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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/
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