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6.2 The Principle of Virtual Work in geometrically nonlinear problems 193 where w w V LM = w LM and w V LM are the components of the incremental second Piola-Kirchho stress tensor; it is important to recognize that the three tensors in Eq. (6.33) are referred to the spatial configuration at time w. Also, w+w w % LM = w % LM = (6.34) because w w % LM =0. Replacing with Eqs. (6.33) and (6.34) in Eq. (6.32), we get Z w Y ¡ w LM + w V LM ¢ ( w % LM ) w dY = w+w Z h{w > (6.35) we can write an incremental constitutive equation referred to the wconfiguration, w V LM = w F LMNO w % NO > (6.36) and get, Z w Y ¡ w LM + w F LMNO w % NO ¢ ( w % LM ) w dY = w+w Z h{w = (6.37) In a fixed Cartesian system we can show that (Bathe 1996), w % = 1 2 ( w x > + w x > + w x > w x > ) (6.38) where w x > = Cx C w } = We can decompose the strain increment into a linear and a nonlinear increment in the unknown incremental displacemen t; that is to say, w % = w h + w w h = 1 2 ( w x > + w x > ) (6.39) w = 1 2 ( w x > w x > ) = Hence we can write Eq. (6.37) as, Z w Y £ w + w F ¡ w h + w ¢¤ ¡ w h + w ¢ w dY = w+w Z h{w = (6.40) The above is the momentum balance equation at time w + w;whichis a nonlinear equation in the incremental displacement vector. Proceeding in the same way as in the total Lagrangian formulation we obtain the linearized momentum balance equation (Bathe 1996): Z w Y w F w h w h w dY + Z w Y w w w dY (6.41) = w+w Z h{w Z w Y w w h w dY= 194 Nonlinear continua It is easy to show that r V LM = r w w V lm ¡ w r [ 1 ¢ L l ¡ w r [ 1 ¢ M m (6.42) r % LM = w % lm w r [ l L w r [ m M (6.43) and therefore if the same material is considered in both formulations the incremen tal constitutive tensors should be related, F LMNO = r w w F pqst ¡ w r [ 1 ¢ L p ¡ w r [ 1 ¢ M q ¡ w r [ 1 ¢ N s ¡ w r [ 1 ¢ O t = (6.44) Any problem can be alternatively solved using either the total or the up- dated Lagrangian formulations and the results should be identical (Bathe 1996). For solving finite-strain elastoplastic problems, in Sect. 5.2.6 we introduced an adhoc incremental form ulation, the total Lagrangian-Hencky formulation. 6.3 The Pr inciple o f Virtual Powe r There are formulations where the primary unknowns are the material veloci- ties rather than the material displacements (e.g. fluid problems, metal-forming Eulerian (Dvorkin, Cavaliere & Goldschmit 1995, Dvorkin & Petöcz 1993) or ALE formulations (Belytschk o, Liu & Moran 2000), etc.). For these cases the momentum conservation leads to, Z w Y w b · w v w w dY + Z w V w t · w v w dV = Z w Y w : w d w dY= (6.45) In the above equation w v is the material velocity at a point and w d is the strain-rate tensor. Of course, we can use, for formulating the Principle of Virtual Power, other energy conjugated stress/strain rate measures, for example: Z Y w b · w v r dY + Z V w t · w v w M V dV = Z Y w : w d dY> (6.46a) Z Y w b · w v r dY + Z w V w t · w v w M V dV = Z Y w r S : w r · % dY> (6.46b) Z Y w b · w v r dY + Z V w t · w v w M V dV = Z Y w r P W : w r · X dY> (6.46c) Z Y w b · w v r dY + Z V w t · w v w M V dV = Z Y w : w r · H dY> (6.46d) 6.4 The Principle of Stationary Potential Energy 195 the last one only being valid for isotropic constitutive relations. 6.4 The Principle of Stationary Potent ial E nergy As we remarked above, the Principle of Virtual Work can be used for any material constitutiv e relation, for any t ype of l oading and for any nonlinearity inthecasetobeanalyzed. In the present section w e will specialize the Principle of Virtual Work for: • Hyperelastic materials. • Conservative external loads. For a hyperelastic material we ha ve seen in Chap. 5 (Eq. (5.3d) that, w r V LM = r C w U( w r %) C w r % LM = (6.47) The external conservative loads are the external loads that can be derived from a potential. Hence, a load field is said to be conservative in a region if the net work done around any closed path in that region is zero (Crandall 1956). A typical conservative load system can be represented as, w f = * = (6.48) Following the definitions introduced above, the load system in Eq. (6.48) is a body attached load system with constant direction. For conservative loads per unit mass, we write w b = C w J ( w u) Cu (6.49) and for conservative surface loads w t = C w j ( w u) Cu = (6.50) Note that the above-defined surface loads are defined as loads per unit reference surface; therefore, its resultant at time w is R V w t dV . We now define a functional of the function w u called the potential energy functional: w r = Z Y ¡ w U + w J ¢ dY + Z V w j dV (6.51) 196 Nonlinear continua Therefore, w = Z Y " C w U C w % : w % + C w J Cu · w u # dY + Z V C w j Cu · w u dV= (6.52) In the above, w u are admissible variations ¡ w u = 0 on V x see Fig. 6.1 ¢ and w % is derived from the displacement variations. Therefore, w = Z Y h w S: w % w b · w u i dY Z V w t · w u dV= (6.53) Hence, for a hyperelastic material under a conservative load system, the principle of virtual work, in Eq. (6.18), can be written as w =0= (6.54) The abov e equation s tates that when the wconfiguration is in equilibrium the potential energy functional reaches a stationary value; i.e. it fulfills the necessary requirements for being an extreme (Fung 1965). In what follows we show that in the case of infinitesimal strains the po- ten tial energy not only is stationary at the equilibrium configuration but it actually attains there a min imum. Using the nomenclature introduced in Eq. (6.1) we write the potential energy functional for an admissible configuration close to the equilibrium one as w r 0 = Z Y £ w U( w % + w %) w b · ¡ w u + w u ¢¤ dY (6.55) Z V w t · ¡ w u + w u ¢ dV= Using a Taylor expansion, w U( w % + w %)= w U( w %)+ C w U C w % ¯ ¯ ¯ ¯ ¯ w % : w % + 1 2 w % : C 2w U C w % C w % ¯ ¯ ¯ ¯ ¯ w % : w % + ··· = (6.56) Hence, w r 0 w r = w + Z Y 1 2 w % : C 2w U C w % C w % ¯ ¯ ¯ ¯ ¯ w % : w % dY + ··· = (6.57) Since at equilibrium w =0, the sign of the l.h.s. is the sign of the in tegrand on the r.h.s 6.4 The Principle of Stationary Potential Energy 197 In the case of infinitesimal strains case we can a ssume t hat w % 0 and we have w U(0)=0(convention) and w S ¯ ¯ 0 = C w U C w % ¯ ¯ ¯ 0 = 0; hence, from Eq. (6=56) w U( w %)= 1 2 w % : C 2w U C w % C w % ¯ ¯ ¯ ¯ ¯ 0 : w % + ··· = (6.58) Since, in a stable material the value of the elastic strain energy is positive for any strain tensor (the elastic strain energy is a positive-definite function) we conclude that, w r 0 w r A0 > (6.59) and the potential energy is a local minimum at the equilibrium configura- tion. In the infinitesimal strains case we call it the minimun potential energy principle (Washizu 1982). Example 6.3. JJJJJ Conservative and nonconservative loading. (a) Conservative loading Let us consider a linear elastic, cantilever beam under the conservative end- load shown in the figure, Conservative load The elastic energy stored in the beam is, w U = Z O 0 HL 2 µ d 2 w x 2 d w } 2 1 ¶ 2 d w } 1 where H is Young’s modulus and L is the beam cross section moment of inertia. The Principle of Virtual Work states, 198 Nonlinear continua w U = w Sx 2 ¡ w U w S w x 2 ¢ =0 where w = w U w S w x 2 is the potential energy of the system. (b) Nonconservative loading We now consider the same linear elastic cantilever beam but under a follower load, as shown in the figure Body-attached follower load The principle of virtual work states, w U = w S sin ¡ w ¢ x 1 + w S cos ¡ w ¢ x 2 = For small displacement derivatives we can approximate sin ¡ w ¢ w µ d w x 2 d w } 1 ¶ O cos ¡ w ¢ 1 hence, w U = w S µ d w x 2 d w } 1 ¶ O w x 1 + w x 2 ¸ = Since w S µ d w x 2 d w } 1 ¶ O w x 1 + w x 2 ¸ 6= C w G Cu · w u the load is nonconservative and the principle of stationary potential energy cannot be used. JJJJJ 6.4 The Principle of Stationary Potential Energy 199 Example 6.4. JJJJJ Stability of the equilibrium configuration (buckling) (Ho 1956). Let us consider the system shown in the following figure, in equilibrium at time w,inthestraightconfiguration: w S : axial conservative load ; O : length of the rigid bar ; n : stiness of the linear spring; n W : stiness of the torsional spring. Assume that the equilibrium configuration is perturbed with a rotation ??1. The axial load displacement is obtained from the following scheme: 200 Nonlinear continua For ??1 , S O 2 2 and O sin() O. The potential of the external load is, w J = w S S = w SO 2 2 = The only deformable bodies are the springs; hence w U = 1 2 n (O) 2 + 1 2 n W 2 = Therefore the potential energy functional of the system is w r = 1 2 nO 2 2 + 1 2 n W 2 + w SO 2 2 and the equilibrium configuration is defined by w =0 which leads to, £ nO 2 + n W + w SO ¤ =0= Since is arbitrary the bracket has to be zero. Two solutions are possible: (i) =0;thatistosay,thestraight(undeformed)configuration. (ii) w S = ¡ nO+ n W O ¢ . For the second solution is undefined. We call this load value the critical value, S fu , because at this load there are two branching solutions ( =0and 6=0). 6.4 The Principle of Stationary Potential Energy 201 S fu defines the bifurcation or buckling load. The equilibrium path is Since in the above derivation the terms higher than 2 were neglected, we cannot assess anything about the branching equilibrium path. JJJJJ Example 6.5. JJJJJ Postbuckling behavior. We repeat the previous example derivation keeping terms higher than 2 .By doing this, we get S = O (1 cos ) O µ 2 2 4 4! ¶ w J = w S S w SO µ 2 2 4 4! ¶ w U n = 1 2 n (O sin ) 2 1 2 nO 2 µ 3 3! ¶ 2 w U W = 1 2 n W 2 = Hence w r = 1 2 nO 2 µ 2 4 3 + 6 36 ¶ + 1 2 n W 2 + w SO 2 2 w SO 4 24 = For the equilibrium configuration w =0and therefore, nO 2 µ 2 3 3 + 5 12 ¶ + n W + w SO w SO 3 6 ¸ =0= 202 Nonlinear continua Since is arbitrary, we get, neglecting terms higher than 3 , nO 2 µ 1 2 2 3 ¶ + n W + w SO µ 1 2 6 ¶¸ =0 which has again two possible solutions: (i) =0the straight solution (ii) w S = nO ³ 1 2 2 3 ´ + n W O 1 2 6 In the second solution, for =0> w S = S fu = ¡ nO+ n W O ¢ . The bifurcation point is the same as the one calculated in the previous exam- ple; howe ver, now is defined. We see that for A0 Eq. (ii) provides w S = w S (). If we examine the case with n W =0,weget w S = nO ³ 1 2 2 3 ´ 1 2 6 and we can represent If we examine the case with n =0,weget w S = n W O ³ 1 2 6 ´ and we can represent [...]... 1) : moment of inertia of the beam cross section is the beam transversal displacement and using linear beam theory (Ho 1956) 204 Nonlinear continua = à Z 2 Ă Â 0 d2 d ả2 2 2 1 d à d d 2 à 0 d2 d 2 2 1 ả à Ă 2 d2 d  1 2 2 ả d In the rst integral we use (Fung 1965) grating by parts twice, we get à Z 0 d2 d 1 2 2 ả 3 d d At Ă 1 2 d  1 1 2 2 3 Ă 2 2 d2 1 2 1 ả + á  d Z 1 1 à 0 d d d2 d 1 2 2 1 ả á =... = d 1 0 For example ê 2 ( where the parameter dition on 1) = à 1 1 cos 2 ả will be determined by imposing the minimization conê ê = ( ) Using the adopted trial function, we get ê 4 = 64 3 2 2 206 Nonlinear continua The minimum value that can attain the above functional is, within the considered set of trial functions, our best approximation to the equilibrium conguration Imposing ê =0 we get = 2 16... Lagrange multipliers technique (Fung 1965, Fung & Tong 2001), we dene a new functional ( ) and we perform on it an unconstrained minimization: = where 1 2 2 is the Lagrange multiplier + Ă 2  (6.62) 208 Nonlinear continua Fig 6.4 Kinematics constraints We need to determine the set ( necessary conditions are, ) that satises = 0 The =0 (6.63a) =0 (6.63b) =0 (6.63c) From the above we get, = = = (6.64a) 2 (6.64b)... we consider in Eq (6.51) a strain tensor that can not necessarily be derived from the displacements eld We now dene the functional: Z Z Ê Ă Â Ô Ă Â + = U ( u) d + u d (6.65) Z Ă Â :( u ) d + 210 Nonlinear continua and we search for the equilibrium conguration imposing =0 we have as independent variables: u Z + Z S: : ( d Z ) d + (6.66) and Z b ã U ; with S = u d Z :( tã ) d Considering that the variations... necessarilly in the nite element approximations Adding Eqs (6.79) and (6.80), we get Z Z S: d + :( S S) d Z Z b ã u d + tã u d (6.83) = From the above, we can state the Principle of Virtual Work as, 212 Nonlinear continua Z S: d = Z b ã as long as we fulll the condition, Z :( S u d + S) d Z =0 tã u d (6.84) (6.85) which is obviously fullled for the continuum problem Other constraints can also be considered... long as we fulll the condition of variational consistency (Simo & Hughes 1986), Z :( ) d =0 (6.75) which is obviously fullled for the continuum problem 6.6 Veubeke-Hu-Washizu variational principles 211 6.6.2 Constitutive constraints via the V-H-W principles Let us now assume that we consider in Eq (6.51) a stress tensor S that is not necessarily derived from the kinematically consistent strain eld... for example à ả à ả Ă Â 1 1 = 1 cos + 1 cos 1 2 2 It is important to note that the above dened trial function: Fullls the essential (rigid) boundary conditions Contains the previous one, 2 ê ( 1 ), as a particular case ( = 0) Since we will determine the values of both constants by imposing on the necessary conditions for attaining a minimum, it is obvious that 6 ê That is to say, we will either nd the . w h w h w dY + Z w Y w w w dY (6.41) = w+w Z h{w Z w Y w w h w dY= 194 Nonlinear continua It is easy to show that r V LM = r w w V lm ¡ w r [ 1 ¢ L l ¡ w r [ 1 ¢ M m (6.42) r % LM = w % lm w r [ l L w r [ m M (6.43) and. potential energy functional: w r = Z Y ¡ w U + w J ¢ dY + Z V w j dV (6.51) 196 Nonlinear continua Therefore, w = Z Y " C w U C w % : w % + C w J Cu · w u # dY. and L is the beam cross section moment of inertia. The Principle of Virtual Work states, 198 Nonlinear continua w U = w Sx 2 ¡ w U w S w x 2 ¢ =0 where w = w U w S w x 2 is the potential