2 2dr dr dr u= + • ∇r r r r

Consider the dot prod(#t of these t!o "e#tors $a scaler res(ltB%




>

6 6 $ % $ %1 2 1 1 2 2

$ %1 2 1 2

$ % $ % $ %1 2 1 2

$ % $ % $1 2 1

dr dr dr dr u dr dr u

dr dr dr dr u

dr u dr dr u dr u

T dr dr dr u u

=

• = + • ∇ • + • ∇

• + • • ∇

+ • ∇ • + • ∇ • • ∇

• + • ∇ + ∇ + ∇

r r r r r r r r

rr r r r r

r r rr r r r r r r

r r rr r r r r% $ %

2

T u u dr

• ∇ •

rr r r

Ge#all that $ %u∇ is a 44 matri. 'he 'I in $ %T u∇ r means

that the matri $ %u∇ is transposed. 'he (nderlined term is

defined as 2 E so that 6 6 $2 %

1 2 1 2 1 2

dr dr dr dr dr E dr • = • + • •r r r r r r

and

1 E$ % $ % $ % $ % 2

T T E u u u u≡ ∇ + ∇ + ∇ • ∇r r r r ⇐ Finite Strain Tensor

@ote that the epression for the finite strain tensor




1E$ % $ % $ % $ % 2

T T

E u u u u= ∇ + ∇ + ∇ • ∇

r r r r

#ontains t!o distin#ti"e terms:

1

E$ % $ % 2T

u u∇ + ∇r r

⇒ linear in displa#ement radients

1 E$ % $ % 2

T u u∇ • ∇r r ⇒ (adrati# in displa#ement

radients




nfinitesimal Strain 'ensor

f the hiher order terms are nele#ted from the finite straintensor E $i.e., /eep onl& linear displa#ement radient

terms%, !e obtain the infinitesimal strain tensor, ε:

1 E$ % $ % 2 T u uε = ∇ + ∇r r r

!here ε is a 44 matri i"en b&:




[ ]

1 1$ % $ %2 2

1 1 $ % $ %

2 2

1 1$ % $ %

2 2

xx xy xz

yx yy yz

zx zy zz

uu u uu y x x x z x x y x z

u u uu u y y y x z

y x y y z

uu u u u y x z z z z x z y z

ε ε ε

ε ε ε ε

ε ε ε

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

∂ ∂ ∂ ∂ ∂

=

+ +

= + +

+ +




@ote: ;oth E and ε are s&mmetri# matri#es.




J 2-D eometri#al loo/ at xxε $defined to be the #hane inlenth of a line sement d !hi#h is oriinall& oriented inthe dire#tion% and (nderoes displa#ements ( and (&:

d

( $% (

& $% (

$3d%

( & $3d%

d6

&

$3d%

<

<6

=

=6



©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids 1?

6dx dx xx

dxε

−≡ > $#hane in lenth%$oriinal lenth% .

{ } { }

{ }

{ }

{ }

1 222

1 222

1 222

1 2

2 22 2

6 $ % $ % $ % $ %

$1 % $ %

1 1 2

x x y y

y x

y x

y y x x x

dx dx u x dx u x u x dx u x

uudx dx dx

x x

uudx dx

x x

u uu u udx dx

x x x x x

∂ ∂

∂ ∂ ∂ ∂

∂ ∂

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= + + − + + −

= + +

= + +

= + + = + + +1 2

@ote: 1 1 $1 2%a a+ ≅ + $for small a%. 'h(s, the last res(lt is

approimatel&:



©2001, W. E. Haisler Chapter 8: Stress, Strain and Deformation in Solids 1A

221 1

6 1 2 2

uu u y x x

dx dx x x x

∂ ∂ ∂

∂ ∂ ∂

= + + +

'he strain no! be#omes

221 1

12 2

6

uu u y x xdx dx x x x

dx dx xx

dx dx

∂ ∂ ∂

∂ ∂ ∂

ε

=

+ + + −−=




Jfter #an#elin the d terms, the finite strain term be#omes

221 1

2 2

uu u y x x xx x x x

∂ ∂ ∂ ε

∂ ∂ ∂

= + + ⇐ finite strain

For small radients, !e ass(me that the s(ared terms aresmall #ompared to the (n-s(ared terms and !e obtain thelinear, or small strain epression:

u x xx x

∂

ε ∂ = ⇐ small strain




Shear strain $rotation of line sements%: @ote that

εxy >1

$ %2

u u y x x y

∂ ∂

∂ ∂ + . Keometri#all&, ea#h of the t!o terms is an

anle as sho!n at the left. ε& is #alled a shear strain and

eometri#all& is 12 $a"erae% of the an(lar rotation of line

&

d

d&

d( &

d(

θ=π/2

θ∗

d

d&

d6 d&6




sements d d& !hi#h oriinall& form a riht anle. n

#ontrast, the engineering shear strain xy is defined as the s(m

of these t!o anles, ie, γ & > 2 ε& >

u u y x x y

∂ ∂

∂ ∂ + .

'he definition of the enineerin shear strain γ & from a raphi#al

"ie!point is an approimation $similar to the s(are rootapproimation made in ε %. From the eometr& abo"e,

6 6 #os 6 6 6dx dy dx dyθ • =r r r r. Define γ & to be the s(m of the

an(lar rotations, ie, γ & >

u u y x x y

∂ ∂

∂ ∂ + . $enineerin shear strain%

#os 6 #os$ 2 % sin 2u u y x

xy xy x y xy xy

∂ ∂ θ π γ γ γ ε

∂ ∂

= − = ≈ = + =




Js in the s(are root approimation made for ε $for the

eometri#al interpretation of strain%, an ass(mption of smallIrotations has been made in definin the shear strain ε& .




Some eamplesConsider a bar of lenht ),

fied at the left end, and!ith a for#e < applied at theriht end. 'he bar stret#hesan amo(nt () at the rihtend. f the enineerinI

strain in the dire#tion is defined to be $#hane in lenth%$oriinal lenth%, then εxx > (L).

We #o(ld also find the epresssion for ($% and appl& the

mathemati#al definition of εxx . We ha"e t!o bo(ndar&#onditions on ($%: ($0%>0 and ($)%> (L. Jss(me

1 2$ %u x C C x= + !here C1 and C2 are #onstants. Jppl&in ;.C.s

)

<

($%

( )




i"es C1 > 0 and C2 > (L) so that ($% > $(L)% . 'h(s the strain

is i"en b&

u x xx x

∂

ε ∂ = > $())%.




Eample: Jt the point LJL sho!n on the !heel, the displa#ementfield has been determined to be as

follo!s $(sin the finite elementmethod%:

2 2

$ , % $0.48 1.41 0.0

2.22 8.1 %10

xu x y x xy y

x y in

−

= + −

− +2 2

$ , % $ 0.2 1.?2 1.?

34. A.04 %10

yu x y x xy y

x y in−

= − − +

−

$ , % 0 z u x y ≈

Determine the strains at point LJL !hi#h is lo#ated at >in,&>in.

<oint J




2 2

$ , % $0.48 1.41 0.0

2.22 8.1 %10

xu x y x xy y

x y in−

= + −

− +2 2

$ , % $ 0.2 1.?2 1.?

34. A.04 %10

yu x y x xy y

x y in−

= − − +

−

L, :L

L, L

$0.A? 1.41 2.22%10 A.4A 10 x y

x xx

x y

u x y in in x in in

xε

= =

− −

= =

∂= = + − =∂

8

8L, :L

2.01 10 y

yy

x y

u x in in

yε −

= =

∂= = −

∂

8

8L, :L

14.2A 10

2

y x xy

x y

uu x in in

y xε −

= =

∂ ∂= + = ÷∂ ∂

, @ote: 2 xy xyγ ε =

0 xz yz zz ε ε ε = = =




@o!, do aain, b(t #onsider finite strain.

Comparin the t!o res(lts, is it M to ass(me small strains forthis problem5



undeformed

deformed


Some 'ho(ht Eer#ises

S(ppose !e ha"e an 88 s(are area o(tlined on a larer #h(n/ of planar material $li/e a plate%. S(ppose the plate is loaded in the -& plane so that the s(are is displa#ed and deformed to a re#tanle assho!n belo! ne! -& #oordinates startin at lo!er left #orner andoin CCW are: $8,%, $20,%,

$20,12% and $8,12%. <i#t(re sho!sdeformedI and (ndeformedIarea. What are &o(r (esses for the strains5 'here is stret#hin in

the dire#tion, so 0 xxε > . 'here

is no stret#hin in the & dire#tion,so 0 yyε = . Jll riht anles

remain riht anles, hen#e there is

no shear strain and 0 xyε = .




We #an a#t(all& sol"e for the displa#ements and strains sin#e !e/no! the initial and final positions of the #orner points.

Jss(me that the displa#ements are i"en b& $a 2-D #(r"e fit%:1 2 4 $ , % xu x y C C x C y C xy= + + +: ? A 8$ , % yu x y C C x C y C xy= + + +

Fo(r #onstants are #hosen be#a(se !e /no! information at

points. @o! appl& /no! #onditions for the fo(r #orner points:1 2 4 $0,0% 8 0 $0% $0% $0%$0% xu C C C C = − = + + +

1 2 4 $8,0% 20 8 $8% $0% $8%$0% xu C C C C = − = + + +1 2 4 $8,8% 20 8 $8% $8% $8%$8% xu C C C C = − = + + +

1 2 4 $0,8% 8 0 $0% $8% $0%$8% xu C C C C = − = + + +: ? A 8$0,0% 0 $0% $0% $0%$0% yu C C C C = − = + + +: ? A 8$8,0% 0 $8% $0% $8%$0% yu C C C C = − = + + +

: ? A 8$8,8% 12 8 $8% $8% $8%$8% yu C C C C = − = + + +

: ? A 8$0,8% 12 8 $0% $8% $0%$8% yu C C C C = − = + + +




n ea#h of the abo"e, the displa#ement at a point is set e(al tothe final positionI N initial positionI of that point. Sol"e for

the #onstants and s(bstit(te ba#/ into xu and yu to obtain:$ , % 8 0. 0 0 8 0. xu x y x y xy x= + + + = +

$ , % 0 0 0 yu x y x y xy= + + + =

)ast e(ation sa&s that displa#ement of all points in & dire#tion $

yu % is a #onstant, !hi#h is #onsistent !ith the pi#t(reI of

deformed and (ndeformed area. @o! #al#(late the strains:

0.: x xx

uin in

xε

∂= =

∂ $a positi"e "al(e indi#ates stret#hin%

0 y yy

u y

ε ∂= =∂

$no stret#hin in & dire#tion%

10 0 0

2

y x xy

uu

y xε

∂ ∂= + = + = ÷∂ ∂

$no shear strain%



undeformed deformed


S(ppose !e ha"e an 88 s(are area o(tlined on a larer #h(n/of planar material $li/e a plate%. S(ppose the plate is loaded in

the -& plane so that the s(are is displa#ed and deformed to a paralleloram as sho!n belo! ne! -& #oordinates startin atlo!er left #orner and oin CCW are: $8,%, $1?,%, $17,12% and$11,12%. <i#t(re sho!sdeformedI and (ndeformedI

area. What are &o(r (esses for the strains5 'here is nostret#hin in the dire#tion, so

0 xxε = . 'here is no stret#hin in

the & dire#tion, so 0 yyε = .

Ho!e"er, oriinal riht anlesare no loner riht anles, hen#ethere is some shear strain and

0 xyε ≠ .




S(ppose !e ha"e an 88 s(are area o(tlined on a larer #h(n/of planar material $li/e a plate%. S(ppose the plate is loaded in

the -& plane so that the s(are is displa#ed and deformed to a(adrilateral as sho!n belo! ne! -& #oordinates startin atlo!er left #orner and oinCCW are: $7,%, $1?.8,?%,$1A., 1.% and $10.8,12.%.

<i#t(re sho!s deformedI and(ndeformedI area. What are&o(r (esses for the strains5 tappears that there is shorteninor stret#hin in the and &

dire#tion hen#e, 0 xxε ≠ and0 yyε ≠ . riinal riht anles

are no loner riht anles,

hen#e there is some shear strain and 0 xyε ≠ .




Note on Tensor Stress & Strain Transformation

For 2-D, Ca(#h&Os form(la pro"ided the follo!in relation:

$ %nt n σ = •$4.41%

'he #omponent of $ %nt in the

dire#tion of the (nit o(t!ardnormal n of a s(rfa#e $or in O

dire#tion is:

O O $ %n x x nt nσ σ = = •

n’

x’

y’

xx

σ

yyσ

xyσ

yxσ

θ

n

$ %nt

nσ sτ

$ %nt r




S(bstit(te Ca(#h&Os form(la into the abo"e, and !rite in both "e#tor and matri notation:

O O

$1 2% $2 2% $2 1%

E E E

n x x

x x x

n n

n n

σ σ σ

σ = = • •

=

'he epressionE E E

n n nσ σ =

i"es the #omponent of stress,nσ , in the dire#tion of the (nit normal n $or in the dire#tion of

the O-ais !hi#h ma/es an anle θ CCW from the -ais%.

)ets do both the "e#tor and matri operations to sho! that

the& are the same. First, Ca(#h&Os form(la is




( )$ % $#os sin %

$ #os sin % $ #os sin %

$ #os sin % $ #os sin %

n xx yx xy yy

xx xy yx yy

xx xy xy yy

t n i j ii ji ij jj

i j

i j

σ θ θ σ σ σ σ

σ θ σ θ σ θ σ θ

σ θ σ θ σ θ σ θ

= • = + • + + +

= + + += + + +

r r

r r

@ote yx xyσ σ = . @o! do the remainin "e#tor operation for

nσ to obtain:

( )O O $ %

$ #os sin %

$ #os sin %

2#os 2 sin #os sin

#os sin

xx xy

n x x n xy yy

xx xy yy

it n

ji j

σ θ σ θ σ σ

σ θ σ θ

σ θ σ θ θ σ θ

θ θ +

= • = • + +

+ +

+=

= 2

r rr

r r

f !e do this in matri notion !e obtain:




[ ]#os

E E E #os sinsin

2 #os 2 sin #os sin

xx xy

n

yx yy

xx xy yy

n nσ σ θ

σ σ θ θ σ σ θ

σ θ σ θ θ σ θ

= =

= + +2

$.4%

'he abo"e is #alled the stress transformation equation $see

e. .4 in the notes%. 'he stress transformation transformsstresses from an $,&% #oordinate s&stem to an $O,&O% s&stem

!here O is rotated b& an anle θ CCW from the -ais.

We #an similarl& sho! that the strain transformation isi"en b&:

O O

$1 2% $2 2% $2 1%

E E E

n x x

x x x

n n

n n

ε ε ε

ε

= = • •=




'he (antit& nε is the #omponent of strain in the dire#tionof a (nit normal n . nε is often #alled the unit elongation in

the n dire#tion $(st as xxε is the (nit elonation in the i or-#oordinate dire#tion%. @otes:1. ;oth E σ and E ε are se#ond order tensors.

2. Jll se#ond order tensors follo! the same transformationform in transformin from $,&,9% to another orthoonal#oordinate s&stem $O,&O,9O%, i.e.,

O O

E E E

n x x n n

n n

σ σ σ

σ = = • •

=and




O O

n x x n n

n n

ε ε ε

ε

= = • •=

4. 'he same transformation applies to

moments of inertia of a #ross-se#tionJ $!hi#h is also a se#ond order tensor%:

2 xx A

y dA≡ ∫ , xy A xydA≡ ∫ ,

2 yy A

x dA≡ ∫ With respe#t to the O-&O #oordinate

s&stem at some anle θ, !e ha"e:2

O O $ O% x x A y dA≡ ∫

x

y

θ

x’

y’




Can also et O O x x b& appl&in the #oordinate transformationto the -& moments of inertia !ritten as a matri: $n is (nit

"e#tor in O dire#tion%:

E xx xy

yx yy

≡

, then O O n x x n n= =

Stress, Strain and Deformation in Solids

Documents

Transcript of Stress, Strain and Deformation in Solids