# Finite Anticlastic Bending of Hyperelastic Solids and Beams

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## Abstract

This paper deals with the equilibrium problem in nonlinear elasticity of hyperelastic solids under anticlastic bending. A three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated. Following a semi-inverse approach, the displacement field prescribed by the above kinematic model contains three unknown parameters. A Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. An Eulerian analysis is then accomplished to evaluating stretches and stresses in the deformed configuration. Cauchy stress distributions are investigated and it is shown how, for wide ranges of constitutive parameters, the obtained solution is quite accurate. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.

## Keywords

Finite elasticity Hyperelasticity Equilibrium Solids Beams Anticlastic bending## Mathematics Subject Classification

74B20## 1 Introduction

The flexure of an elastic body is a classical problem of elastostatics that has been widely investigated in literature because of its great relevance in many practical tasks. The majority of studies has been performed by assuming infinitesimal strains and small displacements of the body under bending (see, among the others, Bernoulli Jacob [1], Bernoulli Jacques [2], Parent [3], Euler [4, 5, 6, 7], Navier [8], Barré de Saint Venant [9], Bresse [10], Lamb [11], Kelvin and Tait [12] and Love [13]).

One of the first investigations dealing with the above equilibrium problem in the framework of finite elasticity was carried out by Seth [14], who studied a plate under flexure in the absence of body forces. Based on the semi-inverse method, he assumed the deformed configuration of the plate like a circular cylindrical shell, keeping valid the Bernoulli-Navier hypothesis for cross sections and assuming a linear dependence for the displacement field. Moreover, he assumed that the stress depends on the strain according to the linearized theory of elasticity. In his work, the bending couples needed to induce the hypothesized configuration of the plate together with the position of the unstretched fibers within the plate thickness (neutral axis) were also assessed.

The flexion problem of an elastic block has been studied by Rivlin [15], considering the deformation that transforms the elastic block into a cylinder having the base in the shape of a circular crown sector. No displacements along the axis of the cylinder have been taken into account, making the problem as a matter of fact two-dimensional. Surface tractions necessary to induced the assumed displacement field have been determined, showing that in the case of an incompressible neo-Hookean material, these surface tractions are equivalent to two equal and opposite couples acting at the end faces. Subsequently, Rivlin generalized his study formulating the equilibrium problem without specifying the form of the stored energy function [16].

Shield [17] investigated the problem of a beam under pure bending by assuming small strain but large displacements. He retrieves the Lamb’s solution [11] for the deflection of the middle surface of the beam. As remarked in that work, for large values of the width-to-thickness ratios, the deflection profile is flat in the central portion of the cross section and oscillatory near the edges.

A closed-form solution of a compressible rectangular body made of Hencky material under finite plane bending has been obtained by Bruhns et al. [18], giving explicit relationships for the bending angle and bending moment as functions of the circumferential stretch.

It is important to note that all the aforementioned works address the bending problem in a two-dimensional context, neglecting systematically the deformation in the direction perpendicular to the inflexion plane. Proceeding in this way, the modeling is simplified substantially, since the displacement field is assumed to be plane, renouncing to describe a phenomenon which in reality is purely three-dimensional. In particular, the transversal deformation, always coupled with the longitudinal inflexion of a solid, is known as the *anticlastic effect*.

This paper presents a fully nonlinear analysis of solids under anticlastic bending. In the next section, a three-dimensional kinematic model, where the longitudinal bending is accompanied by the transversal deformation of cross sections, is formulated by introducing three basic hypotheses. Following a semi-inverse approach, the displacement field prescribed by the kinematic model contains three unknown parameters. In Sect. 3, a Lagrangian analysis is performed and the compressible Mooney-Rivlin law is assumed for the stored energy function. Once evaluated the Piola-Kirchhoff stresses, the free parameters of the kinematic model are determined by using the equilibrium equations and the boundary conditions. With the purpose of evaluating stretches and stresses in the deformed configuration, an Eulerian analysis is accomplished in Sect. 4. Cauchy stress distributions are investigated and intervals for the constitutive parameters, in which the solution is particularly accurate, are established. The whole formulation proposed for the finite anticlastic bending of hyperelastic solids is linearized in Sect. 5, by introducing the hypothesis of smallness of the displacement and strain fields. With this linearization procedure, the classical solution for the infinitesimal bending of beams is fully recovered.

## 2 Kinematics

*deformation*\(\mathbf{f} : \bar{\mathcal{B}}\rightarrow \mathcal{V}\),

^{1}that is a smooth enough, injective and orientation-preserving (in the sense that \(\det \mathbf{Df}>0\)) vector field. The deformation of a generic material point \(P\) can be expressed by the well-known relationship

^{2}\(\mathbf{I}\) is the identity tensor.

Fixed notation, hereinafter the formulation of the equilibrium problem of an inflexed solid will be performed. Solving such a problem means getting the displacement field. But, in general, this direct computation is a daunting task. Thus, to simplify the problem, some hypotheses, more or less based on the physical behavior of the solid, are generally postulated in the literature. In this work, the three-dimensionality of the problem is maintained, without renouncing to study none of the three components of the displacement field.^{3} This allows to examine, in addition to the longitudinal inflexion along the \(Z\) axis of the solid, also the deformation of cross sections, which are initially parallel to the \(XY\) plane. As show experimental evidences (see, e.g., [19] and [20]) the solid transversely undergoes a second inflexion, whose sign is opposite to that longitudinal, known as *anticlastic effect*. Although the longitudinal curvature is generally larger, the two curvatures may have comparable magnitudes.

Taking into account the above considerations, the displacement field will be partially defined by adopting a semi-inverse approach. To this aim, the following three basic hypotheses are introduced.

1. The solid is inflexed longitudinally with constant curvature. Namely, each rectilinear segment of the solid, parallel to the \(Z\) axis, is transformed into an arc of circumference.

2. Plane cross sections, orthogonal to the \(Z\) axis, remain as such after the solid has been inflexed. Cross sections can deform only in their own plane and all in the same way.

3. As a result of longitudinal inflexion, the solid is inflexed also transversally with constant curvature, in such a way that any horizontal plane of the solid is transformed into a toroidal open surface.

The longitudinal inflexion can be thought of as generated by the application of a pair of self-balanced bending moments or by a geometric boundary condition which imposes a prescribed relative rotation between the two end faces of the solid.

In this paper, on the basis of the three previous assumptions, a kinematic model containing some unknown deformation parameters is derived. The model describes the displacement field (2) of the solid and it is the outcome of coupled effects generated by the longitudinal and transversal curvatures. Through this kinematic model and relationship (1), the shape assumed by the solid in the deformed configuration is obtained (cf. Fig. 1b).

Figure 1b shows that material fibers at the top edge of the solid are elongated in the \(Z\) direction (the corresponding longitudinal stretch \(\lambda_{Z}\) will therefore be greater than one), while material fibers are shortened at the bottom edge (\(\lambda_{Z}<1\)). Accordingly, in the vertical \(YZ\) plane, an intermediate curve, where there is no longitudinal deformation (\(\lambda_{Z}=1\)) exists. In Fig. 2 this special curve is represented by the circumferential arc drawn with a dotted line.^{4} On the other hand, fibers at the top edge of the solid are shortened in the \(X\) and \(Y\) directions, while they are elongated at the bottom edge (cf. Fig. 1b). This is the *anticlastic effect* associated with the inflexion of the solid in the \(Z\) direction. In particular, in the generic cross section there will be a smooth curve with \(\lambda_{Y}=1\). In the vertical \(YZ\) plane, each of these curves has a point of intersection. Since cross sections are deformed all in the same way, these points longitudinally form the arc shown in Fig. 2. Points of this arc are images of points belonging to the horizontal rectilinear segment traced for the point \(A\) of the undeformed configuration (cf. Fig. 2). The point \(A\) is fixed, \(A=A'\). The two longitudinal arcs, for which \(\lambda_{Y}=1\) and \(\lambda_{Z}=1\), are distinct. Furthermore, there is no a correlation between the points of these curves and the centroids of the deformed cross sections. These two arcs are crucial to describe the displacement field. The radius \(R_{0}\) of the arc with \(\lambda _{Z}=1\) is known, because it can be determined by using the geometric boundary condition which prescribes the angle \(\alpha_{0}\), \(R_{0}=L/2\alpha_{0}\).^{5} The radius \(R\) of the arc with \(\lambda_{Y}=1\) is instead unknown and its determination depends on resolving the equilibrium problem.

^{6}of the solid can be used:

**R**and

**U**are obtained by the polar decomposition of the deformation gradient \(\mathbf{F}= \mathbf{RU}\).

**R**is a proper orthogonal tensor and denotes the rotation tensor, whereas

**U**is a symmetric and positive definite tensor that indicates the right stretch tensor. For the state of deformation derived from (13), tensor

**U**is diagonal, because the reference system \(\{ O, X, Y, Z \} \) is principal. Diagonal components of

**U**are the principal stretches. Therefore, the diagonal components of

**C**are

^{7}

^{8}

**F**are calculated

## 3 Lagrangian Analysis

^{9}

**S**is a diagonal tensor

**b**and computing the scalar components of the material divergence of \(\mathbf{T}_{R}\), a system of three partial differential equations is obtained

^{10}

^{11}

^{12}since all its terms are null. With (36), (79) and (40), the following terms can be evaluated:

If relationship (44) is satisfied, all points of the longitudinal basic line are in equilibrium. Although in our model there are still two relationships among the free parameters to be determined, for all points of the basic line the displacement field is correct when (44) is satisfied, being it kinematically compatible and equilibrated.

When the basic line is abandoned, the equilibrium equations become much more complicated. Nevertheless, it is reasonable to expect, as a result of the continuity of the displacement field, that the above solution will be yet accurate in a neighborhood of each single point belonging to the basic line (to show this particular aspect of the problem, in the next section, a specific numerical analysis will be performed).

Ultimately, through the relations (44), (49) and (51), the three kinematic parameters \(r\), \(R\) and \(QN\) may be evaluated and the displacement field (24) can be considered completely defined.

## 4 Eulerian Analysis

^{13}

^{14}

The practically straight developments obtained for stretches and for the neutral axis are consistent with the proposed kinematic model, based on the hypothesis of the planarity preservation for all cross sections after deformation. Such a hypothesis, as mentioned at the beginning, requires that cross sections maintain plane and rotate rigidly around the neutral axis. The rotation is finite. Within each cross section, it can then be observed the deformation generated by the anticlastic bending and described by the curvature \(1/r\). These kinematic aspects become evident only when the deformed configurations of cross sections are considered. In other words, these aspects are not perceived with the use of Lagrangian coordinates.

**T**is obtained from the Piola-Kirchhoff stress tensor \(\mathbf{T}_{R}\) through the well-known transformation

**T**is symmetric. The matrix (57) can be rewritten in diagonal form by evaluating its eigenvalues. The resolution of the characteristic polynomial allows the determination of the principal stresses in the deformed configuration

^{15}The constant \(a\), \(b\) and \(c\) have been considered dimensionless. From Fig. 11, it can be observed that there are no qualitative differences between the two diagrams: the stress \(T_{3}\) grows by increasing \(b\), whereas \(T _{1}\) remains substantially unchanged.

The diagrams of Fig. 14(a) and 14(b), relating to the case \(c=0.1\), show that the equilibrium equations along the \(X\) and \(Y\) axes can be considered fulfilled in all points of the cross section, being the numerical values very close to zero.^{16} Fig. 14(d) highlights that the Cauchy principal stresses \(T_{1}= T_{2}\) are practically null. In particular, these values are also null at the boundary of the cross section, thus satisfying the boundary condition which requires that the lateral surface of the solid is unloaded. Definitively, the equilibrium solution for \(c=0.1\) is very accurate.

Figure 15 illustrates the case \(c=1\), that is the case in which the two non-normalized constants \(a\) and \(c\) coincide. The equilibrium along the \(X\) axis is well satisfied for all points of the cross section (cf. Fig. 15(a)), while the equilibrium along the \(Y\) axis shows some slight approximations approaching the upper and the lower edge (cf. Fig. 15(b)). Stresses \(T_{1}=T_{2}\) are still close to zero (cf. Fig. 15(d)) and overall the solution can be considered still accurate.

In the case with \(c=5\), shown in Fig. 16, numerical approximations become more evident. The equilibrium equations are well satisfied only in the central core of the cross section. Stresses \(T_{1}=T_{2}\) remain small only at the left and right edges of cross section (cf. Fig. 16(d)). Therefore, the value \(c=5\) for the reference case can be regarded as the upper limit for the numerical accuracy of the solution.^{17}

_{2}, the diagram plotted in Fig. 20 has been obtained. In this figure, for the middle cross section

^{18}and for the reference case, the moment \(m_{x}\) is assessed numerically by varying the angle \(\alpha_{0}\). On the basis of this result, the boundary conditions at the two end faces can be set geometrically by means of the angle \(\alpha_{0}\) or statically through the application of the corresponding moment \(m_{x}\).

## 5 Infinitesimal Fields

From a finite (or nonlinear) theory an infinitesimal (or linearized) theory can be derived by introducing two fundamental hypotheses, which require that both the displacement and the displacement gradient fields are small.

^{19}

^{20}

^{21}

**H**,

**E**and the skew-symmetric tensor of infinitesimal rigid rotation

**W**are obtained

**W**, the rotation \(\alpha \) around the \(X\) axis and \(\beta \) around the \(Z\) axis are recognizable. Using (62), the deformation gradient (25) can be rewritten in the following approximated form:

^{22}

^{23}

^{24}

^{25}

_{1}of \(S\), the following relations are obtained:

**E**, in the form specified by (67), into the constitutive law (74) and using relations (64)

_{2}and (76), the stress tensor \(\bar{\mathbf{T}}\) assumes the following form:

^{26}

With this last step, the linearization procedure which leads from the finite anticlastic bending of a solid to the well-know infinitesimal bending of a beam is completed.

## 6 Conclusions

The most popular mathematical model for inflexed solids, with both large deformation and displacement fields, was proposed by Rivlin [15]. This model, formulated under the plane displacement condition, does not allow to describe the transversal deformation of cross sections, which in such a model preserve their shape and size and undergo only finite rotations. On the contrary, from a purely physical point of view, the longitudinal inflexion of a solid is always associated with a traversal inflexion, known as anticlastic effect (cf. Fig. 1b).

In the fully nonlinear context of finite elasticity, this paper extends the aforementioned models, because the bending of a solid is described by a three-dimensional displacement field which takes into account the transversal pure deformation of cross sections.

Adopting a semi-inverse approach, this three-dimensional displacement field is defined by a kinematical model based on the following assumptions: the solid is inflexed longitudinally with constant curvature; after deformation, cross sections maintain their planarity and are inflexed in their plane with constant curvature. The kinematic model has three free parameters, which are determined imposing the equilibrium conditions.

Despite the elaborate shape of the displacement field, the stretches resulting from it have simple and compact expressions in terms of exponential functions (cf. Eq. (22)). The deformation state is triaxial and non-homogeneous.

The compressible Mooney-Rivlin law has been assigned to the stored energy function. Once determined the Piola-Kirchhoff stress tensor \(\mathbf{T}_{R}\), the equilibrium equations have been derived. But, having adopted the semi-inverse method, i.e., in practice having assigned for hypothesis the shape of solution a priori, it is inconceivable that the equilibrium equations can be correctly solved for all points of the solid. Nevertheless, a basic longitudinal line has been recognized, where the compatibility and equilibrium conditions are fully satisfied.

To assess the accuracy of the displacement field in correspondence of the other points, the equilibrium equations were normalized and, by means of numerical analyses, it was estimated how much they deviate from zero as one moves away from the basic line. Through a set of diagrams, the existence of a central core surrounding the basic line, where the solution proposed can be considered acceptable, has been highlighted. The extension of this central core depends on the parameters involved. The most important geometric parameters are the length of the solid \(L\) and the angle of inflexion \(\alpha_{0}\), because the central core becomes wider as \(L\) increases and as \(\alpha_{0}\) decreases.

Once completed the Lagrangian analysis, the Eulerian analysis was conducted with the purpose of evaluating stretches and stresses in the deformed configuration. The expressions of the Cauchy principal stresses have been obtained (cf. Eq. (60)) and the effective stress distributions in the inflexed solid are shown by some diagrams. Successively, the influence exerted on these stresses by the normalized constitutive parameters \(b\) and \(c\) have been investigated (cf. Figs. 11 and 13).

The parameter \(b\) does not exert a qualitative influence on the stress distributions in cross sections. More care is needed for the parameter \(c\). When the parameter \(c\) is small or at most equal to the first constitutive parameter \(a\), the obtained solution is numerically accurate, since the equilibrium equations are well satisfied for all points of the cross sections and the lateral surface of the solid is unloaded (cf. Figs. 14 and 15). Increasing the value of the parameter \(c\), the approximations become more evident. The case \(c=5\) may be considered the upper limit for a numerically accurate solution (cf. Fig. 16).

By varying the parameter \(c\), a comparison was made between the neutral line for the stress (\(T_{3}=0\)) and the neutral axis for the deformation (\(\lambda_{z}=1\)). For low values of \(c\), the two lines coincide. For average values of \(c\), the two lines differ. In particular, the neutral line for the stress assumes a curved shape. For high values of \(c\) (cases physically unrealistic), two distinct neutral lines for the stress even appear (cf. Fig. 17).

Knowing the stress distributions, explicit formulae to calculate the normal force and the bending moment in the deformed configuration were given (cf. Eq. (61)). A further verification of the obtained solution was made by checking that the normal force is close to zero. Being available the expression for the bending moment, the value of the moment needed to produce a specific inflexion angle \(\alpha_{0}\) can be assessed. This allows to impose the boundary conditions statically, through the application on the two end faces of the solid of a pair of bending moments (cf. Fig. 20).

The whole formulation exposed in the paper for the finite anticlastic bending of hyperelastic solids was linearized by introducing the hypothesis of smallness of the displacement and strain fields. All derived formulae were rewritten as power series. These series, which depend on the radii \(r\) and \(R_{0}\), were truncated by preserving the first order infinitesimals as \(r\rightarrow \infty \) and \(R_{0}\rightarrow \infty \).

Operating in this way, the nonlinear displacement field (24) was linearized getting exactly the well-known displacement field of the linear theory of inflexed beams (cf. Eq. (65)).

Unlike to what was obtained using the finite theory, in the infinitesimal theory the neutral axis of strain coincides with the neutral line of the stress and they pass through the centroid of the cross section.

The linearization procedure, based on the assumptions of smallness, has shown the transition from the proposed solution for the finite anticlastic bending of solids to the classical solution for the infinitesimal bending of beams.

## Footnotes

- 1.
\(\mathcal{V}\) is the vector space associated with ℰ.

- 2.
\(\mathit{Lin}\) is the set of all (second order) tensors whereas \(\mathit{Lin}^{+}\) is the subset of tensors with positive determinant.

- 3.
This unlike many existing models in literature, in which a component of displacement is systematically annulled, and the inflexed solid is considered under plane displacement condition (see, e.g., [15]).

- 4.
The existence of at least one arc, for which \(\lambda_{Z}=1\), is ensured by the continuity of deformation. Uniqueness is guaranteed by the hypothesis 2, which states the conservation of the planarity of the cross sections.

- 5.
In the sequel, it will be found the relationship between this angle \(\alpha_{0}\) and the pair of self-balanced bending moment to apply to the end faces of the solid.

- 6.
It is assumed that the isotropy property is preserved in the deformed configuration.

- 7.
The other components of tensor

**C**are zero. - 8.
- 9.
The following notations: \(\| \mathbf{A}\| = ( \mathrm{tr}\mathbf{A}^{T}\mathbf{A} ) \) for the tensor norm in the linear tensor space

*Lin*and \(\mathbf{A} ^{\star }=(\mathrm{det}\mathbf{A})\mathbf{A}^{-T}\) for the cofactor of the tensor \(\mathbf{A}\) (if \(\mathbf{A}\) is invertible) are used. - 10.
- 11.
- 12.
The first equation of system (38) is also verified for all points of the vertical plane \(X=0\).

- 13.
- 14.
Plotting the line \(\lambda_{Z}=1\) in the undeformed configuration, namely in terms of Lagrangian coordinates, a curve line with the concavity facing downward is obtained.

- 15.
- 16.
Having made dimensionless the elastic constants also stresses are dimensionless. In addition, even the geometrical dimensions of the solid, as well as the variable \(X\), \(Y\) and \(Z\), are normalized. Consequently, the equilibrium equations become dimensionless and their comparison with the scalar zero takes full meaning.

- 17.
Of course, the accuracy can improve on the basis of geometrical factors, as for example by increasing \(L\) or decreasing \(\alpha_{0}\).

- 18.
If the normal force is exactly zero, then the bending moment \(m_{x}\) is constant along the curved solid.

- 19.
The Landau symbols are used.

- 20.
- 21.
In the sequel, the infinitesimal terms of higher order are omitted definitively.

- 22.
- 23.
- 24.Using the Taylor series expansions, the following approximation is applied:$$ \frac{1}{\lambda }\simeq 1+\frac{Y}{r}+o \bigl(r^{-1} \bigr). $$
- 25.Using the Taylor series expansions, the following approximation is applied:$$ \frac{1}{\lambda^{2}\lambda_{Z}}\simeq 1+\frac{2Y}{r}-\frac{Y}{R_{0}}+o \bigl(r ^{-1} \bigr)+o \bigl(R_{0}^{-1} \bigr). $$
- 26.The same result can be achieved with a compressible Mooney-Rivlin material that satisfies the conditions (75). In fact, substituting (76) in (72), it is found$$\begin{aligned} & S= \biggl[ -(4a+12b+8c)+(4b+4c) \frac{1}{\nu } \biggr] \frac{Y}{r}=0, \\ & S_{Z}= \bigl[ -(8b+8c) \nu +(4a+8b+4c) \bigr] \frac{Y}{R_{0}}=E \varepsilon_{z}. \end{aligned}$$
- 27.
In this formula, \(R\) does not show an explicit dependence of \(H\). However, bearing in mind that \(R\) measures, in the vertical plane \(X=0\), the radius of curvature of the longitudinal line characterized by \(\lambda_{Y}=1\), it can be noted that this line maintains an almost central position when \(H\) varies. A similar remark holds for \(r\), which is linked to \(R\) by the (basically constitutive) formula (44).

## Notes

### Acknowledgements

Authors acknowledge funding from Italian Ministry MIUR-PRIN voce COAN 5.50.16.01 code 2015JW9NJT.

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