1、12.1 剑桥模型第十二章岩土材料本构模型 Roscoe K H, Schofield A N and Wroch C P (1958),On the yielding of soils,Geotechnigue,8(1),22-53 Roscoe K H, Schofield A N and Thurairajah A H (1963), Yielding of soils in states wetter than critical, Geotechnique, 13, 211-240 Roscoe K H and Burland T B (1968), On the generalise
2、d stress-strain behaviour of wet clay, Eds by J.Heyman and F.A.Lechie, Engineering Plasticity(Cambridge University Press),pp.535-609剑桥模型是英国剑桥大学的Roscoe等人于1958-1968年发展起来的，最初针对流经剑桥大学cam河附近的粘土提出的，所以称为Cam-clay模型。剑桥模型中，土体在剪切试验的大变形阶段，它趋向于最后的临界条件，即体积和应力（总应力和孔隙压力）不变，而剪应变还不断持续的发展和流动的状态。 Schofield A. and Wroth
3、 P (1968), Critical State Soil Mechanics, London: McGRAW-HILL. Wood D M (1990), Soil Behavior and Critical State Soil Mechanics, New York: Cambridge Press.因此剑桥模型又称为临界状态模型，相应的土力学框架称为临界状态土力学。等向压缩曲线设比体积plnv11oxpln0ln p正常固结线(NCL)ev +=1正常固结线（NCL）01 e+00pvevvln1 ppex+=+=abpav ln=回弹线(SL)pbv ln=回弹线(SL)应变量等向
4、固结xp0p00evln1 ppex+=00pvln1 ppex+=在三轴压缩子午面上，相同土样具有唯一的破坏线-临界状态线(critical state line)CSLpqo临界状态线(CSL)CU试验CD试验Mpq =fM为破坏应力比；为破坏剪应力；fqp为有效静水应力peo正常固结线(NCL)NCLCSL正常固结粘土或重塑土，应力状态(p,q)与孔隙比e之间存在唯一对应关系临界状态在q:p:空间中的不排水路径在q:p:空间中的排水路径应力应变关系二维化)(31321p +=213232221)()()(21q +=321vdddd +=213232221d)dd()dd()dd(32d
5、 +=ppppp11 2 2 33 v ddddd pq +=+pp pp11 2 2 33ddddW =+pdpvpddd += qpW三维二维塑性功的二维表达式Cam-clay塑性势面（一般推导方法）pddqddpdpv=由塑性流动法则得到：剪胀方程：pdpdpvpdddd MpqpW =+=Cam-clay模型塑性功：pvpdddqMp= Cam-clay模型的剪胀方程：1pvpdddpq1M塑性势面：pqM =pdpvdddpdqddpdpv=d0dqqMpp +=一阶线性微分方程塑性势表达式oMpq =f( )pdpvd,d ( ),0gpq=pvd, ppdd, qln 0qgM
6、p Cp= +=相关联流动法则：屈服面与塑性势面重合ln 0qgM p Cp= +=塑性势面方程：pqoMpq =fxpconst=pv确定了以塑性体应变为硬化参数，与塑性势面相同的屈服面：pvpv(, ) 0fg pq= =等值线pv0ln =+= CpqpMf塑性势面方程：pqoMpq =f0=fxp初值点：()0, = qppx0lnln =+=xpMpqpMf00pvln1 ppex+=v00 01ln 011pkp k qfgep eMp = + =+以为硬化参数的塑性势面方程：pvconst=pvCam-clay屈服面Cam-clay塑性势面（张锋的方法）q/p与e的关系（相等p下
7、）*00(1 ) (1 )qqem D e D epp= = + = +*000lg(1 ) 1cvCepDeep = +v（p,q下）塑性体积应变（p,q下）*00lg1pcsvCC pDep =+*00ln1pvpDep =+*00ln1pvpfDep = +pv*00ln1pvpfDep =+pvpdd/dffpq = 由塑性流动法则得到*011fDpp e =+*1fDq p=pvpdd() 0dM=*011DeM =+00 0ln1(1)pvpfepMe =+Cam-clay应变计算vvddd d0ppff ffpqpq = + =塑性体应变增量的计算：()qppqMMpedd11d
8、0pv+=pvd塑性剪应变增量的计算：pddpqM =pdpvddpv1pddd =pqM由剪胀方程：() qpqMpMpedd11d1-0pd+=v00 01g ln 011pkp k qfep eMp = =+=+弹性应变增量的计算：ijkkijijEE dd1de+=()qppqMMpedd11d0pv+=() qpqMpMpedd11d1-0pd+=pqoMpq =fBconst=pvconst=v不排水应力路径等值线pv修正剑桥模型屈服面（魏汝龙）plnvo0 =BJGJG1 =2 =3 =HK3 =1 =H BK() () (ln ln )BJ p BKee OBOK= = = (
9、 ) (ln ln ) (ln ln ) ( )(ln ln )JGe OHOK OHOK OHOK = = () () 0BJ JGee + =(ln ln ) ( )(ln ln )OB OH OH OK = ()OHOK OHOB =修正Cam-clay模型1968年，剑桥大学Roscoe和Burland采用一种新的能量方程假设，对初始剑桥模型进行修正pdpdd MpW =Cam-clay模型( ) ( )2pd22pvpddd MpW +=修正Cam-clay模型pqoMpq =fxp2xp01lnln1pv22200=+= pMqppef以为硬化参数的修正Cam-clay屈服面方程:
10、const=pvpvCam-clay模型特点Cam-Clay Model是反应压硬性和剪胀性最简单、物理意义最明确的弹塑性模型意义：局限性：研究条件：限于正常固结粘土的常规三轴试验压硬性：不能反映三轴压缩以外的屈服变形特性剪胀性：只能反映剪切体缩，不能反映剪切体胀软化性：只能反映应变硬化，不能反映应变软化剑桥模型的程序实现ijkkijijEE dd1de+=v0dd1epep =+()pv01ddd1MpqeMp = +()-1pd01ddd1p MqeMp = +剑桥模型修正剑桥模型22pv220()2d1()MdpdqeM p +=+22pd22 2202( )2d1()MdpdqeM M
11、 p +=+ +02(1 )d9(1 2 )(1 )edvdqvep+=+v1233(1 2 )dddd deeeevpE=+=03(1 2 )(1 )veEp +=132(1 )d()3edvddE +=Excel 表格实现汇编语言实现应变施加方法剑桥模型的力学响应v00 01ln 011pkp kqfep eMp = +=+()pijij ij ijffpfqdpq = = +200 011 1()11 1fk kqk qMpepeMp eMp p = = + + +11 22 3313ppp = =12 13 230ppp = =13ijijp=01()13ijijfp k qMpeMp
12、p = +11 22 33 112222221111 22 22 33 33 11 12 13 23112( ) 2( ) ( 1)122()()()6( )3( )2 q pq +=+=011fkq eMp=+22212 23 3122222222211 22 22 33 33 11 12 13 23 21 31 321()()()21= ( ) ( ) ( ) 3( ) 3( )2q =+ij3( )2ijij pqq=122222221211 22 22 33 33 11 12 13 23126122()()()6( )32q q=+=03( )112ij ijij pfq kqeMpq
13、 = +0()3( )1132ijij ijij ij ijqM pffpfq k ppq eMp q =+= + +2202331( ) 132ij ijij ij ij m mmsJffpfq kpq e MMJ =+= + +v00 01ln 011pkp kqfep eMp =+=+2v00 031ln 011pmmJkkfep eM= +=+ij m ij ijs= +vv0pijpijffdd +=v() 0ijkl klepklpijfffDd dp +=v() 0ijkleklpij kl iiffffDd + =vijklijkleklijepii ij klfDdff f
14、fD = + vv()( )( )( )ijkl kl ijklijklijklijklijkl ijklijklijklepeij kl klkleklijekleklpii ij kleeij kleklepii ij klfdDdd DdfDdfDdff f fDffDDDdff f fD = + = + Teeep eTTeffDDDDff fD = +111000 = 式中pvfgAp = ijkl ijklkl ijijklijklij klgfDDDfgAD +应变施加方法修正剑桥模型的力学响应()pijij ij ijffpfqdpq = = +22220()1( )fMpeM p =+ +01lnln1pv22200=+= pMqppef22021( )fq eM p =+ + eTeep eTTeffDDDDff fD = +111000 = 式中
