SLIP-LINE FIELD THEORY OF TRANSVERSELY. Ruan Huai-ning (~'t~'~) (Hehai University. Nanjhrg) Wang Wei-xiang (~_e~)

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1 Applied Mathematics and Mechanics (English Edition, Vol. 15, No. 4, Apr. 1994) Published by SUT, Shanghai, China SLIP-LINE FIELD THEORY OF TRANSVERSELY ISOTROPIC BODY Ruan Huai-ning (~'t~'~) (Hehai University. Nanjhrg) Wang Wei-xiang (~_e~) (Chhur University of Geos&nees. Beo'ing) (Receive June 23, 1992: Communicated by Hsueh Dah-wei) A slip-line.field theory of transversely isotropi~' body is proposed in tire present paper" ia order to deal u'ith problems in geolog)' and geotechniques. The Gol'denhlat- The Kopnov one-dimensional failure criterion problem is emplo)'ed. of the motion The parameters of a rigid flying it are plate treated under as explosive functions attack of has an analytic tepriperature. solution only It is when applicable the to polytropic transverse index isotropic of media detonation in non-un(/'orm products temperature equals to three. In general, a./~eld. numerical The hasic analysis equations is required. of plastic In deformation this paper, are however, developed by while utilizing the associated the "weak" ru- shock behavior of the reflection shock in the explosive products, and applying the small parameter purles of flow are derived. By means oj" characteristic Ihle theory, slip-line slope formulas terbation method, an analytic, first-order approximate solution is obtained for the problem of flying and laws o/ variation of stress and velocity along slip lines are obtahwd. The indentaplate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities tion on of senti-it!finite flying plate media obtained is calculated. agree very Tire well theory with developed numerical in results this paper by computers. ma)' be Thus an analytic shnpl(/'ied formula hlto with many two classical parameters theories of high such explosive as Mises. (i.e. Hill. detonation and Coulomb velocity ones. and This polytropic index) for complicated estimation of theory velocity ma)' be of applied flying to plate geotechniques, is established. geological structures, petroleum hrdustrv, mining enghwering, etc. Key words slip-line field, transversely isotropic, non-uniform temperature Explosive driven flying-plate field, failure technique criterion, ffmds limit its equilibrium important use in the study of behavior of I. cladding Introduction of metals. The method of estimation of flyor velocity and the way of raising it are questions There are many problems in geology and engineering which must be solved by the Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal application approach of of solving theories in problem mathematics of motion and of mechanics. flyor is to Mechanical solve the following properties system of of rock equations and soil and governing geological the flow conditions field of detonation in the earth's products crust behind are very the flyor complicated, (Fig. I): The establishment of comprehensive and practicable theories is of theoretical and practical significance to the development of mechanical theories and the solution of practical problems in engineering applicat!on. ap +u_~_xp + au There are various kinds of faults in the earth's crust. Hafner studied faults by calculation of stress fields. First he calculated au stress au fields 1 y according =0, to the equilibrium conditions and then examined the areas using the failure criterion to find out potential faults t'l. in this method, equilibrium and failure are discussed as separately. as This does not conform to the actual situation. a--t Slip-line field theories are useful tools to the study of faults. In the theories equilibrium equations and failure criteria are handled p =p(p, simultaneously. s), Od~ analysed faults with slip lines I-~J. In geomechanics, limit equilibrium methods are also important to stability analysis of engineering projects. respectively, Sokolovski with the solved trajectory many R of problems reflected concerning shock of detonation foundations wave and D as slopes a boundary by the and applicathe tion trajectory of these F of theories flyor as t~l. another boundary. Both are unknown; the position of R and the state parameters D and by initial stage of motion of flyor also; the position of F and the state parameters of products 335

2 336 "Ruan Huai-ning and Wang Wei-xiang But classical slip-line field theories mainly study metals. These materials are homogeneous, continuous and isotropic. The mechanical properties of rocks and soils are, however, complicated. There are planar structures such as bedding planes and joints in the geological body. Thus it is often transversely isotropic. On the other hand, its behavior is affected by the changing temperature of the earth's crust, There are few slip-line field theories about such complicated media. It is necessary to construct new theories to solve problems encountered in engineering application. II. Failure Criterion of Rocks and Soils The Tresca criterion and the Mises criterion are widely.used in the classical plasticity theory. These two criteria neglect the influence of hydrostatic stresses and therefore tension and compression have the same effect. The Mohr-Coulomb criterion is usually used in rock and soil mechanics. It does not account for the influence of the intermediate principal stress. Hill proposed a criterion for orthogonal anisotropic metal in his plasticity theory. It does not apply to rocks The and one-dimensional soils. In order problem to study of the strength motion theories of a rigid of flying a geological plate under body, explosive some scholars attack has proan posed analytic the theory solution on only systems when with the plane polytropic of weakness index of by detonation generalizing products the Mohr-Coulomb equals to three. crite- In general, rion, and a numerical others proposed analysis three-dimensional is required. In this theories paper, however, by generalizing by utilizing the Mises the "weak" criterion. shock By behavior of the reflection shock in the explosive products, and applying the small parameter purthe analysis of various theories, the author considers that the Pariseau criterion and the Gol'- terbation method, an analytic, first-order approximate solution is obtained for the problem of flying denblat-kopnov plate driven by various criterion high are explosives suitable with for polytropic anisotropic indices rocks other and than soils but t41. nearly Saada equal made to three. experiments Final velocities on anisotropic of flying clays. plate He obtained verified agree that very the Gol'denblat-Kopnov well with numerical results criterion by computers. is vatid 15j. Thus In this criterion, an analytic transverse formula with isotropy, two parameters differences of in high behavior explosive between (i.e. detonation tension velocity and compression, and polytropic and the index) influence for estimation of the of intermediate the velocity principal of flying plate stress is established. are all involved. Thus it is used as the basis for the construction of the theory presented in this paper. The expression of this criterion is l 2 2 [P ( o, + ~, ) + R~, ] + -y [F 1. ( o, Introduction +,o, ) +.Hcr.~ + J" ( o,o, +.,7,o,) Explosive driven flying-plate technique ffmds its important use in the study of behavior of + I~,cr,+ 2M(r;~+ r~,) + 2Nrl,]-:-:-= t (2.1) where cladding cr,,crr,cr,,r,y,v~,,r,, of metals. The method of are estimation the normal of flyor and velocity shear and stresses the way parallel of raising to the it are coordinates, questions P. of common R. F. H. interest. J. I. M. N are parameters characterizing the state of anisotropy. It is applicable for transversely Under isotropic the assumptions materials of one-dimensional with y as axis of plane symmetry. detonation If anisotropy and rigid flying disappears, plate, the then normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow field of detonation P=R, F=H=-J=-I=2M/3=2N/3 products behind the flyor (Fig. I): (2.2) substituting Eq. (2.2)in Eq, (2.1) yields [ l[ 3 P (a, + o, + cr:) + yf+ [ (or, +~y +.or, ) ap +u_~_xp + -- (cr,cr, + or,or, + or,or,) + y 3 (r~,~+ =0, r~,+ rl,)]-~ = 1 (2.3) This is the Drucker-Prager criterion as widely used as in rock and soil mechanics. If hydrostatic stresses have no influence on failure, then a--t p =p(p, P--0 s), (2.4) and Eq. (2.3) becomes respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the 2 O 2 2 ". +F+[(a2-+g,+cr',)-(a,~,+cr,e,+cr,cr,)+o(r,,+Ty,+r,,)J}-~-I (2.5) trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters This D and is by the initial well-known stage of Mises motion criterion. of flyor also; Eq. (2,5) the position is applicable of F and for the metals, state parameters Eq. (2.3) for of products isotropic rocks and soils, and Eq. (2.1) for transversely isotropic rocks and soils. In the non-uniform au

3 Slip-Line Field Theory of Transversely Isotropic Body 337 temperature fields in the earth's crust, the material strength changes with temperature. In Eq. (2. I) the parameters are functions of temperature P=_P(T), R=R(T), F=F(T), H=H(T) J=J(T), I=I(T), M.=M(T), N=N(T) (2.6) where temperature is the function of position T-~.T(x,y,z) (2.7) The following formula is obtained when Eq. (2.6) is substituted in Eq. (2.1) ~ [P(T) (or,+ crz) + R(T)a,] + ~[F(T) (or- ~ +a2,) + H(T)a~ + ](T) (cr,aw+a,az) + I(T)~za, + 2M(T) (r-2, + r~,) + 2N(T)r',,]-t".= 1 (2.3) an Eq.(2.8) analytic is a solution general criterion only when adopted the polytropic in this paper. index It of deals detonation with the products transversely equals isotropie to three. body In general, with non-uniform a numerical temperature analysis is required. fields. For In the this sake paper, of however, convenience, by utilizing Eq. (2.8) the is "weak" also written shock in behavior the form of of the Eq. reflection (2.1), where shock there in the are explosive 8 parameters, products, which and applying may be the determined small parameter by uniaxial purterbation compressive, method, uniaxial an analytic, tensile, first-order biaxial compressive, approximate and solution simple is obtained shear tests for vertical the problem and of parallel flying to plate planar driven structures by various of rocks high and explosives soils. with polytropic indices other than but nearly equal to three. an III. analytic Basic formula Equations with two of parameters Plastic Flow of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established. According to the plastic potential theory, strain-stress relations may be obtained from The criterion Explosive of driven Eq. (2.8) flying-plate is used technique as the potential ffmds its function. important Substituting use in the study Eq. (2.8) of behavior in Eq. (3.1), of materials the associated under rules intense of impulsive flow are derived loading, as shock follows synthesis of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions l ~P +~U [2Fa,,+Ja,+Ia,] Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations governing the flow =yr field +-~U of detonation [2Ha,+J(a:+cr,)] products behind the flyor (Fig. I): (3.2) where ~, I p + lu-4r[2fa,+ ia,+ja,].l -2 ap +u_~_xp + -~-=U-~Mr.,,--~-=U-~Mr,., ~=U-I" y =0, Nr.. as as a--t U=F ( al + a: ) +.Hal + I ( a,a, + a,a,) p =p(p, s), + Ia,a, + 2M(r-2,+ r~,) +2Nr~, (3.3) Eqs. where (3.2) p, p, are S, u the are constitutive pressure, density, equations specific of the entropy present and theory. particle In velocity Eqs. (3.2), of detonation t!,, ~,, ~,, products q,,, ~,,, respectively, '~,, are the with normal the trajectory and shear R strain of reflected rates in shock the of directions detonation of wave the coordinate D as a boundary axes. and The the sum of trajectory the first F three of flyor equations as another of Eqs. boundary. (3.2) is Both are unknown; the position of R and the state parameters D and by initial stage of motion of flyor also; the position of F and the state parameters of products au ~.v=gs+ ~t+ ~s=~,(2p + R) /2 + I~U"~[(2F+'J+]) (a,+ a,) + 2(H+I)a,] (3.4)

4 338 Ruan Huai-ning and Wang Wei-xiang The volume strain rate ~,, is neither zero nor a constant. It is related with stresses, temperature, and material properties. Shear expansion and contraction and volume expansion of rocks and soils are involved in Eq. (3.4). If materials are isotropic and if tension and compression have the same effect, then ~v=0 when Eq. (2.2) and Eq. (2.4) are substituted in Eq. (3.4). The condition of plane strain is taken into consideration, which is common in geotechnical engineering. If plastical flow is parallel to xy-plane, or: is derived from Eqs. (3.2) (2P ~- I)r.(2PR- ])cr,- 4P (3 5) or, = 2F -- 2P ~ In the following, only a special series of the Gol'denb[at-Kopnov criterion are discussed because of the difficulty in mathematics. If,r=-H, I=H-2F (3.6) when Eqs. (3,5) and (3.6) are substituted in Eq. (2.8), then [ (a.--a,)' ]4-1 1 ' 4(I-D) I r:. ----_y Gtcr.+ -~_Gza,+K (3.7) where an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection MF shock in the explosive 2P(2F- products, H/2) and applying the '% small parameter purterbation D=.I method, an H(2F-H/2) analytic, first-order Gt= approximate - solution is obtained for the problem of flying ' [2MF(F-P")]'b i (3.8) plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final velocities of flying 2(RF plate + PH/2) obtained agree very well with 2F numerical results by computers. Thus an analytic G2=. formula... [2MF(F_p2)]4- with two parameters, of K= high explosive [2MF(F_p~)]4 (i.e. detonation, velocity and polytropic Eq. index) (3.7) for is estimation the failure of the criterion velocity of of the flying transversely plate is established. insotropic body under the condition of plane strain. If hydrostatic stresses have no influence,.substituting Eq. (2.4) in Eqs. (3.8) gives D#0, GI=G2=0 (3.9) Eq. (3.7) Explosive can be driven reduced flying-plate to technique ffmds its important use in the study of behavior of cladding of metals. The method of estimation [- (a'--crt')2 +vl, 14- (3 1D) t 4(1-D) of flyor velocity =K and the way of raising it are questions This is Under the Hill the criterion. assumptions If of materials one-dimensional are isotropic plane but detonation hydrostatic and rigid stresses flying have plate, influence, the normal substituting approach Eq. of solving (2.2) in the Eqs. problem (3.8) yields of motion of flyor is to solve the following system of equations D=0, GI=G2=G (3.11) Eq. (3.7) can be reduced to ap +u_~_xp + au [(~176 ]+ 2 +r~, =~G(cr,+a,)+K (3.12) y =0, This is the Coulomb criterion. If materials are isotropic and hydrostatic stresses have no inas as fluence, substituting Eq. (2.2) and Eq. (2.4) in Eqs. (3.8),we have a--t D=O. Gt=Gz=o (3.13) p =p(p, s), Eq. (3.7) can be reduced to respectively, with the trajectory R of reflected shock of detonation +r2-w ]+ wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position =K of R and the state (3.14) parameters This D and is by the initial Mises stage criterionl of motion In of the flyor construction also; the position of the of slip-line F and the field state theory, parameters the of other products basic equations are the.equilibrium equations

5 Slip-Line Field Theory of Transversely Isotropic Body 339 OCt I 0~ I oo. ~-~-+X(x,v)=o --~-+~+Y(x,v)=o Ox + y (3.1s) where X(x, Y) and Y(x, Y ) are body forces. Eqs. (3.5) and Eq. (3.7) are the basic equations for stress fields. If the mean normal Stress is denoted by it=, and the angle between the major principal stress and x-axis by 0, from Eq. (3.7) we have (h (0) - G~cos20) c% + Kcos20 or.= h(o)-o.5(g~-g2)eos20 (h(o) - G, eos20)(7,.- Keos20 (3.16) or,= h(o)-o.5(g~-g~)eos20 r,, = [0.5 (GI+ G2) crm+,k] sin20 h(o)-o.5(g,-g2)cos20 where h(o) = ~ l-dsin220 ]+ [:~ (3.17) an The analytic velocities solution parallel only to when the axes the of polytropic coordinates index are of denoted detonation by products v, and v,. equals ~ is to eliminated three. In in general, Eqs. (3.2) a of numerical the rules analysis of flow. is When required. the In geometric this paper, equations however, and by Eqs. utilizing (3.16) the are "weak" substituted shock in behavior Eq. (3.2), of the the velocity reflection equations shock in are the obtained explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. an analytic formula Ov,/Oy with + Ovw/Ox two parameters of high explosive sin20 (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established. Eqs. (3.18) are the basic equations for 1. velocity Introduction fields. If the angle between the major principal strain rate and x-axis is labelled with O t, from the Mohr circle of strain rate Explosive driven flying-plate technique ffmds its important use in the study of behavior of tg20,= ~,u/, Ov./Oydzj3v,/Ox materials under intense impulsive loading, shock = Ov,/Ox- synthesis Ov,/Oy of diamonds, and explosive welding (3.19) and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions When of common the interest. second equation of Eqs. (3.18) is substituted in Eq. (3.19), it is known that in general Under 0'4= the 0 9 assumptions Only when of materials one-dimensional are isotropic, plane from detonation Eqs. (3.11), and rigid (3.18) flying and plate, (3.19) the normal approach of solving the problem of motion of flyor is to solve the following system of equations tg20' = tg20 ~ (3.20) It means that the principal axes of stress coincide with those of str~/a'n rate. Eqs. (3.7), (3.]5) and (3.18) are the basic equations of plastic fiow of the transversely isotropic body. From the five differential equations the five variables ap +u_~_xp (7,,(7,,r,,,v, + au and v, may be-obtained. IV. Stress Fields y =0, In the solution of stress fields, as u. and as O are used as the basic variables. The stress equations may be derived from Eqs. a--t (3.7) and (3.15). Eqs. (3.16) satisfy the failure criterion of Eq. (3.7). When they are substituted p =p(p, in the s), equilibrium equations of Eqs. (3.15), the stress equations are obtained, which are respectively, with the trajectory A Ot7. Oa. O0 O0 1-~ -~ "JCAg R of reflected ~ -$I-A shock 8 ~ of detonation -{-A 4 WfA5 wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters (4.1) Oct= Oct= O0 O0 D and by initial stage of motion +B, of flyor also; +B, the position +B. of F and the ----B. state parameters of products

6 340 Ruan Huai-ning and Wang Wei-xiang ~vhere I (Gt- G2) cos20] Ate- [h(0) + G2cos20] [h(0) - -~- As = - h(o) ] A, = Lh (0) (1 - - D)".4,=E(a,,,,O) ( ~-~TxTx cos20 +-~,/sin20 ) B,= 1 z - x (~,y)[hc o) - ~-(a,- a,) oo~2o ] an analytic solution only when the polytropic index of Bt = [l(gt+gosin20][h(o)-t(g,- 1 detonation GOcos20] products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, B,--[h(0)- an analytic, G,cos20] first-order [h(0)-+(gi-g2)cos20 approximate solution is obtained ] for the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. (4.2) an analytic formula B,--[ with h(o) 2cos20 two (1 parameters -D) - (Gt- of high G,) explosive IlK +l (i.e. T(G~ detonation + Gz)a., velocity ] and polytropic index) for estimation of the velocity of flying plate is established. 2sin20 [K+ I(Gt+G,)a,~ ] h(o) Bt=E(a.,O)(-aa-Txsin20- ~-~-~TyTy Explosive driven flying-plate technique ffmds its cos20 important ) use in the study of behavior of cladding of metals. The -Y(x,y)[h(O) method of estimation --~- G,)I (Gt of flyor - velocity C0S20] and '~ the way of raising it are questions Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal CO8220 ddj[k+~(g,+goam ] approach of solving E(a.,O)=[ the problem 2h(O)(1_D) of motion of 2 flyor dt is to solve the following system of equations... YaK I/dG~ dgz \ 1 dk G ap +u_~_xp + dg2 dgl +( GI dt d~---g2 as as ) o'~,] a--t Eqs. (4.1) are the first partial differential p =p(p, quasi-linear s), equations with two variables and terms appearing on the right side. They may be solved by the application of the characteristic line where theory. p, Thus p, S, the u are slope pressure, formulas density, for the specific two entropy families and of slip particle lines velocity are obtained. of detonation They are products respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as (l another -D) sin20-t- boundary. { [(1 Both -D)sin20]* are unknown; - the [G,h(O) position (1 -D) of R and the state parameters on it are ml= governed by +0os20] the flow [G~h field (0) I of (1 central -D) rarefaction - cos20] }4- wave behind the detonation wave (4.3) D and by initial m2 stage of motion of flyor G~h(O) also; (1 the -D) position +C0S20 of F and the state parameters of products au y =0,

7 Slip-Line Field Theory of Transversely Isotropic Body 341 The slopes of a and fl lines are denoted by m~ and m: respectively. From Eq. (4.3) the three properties of the slip-line field of the transversely isotropic body are deduced as follows: (l) In general the two families of the slip lines are not perpendicular to each other. The slope of et line times that of/3 line is - G~h(O)(l-D)-cos28 (4.4) mt ",,,Z--Gzh(O) (l -D) +oos20 It is known that m~.mz:~- 1. Only when hydrostatic stresses have no influence, the result is m, m: from the substitution of Eqs. (3.9) in Eq. (4.4). Therefore, it can be seen whether the two families of slip lines are perpendicular or not is related to hydrostatic stresses.. (2) In general the two families of slip lines are not symmetrical to the principal axes of stress. If they are symmetrical, then t;g(b-0) -----i;g(0-a). When this equation is expanded and Eqs. (4.3) are substituted in it, we have Glh(O) () -D) + (1-2D)eos20 The one-dimensional problem Gzh(O)(l-D)+eos20 of the motion of a rigid flying =1 plate under explosive attack has (4.5) an analytic solution only when the polytropic index of detonation products equals to three. In general, It does not a numerical hold generally. analysis Only is required. When In materials this paper, are however, isotropic, by it utilizing holds when the "weak" Eqs. (3.11) shock are behavior substituted, of the in Eq. reflection (4.5). Thus shock whether in explosive the two products, families of and slip applying lines are the small symmetrical parameter or pur- not is concerned terbation method, with material an analytic, anisotropy. first-order approximate solution is obtained for the problem of flying plate (3) driven It is by known various from high the explosives properties with mentioned polytropic indices above other that than slip lines but nearly do not equal coincide to three. with maximum shear stress lines. an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) From for estimation the stress of equations the velocity of of Eqs. flying (4.7), plate the is following established. differential equations of stress along slip lines are obtained d(tm+[tdo=ltdx+gldy along~ line } (4.6) d(xm+fzdo=lzdx+gzdy along/3 line where Explosive.. driven flying-plate technique ffmds its important use in the study of behavior of mj (A3B, - A,B3) ) cladding of metals. The f method of estimation of flyor velocity and the way of raising it are questions Under the assumptions I of ml( one-dimensional AsB~ - A~B~) plane detonation and rigid flying plate, the normal approach of solving the 9, problem = ~ - (i=1,2) (4.7) A2B~ - A~Bz of motion + ms ( of A3B,, flyor - is A.,B:,) to solve the following system of equations ms ( A3Ba- AsB3) gl --- AzB, - A,B2 + ms ( A~Bz - A~B3) From Eqs. (4.3) and (4.6), slip lines ap andcrm +u_~_xp and + au 0 can be all obtained. With (7,, and 0 sube stituted in Eqs. (3.16), the stresses cr,,c" au w and au r,~ 1 can be obtained. The present theory may be y =0, reduced to many classical theories. Consider the homogeneous weightless medium. If hydrosta- tic stresses have no influence, substituting as Eqs. as (3.9) in Eqs. (4.3) and (4.6) yields a--t "m~-= (1- D)sin20-T- '~/ [ ( l- D)sin20]2 + c~ (4 8) p mz =p(p, cos20 s), where p, p, S, u are 2Kd'[ pressure, (1 density, - D) sin20] specific z + entropy cosz20 and particle velocity of detonation products respectively, with the trajectory do-~o along~ line ) h (0) (i R -DsinZ20) of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters on it do,,+, are governed 2Kd[(l-D)sin2012+c~ by the flow field I of central rarefaction wave behind the detonation wave D and by initial stage of motion do= 0 along/3 line h (0) (1 of - flyor DsinZ20) also; the position of F and the state parameters of products (4.9)

8 342 Ruan Huai-ning and Wang Wei-xiang?hey are the results of the Hill theory. If materials are isotropic and hydrostatic stresses have nfluence, substituting Eqs. (3.11) in Eqs. (4.3) and (4.6) gives rnt_ sin2o-t-dl -G 2 m,-- G+COS20 (4.10) da,~ 2(K+Ga,~) do=o alonga line - (4 11) dam + 2(K+Ga=) dl-g ~ do-=o along line J [hey are the results of the Coulomb theory. If hydrostatic stresses have no influence in isoropic materials, substituting Eqs. (3.13) in Eqs. (4.3) and (4.6) yields :'= t~g (8-T- z/4) (4.12) The one-dimensional da,. problem - 2KdO= of the motion 0 along of ct a line rigid flying plate under explosive attack has an analytic solution only when the polytropic index of detonation products equals to three. (4.13) In general, a numerical analysis da,,, is + required. 2KdO~.o In along this paper, fl line however, by utilizing the "weak" shock behavior l'hey are of the the results reflection of the shock 1VIises in theory. explosive products, and applying the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying V. plate Velocity driven by various Fields high explosives with polytropic indices other than but nearly equal to three. From the velocity equations of Eqs. (3.18), the following differential equations of velocity an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic tlong index) slip for estimation lines can be of obtained the velocity of flying plate is established. do. + mldv~ along a line } } (s.l) dv.+m~dvr=o alongflline If the velocities parallel to a and fl lines are denoted by v~ and vp respectively, then we have Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials under intense impulsive v.=v.cosa loading, shock + v synthesis pcosfl of diamonds, and explosive welding and cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions vl=v.sina+vpsin# } (5.2) when Eqs. Under (5.2) the assumptions are substituted of one-dimensional in Eqs. (5.1) and plane the detonation relations and between rigid flying the inclinations plate, the normal of slip lines approach and the of solving slopes are the used, problem then of motion of flyor is to solve the following system of equations governing dv~ the flow field of detonation products behind the flyor along (Fig. a I): line ) (s.a) dvp+eos(fl-a)dv,+v,sin(b-a)da=o alongfl line J" ap +u_~_xp + au If hydrostatic stresses have no influence, it is known from the first property of the slip-line field mentioned above that fl-a=n/2 au and au da'd. 1 y =0, When the equations are substituted in Eqs. (5.3), we obtain as as dr,, - vpda~. a--t 0 along ~ line ~, (5.4) J dv#+v.dfl~o p =p(p, s), along fl line They are the well-known Geiringer equations. From the differential equations of velocity, the three properties of velocity fields are deduced. They are respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory (1) In F the of transversely flyor as another isotropic boundary. body, Both slip are lines unknown; coincide the with position constant of R lines and of the velocity. state parameters (2) on In general, it are governed slip lines by the do not flow coincide field I of with central maximum rarefaction shear wave strain behind rate the lines. detonation Only wave when h3)drostatic D and by initial stresses stage have of motion no influence, of flyor the also; two the kinds position of lines of F and are the coincident. state parameters of products (3) In the slip-lines field with straight lines, velocities may change along each family of slip lines if the two families of slip lines are not perpendicular.

9 VI. Slip-Line Field Theory of Transversely Isotropic Body 343 Numerical Calculation of Foundations under Loads In the numerical analysis, the basic formulas of Eqs. (4.3) and (4.6) are applied. The difference method along slip lines is employed. The computer program is worked out. Some problems of semi-infinite foundations under loads are calculated. Several groups of data are presented. At first, the transversely isotropic body in the uniform temperature field is considered. Let D , Gl=Gz--0, K----1 T=0 } (6.1) X(x,u) = Y(x,y) =0,~,1 ~ l I N~ io _x I an analytic solution "\ only when ~"/ i the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock in the explosive products, and applying the small parameter purterbation method, Fig.1 an analytic, The slip-line first-order field approximate in the transversely solution is obtained isotropic for body the problem of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. Final The network velocities of slip flying lines plate is obtained agree through very calculation well with numerical and it is results shown by in computers. Fig. 1. In Thus the fian gure analytic the radial formula slip with lines two of the parameters fanlike of field high are explosive inclined (i.e. obviously detonation the velocity horizontal and polytropic plane. This index) results for from estimation the action of the of velocity planar of structures flying plate in is rocks established. and soils. Calculation shows that the ultimate loads are uniform. The value is orw Introduction (6.2) The pressure Explosive formula driven flying-plate obtained by technique Hill is 16! ffmds its important use in the study of behavior of p*=2k [ l+"~---~-( D l+-~--r z, sin22v) -k-'-] (6 3) cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions where Under v is the the angle assumptions between of the one-dimensional principal axis plane of anisotropy detonation and and x-axis. rigid flying Substituting plate, the normal v----0, K approach and D=0.5 of solving the Eq. problem (6.3), we of obtain motion p* of = flyor is to The solve result the following is approximately system of equal equations to the governing value Eq. the flow (6.2). field Secondly, of detonation the isotropic products body behind in the the flyor non-uniform (Fig. I): temperature field is taken into account. Let D.fGt~G,----O, K= (T-20) ap +u_~_xp + au T=20-30y } (6.4) X(x,y)--y(x,y)--o y =0, as as a--t p =p(p, s), respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters D and by initial stage of motion of flyor also; the position of F and the state parameters of products Fig. 2 The slip-line field in the non-uniform temperature field

10 344 Ruan Huai-ning and Wang Wei-xtang The network of slip lines obtained through calculation is shown in Fig. 2. The two families of slip lines under the pressure plate are curved. The ultimate loads are non-uniform. This results form the changed strengths of material in the non-uniform temperature field. Finally let's con- sider a general case. Let D , Gt=G~ , T~ V X (x,v)----o, Y (x,v)= - I K== (T- 20) } (6.5) an analytic solution only when the polytropic index of detonation products equals to three. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock behavior of the reflection shock Fig. in 3 the The explosive slip-line products, field in a and general applying case the small parameter purterbation method, an analytic, first-order approximate solution is obtained for the problem of flying The plate network driven by of various slip lines high obtained explosives through with polytropic calculation indices is shown other than in Fig. but 3, nearly in which equal all to three. the slip lines are curved. The two families of them are not perpendicular. The radial slip lines of the an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic fanlike index) for field estimation are inclined. of the velocity In this of network flying plate all is the established. characteristics of the slip-line field of the transversely isotropic body in the non-uniform temperature field are shown. VII. Conclusions (1) A series of equations, formulas and properties of the slip-line field theory of the tran- Explosive driven flying-plate technique ffmds its important use in the study of behavior of materials sversely isotropic under intense body impulsive are deduced loading, in this shock paper. synthesis The present of diamonds, theory and may explosive be applied welding in and engineering cladding practice of metals. and The geology. method of estimation of flyor velocity and the way of raising it are questions of common (2) A interest. failure criterion for rocks and soils is proposed. It is constructed from the Gol'denblat-Kopnov Under the assumptions criterion. of one-dimensional The parameters plane in it detonation change with and temperature. rigid flying plate, This the criterion normal involves approach transverse of solving isotropy the problem of media, of motion differences of flyor in is behavior to solve between the following tension system and of compression, equations non-uniform temperature fields, and the influence of the intermediate principal stress. Many classical criteria are special cases of the present criterion. (3) The associated rules of flow are derived. The ap +u_~_xp + au basic equations of stress fields and velocity fields of plastic flow are obtained. From these formulas it is known that the volume J strain rate-of media is neither zero nor a constant. y It is =0, related to material properties, tempera- ture and stresses. The principal axes as of strain as rate do not coincide with those of stress because of transverse isotropy. a--t (4) The slope formulas of slip p lines =p(p, and s), the differential equations of stress and velocity along slip lines are obtained by means of the characteristic line theory. From these formulas it is where known p, p, that S, u the are pressure, two families density, of specific slip lines entropy are not and particle perpendicular velocity to of detonation each other products and not symmetrical respectively, to with the the principal trajectory axes R of reflected stress generally. shock of Slip detonation lines do wave not D coincide as a boundary with maximum and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parashear stress lines. They do not coincide with maximum shear strain rate lines either. They only meters coincide with constant lines of velocity. D and by initial stage of motion of flyor also; the position of F and the state parameters of products

11 Slip-Line Field Theory of Transversely lsotropic Body 345 (5) In special cases, the present theory may be reduced to the Mises, Coulomb and Hill theories. (6) Some problems of semi-infinite foundations under loads are calculated. In the obtained networks, the slip lines are non-perpendicular, curved, or inclined. The present theory has been applied in the analysis of geological structures of Dongming Depression and some problems in geotechnical engineering. The results are satisfactory. References [ I ] Hafner, W., Stress distributions and faults, Geol. Soc. Am. Bull., 62 (1951), [ 2 ] Ode, H., Faulting as a velocity discontinuity in plastic deformation, Geol. Soc. Am., Memoir 79, Rock Deformation (1960) [ 3 ] Sokolovski, V. V., Statics of Granular Media, translated by Xu Zhi-ying, Geology Press, Beijing (1964), (Chinese version) [ 4 ] Ruan Huai-ning, Research on anisotropic strength theories in geomechanics, Advances in The Science one-dimensional and Technology problem of Hehai of the University, motion of a 3 rigid (1~92), flying (in plate Chinese) under explosive attack has an [ 5 analytic ] Saada, solution A. S., only Strain-stress when the relations polytropic and index failure of of detonation anisotropic products clays, J. equals Soil to Mech. three. Found. In general, a numerical analysis is required. In this paper, however, by utilizing the "weak" shock Div., ASCE, 99, SMI2 (1973). behavior of the reflection shock in the explosive products, and applying the small parameter purterbation [ 6 ] Hill, method, R., The an Mathematical analytic, first-order Theury approximate of Plasticity, solution Clarendon is obtained Press, for Oxford the problem (1950). of flying plate driven by various high explosives with polytropic indices other than but nearly equal to three. an analytic formula with two parameters of high explosive (i.e. detonation velocity and polytropic index) for estimation of the velocity of flying plate is established. Explosive driven flying-plate technique ffmds its important use in the study of behavior of cladding of metals. The method of estimation of flyor velocity and the way of raising it are questions Under the assumptions of one-dimensional plane detonation and rigid flying plate, the normal approach of solving the problem of motion of flyor is to solve the following system of equations ap +u_~_xp + au y =0, as a--t as p =p(p, s), respectively, with the trajectory R of reflected shock of detonation wave D as a boundary and the trajectory F of flyor as another boundary. Both are unknown; the position of R and the state parameters D and by initial stage of motion of flyor also; the position of F and the state parameters of products