彈塑性矩陣推導(dǎo)VIP

1、考慮材料的塑性，具增量形式的本構(gòu)關(guān)系可表達(dá)為d g= (d-dp)d 式(1)中，d為彈性矩陣，d p為塑性矩陣。彈性矩陣d的形式為k 4g k 2g33k g k g 3322k -g k -g33000000k 2g 0 0 0 3k2g0003k-g00030g0000g0000g體樹犍k 廣，剪切模量g卮在應(yīng)變空間內(nèi)，塑性矩陣可表達(dá)為dpad f (d f)t式中,a (-)td(二)tdb(t(t (tof為屈服函數(shù);建為塑性應(yīng)力，/d/；f t ft6一(titf(tf7(-)t(-)1/2(t (ti 1 1 1 0 0 0t ； p為塑性功；p為塑性體應(yīng)變；為等效塑性應(yīng)變;為反

2、映加載歷史的參數(shù)。對(duì)于drucker-prager模型，其屈服條件為(6)j22(s2sy為材料常數(shù)。(tsxsy(8)a (-)t d(t(tszt2szx2sxy2syzd d( i (t1 t st)(3k i2、j229k 2 g(10)1gtg tdp2p 9k 2(3k gi -s)(3k it j2= st)j29k2 2t3kgi st3k gsit9k 2 g(9k 2g)h(9k 2 g)、j2(9k2 mmnml162m1s23m1s13mmn2 nnl1s2n1s23n1s3nmlnll21&2l1 s23l1&3l182m1s2n1 s?l(1s2)21s122 s?

3、31s122s31s23 m1s23n1$23l1 s|22s23(1s23)1s132s2363m1s3n國(guó)1s32s21s32s23( 1s3)dpad (d 2g2g)j2sstf il j22令pkrq k2g 3彈塑性矩陣口表達(dá)為2p mqmnqml1s12m1s23m1s13mq mnp2 nqnl1s12n1s23n1s13ndepd dpq mlqnlpl21s12l1s23l1 si3l21s12m1 si2n1s12lg （島）1s122s231s122s131s23m1s23 n1s23l1s122s232g ( 83)1s132s231s13m1 si3n1sj1 s1

4、32s121s132s232g ( s3)令1g3kam1s112，n1s222 , l1 s33(9k 2g)j229k2gdp 11(9k9k29k 2g2 3k g sg (9k 2 g) 口(9k3k gsxdp 122 g)j29k2 29k 2 ggsx79kka=r)2 (3k g q20(9k 2 g), j2112)23k g (9k 2 gmsx(1s112)( 1s22_3ka,9k-22) mn=)(=g (9k(9k 2 g) jg s 2 2 g)j2sxg2dp 13(9k(1s11dp 14(9k_3ka,9k2-=) gg)j22sxszsxsxyg)j2y9

5、k2 2 9k 2 gg2 g)j22)( 1s333k g(9k 2(_2k(9k 2 g)d3kp 15 (9k 2( 3k(9k 2 g)d3kp 16 (9k 2(3kg2 (9k 2.(9k 2 g)j2g sg)2 yz.(9kgg) . j2sx)g2 (9k 2sx)g)j2g.(9k 2 g)j2sxy1s112)1 2162mdp(9k 2 g) .(9kszx 一(9kg2 g)j219k2 2 3k g9k 2 g (9k 2gsx2xg)j22)( 1s22sxsyz g)j2 yg s .(9k 2 g)j2 yzg2jzsxszxsx).(9k 2 g)j2sxg

6、)、, j23k g1s112)1s23 mszx1s112)1s13163m_3ka 9k 22) mn=)( g2sy(9k 2 g), j2gsy(9k 2 g)j2(9k3kag21 syg)j2 y(9k(1s12=)gdp 22(9k9k2 29k 2g_ 3k gg (9k 2 g)、工sydp23(7=(9k(1s22dp 242 g)j2sy9k2 29k 2 ggsyg)j22)( 1s333k g(9k 2 3k(9k 2 g)d3kp 25(9k 23k(9kdp 262 g)3k(9k 23k_3ka_ )29k 2 g3k g2sz(9k 2 g). j23ka9k

7、2) nl=)(g (9g2 (9k 2(9kgg) j2syz3k g q2 sy(9kg), j2221s222) n3k g(9k 2 g)gsz2zkg)j2g)j2sysxysyg2(9k 2 gug2-2(9k3ka ) gsysz g)j2 y sy) g)j2g2(9k 2(9k 2 gj2sysyz g)j2 y y(9k 2 g)j2 s .(9k g g)j2 syzg - g2g)j2 zx (9k 2 g)j28y zx(9k 2 g) (9kg g)j2 sy)(9k 2 g)j21 s221s222)2)2)1s121s231s131s23 ndp 31(9k(i

8、sii9k2 29k 2g_ 3k_gg (9k 2 g)、j22 g)j22)(1s33c 3kasx -=9k g2) ml)(sx (9kg(9k 23k_g_ s2 g),j2 zszg)j2(9k3ka9k 2dp329k2 29k 2 ggsyg)j22)( 1s333 k_g(9k 2 g)=a)(9k 2 g2) nlsy3k g2= sz 小，/(9k 2 g)j2(9kg s 3ka;(9k 2 g)j2 z 9k 2g22 g)j-szsx2=) gg21 szsyg)j2 y(9k(1s23=)g匕乙一吁氣（9,x6）p飛陽(yáng)少）* oo9乙吧八，口9乙呵z一少，6）.

9、，09 ，叱9乙）6）hax封（。乙6）岫0-axs2r(o 7gx6);=(zs夕（916）廣乙，6: 5i11行乙（乙z 4 濘9 z x6) ax 封(9 x6)計(jì) d99 1沖(9乙16)，人 干(9乙16)19.x6,玲（9乙6）收9乙，6）.q9 x午(9x6)p x夕9 乙x6) 9,6戶 51.封(9-6)9 x9x 4 濘(9 7 16)xq axq c2r (o 7 x 6)ag / cs2r(o96）廣（9 x6）乙（乙|附3r(o乙x6)zs*sd乙r（9乙濘（9乙網(wǎng)9，6)xz s2r(o (zs=s(do-回xz x6)gx(99 da乙 x6）za八夕（9乙16）

10、s&宗乙ax -(9 7 x6) 飛 9ax a 濘(9s&t9 (zsz x6)gzasaxs2(/(99x6)xz x6)g封（9矽以9x6） 甘99洗 金9 xx(91s dax6)/xz x6)g9 y2r(o ,x6) 9 1 x67 7-6g g過(guò)） dadp3k g51 (9k 23ksyz/9k 2syzsx(9k g ) j2dpi (9k 23k gg)j252: (9k 2 g) 口3kgsyz9kdp2 g (9k 23k gg)j2c、g-sx) :2syz(9k 2 g)j2_gss(9k 2 gj2-、 g八sy) .2syz(9k 2 g)j2531s112)2

11、)1s23i8m1 (9k 23k9kg) 口g (9k 2syzgsyzsz(9k 2 g)j2g)j2dp543k gs s2 jxy jyz(9k g).，sz) 一g (9k 2 g)j2g1s332)1s231s23idp3k g55(9k 2 g)、j2s2z( .(9k(9k 2gsxy g)j23k9k 2= syzg1s12g)j2sj(1s23)2dp56-3k_gs s _2yzzx:(9k 2 g),j21(9ksyzg)j2_3k9k 2 g1s132s23dp3k g612(9k 23kszx號(hào)g)j2szxsx9kdp 622 g ,(9k 23k g: (9k 23k9kg)jg (9k 2sx)-g)j2(9ksg2(9k 2 g)j2sy)g)j2dp 633k g(9k21 (9k 23kszx (9k-gdp 649k 2 g .(9k3k g2 g)j22jszxsxy(9k g),.j2sz) (9kg(9k 2szxg)j2szxsyg g)j2 sxszx