International Journal of Non-Linear Mechanics 35 (2000) 103}131
Finite deformation beam models and triality theory in dynamical post-buckling analysis David Yang Gao* Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA Dedicated to the memory of Professor P.D. Panagiotopoulos Received for publication 12 January 1999
Abstract Two new "nitely deformed dynamical beam models are established for serious study on non-linear vibrations of thick beams subjected to arbitrarily given external loads. The total potentials of these beam models are non-convex with double-well structures, which can be used in post-buckling analysis and frictional contact problems. Dual extremum principles in unstable dynamic systems are developed. A pure complementary energy principle (in terms of the second Piola}Kirchho!'s type stress only) in "nite deformation mechanics is actually constructed. An interesting triality theory in post-buckling analysis is proved. This theory shows that if the gap function introduced by Gao and Strang in 1989 in positive, the generalized pure complementary energy has only one saddle point, which gives a global stable buckling state. However, if the gap function is negative, the generalized complementary energy may have two so-called super-critical points: the one which minimizes the pure complementary energy gives another relatively stable buckling state; and the other one which maximizes the complementary energy is a unstable buckling state. Application in unilateral buckling problem is illustrated, and an analytic solution for non-linear complementarity problem is obtained. Moreover, the general duality theory proposed recently is generalized into the non-linear dynamical systems. A pair of dual Du$ng equations are obtained. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Non-linear beam theory; Non-linear vibrations; Post-buckling analysis; Complementary energy principle; Finite deformation; Variational method; Duality theory; Non-linear dynamic system; Dual Du$ng equations
1. Introduction The study of non-linear dynamic beam theory has a long history (see for example Refs. [1,2]). Although many large deformation beam theories have been developed engineers are still looking for some simple but useful models for general engineering problems (cf. e.g. Refs. [3}7]). It is interesting to note that the well The main results in this paper are announced at the 7th Conference on Non-linear Vibrations, Stability and Dynamics of Structures, 26}30 July, 1998, Blacksburg, Virginia. * E-mail:
[email protected]. 0020-7462/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 2 0 - 7 4 6 2 ( 9 8 ) 0 0 0 9 1 - 2
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known von Karman plate theory in one-dimensional case can be written as p "0, EIw !(p w ) "f, (1) V V VVVV V V V where I is the moment of inertia per unit length, and f is the laterally distributed load (in this paper, we use the notation ( ) :"(*/*x) ( ) for convenience). Although, by de"nition, the axial stress p depends non-linearly V V on w, i.e., p "E(u # w ), the "rst equation in Eq. (1) shows that p has to be a constant: p "!j (j'0 V V V V V is a compressive axial load). So the second equation is really a linear equation: EIw #jw "f. For the VVVV VV given external loads f and j, the exact solution of this beam model can be obtained immediately. The reason for this illogical phenomenon is due to the ignoring of the thickness deformation in the von Karman theory. Hence, this linear ordinary equation can only be used for pre-buckling analysis, i.e., the eigenvalue problem * EI (w ) dx VV , j "inf A U * (w V) dx where j '0 is the Euler buckling load. Unfortunately, this one-dimensional von Karman equation (1) is still A considered as a non-linear beam theory (see Ref. [8]), and some very complicated variational formulations for obtaining approximate solutions were proposed recently in [9]. In order to study the buckling phenomena, many mathematical and engineering articles on non-linear vibrations of buckled beams are based on the pre-buckled beam systems (see the recent paper by Nayfeh and Kreider [10]). However, a serious study of the real-life bifurcation phenomena of the non-linear vibration beams is very important for both engineers and applied mathematicians. It was shown recently by the author in [11] that the strain in thickness direction is proportional to w , which cannot be ignored in "nite V deformation analysis. Therefore, a non-linear beam model was proposed in [11,12]: EIw !aw w #jw "f, (2) VVVV V VV VV where a"3h (1!l)'0 is a constant. The cubic non-linear term in this equation is mainly due to the deformation in thickness direction. For a given axial load j'0, the total potential of this beam model is a non-convex functional
2 EI (w VV)#6 (w V)!j(w V) dx! f w dx.
P(w)"
*1
*
a
(3)
Obviously, for any kinetically admissible de#ection w,
2 6 (w V)!jN (w V) dx! fw dx ": PH(w),
P(w)*
*1 a
*
where j "j!j , and j is the Euler pre-buckling load. If j '0, the beam is in the post-buckling state. The N A A N beam may buckle up or down, depending on the transverse load f. The double-well energy P has two local H minimizers, corresponding these two possible stable buckling states, and one local maximizer, corresponding to the unstable buckling state (see Fig. 3). It is surprising that in the phase transitions of the Ericksen's bar subjected to an axial extension load, the total potential is the same as P (w) (see Ref. [13]); while the pertubated problem studied in [14] is the same as H the total potential P(w) with EI"1/3. Non-convex variational problems with double-well energy have been discussed extensively by many mathematicians and engineers in phase transitions (see, for example Refs. [15}18]). Unfortunately, the traditional direct analysis and the so-called relaxation methods for solving such non-convex variational problems are very di$cult, and, therefore, some numerical approaches are studied recently (cf. e.g. Ref. [19]). However, by the non-linear duality transformation method developed recently in [20], such non-convex variational problems with multi-well (more than three) energy have been solved in a very simple way (see Refs. [20}22]).
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In non-linear dynamical buckling problems, an additional term ow should be added in the Eq. (2). If the RR de#ection w (x, t) can be separated into w(x, t)"u(t) v(x) (unfortunately, in non-linear dynamical buckling beam problems, this separation is wrong [23]), then the dynamical model of the non-linear beam equation (2) is equivalent to the well known Du$ng-type equation u "ju!bu#fM . (4) RR For the given parameters j, b and the external load fM , this equation may have chaotic solutions. It is known that the non-linear dynamical systems are extremely sensitive to the parameters. Serious study on the dynamical post-buckling analysis of large deformed beam should consider the e!ects of the transverse shear strain on the bending solutions. Actually, as long ago as 1837 Saint-Venant had realized that beam cross sections do not remain plane in bending. In the Timoshenko beam theory, the shear deformation t does not vary in the lateral beam direction, i.e. plane sections remain plane (but not necessarily normal to the longitudinal axis) after deformation. Therefore, various re"ned or higher-order beam theories have been proposed. (See the review articles by Kapania and Raciti [4], Noor and Burton [5].) It has been pointed out by Duva and Simmonds in [24,25] that in linear vibrations of layered, orthotropic elastic beams, the so-called higher-order theories o!er, in general, no advantages beyond those resulting from an enlargement of the class of admissible functions in a Rayleigh}Ritz procedure unless two-dimensional end e!ects are considered. They concluded that elementary beam theory can be applied successfully to layered, orthotropic beams, possibly weak in shear. However, if the beam is subjected to a shear load on the top or bottom of the beam surfaces, the problem is in two-dimensional space, and, therefore, the shear e!ects have to be considered. In order to study the control of the beam/actuator systems, an extended Timoshenko beam was proposed in [12,26]. The shear load in this beam theory could be the applied control on the top and bottom of the beam surfaces. Applications of this extended beam model in smart materials were discussed in [26,27]. To study the shear e!ects on the post-buckling analysis as well as the frictional contact problems, these beam models have been extended to the large deformation theory in [11,12], and an elastoplastic beam model has been established in [28]. This model is a second-order partial di!erential system, which is much easier than the traditional fourth-order beam equations. A primary goal of the present paper is to generalize the extended beam models proposed in [11,12,28] to a large deformation theory with arbitrary dynamical loading systems. A new second-order non-linear dynamical beam theory is developed in Section 3. The applications to the frictional contact problem and buckling analysis are discussed. A non-linear boundary value problem with one parameter for non-linear dynamic post-buckling analysis is proposed in Section 4. It is shown that if the shear e!ect is ignored, the governing equation for this non-linear beam model is a Du$ng-type equation. In Section 5, the dual variational approaches for this non-convex variational problem are discussed. It is well known that the classical Hellinger}Reissner principle involves both the second Piola}Kirchho! stress and the displacement. It is not considered as a pure complementary energy principle. The extremality condition of this principle has been an open problem for many years. The Levinson}Zubov principle involves only the "rst Piola}Kirchho! stress. But its critical point is not a solution to the associated boundary value problem unless the stored energy = is convex in the deformation gradient F (see Refs. [20,29]). Unfortunately, in "nite deformation theory, = (F) is usually non-convex. In this case, the complementary energy density, obtained by the classical Legendre transformation, does not have a simple algebraic form (see Ref. [30]). The generalized Levinson}Zubov's energy, obtained by the Fenchel}Rockafellar duality theory (see Refs. [29,31,32]), is always concave. But if the total potential is non-convex, there exists a duality gap between the potential energy and the generalized Levinson}Zubov energy [20,29]. In order to recover this duality gap in "nite deformation theory, a so-called complementary gap function was discovered by Gao and Strang in 1989 [33]. They proved that if this gap function is positive on the equilibrium admissible space, the generalized Hellinger}Reissner energy is a saddle functional. A general duality theory was established for convex "nite deformation systems. Recently, the remained open problem
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in non-convex, unstable systems has been solved. It is proved in [20,21] that if this gap function is negative, the generalized complementary energy is a so-called super-critical point functional. A so-called non-linear dual transformation method has been proposed for solving non-convex variational problems. The key idea of this method is to choose a geometrically non-linear operator " such that the stored energy density = is either a convex or a concave function of the generalized strain e""u. It is shown in [20,22,31] that by using this method, the fully non-linear, non-convex variational/boundary value problem in "nite deformation theory can be transformed into a non-linear tensor equation. Therefore, the closed-form solution in general "nite deformation theory can be obtained. The secondary goal of this paper is to generalize this general duality theory into post-buckling analysis and non-linear dynamical systems. A pure complementary energy extremum principle is constructed in Section 5. An interesting triality theorem in buckling analysis is proved in Section 6. This theory shows that in post-buckling systems, if the complementary gap function is positive, the saddle point of the generalized Hellinger}Reissner-type energy is a global stable buckling state. However, if this gap function is negative, this general complementary energy may have two super-critical points: the one which minimizes the pure complementary energy gives another local stable buckling state. The other super-critical point, which maximizes the complementary energy, is a unstable buckling state. Application in unilateral buckling analysis is illustrated and an analytical solution is obtained for non-linear complementarity problem in post-buckling analysis. Moreover, a triality extremum principle in dynamical post-buckling analysis is proposed in the last section, and there, a pair of dual Du$ng equations is obtained.
2. Non-linear dynamic beam theory Let us consider an elastic beam whose domain in x}y plane is a rectangle )"+(x, y) 3 1 " 0)x)¸, !h)y)h,. By w(x, t), we denote the transverse displacement of the elastic axis, which is a function of x and t only; while the horizontal displacement, depends on x, y and t: m (x, y, t)"u(x, t)!yh (x, t)#v(x, y, t),
(5)
where u(x, t) is the horizontal displacement of the middle axis y"0; h is the bending angle, h"tan\
*w/*x , 1#(*u/*x)
measured in the counterclockwise direction; v(x, y, t) is the shear deformation. The shear angle is c"tan\
*v (x, y, t), *y
measured in the positive direction (see Fig. 1). So the net rotation angle is then given by t"h!c. For moderately large beam de#ection problems, we may make the following assumptions: *w h/¸&w (x, t) 3 O(1), u&v& 3 O(e), *x *u *v *v * w & & & 3 O(e), *x *x *y *x
(6)
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Fig. 1. Geometry of the extended beam model.
where the notation&stands for &&same order of magnitude''. By the Taylor expansion, we have h"tan\[w /(1#u )]"w #O (e), neglecting terms higher than O(e), the net rotation angle is approx V V V imated by *m t (x, y, t)"! (x, y, t) *y * "! [!yh(x, t)#v(x, y, t)] *y *w *v " (x, t)! (x, y, t). *x *y If v (x, y, t) is a linear function of y, i.e. plane sections remain plane after deformation, then the shear angle c depends on (x, t) only. This is the Timoshenko beam theory. Let u : );[0, t ]P1 be the displacement vector: m (x, y, t) u(x, t)!yh(x, t)#v(x, y, t) u" " (7) g (x, y, t) w(x, t)
By the Green's strain tensor E" [ u#( u)R#( u)R ( u)], we have e u !yh #v # (u !yh #v )# w , VV V V V V V V V w !w v # v e " , WW V V W W v !h#w # (u v !u h#yhh !yv h #v v !v h) e VW W V X W V V W V V W V
(8)
where we simply denote (w )"w , etc. V V Neglecting terms higher than O(e), and by using the engineering strain tensor notations: e "e e "e , V VV W WW c"2e , we then obtain VW e u !yw #v # w V V VV V V w e " . (9) W V c v W
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We can see clearly that the strain e is proportional to w . Hence ignoring the deformation in thickness W V direction by the von Karman theory is indeed unreasonable. The stress vector in this plane deformation problem can be given by the following elastic constitutive equation:
p 1 l 0 V E p " l 1 0 W 1!l q 0 0 b
e V e , W c
(10)
where b"(1!l)/2. We denote by o(x, y) the mass density per unit area in the x, y plane, thus the kinetic energy at the time t 3(0, t ) becomes 1 ¹ (m, w)" o (x, y) [w#m] d). (11) R R 2 We assume the density to be an even function of y so that, with
F
\F o(x, y) dy,
o (x)" F
F
\F o(x, y)y dy
I (x)" N
(12)
and the deformation model represented in Eq. (5), Eq. (11) becomes 1 * 1 * F ¹ (u, v, w)" [o (x) (w#u)#I (x) w ] dx# o(x, y)[!2yw (u #v )#v] dy dx. F R R M VR VR R R R 2 2 \F (13)
Suppose that the beam is subjected to a laterally distributed load f (x, t) and a horizontal load with magnitude p at the end x"¸ in the negative x-direction, by the virtual work principle, we have the following variational equation:
t
*
t
[p de #p de #qdc]d) dt! V V W W
f (x, t)dw dx dt"
d¹ (u, v, w) dt.
t
t
!pdu (¸, t) dt#
(14)
Substituting the geometrical relations (9) into Eq. (14) and considering (u, v, w) as independent variational arguments, we obtain the following Euler}Lagrange equations: F
\F pV V dy"oF u RR,
(15)
p #q "o(!yw #v ), V V W RRV RR F F [yp #p w #(p #p ) w ] dy#f"o w !(I w ) # y(ov ) dy V VV W V V V W VV F RR M RRV V RR V \F \F and the boundary variational equations:
F
F
\F pV " V dudy"0, \F pV " V* du dy#pdu(¸)"0, F F \F pV " V * du dy"0, \F ypV " V * dwV dy"0,
(16) (17)
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F
\F (ypV V#(pV#pW) wV) " V * dw dy"0, F \F qdv" W F dx"0. $
To cite an example of &&natural'' boundary conditions annihilating the integrated terms in the variational equation, we note that if the beam is simply supported at the ends x"0, ¸, and if E, G and o are constants, the governing equations of motion are the following: u #(1#l)w w "o (x) u , (18) VV V VV RR v #bv "yw #oN [u #(v !yw )], (19) VV WW VVV RR RR RRV F F Iw !2h(1#l) (u #(2!l)w )w ! yv dy"2h fM !o 2hw !Iw # yv dy , (20) RR RRVV VVVV V V VV VVV RRV \F \F where oN "(1!l) o/E, fM "(1!l) f/(2hE). If the horizontal velocities u , v and accelerations u , v can be R R RR RR ignored, integrating Eq. (18) gives
1#l k u "! w ! V V 2h (1#l) 2 with (1#l) (1!l) k" p. E Substituting this into Eqs. (18)}(20) we obtain eventually the dynamic equation of the non-linear beam model: v #bv "yw !oN yw ∀(x, y, t) 3);[0, #R), (21) VV WW VVV RRV F yv dy"2h ( fM !ow )#IoN w ∀(x, t)3[0, ¸];[0, #R), (22) Iw !aw w #kw ! VVVV U VV VV VVV RR RRVV \F where a"3h(1!l) is a positive parameter. Since the shear deformation v should be an odd function of y, this fourth-order partial di!erential system can only be used to handle some special external shear loads. In order to obtain a more general beam model, a second-order beam theory is presented in next section. Ignoring the shear e!ects, this beam model has a simple formulation
1 h oN w " (aw !k) w ! w #fM . RR 2h V VV 3 VVVV
(23)
The total potential energy of this model is a double-well energy (3).
3. Extended second-order dynamic beam model Instead of shear deformation v(x, y, t), if we use directly the axial displacement m(x, y) as the unknown function, then the displacement vector is u(x, y, t)"(m (x, y, t) w(x, t))R. The Green strain in this case is
E"
m # (m #w ) (m #w #m m ) V V V W V V W . (m #w #m m ) m W V V W W
(24)
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Since m(x, y)"u(x)!yh(x)#v(x, y), by assumption (6) we have m "(!h#v )"w #O(e). W W V Neglecting terms higher than O (e) we then have
e m # w V V V w e " . W V c w #m V W Suppose the beam is subjected to the following external loading system (see Fig. 2):
f (x, t)"
q> (x, t) f (x, t)
at y"h,
f (x, t)"
!p 0
at x"¸.
q\ (x, t) 0
(25)
at y"!h,
(26)
By the virtual work principle, we have the following variational equation: *
[ fw#q>m (x, h, )#q\m(x,!h, t)] dx dt F # \F pdm(¸, y, t) dy dt" d¹ (m, w) dt ∀m, w. t
t
[p de #p de #qdc] d) dt! V V W W t
t
(27)
Substituting the geometrical relations (25) and the kinetic energy (11) into Eq. (27) and considering. (m, w) as variational arguments, we obtain the Euler}Lagrange equations p #q "om , V V W RR F [(p #p ) w #q] dy#f (x)"o w , V W V V F RR \F q(x, h, t)"q> (x, t), q(x,!h, t)"!q\ (x, t)
Fig. 2. Frictional contact problem.
(28) (29) (30)
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and the boundary variational conditions: F
\F pV (0, y, t)dm(0, y, t)dy"0, F \F (pV (¸, y, t)#p)dm(¸, y, t) dy"0, F \F [(pV#pW)wV#q] "V * dw dy"0.
(31) (32)
For an elastic beam, substituting the constitutive relation into Eqs. (28)}(30), for any (x, y, t)3); (0,#R), we have m #bm "!(1#l)w w #o m , VV WW V VV RR 1 F ((1#l)m w #bm ) dy#fM "oN w , (3(1#l)w #b)w # V VV 2h V V W V RR \F m (x,$h, t)"!w $qN $ (x, t), W V where
(1!l) f (x) M f" , 2hE
(33) (34) (35)
2(1#l) $ 1!l qN $" q , oN "o . E E
This new beam model is a coupled second-order non-linear partial di!erential system with arbitrary external load (26). For clamped/simply supported beam, the boundary conditions can be given as w(0)"0, w(¸)"0, m(0, y)"0, 1#l w (¸)"!pN ∀y3[!h, h], m (¸, y)# V V 2 where pN "(1!l) p/E. In statical frictional contact problems, if the beam is supported on a rigid obstacle, and the shape of the obstacle is described by a strictly concave (x) (see Fig. 2), the governing equations should be m #bm "!(1#l)w w , VV WW V VV $ m (x,$h)#w "$qN (x), W V 1 F (3(1#l)w #b)w # ((1#l)m w #bm ) dy#fM )0, V VV 2h V V W V \F w(x)! (x)*0,
1 (3(1#l)w #b) w # V VV 2h
\F ((1#l)m V w V#bm W) V dy#fM (w! )"0, F
(36) (37) (38) (39) (40)
in which, q\(x) is the frictional force on the contact surface of the beam. Since the contact region S "+x3[0, ¸]" w(x)" (x), is unknown until the problem is solved, this is a free boundary value problem. General theory and numerical methods for solving contact problems were discussed in the monograph by Kikuchi and Oden [34].
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4. Dynamic buckling analysis In this section, we use the second-order beam model to study the dynamic buckling analysis. Suppose that the beam is clamped at x"0, while at x"¸, the beam is simply supported and is subjected to a compressive load p applied in the negative direction of the x-axis. The distributed load f (x, t) on the top of the beam is prescribed. When the combination of the external loads reach a critical value, the beam may buckle. We assume that the compression p at x"¸ changes neither its direction nor its magnitude during the deformation (i.e., p is a so-called dead-load), then the acceleration m in Eq. (33) can be ignored. Since this RR beam model is linear in the displacement m(x, y, t), we can let m(x, y, t)"!pN x#uN (x, y, t), the governing equations of this beam model can be written as uN #buN "!(1#l)w w , VV WW V VV 1 [(1#l) (3w !pN )#b]w # V VV 2h
(41) F
\F ((1#l)uN V w V#buN W) V dy#f "oN w RR.
uN (x,$h, t)"!w $q$ (x, t), W V 1#l uN (0, y, t)"0, uN (¸, y, t)"! w . V V 2
(42) (43) (44)
For a given w(x, t), Eq. (41) is a non-homogeneous linear partial di!erential equation, the solution should be
uN (x, y, t)"!
1#l w dx#v(x, y, t), V 2
(45)
where v(x, y, t) is a homogeneous solution of the following boundary value problem: v #bv "0, (46) VV WW v (x,$h, t)"!w (x, t)$q$ (x, t), (47) W V v(0, y, t)"0, v (¸, y, t)"0. (48) V For a given w(x, t), this problem has a unique solution. Substituting uN (x, y, t)"!((1#l)/2) w # V V v (x, y, t) in Eq. (42), we have V 3a b w #b!ja w # v (x, h, t)!v (x,!h, t) #fM "oN w ∀x3[0, ¸], (49) VV 2h V V RR 2 V
where a"(1!l, j"(1#l)ap/E. So the dynamic buckling problem for large deformed thick beam theory can be proposed as the following: Problem 1. For the given loads fM (x, t) and j*0, ,nd the shear deformation v(x, y, t) and the de-ection w(x, t) satisfying the governing equations v #bv "0, VV WW 3a b w #b!ja w # v (x, h, t)!v (x,!h, t) #fM "oN w , V VV V RR 2 2h V
v (x,$h, t)#w (x, t)"$q$ (x, t) W V
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in the spacetime space );(0,#R) and the boundary/initial conditions v(0, y, t)"0, v (¸, y, t)"0, V w(0, t)"0, w(¸, t)"0, v(x, y, 0)"0, w(x, 0)"w (x, 0)"0. R This is a coupled, non-linear second-order partial di!erential dynamic problem. Remark. (I) If either q$"0 or q>(x)"!q\(x), the formulation of the solution v should be v(x, y, t)" c (w , t) sin (j x) sinh(j y/(b ), L L L L L where j "n(2n#1)/2¸, w is the coe$cient of Fourier sine series of w(x, t). In this case, the shear L L deformation is an odd function of y, i.e. v(x,!y, t)"!v(x, y, t). (II) If the shear e!ects are ignored, this problem reduced to a one-dimensional non-linear dynamical equation (50) ow " a w w !jaw #fM . V VV VV RR This is a Du$ng-type equation. For a given periodical external load fM (x, t) and j'0, this post-buckling system may have chaotic solution. (III) If a"0, Problem 1 is linear. The complete solution with applications in optimal control of smart structures are given in the recent paper [27].
5. Complementary energy variational principles In this section, we will discuss the variational approaches to the statical post-buckling problem of the new beam model proposed in Section 4. Let U"C (); 1) be a displacement space. An element u"(v, w)3U is a continuous, di!erentiable vector in 1 with domain ). For the clamped/simply supported beam subjected to the transverse load fM and the buckling load at x"¸, the kinematical admissible space U LU can be de"ned as
v v(x,!y)"!v(x, y), w(0)"w(¸)"0, 3U . w v(0, y)"v (¸, y)"0 ∀y3[!h, h] V Then the boundary value problem for the statical post-buckling analysis can be proposed. U :" u"
(51)
Problem 2. For the given external loads fM and j'0, ,nd (v(x, y), w(x))3U such that v #bv "0 ∀(x, y)3), VV WW 3a b w #b!ja w # v (x, h)#fM "0 ∀x3[0, ¸], VV h V 2 V
v (x,$h)"!w (x) ∀x3[0, ¸]. W V The total potential energy of this buckling problem is a non-convex functional 1 P (v, w)" H 2
*
[(v V#2 aw V!j)#b(v W#w V )] d)! fM w dx. 1
Then the extremum variational problem associated with Problem 1 can be proposed as:
(52)
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Problem 3. Find (vN , wN ) such that P (vN , w )"inf +P (v, w) " ∀(v, w)3U , (53) H H The solution of this extremum variational problem (i.e., the global minimizer of P ) is a stable post-buckling H state. However, since P is a non-convex, it may have more than one local minimizers. H As we discussed in Section 1, if the shear e!ect is ignored, then P can be written as H * 1 1 a w !j !fM (x)w(x) dx. (54) P (w)" H 2 2 V It is the same as the Ericksen's double-well energy [12] in the extension of a elastic bar! For a given load fM , P (w) may have two minimizers for two possible stable states, respectively. H Since P (v, w) is a strictly convex functional of v, for each w, the problem inf (P (v, w) has a unique H T H solution vN . But P (v, w) is not convex in w. The solution for inf P (v, w) is not unique. The direct methods for H U H solving this non-convex minimization problem are very di$cult. However, the dual variational approach will provide a powerful alternative method for solving this buckling problem. In order to establish a simple complementary energy variational principle, we need to "nd the right dual variables and the conjugate functional of P . Let e"(e, c) be the generalized strain vector, de"ned by the H geometrical equation:
e"
e ""(u)u" c
* *x * *y
aw *x* *V *x
v(x, y) , w(x)
(55)
where the geometrical operator "(u)""(w) is quadratic, hence, e is a Cauchy}Green-type strain vector. In terms of the strain vector e, the stored energy = is a strictly convex (quadratic) function: = (e)" [(e!j)#bc]. H The conjugate stress vector s depends linearly on e:
s"
p 1 0 *= (e) H " " q *e 0 b
e!j . c
(56)
Using the Legendre}Fenchel transformation, the complementary energy is uniquely de"ned by 1 1 =* (s)"sup +sRe}= (e),"sup +pe#qc!= (e, c)," p#jp# q . H H H 2b 2 C A
(57)
Let F(u) : U P1 be the external potential: *
fM w dx.
F(u)"
(58)
The total potential P can be written in a general form H
=H ("u) d)!F (u).
P (u)" H
Remark. Although the stored energy density = is strictly convex in the general strain vector e, P : U P1 is H not convex due to the non-linearity of the geometrical operator ". If we use the real strain vector e"(e , e , c) V W de"ned in Eq. (25), the dual problem will have three variational arguments: p , p and q. Actually, a key step V W in the so-called non-linear dual transformation method is to choose right geometrical operator " so that the
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stored energy density is convex in e""u, and has a simple formulation. This method was original proposed in [33], and has been generalized recently into the non-convex/non-smooth problems in [20, 22]. According to Gao and Strang [33], the non-linear operator " should be split into two parts """ #" , R L where " is the Ga( teaux derivative of e(u), de"ned by R * * *x aw *x V* " (u)"De (u)" * R *y *x
" ""!" is a complementary operator, de"ned by L R 0 ! aw *x* V , " (u)" L 0 0
(59)
(60)
(61)
which plays an important role in complementary variational principles. We let f 31 be the external load: f " fM for x3(0, ¸), 0 at x"¸, , f " 0 for (x, y)3), 0 at y"$h
(62)
and we write
(pe#qy) d), * (u, f ) :" fM w dx.
1s,e2 :"
Then the virtual work principle can be written in the general abstract form 1s, " (u) du2"("* (u) s, du)"(f , du) ∀u3U , (63) R R where "* is the adjoint operator of " , de"ned by R R F F "* (u)s"! (q#aw p) dy ∀x3(0, ¸), (q#aw p) dy"0 at x"¸ R V V V \F \F "* (u)s"!(p #q )"0 ∀(x, y)3), q(x, y)"0 ∀x3(0, ¸), y"$h R V W In terms of the stress vector s, the equilibrium equations and boundary variational conditions can be written in a general abstract form
F
\F(q#aw V p) V dy"fM
"* (u) s"f N! R
"* (u) s"f N!(p #q )"0 R V W F "* (u) s"f N! (q#aw p) dy"0 R V \F "* (u) s"f N!r (x, y)"0 R
∀x3[0,¸], ∀(x, y)3), at x"¸, at y"$h.
(64)
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By the geometrical operator decomposition (59), we have 1s, e (u)2"1s, " (u) u2!G (u, s), R where
(65)
G (u, s)"1s,!" (u) u2 (66) L is the so-called complementary gap function, introduced in [33]. Let S be the classical equilibrium admissible space: U F (q#aw p) dy#fM "0, q (x, $h)"0, ∀x3(0, ¸), V V . (67) S " (p, q, w) \F U p #q "0, ∀(x, y)3), F q#aw p) dy"0, at x"¸. V V W \F According to the general duality theory developed in [33], the total complementary energy functional P : S P1 for this new beam model should be H U 1 1 1 p#jp# q d)# apw d). (68) P (p, q, w)" =* (s) d)#G(u, s)" V H H 2b 2 2 The complementary gap function in this beam theory is a quadratic functional of w:
s ) (!"L (u) u) d)" 2 apw V d). 1
G(u, s)"
(69)
Since G depends only on w and p, we simply write G(u, s)"G(w, p). Its sign depends on the sign of F p dy. \F Since s is a second Piola}Kirchho!-type stress, the functional P can be considered as a Hellinger}ReissnerH type complementary energy, in which the displacement w is involved. We have the constrained complementary energy variational principle: Theorem 1. Among all (p, q, w)3S , the solution (pN , qN , wN ) of the constrained variational problem U d P (p, q, wN ; p, q, w)"0 ∀ (p, q, w)3S (70) H U solves Problem 2. Moreover, if u "(vN , wN )R is the associated displacement such that "u "D=* (s ), then the H following complementarity condition holds: P (vN , wN )#P (p, qN , wN )"0. (71) H H The proof of this theorem is given by the following generalized variational principle. According to the general duality theory developed in [23], for any given geometrical operator such that the stored energy density =("u) is convex in e""u, the Lagrangian associated with potential variational problem is
s ) "u d)! =* (s) d)!F(u).
¸(u, s)"
(72)
In the constrained complementary variational problem (70), with the Lagrange multiplier u"(v, w) introduced to relax the equilibrium constraints in S , this general Lagrangian form can be written as U 1 1 1 * ¸ (u, s)" p aw #v #q w #v d)! p#jp# q d)! fM wdx. (73) H V V V W 2 2 2b
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Let SLC (); 1) be a range of the constitutive mapping: S"+s3C (); 1) " s"D= ("u) ∀u3U,. (74) H An element s3S is a di!erentiable stress vector "eld on ). A point (u , s )3U;S is said to be a critical point of ¸ if it is a solution of the generalized variational problem: H d¸ (u , s ; u, s)"0 ∀ (u, s)3U;S. (75) H Now, the generalized complementary energy variational principle for the post-buckling problem can be proposed as the following. Theorem 2. For the given external loads fM and j'0, every critical point (u , s ) of ¸ is a solution of the buckling H problem (Problem 2), and the following complementarity conditions hold: P (u )"¸ (u , s )"!P (s , wN ). H H H Proof. The Euler}Lagrange equations associated with the variational problem (75) are
(76)
"(u ) u "D=* (s ), (77) H "* (u )s "f . (78) R Since the stored energy density = (e) is strictly convex, the inverse geometrical-constitutive equation (77) is H equivalent to s "D= ("u ). Thus the critical point (u , s ) of ¸ satis"es all the equations in Problem 2, and H H hence, is a solution. Moreover, by the operator decomposition """ #" , we have R L
=*H (s ) d)
¸ (u , s )"("* (u )s!f , u)#1s , " (u ) u 2! H R L
=*H (s ) d)"!PH (s , wN ).
"!G(u , s )!
On the other hand, by the Fenchel}Young equality: =(e )#=* (s )"1s , e 2, H we have
=H ("u ) d)!F(u )"PH (u ).
¸ (u , s )" H
Hence the complementarity (76) is proved. ) Since the classical Hellinger}Reissner-type complementary energy involves both stress and displacement, P is not considered as a pure complementary energy principle. In continuum mechanics, the existence of the H pure complementary energy has been argued for many years. This problem was solved recently. For quadratic operator ", the pure complementary energy can be obtained by (see Refs. [20, 21]) P (s)"inf ¸(u, s) if G(u, s)*0, u P (s)"sup ¸(u, s) if G(u, s)(0,
(79) where the Lagrangian ¸ is de"ned by Eq. (72). For general geometrically non-linear operator ", the pure complementary energy P is de"ned by u
=* (s) d)!G* (s),
P (s)"!
(80)
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where G* (s) is a pure complementary gap function, obtained by solving the following stationary variational problem: G* (s)"sta+1s, "u2!F(u) " ∀u3U , .
(81)
In the non-linear beam theory, the Lagrangian ¸ de"ned by Eq. (73) is a quadratic functional of w. For H any given s3S, the solution of the stationary variational problem (81) is unique: g(x)!F q dy , w " \F V a F p dy \F where *
g(x)"
(82)
V
fM (s) ds.
fM (s) ds!
(83)
Substituting Eq. (82) into Eq. (81), the pure complementary energy functional P is then obtained H 1 1 * [g(x)!q( (x)] p#jp# q d)! dx, P (s)"! H 2b 2 2ap( (x) where
F
\F q dy,
q( (x)"
(84)
F
\F p dy.
p( (x)"
In non-linear buckling analysis, P is usually non-convex. It may have more than one critical point. By H introducing the statically admissible space S LS:
p #q "0 ∀(x, y)3), F p dyO0 ∀x3(0, ¸), V W , \F q(x,$h)"0 ∀x3(0, ¸), F qdy"F p dy"0 at x"¸, \F \F two extremum dual variational problems should be proposed as the following: (a) The sup-dual problem: For a given fM (x) and j'0, "nd s such that S"
p q
P (s )"sup P (s) ∀s3S . H H (b) The inf-dual problem: For a given fM (x) and j'0, "nd s such that
(85)
(86)
P (s )"inf P (s) ∀s3S . (87) H H Next section, we will show that the solutions of these problems will control di!erent buckling states.
6. Triality theory in post-buckling analysis Recall the abstract Lagrangian in general "nite deformation system (72)
s ) ("u) d)! =* (s) d)!F(u),
¸ (u, s)"
(88)
where =* (s) is convex and F(u) is linear. In geometrically linear systems, where " is a linear operator, the Lagrangian is a saddle functional, i.e., ¸ is concave in s and convex (linear) in u. In this case, the following minimax theory is well known: inf sup ¸ (u, s)"sup inf ¸(u, s). u
s
s
u
(89)
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The point (u , s ) is said to be a saddle point of ¸(u, s) on U;S if ¸(u , s))¸(u , s ))¸(u, s ) ∀(u, s)3U;S.
(90)
However in "nite deformation theory, the geometrical operator is non-linear, and the Lagrangian is usually not a saddle functional. In non-convex analysis, two new critical points were introduced in [20,21]. De5nition 1. Let ¸ : U;SP1 be an arbitrarily given functional. A point (u , s ) is said to be a super-critical point (or *>-critical point) of ¸ on U;S if ¸ (u , s))¸(u , s )*¸(u, s )
∀(u, s)3U;S.
(91)
A point (u , s ) is said to be a sub-critical point (or *\-critical point) of ¸ on U;S if ¸ (u , s)*¸(u , s ))¸(u, s )
∀(u, s)3U;S.
(92)
The following lemma is important in non-convex extremum analysis. Lemma 1. If ¸ : U;SP1 is a super-critical point functional, then inf sup ¸(u, s)"inf sup ¸ (u, s). u
s
s
u
(93)
If ¸ : U;SP1 is a sub-critical point functional, then sup inf ¸(u, s)"sup inf ¸ (u, s). u
s
s
u
(94)
The proof of this lemma is given in [20]. The following theorem shows that the extremality properties of the Lagrangian ¸ in post-buckling analysis depend on the gap function. H Theorem 3. Suppose that (u , s ) is a critical point of ¸ . ¹hen (u , s ) is a saddle point of ¸ if and only if G (u, s )*0 H H ∀u3U ; (u , s ) is a super-critical point of ¸ if and only if G(u, s ))0 ∀u3U . H Moreover, if (u , s ) is either a saddle point of ¸ or a super-critical point of ¸ , then u and s must be respectively H H the critical points of P and P, and the complementarity condition holds: H H P (u )"¸ (u , s )"P (s ). (95) H H H The proof of this theorem is given in the general theory established in [20,21]. Remark. (I) This theorem shows that there is no duality gap between the non-convex total potential energy P and the pure complementary energy P. Since = (e) is convex, the classical Legendre transformation (i.e., H H H the Fenchel}Young equality) leads to P (u )!P (s )"1s , "u 2#G*(s )!(f , u )"0. H H The gap function G(u (s ), s )"G* (s ) recovered the complementarity condition between the two brackets: 1s , e(u )2"(f , u )#G (u , s ). So the primal and dual stationary variational problems are equivalent in the sense that they have the same solution set. The properties of solutions will be clari"ed by the following triality theory. If " is linear, then G,0. In this case,
=*H (s) d)"!PH (s)
P (s)"! H
is the classical complementary energy in linear elasticity.
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(II) The geometrical operator decomposition """ #" (59) is crucial important. The adjoint of " is the R L R equilibrium operator; while " leads to the gap function. However, in a series of publications by Telega split L the Green strain tensor "u"E into a linear part [ u#( u)R] and a non-linear part ( u)R ( u). Using the well known Fenchel}Rockafellar duality theory in geometrically linear optimization, he produced many what he called the truly complementary energy principles in large deformation structural mechanics (see Ref. [35] and the references cited therein). Unfortunately, the duality gap between the total potential and their truly complementary energy is R unless the second Piola}Kirchho! stress T is positive-de"nite almost everywhere in the domain ). As it has been proved in [29,33] that if T is positive de"nite on ), the total potential energy is convex, and the classical complementary energy variational principles are equivalent to the Levinson}Zubov principle, which has the simplest formulation. Also, a key mistake in his method was pointed out in [21]. Applications of the Levinson}Zubov principle have been well studied in large deformation structures (see, for example Refs. [36,37]). In buckling analysis, for some given external loads fM and j'0, the beam may have two stable buckling states. Mathematically speaking, the solution of Problem 2 is not unique and the Lagrangian could have several critical points. The gap function may be positive at one critical point and be negative at others. Suppose (u , s ) is a critical point of ¸ (u, s). Let U ;S be a neighborhood of (u , s ) such that P and P have H H H only one critical point on U and S , respectively, the following interesting triality theorem in buckling analysis was discovered recently in [21]: Theorem 4. Suppose that (u "(vN , wN ), s "(p, q)) is a critical point of ¸ . If G(wN , pN )'0, then (u , s ) is a unique H saddle point of ¸ , u is a global minimizer of P , and the following minimax complementary extremum principle H H holds: P (u )" inf P (u)"sup P (s)"P (s ). H H H H u3U s3S If G(wN , p))0, then we have either the minimum complementary extremum principle:
(96)
P (u )" inf P (u)" inf P (s)"P (s ) H H H H u3U s3S or the maximum complementary extremum principle
(97)
P (u )"sup P (u)"sup P (s)"P (s ). (98) H H H H u3U s3S In this case, if either u is a local minimizer of P or s is a local minimizer of P, then (u , s ) is another local stable H H buckling state. However, if either u is a local maximizer of P or s is a local maximizer of P, then (u , s ) is H H a unstable buckling state. Proof. By Theorem 3, if G(u , s )'0, the critical point (u , s ) must be a saddle point of ¸ on U ;S . Since = H H is strictly convex, = ("u)"sup +("u) ) s!=* (s),. If G (wN , p) is strictly positive, u is unique global H Q H minimizer of P on U (see Ref. [33]). Since G is a quadratic function of w, it is positive if p*0. By Theorem H 3, for all s3S such that p*0, we have inf sup ¸ (u, s)"¸ (u , s )"sup inf ¸ (u, s). (99) H H s3S u3U H If p(0, ¸ is not a saddle functional. So if G(u , s )'0, the critical point (u , s ) is a unique saddle point of ¸ on H H U ;S . If the gap function G(wN , p))0, then (u , s ) is a super-critical point of ¸ on U ;S : H ¸ (u, s ))¸ (u , s )*¸ (u , s) ∀(u, s)3U ;S . H H H If u maximizes P on U , we have H P (u )"sup P (u)" sup sup ¸ (u, s)"sup sup ¸ (u, s)"sup P (s)"P (s ). H H H H H H u3U s3S s 3S u3U s3S u3U s3S
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This shows that P (u )"sup P (u) is equivalent to P (s )"sup P (s). Since both u and s are local maximizers H H H H of P and P, respectively, (u , s ) should be a unstable buckling state. H H If u minimizes P on U , by Theorem 3, H P (u )" inf P (u)" inf sup ¸ (u, s)"¸ (u , s ), H H H H u3U u3U s3S i.e. (u , s ) mini-maximizes ¸ on U ;S . By Lemma 1, for super-critical point we have H inf sup ¸ (u, s)"¸ (u , s )" inf sup ¸ (u, s). (100) H H H u3U s 3S s3S u3U Since G(wN , p))0, on U ;S , Theorem 3 leads to P (s )"¸ (u , s )"inf sup ¸ (u, s)" inf PH (s). H H H s 3S s u It is shown that u minimizes P on U is equivalent to that s minimizes P on S . Since u is a local minimizer H H of the total potential energy, it must be a local stable buckling state. ) As an example, let us consider the following unilateral post-buckling variational problem of the non-linear beam model without the shear e!ect:
* 1 1 aw !j !f w dxPmin ∀u3U . (101) V 2 2 As we mentioned before, this non-convex variational problem has been studied extensively in phase transitions. Suppose that the obstacle is a rigid, smooth, and #at foundation, so that the kinetically admissible space U can be written as P (w)" H
U "+w3C (0, ¸) " w(x)*0 ∀x3(0, ¸), w(0)"0,.
(102)
The Euler}Lagrange equations for this unilateral variational problem with inequality constraint are [aw ( aw !j)] #f"g ∀x3(0, ¸), (103) V V V (104) aw ( aw !j)"0 at x"¸, V V where g31 is a Lagrange multiplier introduced for the inequality constraint w(x)*0. By the Kuhn}Tucker theory, g has to satisfy the following so-called complementarity condition g(x)w(x)"0, ∀g(x))0, w(x)*0 ∀x3(0, ¸).
(105)
Traditional analytic methods for solving this non-linear problem with non-linear complementarity condition are very di$cult. In terms of the linear strain (deformation gradient) e"w , the stored energy V =(e)" ( ae!j) is a double-well energy. Its Legendre conjugate does not have a simple algebraic form. So the traditional Levinson}Zubov complementary energy principle cannot be used for this non-convex variational problem. However, by the non-linear dual transformation method, we are able to solve this problem. Instead of e, we introduce the non-linear geometrical operator e""w" aw . The stored energy density V = (e)" (e!j) (106) H is then strictly convex in the &&"nite strain'' e. The dual variable of e is simply given by p"D= (e)"(e!j). H
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The conjugate function =* can be easily obtained by the Legendre}Fenchel transformation H =* (p)"sup +pe!= (e)," p#jp. H H C For the quadratic operator ", we have
(107)
" (w)w"aw , " (w)w""(w)w!" (w)w"!aw . R V L R V So the gap function is simply a quadratic functional: *1
2 aw V p dx.
G(w, p)"
(108)
In terms of p, the Euler}Lagrange equation can be written as "* (w)p"!a(w p) *f ∀x3(0, ¸), R V V "* (w)p"aw p"0 at x"1. (109) R V Since p"(e!j), and e" aw *0, the range of p should be [!j,#R). Then the statically admissible V space S can be given as S "+p3C (0, ¸) "!j)p(R, p(x)O0 ∀x3(0, ¸), p(¸)"0,. The Lagrangian form in this 1-D problem is
* 1 1 aw p! p#jp!fM w dx. V 2 2 For a given p3S , solving the equilibrium inequality (109) for w , we have V g(x) w " , V ap ¸ (w, p)" H
(110)
(111)
where the function g in this unilateral variational problem is *
V
fM (s) ds! fM (s) ds.
g(x))
(112)
Substituting Eq. (111) into G(w, p) leads to the pure gap function * g(x)
2ap dx.
G* (p)"
(113)
Then the pure complementary energy for this problem is
* 1 g(x) p#jp# dx, 2 2ap which is a non-convex functional de"ned on S . Let P (p)"!
1 g =* (p)"! p#jp ! . T 2 2ap
(114)
(115)
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Fig. 3. Double-well energy = (w) (solid line) and its conjugate =* (p) (dashed line). T T
It graph is shown in Fig. 3. The critical condition dP (p; dp)"0 gives the following dual Euler}Lagrange equation: p
1 1 p#j " g(x). 2 2a
(116)
This is an algebraic equation! For any given external load fM (x) such that g is obtained by Eq. (112), this cubic equation has at most three roots p (x), i"1, 2, 3. By Eq. (111), the analytic solution for this contact G post-buckling problem is
w (x)" G
V g(t) dt*0 ∀x3(0, ¸). ap (t) G
(117)
If we let = (w)" ( aw!j)!gw. (118) T This is the well known van der Waals energy. Its dual function is just =*. By Triality theorem we know that T there exists a j '0 such that when j'j , both = and =* have three critical points (see Fig. 3). In the
T T case of p (x)'0'p (x)'p (x) ∀x3(0, ¸), the triality theorem tells that w (x) is a global minimizer of P , H w (x) is a local minimizer and w (x) is a local maximizer. In the unilateral buckling problem, the solution w G has to be in U . Hence if the beam is subjected to a positive (upward) load f(x)*0 ∀x3(0, ¸), then g(x)*0. In this case, the global minimizer w (x)*0 ∀x3 (0, ¸) is the only one solution to the unilateral buckling problem. However, if f(x))0 ∀x3(0, ¸), the local minimizer w (x)*0 ∀x3 (0, ¸) is a stable post-buckling state; the local maximizer w (x)*0 is a unstable state. The Lagrangian associated with = is T ¸ (w, p)" awp!( p#jp)!gw. T It is a saddle function for p'0. If p(0, it is a super-critical point function (see Fig. 4). For j"a"1, g"6.3, its three critical points are: (w , p )"(1.55, 0.19), (w , p )"(!0.32, !0.95), (w , p )"(!1.23, !0.24).
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Fig. 4. Graph of the Lagrangian ¸ (w, p). U
It is easy to verify that (w , p ) is a saddle point, the rest are super-critical points and by the triality theory, we have = (w )" min = (w)"¸ (w , p )"max =* (p)"=* (p ), T T T T T p*0 "w"*(2 max max = (w )" = (w)"¸ (w , p )" =* (p)"=* (p ), T T T T T !1(p(p w (w((2 = (w )" min = (w)"¸ (w , p )" min =* (p)"=* (p ). T T T T T !1(p(0 "w"((2 7. Dual extremum principle in non-linear dynamics Now we are going to study the duality in dynamic systems. To framework our theory in general "nite deformation dynamic problems, we let U be a general displacement space, U LU a bounded, convex kinetically admissible space, in which, the essential initial/boundary conditions are prescribed. Suppose that " is a pure quadratic "nite deformation operator such that for any given parameter j 3 1, the stored energy density = (e):" = (e, j) is convex in the general strain measure e""u. In general parametric variational H problems, the parameter j can be considered as an internal variable. It could be represented in scalar, vector or tensor forms (see [20]). Physically speaking, the internal variable could be a buckling load in structural analysis, a residual strain in elastoplasticity, or distributed parametric control in dynamical systems. For the given external load f and the parameter j3 1, we let
=H (e) d),
F(u)"(f , u),
; (e)" H
be the internal potential and the external potential, respectively. Then the total potential of the "nite deformation system can be written in general setting
=H ("u) d)!(f , u).
P (u)"; ("u)!F (u)" H H
(119)
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By the Fenchel transformation, the complementary energies of ; and F are de"ned, respectively, by H
;* (s)"sup +1s, e2!; (e)," =* (s) d), e H H H If f"f , F* (f)"sup +(f, u)!F(u),"0
(120)
F* (f)"sup +(f, u)!F(u),"#R if fOf .
(121)
u u
Since ; (e) is convex, the following statical constitutive relations are equivalent: H *=* (s) *= (e) H H 8e"D;* (s)" s"D; (e)" H H *e *s Moreover, the Ga( teaux derivative of F gives the statical equilibrium equation: f "DF(u), u 3 U .
(122)
For Newtonian systems, the kinetic energy ¹(v) is a quadratic functional of the velocity v"u . Its Ga( teaux derivative gives the conjugate variable of the velocity v, i.e., the momentum p"D¹ (v).
(123)
The complementary kinetic energy is uniquely de"ned by the Legendre transformation: ¹*(p)"1p, v2!¹ (v)
(124)
and the dynamical constitutive relation (123) is invertible: v"D¹* (p).
(125)
The total action of the general "nite deformation dynamical system is then a non-convex functional with one parameter j:
[¹ (u )!PH (u)] dt. t
% (u)" H
(126)
The solution u of the dynamical variational problem d% (u ; u)"0 ∀u3U
(127)
should satisfy the following abstract dynamic equation: d "* (u ) D=("u )# D¹ (u )"f . R dt
(128)
For the non-linear dynamical post-buckling problem discussed in Section 4, the kinetic energy ¹ associated with Problem 1 should be understood as *1 o w dx, (129) R 2 since the velocity in the axial direction is ignored. In terms of notations de"ned in Section 5, the abstract dynamical equation (128) is equivalent to all the governing equations in Problem 1. We are interested in duality theory of the following least action problem.
¹ (u )"
126
D.Y. Gao / International Journal of Non-Linear Mechanics 35 (2000) 103}131
Problem 4. For the given external load f and the parameter j, ,nd u 3U such that % (u )" inf % (u). H H u3U
(130)
The generalized Lagrangian associated with this problem is
[1p, u 2!1"u, s2!¹ * (p)#;*H (s)#F(u)] dt. t
L (u, s, p)" H
(131)
The critical condition DL (u , s , p )"0 leads to the Euler}Lagrange equations: H "u "D;* (s ), H d u "D¹* (p ), dt "* (u ) s #p "DF (u ). R
(132)
Since ; (e) and ¹ (v) are convex, the Euler}Lagrange equations in Eq. (132) are equivalent to the abstract H dynamical equation (128). Let PLC be an admissible space of momentum. Let S be a convex subset of the space S. The dual extremum principle in dynamical systems can be proposed. Theorem 5. For any given (u, s, p)3U ;S;P, inf sup inf L "inf inf sup L "inf inf sup L . u p s u s p s u p H H H
(133)
However, if the gap function satis,es G (u, s)*0 ∀(u, s)3U ;S , then inf sup inf L "inf inf L . u p s s p H H
(134)
If G (u, s))0 ∀(u, s)3U ;S , then inf sup inf L "inf sup inf L . u p s s p u H H
(135)
Proof. For any given u 3U , the Lagrangian L is a saddle functional on S;P, and hence H sup inf L (u, s, p)"inf sup L (u, s, p) p s s p H H
∀u3U .
(136)
Then Eq. (133) can be easily proved as we can take the in"mum of L in either order on U ;S. H Since " is a pure quadratic operator, if the gap function is positive on U ;S , then for any given s 3 S , 1s, "u2 is a convex functional of u (see Ref. [33]). The Lagrangian L is then a super-critical point functional H of (u, p). By Lemma 1, we have inf sup L (u, s, p)"inf sup L (u, s, p) u p p u H H
∀s3 S .
However, if the gap function is negative on U ;S , then for any given s 3S , the Lagrangian L is a saddle H point functional of (u, p). So inf sup L (u, s, p)"sup inf L (u, s, p) ∀s3S . u p p u H H Statements (134) and (135) can be easily proved by combining Lemma 1 and Eq. (133). )
D.Y. Gao / International Journal of Non-Linear Mechanics 35 (2000) 103}131
127
It is easy to check that % (u)"sup inf L (u, s, p) ∀u3 U . p s H H
(137)
The total complementary action for the "nite deformation dynamical systems can be de"ned as
[;*H (s)#G (s, p)!¹ * (p)] dt, t
% (s, p)"
(138)
where G (s, p) is a pure complementary gap function in dynamical system, obtained by solving the following stationary variational problem: G (s, p)"sta +1p, u 2!1s, "u2#F(u)"∀u3U ,.
(139)
It is clear that for the pure quadratic operator ", we have % (s, p)"sup L (u, s, p) H u3U
if G(u, s)*0,
% (s, p)" inf L (u, s, p) if G(u, s)(0. H u3U
(140)
If " is a linear operator, then G (s, p)"0 s.t. "* s#p "f .
(141)
In this case,
[;*H (s)!¹ * (p)] dt t
% (s, p)"
(142)
is the classical complementary action in linear elastodynamics [38]. To illustrate the application of this duality theory in non-convex dynamical systems, we consider the following one-dimensional problem:
t
% (w)" H
1 1 ow! aw!j dtPmin ∀w3U . R 2 2
(143)
The kinetically admissible space U in this one-dimensional dynamical system is simply given as U "+w3C (0, t ) " w(0)"w , w (0)"v ,. R
(144)
The critical condition of % leads to a non-linear ordinary di!erential equation H oN w "jaw! a w, ∀t3(0, t ). RR
(145)
This is the well-known Du$ng equation. For some given parameters j, a'0, this non-linear equation may have chaotic solutions. By introducing a pure quadratic operator e""w" aw, the stored energy density = (e) and its conjugate =* (p) are the same as those given in Eqs. (106) and (107), respectively. The gap H H function is simply a quadratic functional
2 aw p dt.
G(w, p)"
t
1
(146)
128
D.Y. Gao / International Journal of Non-Linear Mechanics 35 (2000) 103}131
Fig. 5. x"3, v "0.4.
D.Y. Gao / International Journal of Non-Linear Mechanics 35 (2000) 103}131
129
The dynamical Lagrangian form is then
p w R!2 aw p!2oN p#2 p#jp dt. t
L (w, p, p)" H
1
1
1
(147)
Here p"oN w stands for the momentum. Fixing p and p, the variationl problem sta L (w, p, p) gives R U H pR awp"!pR N w"! . (148) ap So the dual action is obtained
% (p, p)" H
t
1 pR 1 p#jp# ! p dt. 2 2ap 2o
(149)
The statically admissible space S is S "+p3C(0, t ) "!j)p(t)(#R, p(t)O0 ∀t3(0, t ), p(0)" aw!j,. Let P "+p3C (0, t ) " p(0)"o v ,. Dual least action problem is then % (p, p)Pmin ∀(p, p)3S ;P . H The critical condition D% (p, p)"0 leads to the following dual Euler}Lagrange equations: H 1 p (p#j)" pR ∀t3(0, t ), 2a
d pR 1 # p"0 ∀t3(0, t ). dt ap oN
(151) (152)
This coupled non-linear algebraic/di!erential system is the dual of the Du$ng equation. In algebraic geometry, Eq. (151) in the dual phase space p-pR is the so-called singula cubic curve [see Fig. 5(g)]. It has some very important algebraic properties. By Theorem 5, for the same initial conditions, problems (143) and (150) have the same solution set. Since the Du$ng equation is very sensitive to the parameter j, di!erent methods give di!erent (or chaotic) results. However, the primal}dual method provides a powerful complementary bounding approaches to the real solution of the non-linear dynamical systems. For the given data a"1, w "0, v "0.2, Fig. 5 shows that the di!erences between the primal results (solid line) and the dual results (dots) vary with the parameter j. Detailed study on the primal}dual approaches for solving forced Du$ng equation and more interesting results will be given in another paper. 8. Conclusions Two dynamical beam models are developed in this paper for serious study of the post-buckling behavior in "nite deformations. As we can see that in both models, the deformations in thickness direction are proportional to w , which cannot be ignored when the beam is subjected to moderate large rotation w . V V Since the total potentials of these two beam models are non-convex with double-well structures, these two models can be used in the post-buckling analysis. The second order beam model can also be used to study frictional contact problems. An analytic solution for unilateral post-buckling problem is obtained. The properties of this solution are clearly clari"ed by the triality theory.
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D.Y. Gao / International Journal of Non-Linear Mechanics 35 (2000) 103}131
Duality theory and methods play very important roles in "nite deformation mechanics. A uni"ed approach and survey on the recent developments of the complementary variational problems in "nite deformation theory are given in [31]. A detailed study on the non-linear dual transformation method and the duality/triality theory in non-convex, parametrical variational problems are given in [20]. Based on the general theory proposed in [21], a pure complementary energy principle in three-dimensional "nite deformation mechanics is proposed in [20,22,31]. It is proved that by using this pure complementary energy principle, the non-linear partial di!erential equations can be transformed into non-linear tensor equations. Therefore, the general closed-form solutions are obtained for mixed boundary value problems in "nite deformation theory [20,22]. The analytic solutions for non-convex variational problems with multi-well (more than three) energies are also given in [20]. These results have been generalized into non-smooth variational problems [31]. The two beam models proposed in this paper can be easily generalized into non-linear plate theory. The dynamical triality extremum principle can be used to study some very interesting problems in non-linear dynamical systems. Acknowledgements Professor J. Simmonds' valuable suggestion and the helpful comments from an anonymous referee are gratefully acknowledged.
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