# Generalized reciprocity theorems for infinitesimal deformations superimposed upon finite deformations of rods: the plane problem

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## Abstract

The present paper is concerned with bending of extensible and shear-deformable rods, where we treat the case of plane deformations. Our emphasis is laid upon infinitesimal deformations superimposed upon finite deformations. In this framework, we focus on deriving generalized reciprocity theorems that must hold with respect to the superimposed infinitesimal deformations, when produced by different types of loadings. Besides imposed distributed and concentrated force and moment loadings, we consider the action of eigenstrains. Moreover, we also take into account the presence of jumps in kinematic entities, particularly concerning jumps of axial rod positions and of cross-sectional rotations. We first derive local and global universal reciprocity relations, which must hold for rods made of any material. The global universal relation then is specialized to the case of hyperelastic rods. From the latter relation, we prove that reciprocity theorems known from the linear theory, such as the theorems of Betti, Maxwell and Land, as well as Maysel’s formula, formally do hold for infinitesimal deformations superimposed upon finite deformations. As a formulation that appears to be novel also with respect to the purely linear theory, we present a combination of Maysel’s formula with the theory of Land, the former dealing with eigenstrains, the latter involving jumps in kinematic entities.

## 1 Introduction

The present paper is concerned with bending of extensible and shear-deformable rods. For a comprehensive presentation of the theory of rods, see the book on the mechanics of deformable solids [1] by V.V. Eliseev, to whom our contribution is devoted in grateful memory.

Subsequently, we treat the case of plane deformations of rods, the finite strain theory of which dates back to Reissner [2]. Our emphasis is laid upon infinitesimal deformations superimposed upon finite deformations. For fundamentals of this type of superposition, see Knops and Wilkes [3]. In the present contribution, we focus on deriving generalized reciprocity theorems that must hold with respect to the superimposed infinitesimal deformations, when produced by different loadings. Besides imposed distributed and concentrated force and moment loadings, we also consider the action of eigenstrains. Moreover, we also take into account the presence of jumps in kinematic quantities, particularly concerning jumps of axial rod positions and of cross-sectional rotations.

Our contribution is organized as follows. In Sect. 2, we present the local relations of equilibrium for the finite strain case, as well as the local incremental equilibrium relations that hold for superposed infinitesimal deformations. Following and extending the procedure for rods rigid in shear presented by DaDeppo [4], the incremental equilibrium relations are derived as time rate forms of the equilibrium relations for the finite strain case. In Sect. 3, we derive a local virtual work relation by multiplying the local incremental equilibrium relations with proper infinitesimal virtual deformation rates. This relation is further simplified by introducing the rates of generalized strain measures that were introduced by Reissner [2]. Local and global universal reciprocity relations are eventually derived in Sect. 4. We talk about a universal relation, when it does hold for any constitutive relation of the rod under consideration. In our derivation, we consider the virtual deformation quantities to be produced by a set of virtual internal and external forces and moments, which forms an equilibrium system, likewise to the original incremental problem. We are thus allowed to interchange original and virtual quantities in the local virtual work relation derived before in Sect. 3. This yields an adjoint virtual work relation. Subtracting the two virtual work relations provides the local universal reciprocity relation. The global form of the latter follows by integration along the rod segment under consideration in the reference configuration. In doing so, we also take into account the presence of concentrated external forces and external moments, which correspond to equilibrium relations of jump of the internal forces and moments. Besides of these static jump relations, we also consider kinematic jumps to be present in the infinitesimal superimposed deformations, namely jumps of the position vector or of the cross-sectional rotation at a certain location. Jumps may be present in both, the original and the virtual problem context. The static and kinematic relations of jump enter the universal global reciprocity relation, since integration must be performed piecewise, due to the presence of jumps in the local virtual work relations.

In Sect. 5, we specialize the universal global reciprocity relation derived in Sect. 4 to the case of hyperelastic rods, where we involve the generalized strain measures of Reissner [2], as well as proper integrability relations. We particularly allow the presence of incremental superimposed eigenstrains in the original problem. The generalized reciprocity theorem so obtained includes not only distributed forces and moments as well as jumps in forces and moments, but also eigenstrains and jumps in the kinematic relations. As special forms of this generalized formulation, various reciprocity relations known from the purely linear theory are proved to hold also for infinitesimal deformations superimposed upon finite deformations of shear-deformable and extensible rods, such as the Betti and Maxwell reciprocity theorems, see DaDeppo [4] for rods rigid in shear, and see Ziegler [5] for the linear theory. We further deal with an extension of a reciprocity theorem that involves jumps in the kinematic entities, and which in the purely linear context dates back to Land, see Kurrer [6] for a contemporary exposition. Concerning eigenstrains, we derive an extension of Maysel’s formula of the linear theory of thermoelasticity, see Ziegler and Irschik [7] for the linear theory. As a topic which appears to be novel also with respect to the purely linear theory, we derive a reciprocity formulation that connects superimposed jumps of the virtual kinematic entities with the presence of superimposed eigenstrains in the original problem, and which thus allows to compute eigenstrain-induced internal forces and moments, when the solution due to the superimposed jumps in the kinematic entities is known. In Sect. 6, we finally present numerical and analytic examples for the correctness of the latter formulation.

## 2 Relations of equilibrium and incremental equilibrium relations

*x*,

*z*)-plane of a global Cartesian coordinate system with unit base vectors \(e_x \), \(e_y \), and \(e_z \), such that \(e_y \) is perpendicular to the plane of deformation. The curved axial coordinate of the rod in the undeformed reference configuration is denoted as \(s_0 \). The rod may have a curved axis already in the reference configuration. The axial coordinate of the rod in the deformed configuration is called

*s*, see Fig. 1. Studying equilibrium of a rod element of differential axial length d

*s*, but formulating any entities as functions of the reference coordinate \(s_0 \), one obtains the local relations of equilibrium for the deformed configuration of the rod as, see [1, 2],

*f*with respect to \(s_0\) is denoted by a prime:

*y*axis of the global system by the angle \(\varphi \), such that

*N*and

*Q*normal and shear force, respectively, and

*M*is the bending moment. An analogous decomposition for the imposed forces and couples reads:

*t*. Taking into account an infinitesimal deformation with infinitesimal displacements superimposed upon the intermediate configuration, we are seeking for the rate form of Eqs. (13) and (14). Recall that such a superposition makes practical sense, since we have required the intermediate configuration to be infinitesimally superstable [3]. Following considerations by Da Deppo for rods rigid in shear [4], the rate form can be obtained from the equilibrium conditions in Eqs. (13) and (14) by differentiation with respect to the parameter

*t*. Subsequently, the derivative of some entity

*f*with respect to

*t*, holding the place \(s_0 \) fixed, is denoted by a superimposed dot:

*f*that corresponds to the superimposed infinitesimal deformation under consideration, being expressed as a function of the place \(s_0 \) in the reference configuration. Analogous to Eqs. (10) and (11), we now have

## 3 Local virtual work relations

## 4 Local and global universal reciprocity relations

## 5 Global reciprocity relations for hyperelastic rod

### 5.1 Betti’s reciprocal theorem

### 5.2 Maxwell’s reciprocal theorem

### 5.3 Reciprocity theorem dating back to Land

### 5.4 Extension of Maysel’s formula

### 5.5 An extension of Maysel’s formula motivated by the theorem of Land

## 6 Example problem

*EI*denotes the bending stiffness,

*EA*is the tensile stiffness, and

*kGA*stands for the shear stiffness,

*k*being the shear factor, and

*E*denoting Young’s modulus. The cross-sectional area is

*A*, and

*I*denotes the cross-sectional moment of inertia.

*u*, and

*w*is the deflection. In order to close the problem, the following kinematic relations are considered, which follow by representing the deformation gradient vector in the form of Eq. (12) in the global coordinate system, using Eqs. (4) and (5), and resolving for the displacement gradients:

*u*,

*w*, \(\varphi \),

*N*,

*Q*, and

*M*in the intermediate configuration. This set is accompanied by the following six boundary conditions:

For a validation, numerical solutions of the three exemplary boundary-value problems (the intermediate, the original, and the virtual problem) stated in Eqs. (52)–(73) have been obtained by means of the symbolic computer code Maple [10], version 2017. For the nonlinear intermediate problem, the command *dsolve*, together with the options *numeric* and *output=listprocedure*, was utilized. The numerically obtained point-wise results for the intermediate configuration were introduced into the Maple-formulations for the linear original and virtual problems via the additional option *known=()* in the command *dsolve*. Default settings of Maple [10] were used, e.g. concerning a selection of integration procedures. It turned out that, as long as the procedures did converge, results were obtained with a small computational effort, “just at fingertips”, despite the problems at hand appear to be comparatively involved. In our numerical computations, differences between the numerical outcomes of the original problem and of Eq. (77), the latter involving the virtual problem, remained comparatively small. For instance, we considered a thick rod of circular cross section with diameter \(d=0.2\,\hbox {m}\), length \(l=1\,\hbox {m}\), and Young’s modulus \(E=100\,\hbox {GPa}\). Bending stiffness then is \(EI=2.5\,\uppi \hbox {MPa}\), tensile stiffness is \(EA=\uppi \hbox {MPa}\), and we set \(kGA=0.5\,\uppi \hbox {MPa}\). For the boundary conditions of the intermediate problem, Eq. (63), the end moment was taken as \(M_0 =10 \pi ^{2}\hbox {KNm}\), and an axial compressive force \(R_{x0} \) was applied, which was 95% of the classical Euler buckling force for the problem at hand, not taking into account shear and extension, see e.g. [5]. Having obtained the solutions for the intermediate problem, Eqs. (52), (53), (57), and (59)–(64), the original problem in Eqs. (65)–(72) was solved directly for a unit incremental imposed curvature, \({\dot{\varphi }'}_e =1\,\hbox {m}^{-1}\). Afterwards, the solution of the virtual problem was computed and used to evaluate Eq. (77) for obtaining the bending moment \(\left. {{\dot{M}} } \right| _{x=l} \) at the clamped end. The numerical algorithms implemented in Maple in our hands delivered a difference of 0.0525% between Eq. (77) and the outcome of the original problem in Eqs. (65)–(72). When the diameter of the beam was decreased to \(d=0.01\,\hbox {m}\), this difference increased to 0.4203%. The following explanation has been provided:

As already mentioned, the above Land-type extension of Maysel’s formula in the context of an intermediate state with a large deformation, Eq. (51), appears to represent a novel contribution to the eigenstrain theory. Extended numerical studies, concerning initially curved beams and eigenstrain loadings that are more complex than the constant bending-type one acting on an initially straight beam, which has been treated in the above exemplary numerical study with respect to Eq. (77), will be published in a forthcoming paper.

*H*. The symbols \(T_1 \) and \(T_2 \) denote the temperatures on the top and bottom surfaces, respectively, \(\gamma \) is the coefficient of temperature expansion, and

*t*stands for the thickness of the rod. This case is studied in Table 8.1: shear, moment, slope, and deflection formulas for elastic straight beams, Example 6c of Roark’s Formulas for Stress and Strain [11]. We are interested in the corresponding bending moment at \(x=a\), where the Heaviside-box begins. From Table 8.1, Example 6c of [11], we learn that

## Notes

### Acknowledgements

Open access funding provided by Johannes Kepler University Linz. Support of the present work in the framework of the strategic research of the COMET-K2 Center “Linz Center of Mechatronics (LCM)” is gratefully acknowledged.

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