弹塑性矩阵推导

1、弹塑性矩阵推导考虑材料的塑性，其增量形式的本构关系可表达为(1)d = (D -Dp )d 式(1)中， D为弹性矩阵， Dp 为塑性矩阵。弹性矩阵 D 的形式为KGKGKG000333K2GK4GK2G000333K2GK2GK4G000333000G000000G000000GDGE 2(1 )(2)体积模量 K 3(1E2 ) ，剪切模量在应变空间内，塑性矩阵可表达为1DADp(D)T(3)式中，A ( f)T D f 为屈服函数；fp )TD f(p为塑性应力，D p ；(4)p时p时T p fpI T fp fp( f )T( f )1/2 (5)I 1 1 1 0 0 0T ； p

2、 为塑性功；p 为塑性体应变； p 为等效塑性应变；为反映加载历史的参数。对于 Drucker-Prager 模型，其屈服条件为I1 J2 0(6)1 2 2 2 2 2 2I1 x yz ， J22(SxSySz )SxySyzSzx ，为材料常数。Dp1DADpD(2 J2 S(7)SxSySz2Sxy2Syz2Szx(8)I 2 J2 S)A ( f )T D(D)T12(3K9K 2 G9K 2 29K 2IITGJ2 3K G (9K 23K I G SJ2T 1 T2 J2 S )(3K IJ2S)29K 2 GS)(3KIT(9)(10)G) J2 Imn ml 1S12m 1S

3、23m 1S13mmn2n nl 1S12n 1S23n 1S13nml nl l2 1S12l 1S23l 1S13l1S12m1S12n1S12l( 1S12 )1S121S132S232S12G ST )J23K G(9K 2 G) J21S23m 1S23n 1S23l 1S12 2S23 ( 1S23) 2S23SI TG2 (9K 2G)J2SST1S131S121S13 (1S13m1S13n1S13l2S132S231S13)443G，弹塑性矩阵可表达为p2 mqmnqml1S12m1S23m1S13mqmnp2 nqnl1S12n1S23n1S13nDep D Dpqmlqn

4、lpl21S12l1S23l1S13l1S12m1S12n1S12l2G ( 1S12 )1S12 2S231S12 2S11S23m1S23n1S23l1S12 2S232G ( 1S23)21S13 2S21S13m1S13n1S13l1S13 2S121S13 2S23G ( 1S13)23令13KaDp 11(9K 2 G) J29K 2 2 3K9K 2 G1S11 2 ，n 1S22 2 ，(9K 2 G) J2 Sx3Ka3K(9K9K 2 G ) (2 G Sx2 G) J2 x221S112 )mG2(9K 2 G) J29K22p 129K 2GG(9K2G)J21S112

5、)(1S229K22DSx(9K 2 G GDp 13GSx2x2 G)J23K G(9K 2 G) J2 SyG )( (9K9K 22 ) mn3K G S(9K 2SxG2 G)J2G) J 2Sy9KG2S S(9K 2 G) J2 SxSy 3Ka ) 222G(9K 2 G)J2( 1S112 )( 1S333K G (9K 2Sx9K2) ml3K G (9K 2 ( 3K(9K 2 G)3K 15 (9K 2Dp 14Dp( 3K(9K 2 G)3KDp 163K3K G2(9K 2GG (9K 2 G) J2G) J2 Sz 3Ka )(G) J2 SxSzG2SxSz(9K

6、 2 G)J 2 x z 3Ka 9K 2G)SG) J2 SxyGG2 (9K 2SxSxyG)J2 x xyG2(9K 2 G) J2 x (9K 2 G)J2 S GSyzG) J2 SyzG(9 K 2GSx)xy1S112)1S121S12mG2 (9K 2Sx)G)J2SzxG(9 K 2 G) J2 zx (9KG2Sx )K 2 G) J2( (9K 2 G) (9G)J2 SxSyzG2Syz(9 K 2 G)J21S112)1S231S23mG)J2 SxSzx(9K 2 G)J2 Szx1S112)1S131S13mDp21(9K 2( 1S11Dp 22Dp(9K23(

7、9K( 1S229K 2 2 3K G S2 2 Sx 9K 2 G (9K 2 G) J 2 x GSx3Ka2)(G)J29K 2 G (9 K2 )( 1S222 ) mn229K 2 2 3K G 9K 2 G (9K 2 G2 G)J Sy2 G)J29K 2 29K 2 GG2 G)J Sy2 G)J22 )( 1S33SyG) J2 y 3Ka )2 ( 9K 2 G ) (3K G2Sz(9K 2 G) J 23Ka )(G9K2) nl3K G2Sy(9K 2 G) J 2 yG2 G)J Sy2 G) J2G 2 S S (9K 2 G) J2 SySx 3Ka ) 9K

8、2 G3K G S (9K 2 G) J2 Sy2 )2 n2G2Sy22y (9K 2 G) J2 y1S223K G2(9 K 2G(9 K 2 G)J2G) J2 SySzG22 Sy SZ (9 K 2 G)J2 y Z 3Ka )G9K 23K G(9 K 2 ( 3K(9K 2 G)3K 25 (9K 23K2 G)3K(9 K 2DpDpDp24(9K26( 3K(9K 2 G)Dp 31229K 2 22(9K 2( 1S11G) J2 SxyG(9K 2GSyzG) J2 SyzG(9K 2GG2 (9K 2G)J2 Sy )G(9KSySxyG)J2G(9K 2 G) J2

9、 Sxy2)1S121S12nSySyzG)J2 y yz GS(9 K 2 G)J2 yz SG2G) J2 zx (9K 2 G)J 2 G (9K 2 G) J2G)J2 Sy)Sy SzxSy )Szxy (9K 2 G) J 2 zx2)2)1S131S23n1S13n3K G9K 2 G (9K 2GSxG)J22 )( 1S33Sx3K GSzG ) J2 Sx (9 K 2 G) J2 Sz 3Ka )( G S 9K 2 G (9K 2 G) J2 2) mlG2(9K 2 G) J 3Ka ) 9 K 2 GSzSx2p 329K 2 29K 2GG(9K2 G)J21S2

10、22)(1S339K22p 339K 2GGD(D(9K 23 K G S(9K 2 G) J2 SySy3Ka )(9K 2 G(9 K2 ) nl3K GDp 34 2SzG)J23K G(9K( 3K(9K 2 G)3K 35 (9 K 2 ( 3K(9K 2 G)3K 36 (9K 2 ( 3K(9K 2 G)(9K 2 G) J2 Sz3K2a)2 (2GG2 (9K 29KDpDpG) J2 SxyG(9K 2GSyzG) J2 SyzG(9K 2GSz)G) J2 zG(9KDpDp3K G2S (9K 2 G) J 2G2Sz2 G) J29KG2 S S(9K 2 G) J2

11、 SzSy3Ka )2 G )3K G S (9K 2 G) J2 SzS332) 2 l2Sz SxyG)J2GSxy(9K 2 G)J2 Sxy 2Sz) G)J2SzxG2G) J2 Szx (9K 2G2Sz)(9K 2 G) J2G)J2 SzSyzG(9K 2 G) J2 SyzSzSzxG)J2GSzx(9K 2 G) J2 zx(9K1S331S33G22 G)J2Sz22)2)2)1S121S231S131S12l1S23l1S13l2GSxyG2G) J2 xy (9K 2 G) J2G S ) GG (9K 2 G) J2 x (9K 223K G SG23K41 (9

12、K 2 3K2Sxy Sx9KG)J2 Sxy1S112)1S121S12m42SxySy(9 K 2 G) J2 Sxy (9K 2 G)J23K G S ) G S 2 G (9K 2 G) J2 y (9K 2 G)J 2 xy3K GS G2S S(9K 2 G) J2 Sxy (9K 2SxySz3K G29KDp 43G)J2( S ) G 9K 2 G (9K 2 G) J2 z (9K 2G)J2 Sxy1S221S332)2)1S121S12n1S12lDpDpDpDp51DpDpDpDpDpDp3KG(9K2G ) J23KG(9K2G) J23KG(9K2G ) J23K

13、G444546SxySzx(9K G2(9 K 2G22(9K 2 G)J22 G 2 Sxy ( (9 K 2 G)J2 Sxy )Sxy SyzGSxyxy yz2 xy(9K 2 G) J2SG)J2xy( 1S12 )3K2Syz1S12 2 S239K 2 G3K9K 2 GSzx1S122S13SyzG) J2 yz GG (9K 23K G(9K 23K2(9 K 2 3K2SyzSx5253G)J2SyzG) J2 yzGG (9K 2 G) J23K G(9 K 2Sx )2x (9K 2 G) J2 G2SS(9K 2 G) J2 SyzSyGSyz1S112)1S23m3

14、K9KG) J2 Syz G G(9 K 2(9K2G ) J23KG(9K2G ) J23KG(9K2G) J2545556Syz ( (9K 223 K G S SSxySyz613K G2(9 K 2 3K29K 2 GSy )2Syz(9K 2 G) J 2 G2(9K 2 G)J2SyzSz1S222)1S23nG)J2Sz)(9K 2 G) J22Sxy(9K 2 G) J2G2GSyz)2G)J2S S G SyzSzx(9KG)J2SyzSyz2)1S23l3K9K 2SyzG1S122 S23( 1S23) 23KSzx9K 2 G zx1S132S23SzxG2G) J2 zx (9K 2 G)J2GSx )G(9K 2 G) J2 x (9K 2SzxSxG)J2Szx( 1S112 )1S131S13mDp3K G62 (9 K 23K2SzxG2G) J2 zx (9K 2 G)J2 GG2Sy )2Szx ( 1S22 2 ) 1S131S13n(9K 2 G) J2(9K 2 G)J223K G S G263 (9K 2 G) J2 Szx (9K 3K G2 G(9 K 2( 9K 2 GDp( 9KG)J2SzxSy2 G)J2Sz)