This preview shows pages 1–17. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: I. p; ’/’ i /
Exam #2 ME572 (Prof. Pan) Name I 4' / a”; K.
10:00AM ‘ 12300 PM, December 14, 2007 or 4:00PM  6:00 PM, December 17, 2007 Open book, open note, and no homework solutions. Box your answers and include units. 1. Consider a slab of sheet metal subject to equal biaxial resultant tensile force L in the
x1 and x2 directions as shown. The dimensions in the x1 and x2 directions are denoted as w and the thickness of the sheet is denoted as t. The original dimensions
of the sheet are denoted as we and to. The Mises yield criterion is used to describe the plastic behavior of the sheet. Since the plastic deformation is large, rigid plastic
behangL is assumed here. The relation of the equivalent tensile stress 6 and the equivalent tensile strain E is given as 6 = K?” . Determine the maximum load Lmx in terms of K, n, Wu and to. 2. We consider a polymeric material under uniaxial loading conditions as shown. The uniaxial tensile yield stress 0', is 60 MPa but the uniaxial compressive yield stress 0‘ is 75 MPa. We can model the yielding of the polymer by the DruckerPrager C yield function as
f =0. +f3lwm 0ge =0 where 0e is the effective stress, ,u is the pressure sensitivity, 0' is the mean stress, I" and age is the generalized effective stress. Here, the effective stress 0e is deﬁned as
3 0‘2 : (—2— 01710;)” 2 where 0;. are the deviatoric stresses. The mean stress am is
.  o . . u . . .
defined as 0',” = For Simpliclty, we consrder this material as a rigid perfectly plastic material. Therefore, age is a constant. (a) Determine the value of the pressure sensitivity ,u . (b) Now we conduct a pure shear test where we apply a pure shear deformation to a
material element, determine the shear yield stress based on the yield function. (0) Consider a material element under uniaxial tensile loading conditions in the x]
direction. Determine the plastic Poisson ratio Vp (= —d82”2 / d611 = —d£3”3/d£f1) with the assumption of the associated (normality) plastic ﬂow rule.
(d) Consider a material element under uniaxial compressive loading conditions in the x1 direction. Determine the plastic Poisson ratio V p (2 —a’€2"§/d£{’l =d83’g/d€{’,) With the assumption of the associated (normality) ' plastic ﬂow rule. ’( In transformation—toughened ceramics, the phase transformation is mainly controlled
by the hydrostatic stress. For simplicity, when the mean stress 0'," equals to a critical value 0",, phase transformation takes place. The phase transformation criterion is
defined as ' 0m = Okk = 0! Assume that the linear elastic asymptotic cracktip field is applicable to determine the
phase transformation size and shape. Consider a crack under plane strain mode I loading conditions. The inplane asymptotic elastic cracktip stresses including the T stress are K’ cos§(1—sin§sin£)+T U”=,/2m 2 2 2 K’ cosg (1+ sin —6—sin E) a” : 1/2727” 2 2 2 K, a. 6’ 36
cos—sm—cos— 0X" 1/ 2727 2 2 2 Derive the analytical expression for the transformation radius r, as a function of 6 for a given set of K I , 0', and T near a crack tip under plane strain mode I loading conditions. Give the expression for the transformation radius r, for T = 0 also. U.) 4. Consider an axisymmetric problem of two centrally loaded circular diaphragms. The
cross sections along the symmetry plane of the connected circular diaphrang are
shown. The two circular diaphragms have a radius a and the thickness h as shown.
The two diaphragms are connected along the circumference. A load P is applied
along the center line of the upper diaphragm and a load P is applied along the center
line of the lower diaphragm as shown. The displacement w at the load application
point for the upper diaphragm and the lower diaphragm can be derived from the plate
bending theory as _ 3Pa2(l—v2) W
47th3 where E is the Young’s modulus and V is the Poisson’s ratio. (a) Derive the expression for the energy release rate in terms of the geometric
parameter, the material parameters, and P. (b) Specify the fracture mode (mode I, II or mixed mode) for the crack in this
diaphragm specimen and give an argument for your answer. Derive the stress
intensity factor solution for the crack. <4? I C Pn’r/f Paﬂ? . H A/
B: : KE’” 3 o
192E€rm¢rVLQ me r77 ferns 470 K, 04,41; gig, Is .» WM; rimfressx’é7/f’fg O = Z 6 3
L 64 M [MD ah:
Mt .
.: O—Moé Xéjofo £29: 42f
Wu 0 W: 4! I
~ (Ta/oi; K2; a 7M mag, 0&5?— "79 8%”);
__ a2 (a a)
— out: >6 8%; :3 9P<59=A7Z
, 192, , g ‘9
\ 4) Ham ‘6) 8
‘Sfress £122,619. ($.41: O— E“: = 330‘
LIZ] U 0 0 EEC/J: écr 0 O
o 6‘ o o (fac7 @
o o 0 0 o —§U I/on R7383 ~a OILS}: g/ Wfﬁ7°”ﬁ/Q 00 5X1".
HajaI — flasﬂc '5 if: s:
S) OIEII=OQ512 : "é'dxgay ' 22(22222"3\:533. L615 5“ £ng = I2 7’ 0 Li]: 8 ° 0
0 dz 9 a 3 °
0 0 #146 O O 9‘18 PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN —I
r MK (AEDM K 2, = 2m KCAEYM  1m I< (22>M" \ }<()_5)”‘
I<(15)W [Dot—~Aal : O. (A: 3 =5. ﬁmm. LMOI OCCM r: PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN gbeeI/f . g—OLJ“ ‘ I‘S
La“) ’bé‘f’c'Y‘mJM/L, @ WWW +emian. ,/ (pm/a
ESE]: 0? O o"? CQ/J': IF EC?
0 o ow; I 3 I ’io‘l
,L
C7 7. Z CI IL L L N
L33 2*I1’3‘7WI I‘TII
(CI—*1: +Wslom U2=I7l
Cow/1F ﬁzzCE
65
W = “TEli = 35:“
O 3.
”fwfmn tart
: m “Iii/g _ 1: O‘
I 3 07/“ 0%
D7 (Hg = Uﬁe
U?
VIE
me‘bresjoé fag—Jr
,0.
:I: “U? "FEDTFOO‘LQ = c). 97(é‘4) : $9. W: “127%,er :07 (VI/Dog
jg. _ PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN W1<l+f1§2 3. age lilacre. Cf“: {acMDak
60
50+ 73 : 66:8
Lam? ’
UT ‘0 '~ 9% LOW QTz’Wp"
7r~fg : 064,
60+ (63 ~ 741 25
5/4“ ’ 6/“
504—7? : ¢f_éo
a;
___ Ii W60) _ V‘Kf;
M {Mar “173: = WM $160+ gxomc; my WA,
: “a gﬁod‘ﬁgf 7 66196 r4?” .L—ﬂk "l‘ PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN do) Pure 9W ﬁes? Cons7ol.2r 5L tomEm fest 0 [Bibi—£3. PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN Cc) Pmch Po?sson‘s mtfo'. VJ’"’0I2f
(I WAVA/Xiaﬂ. “694437814. PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN .L—C —2__ l X‘ S; Y; ._
A \
VIE f ” A 7‘
,__.” _ E I + I; 5+” PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN GI) VF mmfrcsfran +c51‘ n? = 0M. xyi JE F .110)
A8“) I JongvFﬁ’ +3/Ax& WA? and mmfaresion PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN %’DI>I€IM 3 phase "ﬁrmsqgrmk orffeﬁon 5;“ = CE I o
ﬂame sffﬁfm. rmooiz I,
> 532 — Maggy £2%:0~ EICE~WUHUgﬂfo
K\ "’4'
_ V/Cz
.32“ F K1603: : PI K & ’ K; g :g
.3 I J—Dié XlC‘oS: +7” 4 Vim Yléos: +T?
: Ci.” E K1 xLCOngvIT? : @‘t r) Phage {Ta/“5% zon Tjﬂ ' Hi Wmﬂmﬂ I A , I L , Kg:
. _—_ u ‘1 ~ 2
It m; I ((—3) a; m GIMME) W) I
I
PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN 47KB If C41) kviSérIl/e Hue (daftSe mfg f P AWITeoI "9 jeaoI é! £42 maféfr'ej F: 9&5f7f. cmﬂme 4* tag WU. PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN Dial—Ills FfIaftITD/I‘ Ipr (yr?on I °FIW m5 Griz E/—(/<ZS’) e I '  (H71) 2—
sttm/n f1 _ ? KI)
#Ixrkz/mmetr/‘c 5“. p/Mze, shat/32. 9: £453 (5). ’9 57/3) 9:1. PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN PHILIP PARK CIVIL & ENVIRONMENTAL ENGINEERING UNIVERSITY OF MICHIGAN ...
View
Full
Document
This note was uploaded on 10/13/2009 for the course ME 572 taught by Professor Pan during the Spring '07 term at University of MichiganDearborn.
 Spring '07
 pan

Click to edit the document details