When the virtual work is considered as a time integral of
virtual power, a generalized form of the   virtual work principle
is obtained. The Euler-Lagrange equation of it gives an equation for the
divergence of the Truesdell rate of stress. The equation of motion on the
stress rate field is one of the results of this paper

Béda, Gyula

English

Periodica Polytechnica (Budapest University of Technology and Economics)

PERIODICA POLYTECHNICA SER. MECH. ENG. VOL. 48, NO. 1, PP. 5–8 (2004)THE GENERALIZED PRINCIPLE OF VIRTUAL WORKGyula BÉDADepartment of Applied MechanicsBudapest University of Technology and EconomicsH–1521 Budapest, Hungarye-mail: beda@mm.bme.huReceived: March 30, 2004AbstractWhen the virtual work is considered as a time integral of virtual power, a generalized form of the virtualwork principle is obtained. The Euler-Lagrange equation of it gives an equation for the divergenceof the Truesdell rate of stress. The equation of motion on the stress rate field is one of the results ofthis paper.Keywords: virtual power, equation of motion, Truesdell rate.1. IntroductionThe generalization of virtual work principle emerges in the investigation of thirdorder wave or for example in supervision of finite element method. The problemwill be raised in case of third order wave.The investigation of the third order wave necessitates the knowledge of thedynamic compatibility equation. This equation rises from the first equation ofmotion in case of the acceleration wave. Now it needs the time derivative of thefirst equation of motion. The material time derivative isn’t simple in the currentconfiguration. Using the principle of virtual power, namely the principle of virtualwork also for finite deformation, the derivative will be obvious and indisputable.We assume that the integral of the virtual power with respect to time is the virtualwork. Hence, from the principle of virtual work the time derivative of the firstequation of motion can be obtained and then the dynamical compatibility equationcan be calculated. The time derivative of the first equation of motion will be calledthe equation of motion on the stress rate field. Many authors have dealt with thisproblem in the case when the body was in equilibrium [8, 9, 10].2. The Principle of Virtual WorkIn continuum mechanics the principle of virtual power is:∫Vtklv∗k;l dV =∫Vqkv∗k dV +∫App̃kv∗k dA , (1)6 GY. BÉDAwhere tkl , v∗k , v∗k;l and qk denote the Cauchy stress, the virtual velocity, the virtualvelocity gradient and the difference between the body force and the force of inertiain domain V and p̃k is the surface force on boundary surface Ap . A is Av + Ap ,on Av the velocity ṽk is known.The stress tensor in (1) satisfies the second Cauchy equation of motion, thatis, tkl = t lk .Assume as a starting point that the integral of the power for a given period[t1, t2] means the work during this period. Thus, (1) integrated with respect to timet gives ∫ t2t1∫Vtklv∗k;l dV dt =∫ t2t1∫Vqkv∗k dV dt +∫ t2t1∫Ap̃kv∗k dA dt. (2)As it can be seen, the virtual deformation rate v∗kl on the left hand side of theequation has been replaced by virtual velocity gradient v∗k;l . This replacementleaves the product tklv∗kl unaltered since tkl = t lk . The material time derivative ofthe deformation gradient isẋ k,K = vk;px p,Kand from this,vk;p = ẋ k,K X K,p (3)With the displacement vector u used and the derivative thereof with respect to time,then to X K , written in indexed form arevk = u̇k and ẋk,K = u̇k;qxq,K ≡ u̇k:K ,respectively.Thus, (3) becomesvk;p = u̇k,K X K,p = u̇k;p. (4)With the volume integral on the left side of (2) transformed to the initial configu-ration, the integrals with respect to time and over volume V0 can be interchanged:∫V0∫ t2t1t lk u̇∗k,K XK,l J̄ dt dV0 =∫V0∫ t2t1J̄ qk u̇∗k dt dV0 +∫A0p∫ t2t1J̄ tkl u̇∗k XK,l dt dA0K ,(5)where J̄ = dVdV0 .Consider now the integrals with respect to time, one after the other:∫ t2t1J̄ t lk XK,l u̇∗k,K dt =∫ t2t1[(J̄ t lk XK,l u∗k,K) − ( J̄ t lk X K,l ) u∗k,K ] dt.The first integral can be calculated from time t1 to t2 on the right side, that is,∫ t2t1J̄ t lk XK,l u̇∗k,K dt =(J tlku∗k;l)t2t1−∫ t2t1J̄(t pk vq;q + ṫ pk − t lkv p;l)X K,pu∗k,K dt. (6a)THE GENERALIZED PRINCIPLE OF VIRTUAL WORK 7After similar transformations, the first integral with respect to time on the right sideof (5) is as follows:∫ t2t1J̄ qk u̇∗k dt =(Jqku∗k)t2t1−∫ t2t1J̄(q̇k + vs;sqk)u∗k dt (6b)Here also, virtual displacement u∗k = 0 at time t1 and t2 in (6b).Finally, after transformation of the second integral on the right side of (5),∫ t2t1J̄ tkl X K,l u̇∗k dt =(J tkl X K,l u∗k)t2t1−∫ t2t1J̄(tkpvs;s + ṫ kp − tklv p;l)X K,pu∗k dt. (6c)With Eqs. (6a), (6b) and (6c) substituted into (5) and after proper rearrangement,the principle of virtual work is [1].∫ t2t1∫V(ṫ kp − tkqv p;q + tkpvs;s)u∗k;p dV dt =∫V[(t lk;l + qk)u∗k]t2t1dV+∫Ap[(p̃k − tklnl)u∗k]t2t1dA+∫ t2t1∫V(q̇k + qkvs;s)u∗k dV dt−∫ t2t1∫Ap(ṫ kp + tkpvs;s − tklv p;l)n pu∗k dAdt, (7)thereforet lk;l + qk = 0 is the first Cauchy equation of motionandp̃k ≡ tkpn p dynamic boundary condition on Apcan be obtained from the first and secound terms of the right hand side of the Eq. (7).3. The Equation of Motion on the Stress Rate FieldThe Eq. (7) refers to continue and its any part. Otherwise, on the basis of all thathas been mentioned above, the Euler-Lagrange equation given below is obtainedafter the suitable mathematical transformation:t̊ kp;p + tqpvk;qp + q̆k = 0 (8)supposing that the Cauchy equations of motion are satisfied. Here t̊ kp denotes theTruesdell rate of Cauchy’s stress tensor, that is,t̊ kp ≡ ṫ kp − tkqv p;q − tqpvk;q + tkpvs;sand8 GY. BÉDAq̆k = q̇k + qkvs;s − qsvk;sor when q̄k is the body force densityandqk ≡ q̄k − ρv̇k then q̆k = ˙̄qk + qkvs;s − qsvk;s − ρ(v̈k − v̇svk;s),where ρ is the mass density and it satisfies the continuity equation.The (8) is the equation of motion on the stress rate field (7), [8, 9, 10]. Theboundary condition on Ap is t̊ kpn p + tqpn pvk;q = 0.4. ConclusionThe principle of virtual work is extended to continue, which performs finite defor-mation. The deformation depends on time, too. The equation of motion for stressrate is derived from the generalized principle.References[1] BÉDA, GY., Generalization of Clapeyron’s Theorem of Solids, Periodica Polytechnica Ser.Mech. Eng., 44 No. 1, (2000).[2] ERINGEN, A. C. – SUHUBI, E. S., Elastodynamics, Academia Press, New York and London,1974.[3] MARSDEN, J. E. – HUGHES, T. J. R., Mathematical Foundations of Elasticity, Prentice-Hall,Englewood Cliffs, N.Y. 1983.[4] BÉDA, GY., The Possible Fundamental Equations of the Constitutive Equations of Solids,Newsletter TU Budapest, 10 No. 3 (1992), Budapest.[5] BÉDA, GY., The Possible Fundamental Equations of the Continuum Mechanics, PeriodicaPolytechnica Ser. Mech. Eng., 35 No. 1–2 (1991).[6] BISHOP, R. E. D. – GLADWELL, G. M. L. – MICHAELSON, S., The Matrix Analysis ofVibration, Cambridge University Press 1965.[7] BÉDA, GY., The Constitutive Equations of the Moving Plastic Bodies, DSc. thesis, Budapest1982 (in Hungarian).[8] HILL, R., Some Basic Principles in the Mechanics of Solids without a Natural Time, J. of theMech. and Phys. of Solids, 7 (1959).[9] THOMPSON, E. G. – SZU-WEI, YU, A Flow Formulation for Rate Equilibrium Equations, Int.J. for Numerical Methods in Engineering, 30 (1990).[10] DUBEY, R. N., Variation Method for Nonconservative Problems, Trans. ASME Journal ofApplied Mechanics, (1970) pp. 133–136.

THE GENERALIZED PRINCIPLE OF VIRTUAL WORK

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THE GENERALIZED PRINCIPLE OF VIRTUAL WORK

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