Advanced mechanics of materials PDF

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Lecture notes for entire semester. Professor Shen Spring 2018....

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Chapter1:Stress&Strain Stressatapoint Internalsurface (internalforce,causedbythe“removed”upperpart) Stresscomponent Vij =_______ i:directionofoutwardnormalvectortothecutsurface. j:directionofinternalforcecomponent.

Ex Simplepulling F (totalforce)

~Loadingis_ _______throughoutbody;

y

neednotworryabout______ x z

area:A (i → y)

F (j→ y)  1

F(totalforce)

Ex (cont’d) y

i → x,Fx =Fy =Fz =___

x

 Vxx =__, Vxy =__, Vxz =__

z

(right halfactingonleft half)

t

n

n: normal,t: tangential F Fn

45q

Ft

A _____ cos 45$ F  V nn _________ 2 F _________ V nt 2 An

Fn An

F/ 2 2A F/ 2 2A

IfusingMohr’scirclerepresentation: Shearstress

V xx ( 0)

V yy (

F ) A

Normalstress

90q (physically____)

_________ (

1F 1F , ) 2 A 2 A



2

GeneralStressElement Vyy Wyz Wzy

y x z

V zz

Wyx

Wxy Vxx

Wzx Wxz

Signconvention: • Normalstress:tensile “”;compressive “” (textbooktendstousemagnitude,supplemented withsymbolsT orC) • Shearstress:arrowonpositive facepointingin ________axis “”;arrowonnegative facepointingin________axis “”;….

Matrixrepresentation:

Note:Frommomentbalance,_____________________________.  ___independent componentsinstressmatrix.

Whenaplaneisspecified,theforceperunitareaactingonthisplaneiscalled ____________or_________,{S},withcomponents. ____ 3

______normalvectoroftheplane



Or,

… e.g., y x

Wxz

Wxy Vxx

On(100)plane(“x”face):

z

Similarly,on(010)plane(“y”face): Vyy Wyz Wzy Vzz

Wyx

recall

Wxy V xx

Wzx Wxz 4

Ex Considerstressstateatamaterialpoint: (some unit)    



  

[ V]=

y(j) x(i) z(k)

& & & & Onplanewithunitnormal, n 0i  0 j  1k

stressvector

or,______________



& & & & n 1i  2 j  3k On(123) plane,unit normal____ stressvector

 

5

StressTransformation (ChangeofCoordinates) y

t

Vtt T

x

Wnt

n

Unitnormal forthen‐plane:

Vnn Unitnormal forthet‐plane: _________________

ToobtainVnn, Wnt andVtt fromknown[ V],

Vyy

Wxy Vxx

 sx ½ ®s ¾ ¯ y¿

Vnn

Stressvectoractingonn‐plane, Innerproduct

 Inmatrixnotation, Similarly, Wnt =________

Vtt =________

6

y MPa andT =45q.

Ex Givenstressstate

t

Obtain Vnn , Wnt and Vtt .

x

45q

n

 Thenew(innt‐coordinatesystem)isMPa. 

 

y x

=

t



n 7





Ex (cont’d)



y x

=

t

y t



n

x

45q

IfusingMohr’scircle:

n

Shear

( Vtt , Wnt) 



Normal

2u45q

( Vxx ,W xy)

Note:Conventionforplacementofshear onMohr’scircle: Ashearstresscausingaclockwise rotationinthephysicalelementisplotted _______thehorizontalaxis; ashearstresscausingacounterclockwise rotationinthephysicalelementis plotted_______thehorizontalaxis.

 8

PrincipalStresses Stressvectorsperpendiculartotheplanestheyacton,withno ______component.

=

y x

Principal__________ Principalstresses

Ex Usethesameexampleabove, Shear

MPa.

Wmax Radiusofcircleis u50=70.71

V2





V1 

Normal

 PrincipalstressV1 =__________________MPa  PrincipalstressV2 =__________________MPa.

(Vxx , Wxy)

Notethemaximum(possible)shearstressis(orradiusofcircle,70.71MPa). 9

3DStressTransformation {n}:unitnormalvector(directioncosines):

Traction(stressvector){s}=_______,or

Vnn =__________ Theshearpartofthestressvector(tangentialtoplane) isthen____________.

10

3DPrincipalStresses ~Principalstresses are ,whenusingthe three__________________asthecoordinateaxes. Mathematically,principalstressesare_______________of[V ];principaldirectionsare _______________of[V]. or,





“singular”,ifsolutionexists _______________

  Equationisofform:

11

(cont’d) I1 =_____________ I2 = I3 =__________

Show!

~CharacteristicEquationofmatrix[ V];I1,,2,I3:__________of[V]. Theydonotvarywithcoordinatesystem (sincethesevaluesdeterminetheprincipalstresses).

ThreesolutionsofVp aretheeigenvalues. SubstituteVp backto_____________  Get3correspondingeigenvectors{p} (mutuallyperpendicularprincipaldirections)

Note:Itiscommontorepresentthe3principalstressesas V1,V2 andV3,with ____________.

12

Ex Givenstressstate

MPa

 Characteristicequation:_________________________________   ? Principalstresses:_________,_________,_________. Toobtainprincipaldirections,let

    

ForVp =360MPa:

 Cantakepx =___,py =____,pz =___;butneedunitvector



1 3   2 13 0 2



13

Ex (cont’d) ForVp =‐160MPa:  Cantakepx =___,py =____,pz =___



3 2  1 13 0 2



ForVp =‐280MPa:

 px =___,py =____,pz =_________

Notethe3eigenvectorsaremutuallyperpendicular. Iflet becomes

bethenewcoordinateaxes,thenthestressstate .

14

OctahedralStresses 2

Takeprincipaldirectionsasthe1,2,3‐axes; principalstressesare V1,V2, V3. Octahedralplane,withunitnormal{n}= 1

3

Recallstressinvariants         ______________                     ___________________

  det

 _________

1

3  0 0 1 0  0 3 = Stressvectorontheoctahedralplane{s}=________= 0 0  1 3

innerproduct 1

Stressvectornormalto thisplaneis___________ = magnitude

1 3

direction

 ,  , 

3 1 3 1



3 15

(cont’d)

2

1 1

Stressvectornormalto thisplane =

3

 ,  , 

3 1 3 1



3

=______________________

1

Voct ,theoctahedralnormalstress (alsoknownas“hydrostaticstress”)

3

Stressvectortangenttothisplane(shear)is_____________ 1

=

 1   󰇛     󰇜 3 3 

1

3 1 3 1

=

3

1

Itsmagnitude   3   

1   󰇛       󰇜 3 1 1   󰇛       󰇜  3 3 1   󰇛       󰇜 3 

   



 /

   

~octahedralshearstress

Show!

Note:Woct =__________ ButI1 andI2 areinvariants  CanuseI1 andI2 forarbitrary coordinatesystems,and substituteback *Woct isimportantinpredicting______________ofmetals. 16

Strain Considerstretchingofbars: Toproducethesame extentof“deformation”,needto havedisplacement____.

G (displacement)

L0

 Tensilestrain=

2L0

Foranarbitrarybody: “”fortension,“”forcompression y

A

'x

A’

x

In3D, Hxx =____, Hyy =____,Hzz =____~normal straincomponents

Shear strains: y

 

'u 'y

      2  

Similarly,

D

'v

 

 

x

'x 17

Note:1.    ,   ,    2.TheJ’sabovearebasedonengineeringdefinition;insolidmechanics weoftenuse  ≡  ≡  ≡  3.The___independent componentsformstrainmatrix   followsthesametransformationruleasstress. t

y

  

  ,which 

e.g.,unitvectorsofthetwo“new“directions:{n},{t} n T

x

  



   





  

  

     

    4.Principalstrains:____________of[H] (indirectionsnormaltoplaneswith zeroshearstrain) 18

y

Ex

t torsion

   

cos 45°    sin 45 °  0

 

  

1

1

2

2

1

1

2

2

0

0

0

0 0.001 0.001 0 0 0

0 0.001 0.001 0 0 0

0 0 0

0 0 0

n

0 0.001 0 Given   0.001 0 0 0 0 0

45q

x

  0.001

1 2 1  2 0 1 2 1 2 0

1 2

1 2

0

2 0.001 0.001 0  0.001  2 2 2 0 1



 

1

1

2

2

0

0 0.001 0 0.001 0 0 0 0 0

2 1



2 0

Shearstrain(J/2)

UsingMohr’scircleforstrain:

y

0.001

t

n

Normalstrain

0.001

‐0.001

x

19

Chapter 2: Material Description Linear Elastic Model (Generalized Hooke’s Law) ~ Followed by all materials when deformation is sufficiently small; need some “elastic constants” to relate ____ and ____. In 1D (and 1D only), 𝜀 =

𝜎 𝐸

(E: Young’s modulus)

In general, all stress components may influence any strain component. e.g.,

𝜀𝑥𝑥 = 𝑓1 (𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑧𝑧 , 𝜏𝑥𝑦 , 𝜏𝑦𝑧 , 𝜏𝑥𝑧 )

𝜀𝑦𝑦 = 𝑓2 (𝜎𝑥𝑥 , 𝜎𝑦𝑦 , 𝜎𝑧𝑧 , 𝜏𝑥𝑦 , 𝜏𝑦𝑧 , 𝜏𝑥𝑧 )

etc.

linear function



𝐶11 𝜀𝑥𝑥 𝜀𝑦𝑦 𝐶21 𝜀𝑧𝑧 𝐶31 = 𝛾𝑦𝑧 𝐶41 𝛾𝑥𝑧 𝐶51 𝛾𝑥𝑦 𝐶61

𝐶12 𝐶22 𝐶32 𝐶42 𝐶52 𝐶62

𝐶13 𝐶23 𝐶33 𝐶43 𝐶53 𝐶63

𝐶14 𝐶24 𝐶34 𝐶44 𝐶54 𝐶64

𝐶15 𝐶25 𝐶35 𝐶45 𝐶55 𝐶65

__________ matrix

𝐶16 𝐶26 𝐶36 𝐶46 𝐶56 𝐶66

s yy

𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑥𝑦

t yz t zy s zz

y

tyx txy sxx tzx txz

x z 1

To simplify compliance matrix [C], consider strain energy. Uniaxial loading:

F

F

F d , _______ stored in solid

d (displacement)

Or,

s

, energy per volume (__________________); sample unit: N/m2 = J/m3

With e = _____ (under uniaxial loading) ⇒

e

Generalize to multiaxial loading, ⇒

,

and,



___ ; i, j = x, y or z.

same

; generalize:

⇒ ___________ ~ [C ] is symmetric 2

(cont’d) There are ____ independent elastic constants for general ____________ materials. ~ means materials with direction-dependent properties.

Number of constants can be reduced for less general anisotropy (i.e., with certain “symmetry” conditions).

e.g., a fiber composite lamina possesses ___________ symmetry. ⇒ ____ constants y

x z

 C11 C12 C13  C C C 0   12 22 23  C13 C23 C33   C44   0 C55    C 66 

3

For materials displaying cubic symmetry: ___ independent constants Using engineering notation ⇒

y x

same property

z

E: Young’s Modulus or,

n : Poisson ratio (

)

G: Shear Modulus

~ 3 independent elastic constants

For isotropic materials (same property in ____ direction): Can now derive independent constants using an example, based on cubic symmetry. Consider 2D pure shear loading. 4

(cont’d)

t

With any rotation of coordinates, say 45∘ ⇒ [s] becomes

y

, [ e] becomes

=

-t

t

n

x

Show!

(E, n remain the same; szz = 0 in the present case)

Recall 

t

⇒G =

for isotropic materials ~ ___ independent elastic constants

Summary: Hooke’s law for isotropic materials

5

Expressing [s ] in terms of [e ] (for isotropic materials):

6

2D Plane State

y x

Plane stress z

s xx t  xy  0

t xy s yy 0

0 0 0

~traction-free on z-plane

Obtain [e] through ___________. In particular, there is a non-zero

y

Plane strain

e.g., x z

Constraint (prevents z-displacement in specimen); also x- and y-displacements are not a function of z.

e xx  xy   xy e yy  0 0

0 0 0

From  There is a non-zero 7

Failure Theories Elastic description no longer valid when “failure” occurs; could mean: s

• Yielding (plastic deformation) for _______ materials • Fracture for _______ materials

e

Maximum Shear Stress Theory ( ______ yield criterion) ~ for ductile materials Failure (yielding) occurs if max. shear stress reaches shear yield stress ( _____ ) in uniaxial tensile test. tyield =

Note Mohr’s circle: s yield (tensile yield stress)

If principal stresses are defined to be

tmax

, then yielding occurs when

tyield 8

(cont’d) Consider plane stress with

, and

in arbitrary order:

Can draw _______________ on the “stress space”.

s2 syield -syield

If

syield -syield

If

, then

s1

( s3 ≡ 0)

,

tyield

⇒ _____________

~ vertical line in the first quadrant

If

,

⇒ _____________ ~ horizontal line in the first quadrant

⇒ _________________ ~ line in the fourth quadrant

∙∙∙

For stress state inside the envelope ⇒ elastic ; Failure (yielding) occurs if stress state reaches the boundary.

9

Maximum Octahedral Shear Stress (t oct) Theory ( __________ yield criterion) ~ for ductile materials Failure (yielding) occurs when toct reaches ______________in uniaxial tensile test.

Recall

;

Yielding at tensile test ⇒ For general stress state, yielding occurs when

⇒ Define von Mises effective stress Criterion is then ___________ If under plane stress (with s3 = 0) ⇒ ⇒

or,

~ an ellipse in s1s2-space (see next)

10

Comparison of the two criteria in plane stress Max. octahedral shear stress (Mises)

Many metals show yielding behavior between Tresca and Mises, but closer to Mises. Max. shear stress (Tresca) more conservative

Note: If not using principal stresses,

(recall t oct expression in Ch. 1:

)

⇒ Max. toct theory (Mises) is still _______________. 11

Ex Given:yieldstressinuniaxialtensiletestMPa.Forastressstateof ,

,

,willyieldingoccur?

Ifusingmax.shearstresstheory:(needtoidentifymax.shearfirst) ( Vxx , Wxy)



360

 270

y

⇒Yielding_______occur. x

(Vyy , Wxy)

Ifusingmax.octahedralshearstresstheory:

⇒Yielding__________occur. ~max.shearstress󰇛Tresca󰇜ismoreconservative

12

FailureTheoriesforBrittleMaterials Max.NormalStressTheory Failureoccursifmax.normal(_________)stressreachesultimatenormalstress(____) inuniaxial tensiletest. V2 V ult

Inplanestress:

Vult

Vult

V1

Vult *Themainproblemforthistheoryisthat,fortypicalbrittlematerials,theultimatestress incompression (VC)isactually________thanintension ( VT).

ModifiedMohr’sTheory(Mohr‐Coulomb) V2 V1 1  V T VC

V2 VT

VC

VT

V1 V1 V2 1  VT V C

VC

13

StressIntensityFactor Whenasharpcrackexists(orisassumedtoexist)inmaterial:

V Stressnearthe________canbeveryhigh impracticaltouse stress‐based design(todeterminefailure) 2a

 Topreventfailure,needtomakesurecrackdoesnot__________. Introducestressintensityfactor K V Sa (basedon“fracturemechanics”)

½ofcracklength

Crackpropagation occursifK reachesacritical value_____. Amaterialproperty,unit_________or_________ (alsocalled“fracturetoughness”)

Ex Ifgivencracklength2a =0.008m,nominalappliedstress50MPa: stressintensityfactor K

V Sa

50 S  0.004

____________

 HavetouseamaterialwithKcrit ___5.6MPam1/2,to avoidcrackpropagation 14

TypicalKcrit values:

󰇛MPa m󰇜

15

Note: Thereare___possiblemodesofcrackpropagation.

Mode I

Mode II (______ mode)

Mode III (_______ mode)

Needtousenominalshearstress W incalculating___and___

Formixed‐mode fracture,say,combinedmodesI andII,equivalent stressintensity factorKequiv =____________  IfKequiv reachesKcrit,crackbecomes________(willpropagate).

16

Chapter3:BasicStructuralMembers Rods :foraxialloading Beams :for________ Shafts :for________

ReviewofPureBending T R neutralaxis (H xx =0)

Forpointsatdistance___abovetheneutralaxis, L’ ...