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1、BRIEF NOTES 0 . 0 0 . 5 1.0 1.5 2 . 0 2 . 5 Load Ratio Nx/Nx c Fig. 2 Load-frequency relationship (key as for Fig. 1) Results and Discussion. Numerical results were calculated for the static deflection and fundamental natural frequency of vibration for square plates having various degrees of initial
2、 imperfection. The imperfection and subsequent static and dy- namic displacements were all taken to be of the single term form, sin(irx/a)sm(Try/a), corresponding to the fundamental vibration and buckling mode shape, with the central displace- ments Z0, Z, and H, appropriately. For the Airy stress f
3、unction series (5) and (10), the first three symmetrical beam modes were taken (p, q = 1,3,5) and, in equation (6), the first series was summed to infinity and the double series was summed over m, n = 1, 3, 5 . . . 11. A value of 0.3 was used for Poissons ratio. The values calculated for and f were,
4、 respectively, 1.000 and 0.177 for Case (1) and 0.730 and 0.418 for Case (2). These agree closely with those obtained from a Rayleigh-Ritz dis- placement solution (Ilanko and Dickinson, 1987), using an equivalent number of terms, which gives 1.000, 0.178, 0.731, and 0.417, respectively. The results
5、are shown graphically in Figs. 1 and 2, where the considerable effect of the initial im- perfection may be observed. The stiffening effect of the tan- gential edge restraint in Case (2) is also very evident in Fig. 1, where the flat-plate buckling load may be seen to have been increased by 37 percen
6、t and the square of the nondimensional central deflection, x2, is very significantly reduced. The effect of the tangential restraint on the natural frequency parameter (oj/0)2 is rather more complicated since the frequencies tend to increase due to the in-plane stiffening effect but to decrease due
7、to the reduced out-of-plane curvature. References Bassily, S. F., and Dickinson, S. M., 1977, The Plane Stress Problem for Rectangular Regions Treated Using Functions Related to Beam Flexure, Int. Journal of Mechanical Sciences, Vol. 19, No. 11, pp. 639-650. Hui, D., and Leissa, A. W., 1983, Effects
8、 of Geometric Imperfections on Vibrations of Biaxially Compressed Rectangular Flat Plates, ASME JOURNAL OF APPLIED MECHANICS, Vol. 50, pp. 750-756. Ilanko, S., and Dickinson, S. M., 1987, The Vibration and Post-Buckling of Geometrically Imperfect, Simply Supported, Rectangular Plates Under Uni- axia
9、l Loading. Part I: Theoretical Approach, Part II: Experimental Investiga- tion, Journal of Sound and Vibration, Vol. 118, No. 2, pp. 313-351. Kapania, R. K., and Yang, T. Y., 1986, Buckling, Post Buckling and Non- Linear Vibration of Imperfect Laminated Plates, Proceedings of the ASME Pressure Vesse
10、ls and Piping Conference, Chicago. Accurate Dilatation Rates for Spherical Voids in Triaxial Stress Fields Y. Huang4 Introduction In the course of a study of cavitation in elastic-plastic solids (Huang, Hutchinson, and Tvergaard, 1989), it was discovered that the well-known and widely used formula o
11、f Rice and Tracey (1969) significantly underestimates the dilatation rate of an isolated void subject to stress fields with moderate to high triaxiality. The purpose of this Note is to indicate why earlier analyses lead to underestimates and to provide accurate results. Attention is limited to a sph
12、erical void in an infinite rigid- perfectly plastic solid characterized by the Mises yield con- dition, ae = aY, where (Te = (3sijsij/2)l/2 with s,j as the stress deviator. Remote from the void, the nonzero stresses satisfy a, = a2 and T3-(j = 0. The Rice-Tracey high triaxiality approximation for th
13、e effect of the remote mean stress, am= -,-j+ Vjj), 0 is the infinite volume exterior to the void, S is the surface of the void, and ,- is the unit normal vector to S pointing out of the void. The solid is incompressible so the velocity field must satisfy vkik = 0. The deviator stress in equation (2
14、) is that associated with e through the yield condition, i.e., su = oyiu/(3kuku/2)l/2. (3) The remote strain rate ey is prescribed and an additional velocity field y*is defined so that e(j = eTj + etj with efj = - (vfj + vfj). (4) Among all additional fields satisfying v* = o(r3/2) as r00, the exact
15、 field minimizes 1 is clearly For lower triaxiality, i -o (a- (3am 1 ff, -r-= 1.28 ( 4 exp - , - s i (10) e V aY/ 2oy/ 3 aY gives a reasonable fit to the present results in Fig. 1. At remote uniaxial tension, am/aY= 1/3, the error is less than 1 percent. For l/3xm/Tyl, the maximum error of (10) is l
16、ess than 5 percent. 4 Effect of Elasticity on Dilatation Rates Elasticity has a significant effect on the dilatation rate of an isolated void at high triaxiality. Huang et al. (1989) cal- culated the normalized dilatation rate of a void in an elastic- perfectly plastic solid at several levels of yie
17、ld stress to Youngs modulus, aY/E, including the rigid-perfectly plastic limit, aY/E = 0, presented in this work. The elastic-perfectly plastic solid has a cavitation limit in that the dilatation rate becomes unbounded as am/aY approaches a limiting value. For a Pois- sons ratio of 0.3, the cavitati
18、on limit of am/aY is about 3.6 for aY/E= 0.005, 3.9 for aY/E = 0.003, and 4.7 for aY/E= 0.001. The cavitation limit is unbounded as aY/E-0. Elasticity clearly influences the dilatation rate at triaxiality levels above am/uY-2. Further details are given in the paper by Huang et al. (1989). Acknowledg
19、ments The author gratefully appreciates suggestions and comments from Prof. J. W. Hutchinson. Discussions with Prof. J. R. Rice and his comments are also appreciated. This work was supported in part by the Materials Research Laboratory under Grant NSF-DMR-86-14003 and in part by Division of Applied
20、Sciences, Harvard University. References Budiansky, B., Hutchinson, J. W., and Slutsky, S., 1982, Void Growth and Collapse in Viscous Solids, Mechanics of Solids, The Rodney Hill 60th An- niversary Volume, H. G. Hopkins, and M. J. Sewell, eds., Pergamon Press, Oxford, pp. 13-45. Journal of Applied M
21、echanics DECEMBER 1991, Vol. 58 /1085 Downloaded From: http:/appliedmechanics.asmedigitalcollection.asme.org/ on 03/13/2013 Terms of Use: http:/asme.org/terms BRIEF NOTES Huang, Y., Hutchinson, J. W., and Tvergaard, V., 1989, Cavitation In- stabilities in Elastic-Plastic Solids, Division of Applied
22、Sciences, Harvard Uni- versity, Report # Mech-153, to appear in J. Meek. Phys. Solids. Rice, J. R., and Tracey, D. M., 1969, On the Ductile Enlargement of Voids in Triaxial Stress Fields, J. Mech. Phys. Solids, Vol. 17, pp. 201-217. A New Expression of the Energy Theorem in Discrete Mechanical Syste
23、ms Rene Souchet5 Recently, Kane and Levinson proposed testing numerical in- tegrations by a checking function using an expression of the energy theorem involving the well-known Hamiltonian func- tion. This paper deals with a new expression of this energy theorem that gives an alternative to the Kane
24、-Levinson check- ing function. 1 Introduction Kane and Levinson (1988) proposed to test numerical in- tegrations of equations of motion of discrete mechanical sys- tems by using the checking function C defined as: C=H+Z, H=V+K-,-Kn. (1) In this formula, Kis a potential function for given forces, K2
25、and K0 are the usual homogeneous parts of the kinetic energy K, and Z is a function defined by the differential equation: Z=ft, Qu , Qm l up), (2) q = (qu ., q) being generalized coordinates, and uu ., up (p . dt dt so that we have C=V+K2-K0 + Z. Now, if we develop (3), we obtain: d (5) (6) dt V+K2)
26、-mR2QQsm2ql - mR2Q2 QiSinqiCosqi + /nQQsin2 = 0 where two terms cancel out. Therefore, we can define a func- tion Z by Laboratoire de Mecanique Theorique, Universite de Poitiers, 86022 Poitiers, Cedex, France. Manuscript received by the ASME Applied Mechanics Division, Mar. 26, 1990; final revision,
27、 June 15, 1990. so that Zi= mR fl UisinqCOsqi, ui = ql, C, = V+Ki + Z, (7) (8) is the new checking function. It is the purpose of this paper to show how one can directly obtain the functions Z in general cases. In order to perform this work, it is necessary to briefly survey the main features of the
28、 dynamics of nonholonomic systems for both funda- mental and technical reasons. 2 Kinematics We consider a dynamical system E, containing rigid bodies Sa, a=l,.,A, whose configuration in a reference frame N= 0;xiX2X3) can be specified by coordinates q=qr r=,.,n: OP=F(t,q(t),PD) (9) where P0 is the i
29、nitial position of the particle actually situated in P. The velocity v of particle P in N is defined as: frfdqr dt - Y V f l tfl- tf- (10) (11) where ifir and ift are functions of t, q, and P, and are referred to as the holonomic partial velocities of P in N. Following Kanes scheme (Kane and Levinso
30、n, 1985), we introduce generalized speeds ur, r= ,.n, in such a way that: n qs=1Wsrur + Xs,s= n (12) where Wsr and Xs are functions of / and q. Then we can write: v(P) = v + v v=Jvrun (13) r=l where it is possible to have v and v, from (11). Now we suppose that E is a nonholonomic system, subject to
31、 (n-p) constraints expressed as: p uk=YAkrur + Bk, k=p+l,.,n, (14) r=l where Akr and Bk are functions of t and q. Among the relations (14), we have some motion constraints, i.e., some restrictions imposed on the positions and the velocities of rigid bodies Sa, a = 1,. ,A, and the constraints of rigi
32、dity for each body of the system, i.e., some relations between parameters q,.,q when it is made use of hybrid coordinates, e.g., Euler param- eters, for rigid body motions. Using constraint relations (14), we have: v(P) = V+ v,u, V= J Vrur. (15) Following Kane (Kane and Levinson, 1985), V and V, fun
33、c- tions of t, q, and P, are referred to as the nonholonomic partial velocities of P in N. The above decompositions are, of course, applicable to any discrete system E. But, when is made up of rigid bodies, we can write for every particle P of the rigid body S: v(P) = v(G)+o(S) AGP (16) where u(S) i
34、s the angular velocity of S in N, and G is the mass center of S. Now we use the following decomposition: p o(S)=Q(S) + Qut(S), Qu(S) = JQur(S)ur (17) 1086 / Vol. 58, DECEMBER 1991 Transactions of the ASME Downloaded From: http:/appliedmechanics.asmedigitalcollection.asme.org/ on 03/13/2013 Terms of Use: http:/asme.org/terms