Proceedings of Institute of Mathematics of NAS of Ukraine 2000, Vol. 30, Part 1, 223–230. Symmetry Groups and Conservation Laws in Structural Mechanics Vassil M. VASSILEV and Peter A. DJONDJOROV Institute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bontchev St., Bl.4, 1113 Sofia, Bulgaria E-mails: [email protected] and [email protected] Recent results concerning the application of Lie transformation group methods to structural mechanics are presented. Focus is placed on the point Lie symmetries and conservation laws inherent to the Bernoulli–Euler and Timoshenko beam theories as well as to the Marguerre- von K´arm´an equations describing the large deflection of thin elastic shallow shells within the framework of the nonlinear Donnell–Mushtari–Vlasov theory. 1 Introduction The present paper is concerned with the invariance properties (point Lie symmetries) of three classes of self-adjoint partial differential equations arising in structural mechanics – the dynamic beam equations of Bernouli–Euler and Timoshenko type governing vibration of beams on a variable elastic foundation and dynamic stability of fluid conveying pipes, and Marguerre-von K´arm´an equations describing the large deflection of thin isotropic elastic shallow shells subjected to an external transverse load and a nonuniform heating. Once the invariance properties of a given differential equation are established, several impor- tant applications are available. First, it is possible to obtain classes of group-invariant solutions. For a self-adjoint equation another application of its symmetries arises since it is the Euler– Lagrange equation of a certain functional. If a symmetry group of such an equation turned out to be its variational symmetry as well, that is a symmetry of the associated functional, then Noether’s theorem guarantees the existence of a conservation law for the solutions of this equation. Needless to recall the fundamental role of the conserved quantities and conservation laws (or the corresponding balance laws) for the natural sciences, however it is worthy to point out that the available conservation (balance) laws should not be overlooked in the examination of discontinuous solutions (acceleration waves, shock waves, etc.) or in the numerical analysis (when constructing finite difference schemes or verifying numerical results, for instance) of any system of differential equations of physical interest. It should be remarked also that the path- independent integrals (such as the well known J-, L-and M-integrals) related to the conservation laws are basic tools in fracture analysis of solids and structures. Throughout this paper: Greek (Latin) indices have the range 1, 2 (1, 2, 3), unless explicitly stated otherwise, and the usual summation convention over a repeated index is employed. The k-th order partial derivatives of a dependent variable, say w, that is ∂kw/∂xα1 ∂xα2 ...∂xαk 1 2 α α α (k, α1,α2,...,αk =1, 2,...), are denoted either by wα1α2...αk or wx 1 x 2 ...x k , where x ,x ,... are the independent variables. A similar notation is used for the partial derivatives of any other function, say f, of the independent variables but, in this case, the indices indicating the differentiation are preceded by a coma. Dα (α =1, 2,...) denote the total derivative operators. For the basic notions and statements used in the group analysis of differential equations and variational problems see [1] or [2]. 224 V.M. Vassilev and P.A. Djondjorov 2 Symmetries and conservation laws of beam equations Bernoulli–Euler beams. Consider the class of self-adjoint partial differential equations αβ γw1111 + χ wαβ + κ(x)w =0, (1) in two independent variables x =(x1,x2) and one dependent variable w(x), where γ =const=0, χαβ are arbitrary constants, and κ(x) is an arbitrary smooth function. Equations of this special type are used to study problems concerning dynamics and stability of both elastic beams resting on elastic foundations and pipes conveying fluid. In these cases, x1 is associated with the spatial variable along the rod axis, x2 – with the time, and w represents the transversal displacement field. In [3] the point Lie symmetries of (1) are examined and the solution of the corresponding group-classification problem with respect to the arbitrary element {γ,χαβ,κ(x)} is given. Evi- dently, each equation of form (1) is invariant under the point Lie groups generated by the vector fields X0 = w∂/∂w and Xu = u (x) ∂/∂w, where u (x) is any smooth solution of the respective equation. The results of the group-classification are summarized in Table 1 below, where the equations invariant under larger groups are given through their coefficients together with the generators of the associated symmetry groups. Table 1 # Coefficients Generators 2 1 1 2 1 2 1 κ(x)=f(β x − β x ) β X1 + β X2 22 αβ 2 2 −2 1 2 χ =0 , det(χ )=0,κ(x)= β + x f(y), {β +2χβ }X1 2 −1/2 2 y = β2 + x2 β1 + x1 − χx2 +2β X2 + X3 22 αβ 2 2 −4/3 1 11 12 2 χ =0, det(χ ) =0 ,κ(x)= β + x f(y), {β +3(χ /χ )β }X1 3 −1/3 2 y = β2 + x2 β1 +2x1 − χ11/χ12 x2 +6β X2 +2X˜3 22 αβ 2 −2 4 χ =0 , det(χ )=0,κ(x)=κ0 β + x , X1,2βX2 + X3 22 αβ 2 −4/3 5 χ =0, det(χ ) =0 ,κ(x)=κ0 β + x X1,3βX2 + X˜3 22 αβ χ =0, det( χ )=0, βX1 + X3, 6 1 2 −4 κ(x)=κ0 β + x − χx χX1 + X2 22 αβ ˜ χ =0, det( χ ) =0, βX1 +2X3, 7 1 11 12 2 −4 11 12 κ(x)=κ0 β +2x − χ /χ x (χ /χ )X1 +2X2 22 αβ 8 χ det(χ ) =0 ,κ(x)=const X1, X2 22 αβ 9 χ det(χ )=0,κ(x)=const=0 X1, X2 22 αβ 10 χ =0 , det(χ )=0,κ(x)=0 X1, X2, X3 22 αβ 11 χ =0, det(χ ) =0 ,κ(x)=0 X1, X2, X˜3 1 2 12 22 Here f is an arbitrary function, β,β ,β are arbitrary real constants, χ = χ /χ and Xα = α 1 2 1 2 2 1 2 1 2 2 ∂/∂x ,X3 = x + χx ∂/∂x +2x ∂/∂x , X˜3 = x + χx ∂/∂x +3x ∂/∂x . It is found [3] that all vector fields quoted under numbers 1, 3, 5, 7, 8, 9 and 11 generate variational symmetries of the respective equations of form (1), while in case # 2 variational 1 2 2 symmetries are associated with {β +2χβ }X1 +2β X2 + X3 +(1/2)X0,incase#4–withX1 Symmetry Groups and Conservation Laws in Structural Mechanics 225 and 2βX2 + X3 +(1/2)X0, in case # 6 – with χX1 + X2 and βX1 + X3 +(1/2)X0,andincase #10–withX1, X2 and X3 +(1/2)X0. Once the variational symmetries are identified, we derive the corresponding conservation laws. They are listed in Table 2 in the same order as in Table 1 using the notation: 1 2 11 2 22 2 2 2µ B(1) = −(1/2) γ(2w1w111 − w11)+χ w1 − χ w2 + κw − (1/2)(χ wwµ),2 , 2 2µ 2µ B(1) = −χ w1wµ +(1/2)(χ wwµ),1 , 1 1µ 1µ B(2) = −χ w2wµ + γ(w2w111 − w11w12) − (1/2)(γw1w11 − χ wwµ),2 , 2 2 22 2 11 2 2 1µ B(2) = −(1/2) γw11 + χ w2 − χ w1 + κw +(1/2)(γw1w11 − χ wwµ),1 , α 1 2 α 2 α αµ 1α B(3) = x + χx B(1) +2x B(2) + χ wwµ +(1/2)γδ (ww111 − w1w11), ˜α 1 11 12 2 α 2 α B(3) = x +(χ /χ )x B(1) +3x B(2) αµ 1α 11 12 +(1/2) χ wwµ + δ (χ ww1 +2χ ww2 − γw1w11) . Table 2 # Conservation laws 1 α 2 α 1 Dα β B(1) + β B(2) =0 1 2 α 2 α α 2 Dα β +2χβ B(1) +2β B(2) + B(3) =0 1 2 α 2 α ˜α 3 Dα β +3χβ B(1) +6β B(2) +2B(3) =0 α α α 4 DαB(1) =0, Dα 2βB(2) + B(3) =0 α α ˜α 5 DαB(1) =0, Dα 3βB(2) + B(3) =0 α α α α 6 Dα βB(1) + B(3) =0, Dα χB(1) + B(2) =0 α ˜α 11 12 α α 7 Dα βB(1) +2B(3) =0, Dα (χ /χ )B(1) +2B(2) =0 α α 8 DαB(1) =0, DαB(2) =0 α α 9 DαB(1) =0, DαB(2) =0 α α α 10 DαB(1) =0, DαB(2) =0, DαB(3) =0 α α ˜α 11 DαB(1) =0, DαB(2) =0, DαB(3) =0 In addition, each equation (1) admits conservation laws of form αµ 1α Dα{χ (uwµ − u,µ w)+δ γ(uw111 + u,11 w1 − u,111 w − u,1 w11)} =0, u(x) being any solution of the equation considered. Timoshenko beams. The Timoshenko beam equations Jϕtt = EJϕxx + nGA(wx − ϕ),Awtt = nGA(wxx − ϕx), (2) describe the motion of beams accounting for the buckling of the beam cross-section. They are two coupled second order linear partial differential equations in two independent variables – the time t and the coordinate along the beam axis x, the dependent variables being w(x, t)and 226 V.M. Vassilev and P.A. Djondjorov ϕ(x, t), associated with the transversal displacement of the beam axis and the rotation angle, respectively. In these equations is the density of the beam material, E – the modulus of elasticity, G – the shear modulus, J and A – the moment of inertia and the area of the beam cross-section, n – a coefficient related to the buckling of the cross-section.