The problems of beams and deformable solids refer both to the mechanical response of bodies when subjected to the external actions, including forces, torques, and imposed displacements. However, from the mathematical viewpoint, these two problems are intrinsically very different. Even when restricted to small strains, the kinematics of these two types of bodies are disparate: whereas the former is described by a displacement field on an open set of two or three-dimensional Euclidean space, the latter depends on the displacement and the rotation on an interval of the real line. The equilibrium equations of a deformable solid, moreover, are partial differential equations, in contrast with the ordinary differential equations that describe the equilibrium of forces and momenta in a beam.
Despite the apparent differences between the mathematical description of the mechanics of beams and deformable solids, there are deep relations between them. After all, beams are nothing but a special class of solids whose equations can be obtained from the equations of solid mechanics by exploiting some asymptotic behavior or by constraining the class of admissible kinematics (see, for example, [1, 2] for a description of these two avenues for model reduction).
One specific aspect that is of both theoretical and practical interest is the combination of the equations of beams and solids within a single mechanical system or structure. From the theoretical point of view, the interest lies in the formulation of links between these two types of equations and the well-posedness of the resulting boundary-value problems. From the practical side, joint beam/solid equations lead to numerical methods that can efficiently represent the behavior of (beam) structures with subsets studied as three-dimensional solids.
Recently, the author has presented novel formulations of coupled beam/solid mechanics that lead to numerical methods, both in the linear and nonlinear regimes . These formulations, based on new variational principles, can be easily discretized using, for example, finite elements, and replace commonly employed ad hoc links between beams and solids (e.g., [4, 5, 6]). The latter, often based on constraints on the discrete solution, lack a variational basis and thus neither their well-posedness nor their stability can be ascertained.
In this article we study boundary-value problems of linked, deformable, beams and solids in the context of linearized elasticity, as defined by a constrained variational principle. The main goal is to prove the well-posedness of problems with beams and solids involving the minimum set of boundary conditions, effectively proving that the linking terms provide the right stability to the equations, precluding rigid body motions of the system. The boundary-value problems that will be studied have the structure of saddle-point optimization programs in Hilbert spaces (e.g., ) and standard analysis techniques can be used to study their stability and well-posedness.
In section 2 we summarize the equations that govern deformable solids and beams in the context of small strain kinematics, highlighting the variational statement of these two problems and their essential mathematical properties. Section 3
formulates the simplest problem consisting of a beam and a solid that share an interface with the minimum set of Dirichlet boundary conditions. A joint variational principle, where the kinematic compatibility is introduced with Lagrange multipliers, is presented as well. The well-posedness of the resulting boundary-value problem is analyzed in Section4. The article concludes with a summary of the main results in Section 5.
2 Problem statement
This article analyses boundary value problems of joint continuum solids and beams whose solutions correspond to the mechanical equilibrium of both types of bodies, as well as certain compatibility relations in their shared interfaces. Before formulating the global problem, the governing equations of elasticity and beams are briefly reviewed, and their main mathematical properties are identified.
The choice of boundary conditions in these problems is crucial. To show that the constraints that are later introduced effectively link beams and solids, we will present the pure traction problem of an elastic solid and a mixed traction-displacement problem of a beam. Later, we will prove that these two bodies, when appropriately connected, result in a stable structure.
2.1 The Neumann problem of small strain, elastic solids
We start by describing the continuum solid, and we restrict our presentation to an elastic one that occupies a bounded open set with volume . The boundary of the solid is denoted and we identify a subset that will later be linked to a beam.
In classical elasticity, the unknown is the displacement
, the Hilbert space of vectors fields with (Lebesgue) square-integrable components and square-integrable (weak) first derivatives. The stored energy of the deformable body is given by a scalar function, where
is the infinitesimal strain tensor andis the gradient operator. More specifically, for linear isotropic materials this function takes the form where are the two Lamé constants, the dot product refers to the complete index contraction, and is the trace operator.
Considering that the body might be subject to body forces and surface tractions on , the total potential energy of the body is
for all . We note, in passing, that the potential energy (1) might not have any minimiser in — if the forces are not equilibrated — or alternatively, have an infinite number of them, since the displacement function has no imposed values at the boundary . If studied by itself, the minimization of the potential (1) corresponds to the Neumann problem of elasticity and the right functional analysis setting corresponds to the quotient space of modulo the set of infinitesimal rigid body motions.
To set up the analysis framework for the study of three-dimensional solids, we first recall the norm on the space which has the standard form
The bilinear form (2a) verifies the following continuity and stability bounds
for some positive constants , and all . It bears emphasis that, due to the lack of Dirichlet boundary conditions on the boundary of the body, the bilinear form is not coercive in . Rather, and based on Korn’s second inequality , only the weaker statement (4b) can be made. Also, the linear form is assumed to be continuous, i.e.,
with for all .
2.2 Beam mechanics
A cantilever beam of length is now studied. Its curve of centroids is described by a known smooth curve , with a cross section attached to each point of the curve and oriented according to a known smooth rotation field , the latter referring to the set of proper orthogonal tensors. The points on and sections are parameterized by the arclength and we choose and to correspond, respectively, to the clamped section and free tip.
Let be a Cartesian basis. Then maps to the unit tangent vector to curve of centroids at the point , and to the directions of the principal axis of the cross section at the same point. The displacement of the centroids will be given by the vector field and the incremental rotation vector of the cross sections as . Following our previous notation, refers to the Hilbert space of vectors fields on with vanishing trace at .
Shear deformable, three-dimensional beams employ two deformation measures, namely,
where the prime symbol denotes the derivative with respect to the arc-length. The strain holds the shear and axial deformations, whereas the vector contains the bending curvatures and the torsion deformation.
The simplest section constitutive law for a beam of a linear elastic and isotropic material with Young’s and shear moduli , respectively, is based on a quadratic stored energy function per unit length. It has the form
with section stiffness and , where is the cross section area, are the (shear) reduced sections areas in the two principal directions,
are the two principal moments of inertia, andis the torsional inertia. When the beam is under distributed loads and moments, denoted respectively as and , and subject to a concentrated load and moment at the tip, its total potential energy can be expressed as
for all , and .
To set up the functional setting for the beam problem, we recall the norms on the space of displacements and rotations which are
Also, the product space , the natural setting for the beam problem, has the product norm
The bilinear form (9) verifies the continuity and stability bounds
for some constants and all . In contrast with the bilinear form of the solid, and precisely due to the boundary conditions on the beam, the bilinear form is coercive in . The linear form will be assumed to be continuous as well, i.e., there exists a constant such that for all
3 Joint formulation of solids and beams
We consider now the formulation of a problem in which a beam and a three-dimensional solid, connected at some interface, deform to reach equilibrium under the action of external forces. Two issues need to be discussed. First, the minimal compatibility conditions that can be used to link the kinematics of the beam and the solid on their shared interface. Second, the stability and well-posedness of the global problem under the smallest set of Dirichlet boundary conditions.
The first issue will be addressed in this section, and follows our previous work . The second issue in studied in Section 4. To analyse both of them, we consider the simplest case, an elastic solid as the one described in Section 2.1, devoid of Dirichlet boundary conditions, attached through a surface to the tip of a cantilever beam, of the type defined in Section 2.2. The number of Dirichlet boundary conditions for the global problem is thus six, and it remains to be proven that, when the right links are employed, the former suffice to ensure the stability of the problem. Other, apparently more complex situations (with more beams or solids), are essentially equivalent to this one.
3.1 Link formulation
We define next two constraints relating the displacement and rotation vector of the beam at the free end, denoted respectively as and , with the displacement field of the body on the connected surface . More precisely, the first constraint imposes that the tip displacement is equal to the average displacement of the body on , that is
The second constraint imposes that the rotation at the tip of the beam, indicated as , is identical to the average surface rotation on . To express it, consider curvilinear coordinates on with vectors tangent to the coordinate lines. Following , the sought constraint can be expressed as
where the tensor is given by
the convention of sum over repeated indices is employed, with running from 1 to 2, and being the dual basis of the curvilinear coordinates. See Appendix A for its derivation.
3.2 Global problem statement
In this joint problem, the equilibrium of the structure consisting of the clamped beam, the deformable body and the connecting link is obtained from the stationarity condition of a Lagrangian. To define the latter, consider first the space of Lagrange multipliers
Since the global problem involves two types of bodies, we start by defining one last product space with norm
for all . On this space, we can define the bilinear form and the linear form by
The joint equilibrium of the solid and beam will be obtained as the saddle point of the Lagrangian defined as
where the notation denotes the product on the surface . The optimality conditions of the Lagrangian give the mixed variational problem: find such that
for all in , with
The solvability of problem (22) requires the careful consideration of the properties of both bilinear forms and , as well as the spaces on which they are defined.
Mixed variational problems such as the one described in Eqs. (22) have been extensively studied in the literature [10, 11]. Their well-posedness pivots on two conditions: the ellipticity of the bilinear form on a certain set defined below, and the inf-sup condition of the bilinear form .
for some and all . We note, again, that the bilinear form is not coercive in , as a result of the lack of coercivity of the bilinear form in the problem of the deformable solid.
The set consists of all the functions where the bilinear form vanishes, i.e.,
The well-posedness of the saddle point problem is the result of two theorems that we state and prove next.
The bilinear form is -elliptic on .
Let the function be defined as
for all . We prove first that this function is positive definite on . For , if and only if
The bilinear forms and are positive semidefinite and positive definite, respectively. Hence, must be equal to and must be an infinitesimal rigid body motion. The only rigid body deformation in is
with the position of the center of area of . But, since and , the function must also be identically zero.
To prove next that for some constant , and any , suppose that it is not true. Then there is a sequence with
Since , the sequence is bounded in and, by Rellich’s theorem, there is a subsequence that converges in to a function . But, noting that , this must be a Cauchy sequence in the norm
But this norm is equivalent to due to Korn’s second inequality and the ellipticity of . Hence, the sequence is Cauchy with respect to and since is a Hilbert space, it converges to . The two norms being equivalent proves that
Above we showed that is positive definite in , hence but
Since this is impossible, we conclude that there exists such that .∎∎
The second condition required to guarantee the well-posedness of the mixed problem is the inf-sup condition on the bilinear form .
There exists a constant such that for all ,
To prove this bound, we choose . Thus,
with the inertia
where we have employed the boundedness of and . ∎∎
We have presented the small strain form of a variational principle that governs the collective equilibria of linked beams and deformable solids. This is a remarkable principle in that it combines the mechanical response of two types of bodies with very different kinematic descriptions.
The variational principle rests on two compatibility conditions that link, in the weakest possible way, the kinematics of beams and solids on their common interface. While the compatibility of translations is fairly straightforward, the compatibility of beam rotations and displacements of the solid’s surface is new and based on a recent work of the author .
The optimality conditions of this variational principle give rise to a saddle point problem whose well-posedness is proven. In addition to the mathematical consequences of such a result, it evinces that it can be the basis of convergent numerical discretizations for structural models combining beams and deformable solids.
We close by noting that the well-posedness of the problem does not rely on the elastic response of either the solid or the beam. Rather, only some (weak) coercivity conditions of the bilinear forms of the solid and the beam are required for the proof. Hence, the result obtained can be, in principle, extended to inelastic structures in which the same stability estimates hold, even if just incrementally.
Appendix A Derivation of the rotational constraint
We derive next an intrinsic form of the constraint that links the rotation vector at the tip of the beam, denoted as , with the average rotation of the surface .
Following , we consider first the large strain case. For that, we define the surface deformation gradient. Given a solid with reference configuration and a surface with curvilinear coordinates , and tangent vectors , the surface deformation gradient is the map
where is the deformation of the solid, is a basis of , and with is the dual coordinate basis on the reference surface .
Since the surface deformation gradient has a unique polar decomposition  , with and a rank-two symmetric tensor, the rotation at the tip of the beam is equal to the average rotation of the surface deformation gradient if and only if
To study the form of this constraint in the small strain regime, we linearize the integrand of Eq. (32). Using as a small parameter, we can introduce the expansion
This constraint links the rotation vector with a certain average rotation of a general surface . When this surface is plane, as required to represent the cross section of a beam, we can select the curvilinear coordinates with a constant tangent basis so that Eq. (37) can be written in the more compact way:
-  A E Green, P M Naghdi, and M L Wenner. On the theory of rods I. Derivations from the three dimensional equations. Proceedings of the Royal Society of London, Series A, 337:451–483, 1974.
-  A E Green, P M Naghdi, and M L Wenner. On the theory of rods II. Developments by direct approach. Proceedings of the Royal Society of London, Series A, 337:485–507, 1974.
-  I Romero. Coupling nonlinear beams and continua: Variational principles and finite element approximations. International Journal for Numerical Methods in Engineering, 114:1192–1212, 2018.
-  K S Surana. Transition finite elements for three-dimensional stress analysis. International Journal for Numerical Methods in Engineering, 15:991–1020, 1980.
-  K S Surana. Transition finite elements for axisymmetric stress analysis. International Journal for Numerical Methods in Engineering, 15:809–832, 1980.
-  T C Gmür and R H Kauten. Three-dimensional solid-to-beam transition elements for structural dynamics analysis. International Journal for Numerical Methods in Engineering, 36(9):1429–1444, 1993.
-  D Boffi, F Brezzi, and M Fortin. Mixed Finite Element Methods and Applications, volume 44 of Springer Series in Computational Mathematics. Springer Science & Business Media, Berlin, Heidelberg, July 2013.
-  M. E. Gurtin. The linear theory of elasticity. In S Flügge, editor, Encyclopedia of Physics. Springer, Berlin, 1965.
-  J E Marsden and T J R Hughes. Mathematical foundations of elasticity. Prentice-Hall Englewood Cliffs, 1983.
-  F. Brezzi and M Fortin. Mixed and hybrid finite element methods. Springer, Berlin, 1991.
-  J E Roberts and J M Thomas. Mixed and hybrid methods. In P G Ciarlet and J L Lions, editors, Handbook of Numerical Analysis, pages 523–639. North-Holland, Amsterdam, 1989.
-  I Romero and J J Arribas. A taxonomy of polar decompositions for singular second order tensors in R. Archives of Mechanics, 61(5):383–390, 2009.