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p d σ=(D -D )d ε (1)式(1)中，D 为弹性矩阵，p D 为塑性矩阵。

弹性矩阵D 的形式为422000333242000333224000333000000000000000K G K G K G K G K G K G K G K G K G G G G ⎡⎤+--⎢⎥⎢⎥⎢⎥-+-⎢⎥⎢⎥=⎢⎥--+⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎣⎦D (2) 体积模量3(12)E K μ=-，剪切模量2(1)E G μ=+。

在应变空间内，塑性矩阵可表达为1()T f fA ∂∂=∂∂p D D D σσ(3) 式中，()()T T p f f f f A B ∂∂∂∂=--∂∂∂∂D D σσσσ(4)f为屈服函数；p σ为塑性应力，p p =σD ε；1/2(()())T p T pT p f f f f B f f f ωθε∂∂⎧⎪∂∂⎪∂∂⎪'=⎨∂∂⎪∂∂∂⎪⎪∂∂∂⎩σσI σσσ(5)[111000]T '=I ；p ω为塑性功；p θ为塑性体应变；p ε为等效塑性应变；κ为反映加载历史的参数。

当p κω=时当pκθ=时 当p κε=时对于Drucker-Prager 模型，其屈服条件为120f I J α== (6)1x y z I σσσ=++，22222221()2x y z xy yz zxJ S S S S S S =+++++，α为材料常数。

22f J α∂''=∂I σ (7)222Txyzxyyzzx S S S S S S '⎡⎤=⎣⎦S (8)22()32f K J J αα∂'''==+∂DD I I σ (9)222()()(3)92T T T f f A K K G J J ααα∂∂'''===+∂∂D I I σσ (10)1()T f f A ∂∂=∂∂p D D D σσ22222222222222112123113211212311321121231131121121121(3)(3)999(9)(9)(9)(T TT T T TK K K GJ J K G K G K G J K G J K G J m mn ml S m S m S m mnn nl S n S n S n ml nl l S l S l S l S m S n S l ααααααααβββββββββββββ''=++''''=++++++=p D I I I I S SS 211211222311221321231231231122231231132232113113113113212113223113)()()S S S S S S m S n S lS S S S S S m S n S lS S S S S ββββββββββββββββββββ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⋅⋅⎢⎥⎢⎥⋅⋅⎢⎥⋅⋅⎢⎥⎣⎦令43p K G =+，23q K G =- 弹塑性矩阵可表达为2112123113211212311321121231132112112112112112223112213123123123112223()(p m q mn q ml S m S m S mq mn p n q nl S n S n S n q ml q nl p l S l S l S l S m S n S l G S S S S S S m S n S lS S G βββββββββββββββββββββββ------------------=-=-----⋅-⋅----⋅-ep p D D D 21231132232113113113113212113223113)()S S S S m S n S l S S S S G S ββββββββββ⎡⎤⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥⎢⎥-⋅⎢⎥----⋅-⋅-⎢⎥⎣⎦令122(9)K G J βα=+，229K Gβα=+，1112m S ββ=+，1222n S ββ=+，1332l S ββ=+ 2222222211222222111222299(9)(9)(9)(()(9)9x x x x K G S S S K G K G J K G J K G J S m K G J K Gαααααββαα⎡⎤=++⎣⎦++++==+=++p D22222221222222222221112122299(9)(9)(9)()((9)9(9)9()()y x x yx y K G S S S K G K G J K G J K G J K G J K GK G J K GS S mnαααααααααββββ⎡⎤=+++⎣⎦++++=+++++=++=p D2222222132222222221112133299(9)(9)(9)()((9)9(9)9()()z x x zx z K G S S S K G K G J K G J K G J S S K G J K GK G J K GS S mlαααααααααββββ⎡⎤=+⎣⎦++++=++++++=++=p D2221422111211211222222(9)(9)()()(9)(9)(9)xy x xyx xy G S S K G J K G J S S S S mK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2221522111212312322222(9)(9)()()(9)(9)(9)yz x yzx yz G S S S K G J K G J S S S mK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2221622111211311322222(9)(9)()()(9)(9)(9)zx x zxx zx G S S S K G J K G J S S S S mK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D222221222222222111212229(9)(9)(9)()((9)9(9)9()()x y y xx y S S K G K G J K G J K G J K G J K GK G J K GS S mnααααααααββββ⎡⎤=+⎣⎦++++=+++++=++=p D 2222222222222222122222299(9)(9)(9)(()(9)9y y y y K G S S K G K G J K G J K G J S S n K G J K Gαααααββαα⎡⎤=++⎣⎦++++=+=+=++p D2222222232222222221222133299(9)(9)(9))((9)9(9)9()()z y y Zy z K G S S S K G K G J K G J K G J S K G J K GK G J K GS S nlαααααααααββββ⎡⎤=+++⎣⎦++++=++++++=++=p D2222422122211211222222(9)(9)()()(9)(9)(9)xy y xyy xy G S S K G J K G J S S S nK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2222522122212312322222(9)(9)()()(9)(9)(9)yz y yzy yz G S S K G J K G J S S S S nK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2222622122211311322222(9)(9)()()(9)(9)(9)zx y zxy zx G S S K G J K G J S S S nK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2222222312222222221112133299(9)(9)(9)()((9)9(9)9()()x z z xx z K G S S S K G K G J K G J K G J S S K G J K GK G J K GS S mlαααααααααββββ⎡⎤=+⎣⎦++++=++++++=++=p D222232222222222122213329(9)(9)(9)((9)9(9)9()()y z z yy z S S S K G K G J K G J K G J S K G J K GK G J K GS S nlααααααααββββ⎡⎤=+++⎣⎦++++=++++++=++=p D2222222233222222133222299(9)(9)(9)(()(9)9z z z z K G S S K G K G J K G J K G J S S l K G J K Gαααααββαα⎡⎤=+⎣⎦++++=+=+=++p D2223422133211211222222(9)(9)()()(9)(9)(9)xy z xyz xy G S S K G J K G J S S S S lK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2223522133212312322222(9)(9)()()(9)(9)(9)yz z yzz yz G S S K G J K G J S S S lK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2223622133211311322222(9)(9)()()(9)(9)(9)zx z zxz zx G S S S K G J K G J S S S lK G K G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2224122111211211222222(9)(9)(()9(9)(9)xy xy xx xy G S S K G J K G J S S S S mK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2224222122211211222222(9)(9)(()9(9)(9)xy xy yy xy G S S S K G J K G J S S S nK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2224322133211211222222(9)(9)(()9(9)(9)xy xy zz xy G S S S K G J K G J S S S lK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D222112244222()()(9)(9)xy xy S K G JK G J βαα⎡⎤===⎣⎦++p D1122232452222(9)(9)9xy yz xy yz S S S S S K G J K G J K G ββααα⎡⎤===⋅⎣⎦+++p D 1122132462222(9)(9)9xy zx xy zx S S S S K G J K G J K G ββααα⎡⎤===⋅⎣⎦+++p D 2225122111212312322222(9)(9)()()9(9)(9)yz yz xx yz G S S K G J K G J S S S mK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2225222122212312322222(9)(9)()()9(9)(9)yz yz yy yz G S S S K G J K G J S S S S S nK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2225322133212312322222(9)(9)()()9(9)(9)yz yz zz yz G S S K G J K G J S S S lK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D1122232542222(9)(9)9xy yz xy yz S S S S K G J K G J K G ββααα⎡⎤===⋅⎣⎦+++p D222123255222()()(9)(9)yz yz S K G JK G J βαα⎡⎤===⎣⎦++p D1132232562222(9)(9)9yz zx yz zx S S S S S K G J K G J K G ββααα⎡⎤===⋅⎣⎦+++p D2226122111211311322222(9)(9)()()9(9)(9)zx zx xx zx G S S K G J K G J S S S mK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2226222122211311322222(9)(9)()()9(9)(9)zx zx yy zx G S S K G J K G J S S S nK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D2226322133211311322222(9)(9)()()9(9)(9)zx zx zz zx G S S S K G J K G J S S S S S lK GK G J K G J ααββββααα⎡⎤=+⎣⎦++==+⋅=+++p D1132122642222(9)(9)9zx xy zx xy S S S K G J K G J K G ββααα⎡⎤===⋅⎣⎦+++p D 1132232652222(9)(9)9zx yz zx yz S S S S S K G J K G J K G ββααα⎡⎤===⋅⎣⎦+++p D222113266222()()(9)(9)zxzx S S K G J K G J βαα⎡⎤===⎣⎦++p D。