Consistent Euler–Bernoulli beam theory in statics for gradient elasticity based on Laplacians of stress and strain

Abstract

The Euler–Bernoulli beam bending theory in engineering mechanics assumes that the material behavior is isotropic elastic and that plane cross sections remain plane and rigid. It is well-known that this theory suffers from inconsistencies that, e.g., the shear strain is always vanishing, whereas the shear stress does not vanish. In recent work, consistent Euler–Bernoulli beam theories in classical and explicit gradient elasticities were accomplished by assuming the constitutive response to be anisotropic elastic, subject to internal constraints. This approach is extended in the present paper to get consistent Euler–Bernoulli beam theory for gradient elasticity based on Laplacians of stress and strain. The developed beam theory is employed to discuss bending of cantilever beams.

1. Introduction

Bending of beams is of fundamental importance, e.g., in engineering mechanics, structural analysis, experimental, and computational mechanics. In the past decades, special attention has been devoted to applications in micro- and macro-electromechanical systems (see, e.g., the review work Thai et al. [1]). It is generally accepted that size effects are essential in the material response of small-scale structural elements as, e.g., beams and plates. Experimental evidence is provided, among others, in Lam et al. [2], Farland and Colton [3] and Liebold and Müller [4]. Common elasticity theories, which allow capturing size effects in the material response, are the micropolar, the micromorphic, the gradient and the nonlocal elasticity. There is considerable amount of works concerning bending of beams, which are formulated in the framework of these non-classical elasticity theories. Of interest for our paper are, among others, the works of Papargyri-Beskou et al. [5], Lam et al. [2], Park and Gao [6], Ma et al. [7], Li et al. [8], Kong et al. [9], Lazopoulos and Lazopoulos [10], Reddy [11], Akgöz and Civalek [12], Xu and Shen [13], Li and Batra [14], Liang et al. [15], Ansari et al. [16], Liang et al. [15], Sherafatnia et al. [17], Yaghoubi et al. [18], Dehrouyeh-Semnani et al. [19], Ebrahimi et al. [20], Niiranen et al. [21], and Shaat et al. [22].

Simple Models of gradient elasticity are very attractive, since they deal with symmetric Cauchy stresses and only a few additional material parameters are usually needed in comparison to classical elasticity. Here, we consider models, which are expressed in terms of Laplacians of the stress and the strain tensors and which may be imagined as generalizations of the classical elasticity law

$${\mathrm{\Sigma }}_{\mathrm{ij}}={\mathbb{C}}_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}}.$$

(1)

Particular simple models of explicit and implicit gradient elasticity are, respectively, the laws

$${\mathrm{\Sigma }}_{\mathrm{ij}}={\mathbb{C}}_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}}-c_2{\mathbb{C}}_{\mathrm{ijmn}}(\mathrm{\Delta }\epsilon {)}_{\mathrm{mn}}(\mathrm{KG}-\mathrm{Model})$$

(2)

and

$${\mathrm{\Sigma }}_{\mathrm{ij}}-c_1(\mathrm{\Delta }\mathrm{\Sigma }{)}_{\mathrm{ij}}={\mathbb{C}}_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}}-c_2{\mathbb{C}}_{\mathrm{ijmn}}(\mathrm{\Delta }\epsilon {)}_{\mathrm{mn}}(3-\mathrm{PG}-\mathrm{Model}).$$

(3)

In these equations, $\mathbf{\Sigma }$ , $\mathit{\epsilon }$ and $\mathbb{C}$ are the Cauchy stress, the infinitesimal strain, and the elasticity tensors, respectively; $\mathrm{\Delta }$ is the Laplace Operator; and $c_1,c_2$ are material parameters. It seems that the two gradient elasticity laws (2) and (3) have been introduced and/or become widely known through the works of Altan and Aifantis [23, 24] and Gutkin and Aifantis [25]. The explicit gradient elasticity model (2) is the starting point of many beam theories and can be derived as particular case of Toupin/Mindlin’s gradient elasticity (see Mindlin [26]). Alternatively, using non-classical thermodynamics, Broese et al. [27–29] established an analogy between gradient elasticity models and linear viscoelastic solids. According to this analogy, equation (2) is the gradient elasticity counterpart of the Kelvin viscoelastic solid. (The shorthand notation KG-Model in equation (2) stands for Kelving gradient elasticity model).

Implicit gradient elasticity laws of the form (3) are increasingly interpreted as particular cases of Eringen’s non-local elasticity (see, e.g., Yan et al. [30], Xu et al. [31], Ceballes et al. [32], Lim et al. [33], and the references cited there). In general, Eringen’s nonlocal elasticity theory is often used in order to model bending of beams (see, e.g., Eltaher et al. [34], Fernández-Sáez et al. [35], Koutsoumaris et al. [36], Pisano et al. [37], Shaat et al. [38], Thai et al. [1] and the references cited there). A different way to approach model (3) is proposed by Broese et al. [29]. These authors proved that, on one hand, model (3) can be obtained as particular case of Mindlin’s micro-structured elasticity (equivalently Eringen’s micromorphic elasticity). On the other hand, using non-classical thermodynamics, they established an analogy between equation (3) and the three-parameter-Solids of linear viscoelasticity. (The abbreviation 3-PG-Model in equation (3) stands for Three-Parameter-Gradient elasticity model). As elucidated in Broese et al. [29], on the level of mechanical systems the analogy becomes visible by replacing the dashpot element in viscoelasticity by a non-standard spring in gradient elasticity. On the analytical level the analogy arises by replacing the time derivatives in viscoelasticity by Laplacian derivatives in gradient elasticity. It is also emphasized in Broese et al. [29], that in contrast to the Three-Parameter viscoelastic solid, its gradient counterpart allows energy to be stored in all three springs and it does not allow energy to be dissipated. Therefore, the analogy is a formal mathematical and not a physical one.

The Euler–Bernoulli bending theory of beams in elementary mechanics supposes the material behavior to be isotropic elastic. Furthermore, it assumes that the plane cross sections of the beam remain plane and perpendicular to the deformed beam-axis and that the shape of the cross sections does not change (see, e.g., Bauchau and Craig [[39], sections 5.1 and 5.4.2] or Reddy [40]). That means that cross sections undergo rigid body motions. Unfortunately, this theory suffers from the well-known inconsistency that the strain state according to these assumptions implies, e.g., vanishing shear stress always, which is in contradiction with the equilibrium equations in local form. There are other formulations of Euler–Bernoulli beams, which do not assume the cross section to be rigid, but suffer from similar inconsistencies. Such inconsistencies are accepted by interpreting these to be negligible (see, e.g., Bauchau and Craig [[39], section 5.4.2] and Janecka et al. [41]). Almost all Euler–Bernoulli bending theories of beams in gradient elasticity are inconsistent, or they are formulated on the basis of one-dimensional variational principles, so that such theories deal only with section forces.

A consistent Euler–Bernoulli bending theory of beams, for the classical elasticity law (1) and the gradient elasticity law (2), is proposed in Sideris and Tsakmakis [42] and Broese et al. [43]. The inconsistency is removed by assuming the material response to be transverse isotropic subject to internal constraints. It is shown, that all results of the engineering mechanics approach remain valid also in the framework of the consistent formulation. Furthermore, in the case of the gradient elasticity model (2), the consistent formulation offers the possibility to determine distributions of stress components. In the present paper, this consistent Euler–Bernoulli approach is extended to cover the 3-PG-Model in equation (3). The resulting beam theory is employed to discuss bending of cantilever beams. The predicted responses are compared with those according to the consistent formulations of classical elasticity (1) and KG-Model (2). Hence, the aim of the paper is purely theoretical and in particular description of the behavior of specific real materials is not provided.

2. Notation

Only static problems are considered and the deformations are assumed to be small, so we do not distinguish, as usually done, between reference and actual configuration. Unless explicitly stated, all indices will have the range of integers (1,2,3), while summation over repeated indices is implied. All tensorial components are referred to a Cartesian coordinate system $\{x_i\}$ , which induces the orthonormal basis $\{{\mathbf{e}}_i\}$ . Any tensor $\mathbf{A}$ will be identified by its components $A_{i\dots j}\equiv (\mathbf{A}{)}_{i\dots j}$ and we will write $A_{i..(\mathrm{kl})..p}$ for the symmetric part of A with respect to the indices $k$ and $l$ .

Let $\mathcal{B}$ be a material body, which may be identified by the position vectors $\mathbf{x}=x_i{\mathbf{e}}_i$ and which occupies the space $V$ in the three-dimensional Euclidean point space. We indicate by $\mathbf{n}$ the outward unit normal vector to the surface $\partial V$ bounding the space $V$ . If some tensor component $f$ is function of position $\mathbf{x}$ , then, when no others stated, we use the notation

$${\partial }_{i}f:=\frac{\partial f}{\partial x_i}\equiv f_{,x_i},f^{\prime }:=\frac{\partial f}{\partial x_1}\equiv f_{,x_1}.$$

(4)

The gradient and the divergence operators are indicated by $\mathrm{grad}$ and $\mathrm{div}$ , respectively, so that for second-order tensors $\mathbf{S}$ and third-order tensors $\mathit{\mu }$ , we have

$$(\mathrm{grad}\mathbf{S}{)}_{\mathrm{ijk}}={\partial }_i{S}_{\mathrm{jk}},(\mathrm{div}\mathit{\mu }{)}_{\mathrm{jk}}={\partial }_i{\mu }_{\mathrm{ijk}}.$$

(5)

Thus, for the Laplacian $\mathrm{\Delta }f$ , we have

$$\mathrm{\Delta }f=\mathrm{div}\mathrm{grad}f\equiv {\partial }_i{\partial }_{i}f.$$

(6)

Let $\mathbf{u}=u_i{\mathbf{e}}_i$ be the displacement vector. Then,

$${\epsilon }_{\mathrm{ij}}\equiv {\epsilon }_{(\mathrm{ij})}:=\frac{1}{2}({\partial }_i{u}_j+{\partial }_j{u}_i)={\partial }_{(i}{u}_{j)}.$$

(7)

3. Fundamental equations, concomitant boundary conditions

It has been mentioned in the introduction that the 3-PG-Model may be considered as a particular case of micromorphic elasticity or as counterpart of the Three-Parameter-Solid of linear viscoelasticity. According to this second approach, the 3-PG-Model has to be viewed as constitutive law and the associated boundary conditions as constitutive boundary conditions. In the following, we shall adopt the second interpretation. In any case, the main equations may be summarized as follows (see Broese et al. [29, 44]).

Field and constitutive equations:

$${\partial }_i{\mathrm{\Sigma }}_{\mathrm{ij}}=0,$$

(8)

$${\partial }_i{\mu }_{\mathrm{ijk}}+{\sigma }_{\mathrm{jk}}=0,$$

(9)

where

$${\mathrm{\Sigma }}_{\mathrm{ij}}\equiv {\mathrm{\Sigma }}_{(\mathrm{ij})}:={\tau }_{\mathrm{ij}}+{\sigma }_{\mathrm{ij}}={\mathbb{C}}_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}}+\frac{c_2-c_1}{c_1}{\mathbb{C}}_{\mathrm{ijmn}}{\gamma }_{\mathrm{mn}},$$

(10)

$${\tau }_{\mathrm{ij}}={\tau }_{(\mathrm{ij})}:=\frac{\partial \phi }{\partial {\epsilon }_{\mathrm{ij}}}={\mathbb{C}}_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}},$$

(11)

$${\sigma }_{\mathrm{ij}}\equiv {\sigma }_{(\mathrm{ij})}:=\frac{\partial \phi }{\partial {\gamma }_{\mathrm{ij}}}=\frac{c_2-c_1}{c_1}{\mathbb{C}}_{\mathrm{ijmn}}{\gamma }_{\mathrm{mn}},$$

(12)

$${\mu }_{\mathrm{ijk}}\equiv {\mu }_{i(\mathrm{jk})}:=\frac{\partial \phi }{\partial k_{\mathrm{ijk}}}=(c_2-c_1){\mathbb{C}}_{\mathrm{jkmn}}{\partial }_i{\mathrm{\Psi }}_{\mathrm{mn}},$$

(13)

$${\mathrm{\Psi }}_{\mathrm{ij}}\equiv {\mathrm{\Psi }}_{(\mathrm{ij})},{\gamma }_{\mathrm{ij}}\equiv {\gamma }_{(\mathrm{ij})}:={\epsilon }_{\mathrm{ij}}-{\mathrm{\Psi }}_{\mathrm{ij}},k_{\mathrm{ijk}}\equiv k_{i(\mathrm{jk})}:={\partial }_i{\mathrm{\Psi }}_{\mathrm{jk}},$$

(14)

$$\begin{array}{c}\phi =\phi (\mathit{\epsilon },\mathit{\gamma },\mathbf{k})=\frac{1}{2}{\epsilon }_{\mathrm{ij}}{\mathbb{C}}_{\mathrm{ijmn}}{\epsilon }_{\mathrm{mn}}+\frac{c_2-c_1}{2c_1}{\gamma }_{\mathrm{ij}}{\mathbb{C}}_{\mathrm{ijmn}}{\gamma }_{\mathrm{mn}}\hfill \\ +\frac{c_2-c_1}{2}{k}_{\mathrm{ijk}}{\mathbb{C}}_{\mathrm{jkmn}}{k}_{\mathrm{imn}}.\hfill \end{array}$$

(15)

In these equations, $\mathbf{\Psi }$ and $\mathit{\gamma }$ are internal strains, the so-called micro-deformation and relative deformation tensors, respectively; $\phi$ is the free energy per unit volume; and $\mathit{\tau },\mathit{\sigma },\mathit{\mu }$ are (internal) stress tensors. In particular, $\mathit{\mu }$ is called the double stress tensor. Furthermore, $c_1,c_2$ are material parameters. Assuming $\mathbb{C}$ to be positive definite, it is not difficult to see from equation (15) that $\phi$ will be positive definite whenever

$$c_2>c_1>0.$$

(16)

Equation (8) represents classical equilibrium equations in the absence of body forces, while equations (9)–(13) are viewed as constitutive equations, which lead to the 3-PG-Model (3) (see Broese et al. [29]).

Boundary conditions:

$$\mathrm{Either}{P}_j=n_i{\mathrm{\Sigma }}_{\mathrm{ij}}\mathrm{or}{u}_j(\mathrm{class}.\mathrm{boundary}\mathrm{cond}.),$$

(17)

$$\mathrm{and}\mathrm{either}{T}_{\mathrm{jk}}=n_i{\mu }_{\mathrm{ijk}}\mathrm{or}{\mathrm{\Psi }}_{\mathrm{jk}}(\mathrm{non}-\mathrm{class}.\mathrm{boundary}\mathrm{cond}.)$$

(18)

must be prescribed on $\partial V$ .

Note that a micromorphic continuum with symmetric micro-deformation tensor $\mathbf{\Psi }$ has been introduced by Forest and Sievert [45] under the name of micro-strain continuum.

4. Consistent Euler–Bernoulli beam bending theory for the 3-PG-Model

Consider the rectangular beam shown in Figure 1, which has length $L$ , width $2b$ , height $2c$ , and constant cross sections $A$ . The left, right, top, and bottom boundary planes are $A_l$ , $A_r$ , $A_t$ , and $A_b$ , respectively. The origin of the Cartesian coordinate system $\{x_i\}$ is located at the left boundary plane, the $x_1$ - $x_3$ -plane is a symmetry plane and the $x_1$ -axis is the centroidal axis of the beam. The beam is subject to transverse load $q=q(x_1)$ (force per unit length), which acts in the $x_1$ - $x_3$ -plane, while no external load is assumed to act in $x_2$ -direction. Further loading conditions will be specified below.

Figure 1.

ReYoung’s moduli th $L$ , width $2b$ , and height $2c$ .

It has been remarked in Sideris and Tsakmakis [42], that the Euler–Bernoulli assumptions (cf. Section 1) suggest some kind of anisotropic material response. Actually, assuming transverse isotropic elastic material behavior, subject to internal constraints, these authors established consistent Euler–Bernoulli beam theories for classical and explicit gradient elasticity. This approach will now be used for the 3-PG-Model.

4.1. Transverse isotropic 3-PG-Model with internal constraints

As remarked in Sideris and Tsakmakis [42], vanishing external load in $x_2$ -direction and the Euler–Bernoulli assumption that cross sections are rigid, suggest conditions of plane stress and plane strain. Thus, $u_2={\epsilon }_{2k}={\mathrm{\Sigma }}_{2k}=0$ , and all other components of $\mathbf{u}$ , $\mathit{\epsilon }$ , $\mathbf{\Sigma }$ are functions only of $x_1,x_3$ . We also assume, that ${\mathrm{\Psi }}_{2k}={\tau }_{2k}={\sigma }_{2k}={\mu }_{i2k}=0$ and that all other components of $\mathbf{\Psi }$ , $\mathit{\tau }$ , $\mathit{\sigma }$ , and $\mathit{\mu }$ are functions of $x_1$ , $x_3$ only. Moreover, the assumed material behavior along the beam-axis is different from that along the cross sections. Thus, we suppose transverse isotropy with respect to the chosen coordinate system, with $x_1$ being the axis of symmetry. Furthermore, we set vanishing in-plane Poisson’s ratios ${\nu }_{13},{\nu }_{31}$ , so that ${\mathbb{C}}_{1111}=E$ , ${\mathbb{C}}_{3333}=E^{*}$ , ${\mathbb{C}}_{1133}=0$ , ${\mathbb{C}}_{1313}={\mu }^{*}$ , ${\mathbb{C}}_{1113}=0$ , with $E,E^{*}$ being Young’s moduli, and ${\mu }^{*}$ being shear modulus. Using standard Voigt matrix notation, we find from equations (15)–(13) that

$$\left(\begin{array}{c}{\tau }_{11}\hfill \\ \multicolumn{1}{c}{{\tau }_{33}}\\ \multicolumn{1}{c}{{\tau }_{13}}\end{array}\right)=\left(\begin{array}{ccc}E\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& E^{*}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & {\mu }^{*}\hfill \end{array}\right)\left(\begin{array}{c}{\epsilon }_{11}\hfill \\ \multicolumn{1}{c}{{\epsilon }_{33}}\\ \multicolumn{1}{c}{2{\epsilon }_{13}}\end{array}\right)=\left(\begin{array}{c}E{\epsilon }_{11}\hfill \\ \multicolumn{1}{c}{E^{*}{\epsilon }_{33}}\\ \multicolumn{1}{c}{2{\mu }^{*}{\epsilon }_{13}}\end{array}\right),$$

(19)

$$\left(\begin{array}{c}{\sigma }_{11}\hfill \\ \multicolumn{1}{c}{{\sigma }_{33}}\\ \multicolumn{1}{c}{{\sigma }_{13}}\end{array}\right)=\frac{c_2-c_1}{c_1}\left(\begin{array}{ccc}E\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& E^{*}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & {\mu }^{*}\hfill \end{array}\right)\left(\begin{array}{c}{\gamma }_{11}\hfill \\ \multicolumn{1}{c}{{\gamma }_{33}}\\ \multicolumn{1}{c}{2{\gamma }_{13}}\end{array}\right)=\frac{c_2-c_1}{c_1}\left(\begin{array}{c}E{\gamma }_{11}\hfill \\ \multicolumn{1}{c}{E^{*}{\gamma }_{33}}\\ \multicolumn{1}{c}{2{\mu }^{*}{\gamma }_{13}}\end{array}\right),$$

(20)

$$\begin{array}{c}\left(\begin{array}{c}{\mu }_{i11}\hfill \\ \multicolumn{1}{c}{{\mu }_{i33}}\\ \multicolumn{1}{c}{{\mu }_{i13}}\end{array}\right)=(c_2-c_1)\left(\begin{array}{ccc}E\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& E^{*}\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & {\mu }^{*}\hfill \end{array}\right)\left(\begin{array}{c}{\partial }_i{\mathrm{\Psi }}_{11}\hfill \\ \multicolumn{1}{c}{{\partial }_i{\mathrm{\Psi }}_{33}}\\ \multicolumn{1}{c}{2{\partial }_i{\mathrm{\Psi }}_{13}}\end{array}\right)\hfill \\ =(c_2-c_1)\left(\begin{array}{c}E{\partial }_i{\mathrm{\Psi }}_{11}\hfill \\ \multicolumn{1}{c}{E^{*}{\partial }_i{\mathrm{\Psi }}_{33}}\\ \multicolumn{1}{c}{2{\mu }^{*}{\partial }_i{\mathrm{\Psi }}_{13}}\end{array}\right),\hfill \end{array}$$

(21)

$$\begin{array}{c}\phi =\frac{1}{2}[E{\epsilon }_{11}^2+E^{*}{\epsilon }_{33}^2+4{\mu }^{*}{\epsilon }_{13}^2]\hfill \\ +\frac{c_2-c_1}{2c_1}[E{\gamma }_{11}^2+E^{*}{\gamma }_{33}^2+4{\mu }^{*}{\gamma }_{13}^2]\hfill \\ +\frac{c_2-c_1}{2}\sum _{i=1}^3[E({\partial }_i{\mathrm{\Psi }}_{11}{)}^2+E^{*}({\partial }_i{\mathrm{\Psi }}_{33}{)}^2+4{\mu }^{*}({\partial }_i{\mathrm{\Psi }}_{13}{)}^2].\hfill \end{array}$$

(22)

The Euler–Bernoulli assumption, that cross sections are rigid, suggests infinitely large value $E^{*}$ , and this in turn implies

$$\underset{E^{*}\to \infty }{lim}({\epsilon }_{33},{\gamma }_{33},{\partial }_i{\mathrm{\Psi }}_{33})=0,$$

(23)

in order for the free energy $\phi$ in equation (22) to remain bounded. It follows from equations (19)₂, (20)₂, (21)₂, and (10) that ${\tau }_{33}$ , ${\sigma }_{33}$ , ${\mathrm{\Sigma }}_{33}$ , and ${\mu }_{i33}$ are not determinable by the constitutive law. Generally (cf. Truesdell and Noll [46, section 30]), in order to maintain the geometrical constraints (23), stresses $({\tau }_{33},{\sigma }_{33},{\mu }_{i33})$ are needed, which do not work. That means, ${\tau }_{33}d{\epsilon }_{33}={\sigma }_{33}d{\gamma }_{33}={\mu }_{i33}d({\partial }_i{\mathrm{\Psi }}_{33})=0$ , implying that $({\tau }_{33},{\sigma }_{33},{\mu }_{i33})=\underset{E^{*}\to \infty }{lim}{E}^{*}({\epsilon }_{33},{\gamma }_{33},{\partial }_i{\mathrm{\Psi }}_{33})$ are bounded and therefore $\underset{E^{*}\to \infty }{lim}{E}^{*}({\epsilon }_{33}^2,{\gamma }_{33}^2,({\partial }_i{\mathrm{\Psi }}_{33}{)}^2)=0$ in equation (22). In agreement with engineering mechanics, the requirement $\underset{E^{*}\to \infty }{lim}{\epsilon }_{33}=0$ can always be met by assuming that $u_3\to w(x_1)$ as $E^{*}\to \infty$ .

Another assumption in the Euler–Bernoulli beam theory is that the cross section remains plane after deformation. Thus, there exist axial displacement $U(x_1)$ and rotation $\varphi (x_1)$ , so that, for the limiting case above (see Bauchau and Craig [39, section 5.2]), $u_1=U(x_1)-\varphi (x_1)x_3$ , $u_2=0$ , $u_3=w(x_1)$ , and for the strain component ${\epsilon }_{13}$ we have ${\epsilon }_{13}=\frac{1}{2}(-\varphi +w^{\prime })$ . There is a last assumption in the Euler–Bernoulli beam theory requiring from the cross section to remain normal to the deformed axis of the beam (see Bauchau and Craig [39, section 5.1]). Geometrically, this means that there is no shear deformation in the $x_1-x_3$ -plane, and thus the value of the shear modulus ${\mu }^{*}$ must be infinitely large, ${\mu }^{*}\to \infty$ . The internal constraints

$$\underset{{\mu }^{*}\to \infty }{lim}({\epsilon }_{13},{\gamma }_{13},{\partial }_i{\mathrm{\Psi }}_{13})=0$$

(24)

must hold, in order for $\phi$ in equation (22) to be finite, implying $\varphi \to w^{\prime }$ , $u_1=U(x_1)-w^{\prime }(x_1)x_3$ and ${\epsilon }_{11}=U^{\prime }-w^{\prime \prime }{x}_3$ as ${\mu }^{*}\to \infty$ . In analogy, we set ${\mathrm{\Psi }}_{11}=B(x_1)-{\beta }^{\prime }(x_1)x_3$ with $B$ and ${\beta }^{\prime }$ corresponding to $U^{\prime }$ and $w^{\prime \prime }$ , respectively. In order to maintain the constraints (24), conjugate stresses $({\tau }_{13},{\sigma }_{13},{\mu }_{i13})$ are needed, which do not work, i.e., ${\tau }_{13}d{\epsilon }_{13}={\sigma }_{13}d{\gamma }_{13}={\mu }_{i13}d({\partial }_i{\mathrm{\Psi }}_{13})=0$ . Therefore, the stresses $({\tau }_{13},{\sigma }_{13},{\mu }_{i13})=\underset{{\mu }^{*}\to \infty }{lim}2{\mu }^{*}({\epsilon }_{13},{\gamma }_{13},{\partial }_i{\mathrm{\Psi }}_{13})$ are bounded, but not determinable by constitutive laws, and $\underset{{\mu }^{*}\to \infty }{lim}4{\mu }^{*}({\epsilon }_{13}^2,{\gamma }_{13}^2,({\partial }_i{\mathrm{\Psi }}_{13}{)}^2)=0$ in equation (22). Altogether, we have

$$u_1=U(x_1)-w^{\prime }(x_1)x_3,u_2=0,u_3=w(x_1),$$

(25)

$${\epsilon }_{11}=U^{\prime }(x_1)-w^{\prime \prime }(x_1)x_3,{\epsilon }_{33}={\epsilon }_{13}=0,$$

(26)

$${\mathrm{\Psi }}_{11}=B(x_1)-{\beta }^{\prime }(x_1)x_3,{\mathrm{\Psi }}_{33}={\mathrm{\Psi }}_{13}=0,$$

(27)

$${\gamma }_{11}=U^{\prime }-B-(w^{\prime \prime }-{\beta }^{\prime })x_3,{\gamma }_{33}={\gamma }_{13}=0,$$

(28)

from equations (19)–(22) we find that

$${\tau }_{11}=E{\epsilon }_{11}=E(U^{\prime }-w^{\prime \prime }{x}_3),$$

(29)

$${\sigma }_{11}=\frac{c_2-c_1}{c_1}E{\gamma }_{11}=\frac{c_2-c_1}{c_1}E[U^{\prime }-B-(w^{\prime \prime }-{\beta }^{\prime })x_3],$$

(30)

$${\mathrm{\Sigma }}_{11}={\tau }_{11}+{\sigma }_{11}=\frac{c_2}{c_1}E(U^{\prime }-w^{\prime \prime }{x}_3)-\frac{c_2-c_1}{c_1}E(B-{\beta }^{\prime }{x}_3),$$

(31)

$${\mu }_{111}=(c_2-c_1)E{\partial }_1{\mathrm{\Psi }}_{11}=(c_2-c_1)E(B^{\prime }-{\beta }^{\prime \prime }{x}_3),$$

(32)

$${\mu }_{311}=-(c_2-c_1)E{\beta }^{\prime },$$

(33)

$$\begin{array}{c}{\tau }_{33},{\tau }_{13},{\sigma }_{33},{\sigma }_{13},{\mathrm{\Sigma }}_{33},{\mathrm{\Sigma }}_{13},{\mu }_{133},{\mu }_{113},{\mu }_{333},{\mu }_{313}:\mathrm{not}\mathrm{determinable}\hfill \\ \mathrm{by}\mathrm{constitutive}\mathrm{law},\hfill \end{array}$$

(34)

$$\begin{array}{c}\phi =\frac{1}{2}E{\epsilon }_{11}^2+\frac{1}{2}\frac{c_2-c_1}{c_1}E{\gamma }_{11}^2+\frac{1}{2}(c_2-c_1)E({\partial }_1{\mathrm{\Psi }}_{11}{)}^2\hfill \\ +\frac{1}{2}(c_2-c_1)E({\partial }_3{\mathrm{\Psi }}_{11}{)}^2,\hfill \end{array}$$

(35)

and the field equations (8) and (9) reduce to

$${\partial }_1{\mathrm{\Sigma }}_{11}+{\partial }_3{\mathrm{\Sigma }}_{13}=0,$$

(36)

$${\partial }_1{\mathrm{\Sigma }}_{13}+{\partial }_3{\mathrm{\Sigma }}_{33}=0,$$

(37)

$${\partial }_1{\mu }_{111}+{\sigma }_{11}=0,$$

(38)

$${\partial }_1{\mu }_{113}+{\partial }_3{\mu }_{313}+{\sigma }_{13}=0,$$

(39)

$${\partial }_1{\mu }_{133}+{\partial }_3{\mu }_{333}+{\sigma }_{33}=0.$$

(40)

In an interesting paper, Shaat et al. [22] proposed a micromorphic beam theory with symmetric micro-deformation. Their component ${\mathrm{\Psi }}_{11}$ corresponds to the one in our paper.

Generally, it is not possible to solve analytically the field equations (36)–(40) and to satisfy given boundary conditions (17) and (18). Therefore, Euler–Bernoulli beam theories simplify the mathematical problem by solving a reduced form of these equations, expressed in terms of section and resulting forces. Such theories reduce the three-dimensional material body to one-dimensional continuum along the $x_1$ axis. Points $x_1=0$ and $x_1=L$ are the boundaries, while points $x_1\in (0,L)$ are interior points of the one-dimensional continuum. Boundary conditions of the material body on the planes $x_3=c$ and $x_3=-c$ are accounted for, at any $x_1$ , as resultants of corresponding tractions. These resultants might be viewed as body forces for the one-dimensional beam since they act on interior points $x_1$ . Thus, the forces acting on the one-dimensional beam, and on any subbody of it, are section forces on the boundaries and resultants (body) forces distributed along $x_1$ . If some solutions $U$ , $w$ , $B$ , and $\beta$ have been established, then distributions of the stress components for the three-dimensional material body can be derived on the basis of the elasticity laws (29)–(33) and the field equations (36)–(40). It is clear, that stress components calculated in this way will be compatible only with boundary and loading conditions expressed in terms of section and resultant (body) forces. Summarizing, for the one-dimensional beam continuum there exist only section and resultant (body) forces, while stress components exist only for the three-dimensional material body. The aim is now to introduce section and resultant forces and to derive from the equations above corresponding equations for these forces. A possibility to achieve this is to employ virtual work principles.

4.2. Virtual work principle—standard formulation

Virtual work principles in statics require for the virtual work of the external forces to be equal to the virtual work of the internal forces. We shall determine the virtual work of the external and the internal forces in sections 4.2.1 and 4.2.2, respectively.

4.2.1. Virtual work of external forces

Generally, virtual work principles in elasticity offer the possibility to derive, besides the equilibrium equations and the boundary conditions, also the elasticity law. However, in order to make comprehensible basic features of the traction component $T_{11}$ on the boundary planes $A_b$ , $A_t$ , the elasticity law is assumed to be given.

With regard to the boundary conditions (17), (18) and Figure 1, the virtual work of external forces, $\delta W^{(e)}$ , reads

$$\begin{array}{c}\delta W^{(e)}=\underset{\partial V}{\int }(P_j\delta u_j+T_{\mathrm{jk}}\delta {\mathrm{\Psi }}_{\mathrm{jk}})\mathrm{dV}\hfill \\ \multicolumn{1}{c}{=\underset{A_r\cup A_l}{\int }(P_j\delta u_j+T_{\mathrm{jk}}\delta {\mathrm{\Psi }}_{\mathrm{jk}})\mathrm{dS}+\underset{A_b\cup A_t}{\int }(P_j\delta u_j+T_{\mathrm{jk}}\delta {\mathrm{\Psi }}_{\mathrm{jk}})\mathrm{dS},}\end{array}$$

(41)

where $\delta (·)$ on the right-hand side is the usual variation of (·). We shall use equations (25)–(34) to express $\delta W^{(e)}$ in terms of the following section forces and resultant forces distributed along $x_1$ :

$$N=N(x_1):=\underset{A}{\int }{\mathrm{\Sigma }}_{11}\mathrm{dS},M=M(x_1):=\underset{A}{\int }{\mathrm{\Sigma }}_{11}{x}_3\mathrm{dS},$$

(42)

$$V=V(x_1):=\tilde{V}+\widetilde{\stackrel{~}{V}},$$

(43)

$$\tilde{V}=\tilde{V}(x_1):=\underset{A}{\int }{\mathrm{\Sigma }}_{13}\mathrm{dS},\widetilde{\stackrel{~}{V}}(x_1):=-2b{\left[{\mathrm{\Sigma }}_{13}{x}_3\right]}_{x_3=-c}^{x_3=c},$$

(44)

$$H=H(x_1):=\underset{A}{\int }{\mu }_{111}\mathrm{dS},m=m(x_1):=\underset{A}{\int }{\mu }_{111}{x}_3\mathrm{dS},$$

(45)

$$\chi =\chi (x_1):=\underset{A}{\int }{\sigma }_{11}\mathrm{dS},\mathrm{\Upsilon }=\mathrm{\Upsilon }(x_1):=\underset{A}{\int }{\sigma }_{11}{x}_3\mathrm{dS},$$

(46)

$$q=q(x_1):=\tilde{q}-{\widetilde{\stackrel{~}{V}}}^{\prime },\tilde{q}=\tilde{q}(x_1):=2b[{\mathrm{\Sigma }}_{33}{]}_{x_3=-c}^{x_3=c},$$

(47)

$$p=p(x_1):=2b{\left[{\mathrm{\Sigma }}_{13}\right]}_{x_3=-c}^{x_3=c}.$$

(48)

By invoking the constitutive laws (30)–(32), we obtain the sectional elasticity laws

$$N=g_1(x_1):=\mathrm{EA}(\frac{c_2}{c_1}{U}^{\prime }-\frac{c_2-c_1}{c_1}B),$$

(49)

$$M=g_2(x_1):=-\mathrm{EI}(\frac{c_2}{c_1}{w}^{\prime \prime }-\frac{c_2-c_1}{c_1}{\beta }^{\prime }),$$

(50)

$$H=g_5(x_1):=(c_2-c_1){\mathrm{EAB}}^{\prime },$$

(51)

$$m=g_6(x_1):=-(c_2-c_1)\mathrm{EI}{\beta }^{\prime \prime },$$

(52)

$$\chi =g_3(x_1):=\frac{c_2-c_1}{c_1}\mathrm{EA}(U^{\prime }-B),$$

(53)

$$\mathrm{\Upsilon }=g_4(x_1):=-\frac{c_2-c_1}{c_1}\mathrm{EI}(w^{\prime \prime }-{\beta }^{\prime }),$$

(54)

where

$$\underset{A}{\int }{x}_3\mathrm{dS}=0,I=\underset{A}{\int }{x}_3^2\mathrm{dS}.$$

(55)

As we shall see below, the section force $V$ will be determined from the equilibrium equations and the boundary conditions, while the distributed forces $p$ , $q$ will be given as external loads.

We shall determine first the part of $\delta W^{(e)}$ on $A_r\cup A_l$ and then the part on $A_b\cup A_t$ .

$$\underset{¯}{\mathrm{Boundary}\mathrm{planes}{A}_r,A_l}:x_1=L,0,\mathbf{n}=\pm {\mathbf{e}}_1,n_1\pm 1,$$

(56)

$$P_1=n_1{\mathrm{\Sigma }}_{11},P_3=n_1{\mathrm{\Sigma }}_{13},$$

(57)

$$T_{11}=n_1{\mu }_{111},T_{33}=n_1{\mu }_{133},T_{13}=n_1{\mu }_{113},$$

(58)

$$\delta u_1=\delta U+x_3\delta (-w^{\prime }),\delta u_3=\delta w,$$

(59)

$$\delta {\mathrm{\Psi }}_{11}=\delta B+x_3\delta (-{\beta }^{\prime }),\delta {\mathrm{\Psi }}_{33}=\delta {\mathrm{\Psi }}_{13}=0.$$

(60)

All other components of $\mathbf{n},\mathbf{u},\mathbf{\Psi },\mathbf{P},\mathbf{T}$ vanish. Thus,

$$\begin{array}{c}\underset{A_r\cup A_l}{\int }(P_j\delta u_j+T_{\mathrm{jk}}\delta {\mathrm{\Psi }}_{\mathrm{jk}})\mathrm{dS}=\underset{A_r\cup A_l}{\int }{P}_1\delta \mathrm{UdS}+\underset{A_r\cup A_l}{\int }{P}_1{x}_3\delta (-w^{\prime })\mathrm{dS}\hfill \\ \multicolumn{1}{c}{+\underset{A_r\cup A_l}{\int }{P}_3\delta w\mathrm{dS}+\underset{A_r\cup A_l}{\int }{T}_{11}\delta B\mathrm{dS}+\underset{A_r\cup A_l}{\int }{T}_{11}{x}_3\delta (-{\beta }^{\prime })\mathrm{dS}}\\ \multicolumn{1}{c}{={\left[N\delta U\right]}_{x_1=0}^{x_1=L}+{[M\delta (-w^{\prime })]}_{x_1=0}^{x_1=L}+{\left[\tilde{V}\delta w\right]}_{x_1=0}^{x_1=L}}\\ \multicolumn{1}{c}{+{\left[H\delta B\right]}_{x_1=0}^{x_1=L}+{[m\delta (-{\beta }^{\prime })]}_{x_1=0}^{x_1=L},}\end{array}$$

(61)

where the definitions (42)–(48) have been used.

When calculating the virtual work expended by $\mathbf{P}$ and $\mathbf{T}$ on $A_b\cup A_t$ , it suffices to consider resultant loads by adding corresponding values at $x_3=c$ and $x_3=-c$ .

$$\underset{¯}{\mathrm{Boundary}\mathrm{planes}{A}_{\mathrm{b}},A_{\mathrm{t}}}:x_3=\pm c,\mathbf{n}=\pm {\mathbf{e}}_3,n_3=\pm 1,$$

(62)

$$P_1=n_3{\mathrm{\Sigma }}_{31},P_3=n_3{\mathrm{\Sigma }}_{33},$$

(63)

$$T_{11}=n_3{\mu }_{311},T_{33}=n_3{\mu }_{333},T_{13}=n_3{\mu }_{313},$$

(64)

$$\delta u_1=\delta U+x_3\delta (-w^{\prime }),\delta u_3=\delta w,$$

(65)

$$\delta {\mathrm{\Psi }}_{11}=\delta B+x_3\delta (-{\beta }^{\prime }),\delta {\mathrm{\Psi }}_{33}=\delta {\mathrm{\Psi }}_{13}=0.$$

(66)

It follows that

$$\begin{array}{c}\underset{A_b\cup A_t}{\int }(P_j\delta u_j+T_{\mathrm{jk}}\delta {\mathrm{\Psi }}_{\mathrm{jk}})\mathrm{dS}=\underset{A_b\cup A_t}{\int }{P}_1\delta U\mathrm{dS}+\underset{A_b\cup A_t}{\int }{P}_1{x}_3\delta (-w^{\prime })\mathrm{dS}\hfill \\ \multicolumn{1}{c}{+\underset{A_b\cup A_t}{\int }{P}_3\delta w\mathrm{dS}+\underset{A_b\cup A_t}{\int }{T}_{11}\delta B\mathrm{dS}+\underset{A_b\cup A_t}{\int }{T}_{11}{x}_3\delta (-{\beta }^{\prime })\mathrm{dS}.}\end{array}$$

(67)

For the first three integrals on the right-hand side of equation (67) we find, with the help of equations (62), (63), (42)–(48) and by using partial integration, that

$$\underset{A_b\cup A_t}{\int }{P}_1\delta U\mathrm{dS}=\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{13}\right]}_{x_3=-c}^{x_3=c}\delta Ud{x}_1=\underset{0}{\overset{L}{\int }}p\delta Ud{x}_1,$$

(68)

$$\begin{array}{c}\underset{A_b\cup A_t}{\int }{P}_1{x}_3\delta (-w^{\prime })\mathrm{dS}=-\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{13}{x}_3\right]}_{x_3=-c}^{x_3=c}\delta w^{\prime }d{x}_1\hfill \\ ={\left[\widetilde{\stackrel{~}{V}}\delta w\right]}_{x_1=0}^{x_1=L}-\underset{0}{\overset{L}{\int }}{\widetilde{\stackrel{~}{V}}}^{\prime }\delta wd{x}_1,\hfill \end{array}$$

(69)

$$\underset{A_b\cup A_t}{\int }{P}_3\delta w\mathrm{dS}=\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{33}\right]}_{x_3=-c}^{x_3=c}\delta wd{x}_1=\underset{0}{\overset{L}{\int }}\tilde{q}\delta wd{x}_1.$$

(70)

The last two integrals on the right-hand side of equation (67) include the traction $T_{11}$ , which may be written, with the help of equations (64)₁ and (33), as

$${\left[T_{11}\right]}_{x_3=\pm c}={[\pm {\mu }_{311}]}_{x_3=\pm c}=\mp (c_2-c_1)E{\beta }^{\prime }(x_1).$$

(71)

It is worth noticing that $p$ , $\tilde{q}$ and $\widetilde{\stackrel{~}{V}}$ in the results (68)–(70) are resultant (body) forces acting on points $x_1$ in the interior of the one-dimensional continuum. These forces can be prescribed at every $x_1\in (0,L)$ , as they are expressed in terms of stresses not determinable by constitutive law. Unlike such forces, resultant forces in terms of the traction $T_{11}$ cannot be prescribed in the interval $(0,L)$ , as $T_{11}$ is expressed in terms of the stress component ${\mu }_{311}$ , which is determinable by the constitutive law (33). Therefore, specifying $T_{11}$ on $A_b\cup A_t$ is the same as specifying the solution $\beta (x_1)$ we are looking for. The virtual work principle will be well established only when such terms will be cancel out. That is exactly what will happen next. In other words, resultant (body) forces due to $T_{11}$ are irrelevant for the one-dimensional continuum. Traction forces $T_{11}$ have meaning only for the three-dimensional material body. This view is based on the fact that the stress component ${\mu }_{311}$ is not present in the field equations (36)–(40) and hence it does not contribute effectively to the governing (equilibrium) equations. Therefore we shall denote the two last integrals on the right-hand side of equation (67) by $\delta W_{\mathrm{noneff}}^{(e)}$ :

$$\delta W_{\mathrm{noneff}}^{(e)}:=\underset{A_b\cup A_t}{\int }{T}_{11}\delta \mathrm{BdS}+\int T_{11}{x}_3\delta (-{\beta }^{\prime })\mathrm{dS}.$$

(72)

That ${\mu }_{311}$ does not contribute to the equilibrium equations is also reflected by the fact, that $T_{11}$ is self-equilibriated on $A_b\cup A_t$ , which might be inferred directly from equation (71):

$$\underset{A_b\cup A_t}{\int }{T}_{11}\mathrm{dS}=0.$$

(73)

An immediate consequence of the last equation is that, the first integral of equation (72) vanishes,

$$\underset{A_b\cup A_t}{\int }{T}_{11}\delta \mathrm{BdS}=0.$$

(74)

Thus,

$$\delta W_{\mathrm{noneff}}^{(e)}=\underset{A_b\cup A_t}{\int }{T}_{11}{x}_3\delta (-{\beta }^{\prime })\mathrm{dS},$$

(75)

and by virtue of equations (71), (21), (27)₁, and (33),

$$\begin{array}{c}\delta W_{\mathrm{noneff}}^{(e)}=2b\underset{0}{\overset{L}{\int }}[{\mu }_{311}{x}_3{]}_{x_3=-c}^{x_3=c}\delta (-{\beta }^{\prime })d{x}_1\hfill \\ =4\mathrm{bc}\underset{0}{\overset{L}{\int }}(c_2-c_1)E(-{\beta }^{\prime })\delta (-{\beta }^{\prime })d{x}_1\hfill \\ =\underset{0}{\overset{L}{\int }}\frac{1}{2}(c_2-c_1)E\delta ({\beta }^{\prime }{)}^2\mathrm{Ad}{x}_1\hfill \\ =\delta \underset{V}{\int }\frac{1}{2}(c_2-c_1)E{\left({\partial }_3{\mathrm{\Psi }}_{11}\right)}^2\mathrm{dV}\hfill \\ =\delta \underset{V}{\int }\frac{1}{2}{\mu }_{311}\left({\partial }_3{\mathrm{\Psi }}_{11}\right)\mathrm{dV}.\hfill \end{array}$$

(76)

It follows from equations (67)–(70), (47) and (72) that

$$\begin{array}{c}\underset{A_b\cup A_t}{\int }(P_j\delta u_j+T_{\mathrm{jk}}\delta {\mathrm{\Psi }}_{\mathrm{jk}})\mathrm{dS}=\underset{0}{\overset{L}{\int }}p\delta Ud{x}_1+[\widetilde{\stackrel{~}{V}}\delta w{]}_{x_1=0}^{x_1=L}\hfill \\ \multicolumn{1}{c}{+\underset{0}{\overset{L}{\int }}q\delta wd{x}_1+\delta W_{\mathrm{noneff}}^{(e)}.}\end{array}$$

(77)

After substitution of equations (77) and (61) into equation (41), keeping in mind definition (43), we can express $\delta W^{(e)}$ in terms of section and resultant forces and related section displacements. We decompose $\delta W^{(e)}$ as

$$\delta W^{(e)}=\delta W_{\mathrm{eff}}^{(e)}+\delta W_{\mathrm{noneff}}^{(e)},$$

(78)

and call

$$\begin{array}{c}\delta W_{\mathrm{eff}}^{(e)}:={\left[N\delta U\right]}_{x_1=0}^{x_1=L}+{\left[H\delta B\right]}_{x_1=0}^{x_1=L}+{\left[V\delta w\right]}_{x_1=0}^{x_1=L}\hfill \\ \multicolumn{1}{c}{+{[M\delta (-w^{\prime })]}_{x_1=0}^{x_1=L}+{[m\delta (-{\beta }^{\prime })]}_{x_1=0}^{x_1=L}}\\ \multicolumn{1}{c}{+\underset{0}{\overset{L}{\int }}p\delta Ud{x}_1+\underset{0}{\overset{L}{\int }}q\delta wd{x}_1}\end{array}$$

(79)

as the effective virtual work of external forces.

4.2.2. Virtual work of internal forces

The virtual work of the internal forces, $\delta W^{(i)}$ , follows from equation (35) to be given by

$$\begin{array}{c}\delta W^{(i)}=\delta \underset{V}{\int }\phi \mathrm{dV}=\delta \underset{V}{\int }\frac{1}{2}E[{\epsilon }_{11}^2+\frac{c_2-c_1}{c_1}{\gamma }_{11}^2+(c_2-c_1)({\partial }_1{\mathrm{\Psi }}_{11}{)}^2\hfill \\ \multicolumn{1}{c}{+(c_2-c_1)({\partial }_3{\mathrm{\Psi }}_{11}{)}^2]\mathrm{dV}.}\end{array}$$

(80)

In analogy to equation (78), and recalling from equations (27) and (33) that ${\mu }_{311}=(c_2-c_1)E({\partial }_3{\mathrm{\Psi }}_{11})$ , we decompose $\delta W^{(i)}$ into two parts,

$$\delta W^{(i)}=\delta W_{\mathrm{eff}}^{(i)}+\delta W_{\mathrm{noneff}}^{(i)},$$

(81)

where the effective virtual work of the internal forces, $\delta W_{\mathrm{eff}}^{(i)}$ , is defined through

$$\delta W_{\mathrm{eff}}^{(i)}:=\underset{V}{\int }[E{\epsilon }_{11}\delta {\epsilon }_{11}+\frac{c_2-c_1}{c_1}E{\gamma }_{11}\delta {\gamma }_{11}+(c_2-c_1)E{{\mathrm{\Psi }}^{\prime }}_{11}\delta {{\mathrm{\Psi }}^{\prime }}_{11}]\mathrm{dV},$$

(82)

and

$$\delta W_{\mathrm{noneff}}^{(i)}:=\delta \underset{V}{\int }\frac{1}{2}(c_2-c_1)E{\left({\partial }_3{\mathrm{\Psi }}_{11}\right)}^2\mathrm{dV}=\delta \underset{V}{\int }\frac{1}{2}{\mu }_{311}\left({\partial }_3{\mathrm{\Psi }}_{11}\right)\mathrm{dV}.$$

(83)

For later reference, we use relations (29), (30) and (32) to rewrite $\delta W_{\mathrm{eff}}^{(i)}$ as

$$\delta W_{\mathrm{eff}}^{(i)}=\underset{V}{\int }({\tau }_{11}\delta {\epsilon }_{11}+{\sigma }_{11}\delta {\gamma }_{11}+{\mu }_{111}\delta {{\mathrm{\Psi }}^{\prime }}_{11})\mathrm{dV}.$$

(84)

By invoking equation (14)₂, we find from equation (82) that

$$\begin{array}{c}\delta W_{\mathrm{eff}}^{(i)}=\underset{V}{\int }[\frac{c_2}{c_1}E{\epsilon }_{11}-\frac{c_2-c_1}{c_1}E{\mathrm{\Psi }}_{11}]\delta {\epsilon }_{11}\mathrm{dV}\hfill \\ \multicolumn{1}{c}{-\underset{V}{\int }\frac{c_2-c_1}{c_1}E{\gamma }_{11}\delta {\mathrm{\Psi }}_{11}\mathrm{dV}+\underset{V}{\int }(c_2-c_1)E{{\mathrm{\Psi }}^{\prime }}_{11}\delta {{\mathrm{\Psi }}^{\prime }}_{11}\mathrm{dV}.}\end{array}$$

(85)

This virtual work can be recast by appealing to equations (26) and (27) for ${\epsilon }_{11}$ and ${\mathrm{\Psi }}_{11}$ , and by employing the relations (49)–(55). After lengthy manipulations (see Appendix 1), we arrive at

$$\begin{array}{c}\delta W_{\mathrm{eff}}^{(i)}={\left[g_1\delta U\right]}_{x_1=0}^{x_1=L}+{\left[g_5\delta B\right]}_{x_1=0}^{x_1=L}+{\left[{g^{\prime }}_2\delta w\right]}_{x_1=0}^{x_1=L}\hfill \\ \multicolumn{1}{c}{+{[g_2\delta (-w^{\prime })]}_{x_1=0}^{x_1=L}+{[(g_4+{g^{\prime }}_6)\delta \beta ]}_{x_1=0}^{x_1=L}}\\ \multicolumn{1}{c}{+{[g_6\delta (-{\beta }^{\prime })]}_{x_1=0}^{x_1=L}-\underset{0}{\overset{L}{\int }}{g^{\prime }}_1\delta Ud{x}_1-\underset{0}{\overset{L}{\int }}{g^{\prime \prime }}_2\delta wd{x}_1}\\ \multicolumn{1}{c}{-\underset{0}{\overset{L}{\int }}(g_3+{g^{\prime }}_5)\delta Bd{x}_1-\underset{0}{\overset{L}{\int }}({g^{\prime }}_4+{g^{\prime \prime }}_6)\delta \beta d{x}_1.}\end{array}$$

(86)

4.2.3. Virtual work principle, governing equations, boundary conditions

The virtual work principle states for the beam, and for any subbody of it, that

$$\delta W^{(e)}=\delta W^{(i)}.$$

(87)

We recognize from equations (83) and (76) that

$$\delta W_{\mathrm{noneff}}^{(e)}=\delta W_{\mathrm{noneff}}^{(i)},$$

(88)

so that the virtual work principle (87) turns out to be equivalent to the virtual work principle

$$\delta W_{\mathrm{eff}}^{(e)}=\delta W_{\mathrm{eff}}^{(i)}.$$

(89)

Bearing in mind equations (79) and (86), we deduce from equation (89) the equations

$$g_1^{\prime }+p=0,g_3+g_5^{\prime }=0,$$

(90)

$$g_2^{\prime \prime }+q=0,g_4+g_6^{\prime }=0,$$

(91)

the relations

$$N=g_1,H=g_5,V=g_2^{\prime },M=g_2,m=g_6$$

(92)

and the boundary conditions for section forces

$$\mathrm{either}N\mathrm{or}U,\mathrm{either}H\mathrm{or}B,$$

(93)

$$\mathrm{either}V\mathrm{or}w,\mathrm{either}M\mathrm{or}{w}^{\prime },\mathrm{and}$$

(94)

$$\mathrm{either}m\mathrm{or}{\beta }^{\prime }$$

(95)

have to be prescribed at $x_1=0,L$ .

It is not difficult to verify that equations (90), (91), and relations (92) give

$$N^{\prime }+p=0,V^{\prime }+q=0,M^{\prime }-V=0,$$

(96)

$$H^{\prime }+\chi =0,m^{\prime }+\mathrm{\Upsilon }=0,$$

(97)

which are expressed solely in terms of section forces and resultant forces.

While equations (96) are classical, equations (97) are not. Note also that equations (92)₁, (92)₂, (92)₄, and (92)₅ are nothing but the sectional elasticity laws (49)–(52).

We now can summarize the two interpretations of the 3-PG-Model. The first one assumes the material body as particular case of a micromorphic elastic material, so that equations (9) represent non-classical equilibrium equations and ${\sigma }_{\mathrm{jk}}$ are components of deformation-dependent non-classical body forces. Both $u_i$ and ${\mathrm{\Psi }}_{\mathrm{ij}}$ are independent kinematical degrees of freedom. That means for the one-dimensional beam continuum, that $u$ , $w$ , $B$ , $\beta$ are independent kinematical degrees of freedom, and that equations (96), (97) are equilibrium equations. Forces $\chi$ and $\mathrm{\Upsilon }$ are deformation-dependent resultant body forces obeying the sectional constitutive laws $\chi =g_3$ and $\mathrm{\Upsilon }=g_4$ (see equations (53) and (54)). Forces $N$ , $H$ , $M$ , and $m$ are section forces obeying the sectional constitutive laws (49)–(52) and all boundary conditions in equations (93)–(95) are physical boundary conditions.

The second interpretation considers equations (9), together with equations (10)–(13), as non-classical constitutive equations. In this case, ${\sigma }_{\mathrm{ij}}$ , ${\tau }_{\mathrm{ij}}$ , and ${\mu }_{\mathrm{ijk}}$ are components of internal stresses, and ${\mathrm{\Psi }}_{\mathrm{ij}}$ are components of internal strain, the only independent kinematical degrees of freedom being $u_i$ . In particular, equations (9) are constitutive equations allowing to determine ${\mathrm{\Psi }}_{\mathrm{ij}}$ for given $u_i$ . Accordingly, $U$ , $w$ are independent kinematical degrees of freedom for the one-dimensional beam continuum, whereas $B$ , $\beta$ are internal state variables. Moreover, $\chi$ , $\mathrm{\Upsilon }$ , $H$ , $m$ are also internal state variables and all equations in (92), (53), (54), and (97) are constitutive equations. For given $U$ , $w$ , equations (97) allow to determine $B$ and $\beta$ . In this context, equations (93)₁ and (94) are physical boundary conditions, whereas equations (93)₂ and (95) are constitutive boundary conditions.

4.3. Alternative and equivalent approach

Evidently, the virtual work principle (88), and the governing equations (96), (97), implied by this principle, should also be derivable from the field equations (36)–(40). Indeed, we shall prove in the two following sections that 1) equations (96), (97) might be derived directly from the field equations (36)–(40) and that 2) the virtual work principle (88) might also be derived from the field equations (36)–(40).

4.3.1. Alternative derivation of the governing equations for the section forces

The procedure for deriving equations (96), (97) directly from the field equations (36)–(40), is the same as in engineering mechanics. First we take the integral of equation (36) over $A$ ,

$$\underset{A}{\int }\left({\partial }_1{\mathrm{\Sigma }}_{11}\right)\mathrm{dS}+\underset{A}{\int }{\partial }_3{\mathrm{\Sigma }}_{13}\mathrm{dS}={\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{11}\mathrm{dS}+2b{\left[{\mathrm{\Sigma }}_{13}\right]}_{x_3=-c}^{x_3=c}=0.$$

(98)

Then, using the definitions (42)₁, and (48), equation (98) furnishes equation (96)₁.

Next, we integrate equation (37) over $A$ ,

$${\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{13}\mathrm{dS}+\underset{A}{\int }{\partial }_3{\mathrm{\Sigma }}_{33}\mathrm{dS}={\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{13}\mathrm{dS}+2b{\left[{\mathrm{\Sigma }}_{33}\right]}_{x_3=-c}^{x_3=c}=0,$$

(99)

and use definitions (44)₁, (47)₂, to obtain

$${\tilde{V}}^{\prime }+\tilde{q}=0.$$

(100)

In this equation, we set ${\tilde{V}}^{\prime }=V^{\prime }-{\widetilde{\stackrel{~}{V}}}^{\prime }$ , from equation (43),

$$V^{\prime }-{\widetilde{\stackrel{~}{V}}}^{\prime }+\tilde{q}=0,$$

(101)

and by invoking equation (47)₁, we arrive at equation (96)₂.

In order to derive equation (96)₃, we multiply equation (36) by $x_3$ and integrate the result over $A$ :

$$\underset{A}{\int }\left({\partial }_1{\mathrm{\Sigma }}_{11}\right)x_3\mathrm{dS}+\underset{A}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)x_3\mathrm{dS}=0,$$

(102)

or

$${\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{11}{x}_3\mathrm{dS}+2b\underset{-c}{\overset{c}{\int }}\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)x_{3}d{x}_3=0.$$

(103)

By applying partial integration to the second integral in equation (103),

$${\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{11}{x}_3\mathrm{dS}+2b[{\mathrm{\Sigma }}_{13}{x}_3{]}_{x_3=-c}^{x_3=c}-\underset{A}{\int }{\mathrm{\Sigma }}_{13}\mathrm{dS}=0$$

(104)

If we replace the first term with $M^{\prime }$ (cf. definition (42)₂), the second term with $-\widetilde{\stackrel{~}{V}}$ (cf. definition (44)₂), and the third term with $\tilde{V}$ (cf. definition (44)₁), then

$$M^{\prime }-\widetilde{\stackrel{~}{V}}-\tilde{V}=0.$$

(105)

From this, we obtain equation (96)₃ by taking into account the definition of $V$ in equation (43).

Before going to establish equations (97) from the field equations, we would like to remark the following. Equations (39), (40) involve only stresses not determinable by constitutive law, and hence these stresses cannot affect the proper governing equations of the beam. Therefore, in order to establish equations (97), we need to elaborate only equation (38). More precisely, take the integral of equation (38) over $A$ ,

$${\partial }_1\underset{A}{\int }{\mu }_{111}\mathrm{dS}+\underset{A}{\int }{\sigma }_{11}\mathrm{dS}=0,$$

(106)

and use definitions (45)₁, and (46)₁, to obtain equation (97)₁. Finally, multiply equation (38) by $x_3$ and take the integral over $A$ ,

$${\partial }_1\underset{A}{\int }{\mu }_{111}{x}_3\mathrm{dS}+\underset{A}{\int }{\sigma }_{11}{x}_3\mathrm{dS}=0.$$

(107)

This equation, together with definitions (45)₂ and (46)₂ lead to equation (97)₂, which completes the derivation of equations (96), (97) directly from the field equations.

4.3.2. Alternative derivation of the virtual work principle $\delta W_{\mathrm{eff}}^{(e)}$ = $\delta W_{\mathrm{eff}}^{(i)}$

It has been clear in the last section that the relevant field equations for the beam are equations (36)–(38), so we shall now derive the virtual work principle (88) from these equations. We start with the equilibrium equations (36), (37), by forming the scalar product with $\delta u$ , and integrating the result over $V$ ,

$$\underset{V}{\int }[\left({\partial }_1{\mathrm{\Sigma }}_{11}\right)\delta u_1+\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)\delta u_1+\left({\partial }_1{\mathrm{\Sigma }}_{13}\right)\delta u_3+\left({\partial }_3{\mathrm{\Sigma }}_{33}\right)\delta u_3]\mathrm{dV}=0.$$

(108)

We shall recast this equation by employing equations (25), (26), (31), definitions (42)–(48), as well as partial integration:

$$\begin{array}{c}\underset{V}{\int }\left({\partial }_1{\mathrm{\Sigma }}_{11}\right)\delta u_1\mathrm{dV}=\underset{V}{\int }{\partial }_1\left({\mathrm{\Sigma }}_{11}\delta u_1\right)\mathrm{dV}-\underset{V}{\int }{\mathrm{\Sigma }}_{11}\delta {\epsilon }_{11}\mathrm{dV}\hfill \\ ={\left[\underset{A}{\int }{\mathrm{\Sigma }}_{11}(\delta U-x_3\delta w^{\prime })\mathrm{dS}\right]}_{x_1=0}^{x_1=L}-\underset{V}{\int }({\tau }_{11}+{\sigma }_{11})d{\epsilon }_{11}\mathrm{dV}\hfill \\ ={[N\delta U{]}_{x_1=0}^{x_1=L}+[M\delta (-w^{\prime })]}_{x_1=0}^{x_1=L}-\underset{V}{\int }{\tau }_{11}\delta {\epsilon }_{11}\mathrm{dV}-{\int }_V{\sigma }_{11}\delta {\epsilon }_{11}\mathrm{dV},\hfill \end{array}$$

(109)

$$\begin{array}{c}\underset{V}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)\delta u_1\mathrm{dV}+\underset{V}{\int }\left({\partial }_1{\mathrm{\Sigma }}_{13}\right)\delta u_3\mathrm{dV}+\underset{V}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{33}\right)\delta u_3\mathrm{dV}\hfill \\ =\underset{V}{\int }{\partial }_3\left({\mathrm{\Sigma }}_{13}\delta U\right)\mathrm{dV}+\underset{V}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)x_3\delta (-w^{\prime })\mathrm{dV}+\underset{V}{\int }\left({\partial }_1{\mathrm{\Sigma }}_{13}\right)\delta \mathrm{wdV}+\underset{V}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{33}\right)\delta \mathrm{wdV}\hfill \\ =\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{13}\right]}_{x_3=-c}^{x_3=c}\delta \mathrm{Ud}{x}_1+\underset{0}{\overset{L}{\int }}\left(\underset{A}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)x_3\mathrm{dS}\right)\delta (-w^{\prime })d{x}_1\hfill \\ +\underset{0}{\overset{L}{\int }}({\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{13}\mathrm{dS})\delta \mathrm{wd}{x}_1+\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{33}\right]}_{x_3=-c}^{x_3=c}\delta \mathrm{wd}{x}_1\hfill \\ =\underset{0}{\overset{L}{\int }}p\delta \mathrm{Ud}{x}_1+{\left[(-\underset{A}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)x_3\mathrm{dS})\delta w\right]}_{x_1=0}^{x_1=L}+\underset{0}{\overset{L}{\int }}\left({\partial }_1\underset{A}{\int }\left({\partial }_3{\mathrm{\Sigma }}_{13}\right)x_3\mathrm{dS}\right)\delta \mathrm{wd}{x}_1\hfill \\ +\underset{0}{\overset{L}{\int }}\left({\partial }_1\underset{A}{\int }{\mathrm{\Sigma }}_{13}\mathrm{dS}\right)\delta \mathrm{wd}{x}_1+\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{33}\right]}_{x_3=-c}^{x_3=c}\delta \mathrm{wd}{x}_1\hfill \\ =\underset{0}{\overset{L}{\int }}p\delta \mathrm{Ud}{x}_1+{\left[V\delta w\right]}_{x_1=0}^{x_1=L}+\underset{0}{\overset{L}{\int }}\left({\partial }_1\underset{A}{\int }{\partial }_3\left({\mathrm{\Sigma }}_{13}{x}_3\right)\mathrm{dS}\right)\delta \mathrm{wd}{x}_1+\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{33}\right]}_{x_3=-c}^{x_3=c}\delta \mathrm{wd}{x}_1\hfill \\ =\underset{0}{\overset{L}{\int }}p\delta \mathrm{Ud}{x}_1+{\left[V\delta w\right]}_{x_1=0}^{x_1=L}+\underset{0}{\overset{L}{\int }}2b{\left[\left({\partial }_1{\mathrm{\Sigma }}_{13}\right)x_3\right]}_{x_3=-c}^{x_3=c}\delta \mathrm{wd}{x}_1+\underset{0}{\overset{L}{\int }}2b{\left[{\mathrm{\Sigma }}_{33}\right]}_{x_3=-c}^{x_3=c}\delta \mathrm{wd}{x}_1\hfill \\ =\underset{0}{\overset{L}{\int }}p\delta \mathrm{Ud}{x}_1+{\left[V\delta w\right]}_{x_1=0}^{x_1=L}+\underset{0}{\overset{L}{\int }}q\delta \mathrm{wd}{x}_1.\hfill \end{array}$$

(110)

After substituting of equations (109), (110) in equation (108), we find that

$$\begin{array}{c}{\left[N\delta U\right]}_{x_1=0}^{x_1=L}+{\left[V\delta w\right]}_{x_1=0}^{x_1=L}+{\left[M\delta (-w^{\prime })\right]}_{x_1=0}^{x_1=L}+\underset{0}{\overset{L}{\int }}p\delta \mathrm{Ud}{x}_1+\underset{0}{\overset{L}{\int }}q\delta \mathrm{wd}{x}_1\hfill \\ =\underset{V}{\int }{\tau }_{11}\delta {\epsilon }_{11}\mathrm{dV}+\underset{V}{\int }{\sigma }_{11}\delta {\epsilon }_{11}\mathrm{dV}.\hfill \end{array}$$

(111)

Similarly, multiplication of equation (38) with $\delta {\mathrm{\Psi }}_{11}$ , then integration over $V$ and application of partial integration, gives

$$\begin{array}{c}\underset{V}{\int }\left({\partial }_1{\mu }_{111}\right)\delta {\mathrm{\Psi }}_{11}\mathrm{dV}+\underset{V}{\int }{\sigma }_{11}\delta {\mathrm{\Psi }}_{11}\mathrm{dV}\hfill \\ =\underset{V}{\int }{\partial }_1\left({\mu }_{111}\delta {\mathrm{\Psi }}_{11}\right)\mathrm{dV}-\underset{V}{\int }{\mu }_{111}\delta {{\mathrm{\Psi }}^{\prime }}_{11}\mathrm{dV}+\underset{V}{\int }{\sigma }_{11}\delta {\mathrm{\Psi }}_{11}\mathrm{dV}=0.\hfill \end{array}$$

(112)

We may use the kinematical relation (27) and the definitions (45) to rewrite the first integral on the right-hand side of equation (112) as

$$\begin{array}{c}\underset{V}{\int }{\partial }_1\left({\mu }_{111}\delta {\mathrm{\Psi }}_{11}\right)\mathrm{dV}={\left[\underset{A}{\int }{\mu }_{111}(\delta B-x_3\delta {\beta }^{\prime })\mathrm{dS}\right]}_{x_1=0}^{x_1=L}\hfill \\ ={\left[\left(\underset{A}{\int }{\mu }_{111}\mathrm{dS}\right)\delta B\right]}_{x_1=0}^{x_1=L}+{\left[\left(\underset{A}{\int }{\mu }_{111}{x}_3\mathrm{dS}\right)\delta (-{\beta }^{\prime })\right]}_{x_1=0}^{x_1=L}\hfill \\ ={\left[H\delta B\right]}_{x_1=0}^{x_1=L}+{\left[m\delta (-{\beta }^{\prime })\right]}_{x_1=0}^{x_1=L}.\hfill \end{array}$$

(113)

Using this result, we find from equation (112) that

$${\left[H\delta B\right]}_{x_1=0}^{x_1=L}+{\left[m\delta (-{\beta }^{\prime })\right]}_{x_1=0}^{x_1=L}=\underset{V}{\int }{\mu }_{111}\delta {\mathrm{\Psi }}_{11}^{\prime }\mathrm{dV}-\underset{V}{\int }{\sigma }_{11}\delta {\mathrm{\Psi }}_{11}\mathrm{dV}.$$

(114)

Adding equations (111) and (114), we have

$$\begin{array}{c}{\left[N\delta U\right]}_{x_1=0}^{x_1=L}+{\left[H\delta B\right]}_{x_1=0}^{x_1=L}+{\left[V\delta w\right]}_{x_1=0}^{x_1=L}+{\left[M\delta (-w^{\prime })\right]}_{x_1=0}^{x_1=L}\hfill \\ +{\left[m\delta (-{\beta }^{\prime })\right]}_{x_1=0}^{x_1=L}+\underset{0}{\overset{L}{\int }}p\delta \mathrm{Ud}{x}_1+\underset{0}{\overset{L}{\int }}q\delta \mathrm{wd}{x}_1\hfill \\ =\underset{V}{\int }{\tau }_{11}\delta {\epsilon }_{11}\mathrm{dV}+\underset{V}{\int }{\sigma }_{11}\delta ({\epsilon }_{11}-{\mathrm{\Psi }}_{11})\mathrm{dV}+\underset{V}{\int }{\mu }_{111}\delta {{\mathrm{\Psi }}^{\prime }}_{11}\mathrm{dV}.\hfill \end{array}$$

(115)

Recalling the kinematical relation (14)₂ and the relations (79), (84), we infer from equation (115) that $\delta W_{\mathrm{eff}}^{(e)}=\delta W_{\mathrm{eff}}^{(i)}$ , which completes the proof.

4.4. Discussion

As stated in the introduction, the Euler–Bernoulli beam theory in common use is based on two fundamental assumptions: 1) The material behavior is isotropic elastic. 2) The plane cross sections of the beam remain plane and perpendicular to the deformed beam-axis and additionally the shape of the cross sections does not change (no deformation).

The two assumptions plainly contradict each other, because according to assumption 2 the material behavior along the cross sections is rigid body like, while along the beam-axis it is elastic. Evidently, assumption 2 presupposes anisotropic material response subject to geometrical constrains, which contradicts assumption 1. The consequence in traditional approaches is that if an isotropic elasticity law holds, than the equilibrium equations cannot be satisfied (inconsistency). Even more, it is not possible to determine uniquely stress components like ${\mathrm{\Sigma }}_{13}$ , or to formulate boundary conditions in terms of stress components. Therefore, only one-dimensional virtual work approaches, without reference to stress components, are sound. Equilibrium equations for stresses, however, are important, e.g., in the case of the KG-Model, as they allow to establish the equivalence between two different one-dimensional virtual work formulations and the associated boundary conditions for section forces (see Broese et al. [43]).

There are two possibilities to abolish the inconsistency problem: To modify either assumption 1 or assumption 2. Assumption 2 is very attractive as it leads to a simple deformation geometry. Therefore, assumption 1 has been replaced in Sideris and Tsakmakis [42], Broese et al. [43], and in the present paper, by the assumption of transverse isotropy subject to geometrical constraints. This offered the possibility in the present paper, to derive the one-dimensional virtual work statements by employing both equilibrium equations for stresses and boundary conditions expressed in terms of stresses. This approach made also possible to cancel out the term $\delta W_{\mathrm{noneff}}^{(i)}$ in the elaborated virtual work principles.

4.5. Specific boundary conditions, examples

4.5.1. Vanishing axial load

For the beam in Figure 1, we set (cf. equation (48))

$$p=0\Rightarrow {\left[{\mathrm{\Sigma }}_{13}\right]}_{x_3=-c}^{x_3=c}=0,$$

(116)

so that, from equations (96)₁, (92)₁,

$$N^{\prime }=g_1^{\prime }=0.$$

(117)

Furthermore, we chose for the boundary conditions in equation (93)₁

$$U(0)=0,N(L)=0.$$

(118)

Consequently,

$$N=g_1=0$$

(119)

everywhere. Assume now that for some boundary conditions, solutions $w(x_1)$ and $\beta (x_1)$ have been established. In view of equation (49), the solution (119) can always hold if and only if

$$c_2{U}^{\prime }-(c_2-c_1)B=0.$$

(120)

On the other hand, from equations (90)_2, (51), and (53), we conclude that

$$U^{\prime }-B+c_1{B}^{\prime \prime }=0.$$

(121)

We can eliminate $U$ between the two equations (120) and (121), to obtain

$$c_2{B}^{\prime \prime }-B=0.$$

(122)

By choosing (cf. equations (93)₂, (92)₂, (51))

$$H(0)=H(L)=0\Rightarrow B^{\prime }(0)=B^{\prime }(L)=0,$$

(123)

We deduce from the solution of (122), that $B$ vanishes, and from the solution of equation (120) subject to equation (118)₁, that $U$ vanishes as well:

$$B=U=0.$$

(124)

Altogether, for the bending problems, we deal with in this section, it remains to solve the governing equations (91),

$$g_2^{\prime \prime }+q=0\iff -\frac{c_2}{c_1}{\mathrm{EIw}}^{\prime \prime \prime \prime }+\frac{c_2-c_1}{c_1}\mathrm{EI}{\beta }^{\prime \prime \prime }+q=0,$$

(125)

$$g_4+g_6^{\prime }=0\iff -\frac{c_2-c_1}{c_1}{w}^{\prime \prime }+\frac{c_2-c_1}{c_1}{\beta }^{\prime }-(c_2-c_1){\beta }^{\prime \prime \prime }=0,$$

(126)

subject to the boundary conditions (94) and (95),

$$\mathrm{either}V=g_2^{\prime }=-\frac{c_2}{c_1}{\mathrm{EIw}}^{\prime \prime \prime }+\frac{c_2-c_1}{c_1}\mathrm{EI}{\beta }^{\prime \prime }\mathrm{or}w,$$

(127)

$$\mathrm{either}M=g_2=-\frac{c_2}{c_1}{\mathrm{EIw}}^{\prime \prime }+\frac{c_2-c_1}{c_1}\mathrm{EI}{\beta }^{\prime }\mathrm{or}{w}^{\prime }\mathrm{and}$$

(128)

$$\mathrm{either}m=g_6=-(c_2-c_1)\mathrm{EI}{\beta }^{\prime \prime }\mathrm{or}{\beta }^{\prime }$$

(129)

have to be prescribed at $x_1=0,L$ .

4.5.2. Distributions of stresses

Suppose that some solutions $w(x_1)$ and $\beta (x_1)$ have been determined according to the assumptions made in the last section. Then, ${\mathrm{\Sigma }}_{11}$ can be calculated from equations (31), (124):

$${\mathrm{\Sigma }}_{11}=E(-\frac{c_2}{c_1}{w}^{\prime \prime }+\frac{c_2-c_1}{c_1}{\beta }^{\prime })x_3.$$

(130)

By comparison with the response function of $M$ in equation (128), we can derive for ${\mathrm{\Sigma }}_{11}$ the distribution

$${\mathrm{\Sigma }}_{11}=\frac{M}{I}{x}_3,$$

(131)

which is the same as in engineering mechanics. Therefore, the distributions of ${\mathrm{\Sigma }}_{13}$ and ${\mathrm{\Sigma }}_{33}$ are also the same as in engineering mechanics. This can be verified by inserting the result (131) into equilibrium equation (36) and solving it for ${\mathrm{\Sigma }}_{13}$ . The solution ${\mathrm{\Sigma }}_{13}$ can then be inserted into equilibrium equation (37) to determine the distribution of ${\mathrm{\Sigma }}_{33}$ . Since the stress $\mathit{\mu }$ is regarded here as internal state variable, there is no need to discuss distributions of its components ${\mu }_{\mathrm{ijk}}$ .

4.5.3. Cantilever beam subject to uniform transverse load

Under the assumptions made above, we seek solutions $w(x_1)$ , $\beta (x_1)$ of equations (125) and (126) for a cantilever beam subject to uniform transverse load $q(x_1)=q_0=\mathrm{const}.$ (see Figure 2). The classical boundary conditions (127) and (128) are supposed to be

$$w(0)=w^{\prime }(0)=V(L)=M(L)=0,$$

(132)

while homogeneous tractions are chosen for the non-classical boundary conditions (129),

$$m(0)=m(L)=0.$$

(133)

Figure 2.

Cantilever beam subject to transverse load $q_0$ .

In equations (132) and (133), we can replace the section forces using the sectional constitutive laws (50), (52), and the solution (92)₃. An equivalent form of the boundary conditions (132) and (133) is therefore

$$w(0)=0,w^{\prime }(0)=0,{\beta }^{\prime \prime }(0)=0,{\beta }^{\prime \prime }(L)=0,$$

(134)

$$-c_2{w}^{\prime \prime \prime }(L)+(c_2-c_1){\beta }^{\prime \prime }(L)=0,-c_2{w}^{\prime \prime }(L)+(c_2-c_1){\beta }^{\prime }(L)=0.$$

(135)

It is also convenient to refer the governing equations and the concomitant boundary conditions to dimensionless variables. To this end, we introduce the following definitions:

$${\overline{x}}_1:=\frac{x_1}{L},{\overline{x}}_3:=\frac{x_3}{L},\overline{w}:=\frac{w}{L},\overline{\beta }\equiv \beta ,\overline{b}:=\frac{b}{L},\overline{c}:=\frac{c}{L},$$

(136)

$${\overline{q}}_0:=\frac{q_0}{\mathrm{EL}},{\overline{q}}^{*}:=\frac{3}{4}\frac{{\overline{q}}_0}{\overline{b}{\overline{c}}^3},{\overline{c}}_1:=\frac{c_1}{L^2},{\overline{c}}_2:=\frac{c_2}{L^2}.$$

(137)

Changing notation in the remainder of the paper, we set

$$({)}^{\prime }:=\frac{d()}{d\overline{x}}.$$

(138)

The dimensionless forms of the differential equations (125) and (126) become then

$$-\frac{{\overline{c}}_2}{{\overline{c}}_1}{\overline{w}}^{\prime \prime \prime \prime }+\frac{{\overline{c}}_2-{\overline{c}}_1}{{\overline{c}}_1}{\overline{\beta }}^{\prime \prime \prime }+{\overline{q}}^{*}=0,$$

(139)

$$-{\overline{w}}^{\prime \prime }+{\overline{\beta }}^{\prime }-{\overline{c}}_1{\overline{\beta }}^{\prime \prime \prime }=0,$$

(140)

while the boundary conditions (134) and (135) take the forms

$$\overline{w}(0)=0,{\overline{w}}^{\prime }(0)=0,{\beta }^{\prime \prime }(0)=0,{\beta }^{\prime \prime }(1)=0,$$

(141)

$$-{\overline{c}}_2{\overline{w}}^{\prime \prime \prime }(1)+({\overline{c}}_2-{\overline{c}}_1){\beta }^{\prime \prime }(1)=0,-{\overline{c}}_2{\overline{w}}^{\prime \prime }(1)+({\overline{c}}_2-{\overline{c}}_1){\beta }^{\prime }(1)=0.$$

(142)

Now, a look at the KG- and the 3-PG-Model in equations (2) and (3) could give the impression, that the KG-Model might be viewed as particular case of the 3-PG-Model for $c_1\to 0$ . That this is generally not correct, is discussed among others in Broese et al. [44]. The reason is that the two models are described by partial differential equations. Therefore, concomitant boundary conditions are also parts of the models, which are for the KG- and the 3-PG-Models quite different. Similar remarks hold also true between the 3-PG-Model and classical elasticity in the limiting case $c_1\to c_2$ . Nevertheless, it is of interest to compare predicted responses by the two gradient elasticity models with each other, as well as with responses predicted by classical elasticity, for the particular case of the cantilever beam in Figure 2. Description of the cantilever beam problem in Figure 2 with the KG-Model has been addressed in Sideris and Tsakmakis [42] and Broese et al. [43]. It was shown (cf. Broese et al. [43]), that two different versions of loading and boundary conditions, termed KG-Model-A and KG-Model-NA, might be attributed to the cantilever beam in Figure 2. Hence, there arise several questions for the responses predicted by the 3-PG-Model in the special case of the cantilever beam problem. Do these responses converge for $c_1\to 0$ , and if yes, do coincide the limiting responses with those predicted by the KG-Model-A or the KG-Model-NA? Additionally, do converge the responses for $c_1\to c_2$ , and if yes, are the limiting responses the same as in the case of classical elasticity? The best way to answer these questions is to elaborate analytical solutions of the system (139)–(142).

After lengthy, but straightforward manipulations, it can be verified that the system of differential equations (139)–(142) leads to two differential equations, one for $\overline{w}({\overline{x}}_1)$ ,

$${\overline{c}}_2{\overline{w}}^{\prime \prime \prime \prime \prime \prime }-{\overline{w}}^{\prime \prime \prime \prime }+{\overline{q}}^{*}=0,$$

(143)

with boundary conditions

$$\begin{array}{c}\overline{w}(0)=0,{\overline{w}}^{\prime }(0)=0,\hfill \\ [{\overline{c}}_2{\overline{c}}_1{\overline{w}}^{\prime \prime \prime \prime \prime }+({\overline{c}}_2-{\overline{c}}_1){\overline{w}}^{\prime \prime \prime }{]}_{{\overline{x}}_1=0}=0,\hfill \\ [{\overline{c}}_2{\overline{c}}_1{\overline{w}}^{\prime \prime \prime \prime \prime }+({\overline{c}}_2-{\overline{c}}_1){\overline{w}}^{\prime \prime \prime }{]}_{{\overline{x}}_1=1}=0,\hfill \end{array}$$

(144)

$${[-{\overline{w}}^{\prime \prime \prime }+{\overline{c}}_2{\overline{w}}^{\prime \prime \prime \prime \prime }]}_{{\overline{x}}_1=1}=0,{[-{\overline{w}}^{\prime \prime }+{\overline{c}}_2{\overline{w}}^{\prime \prime \prime \prime }-{\overline{c}}_1{\overline{q}}^{*}]}_{{\overline{x}}_1=1}=0,$$

(145)

and one for $\beta ({\overline{x}}_1)$ ,

$${\overline{c}}_2{\beta }^{\prime \prime \prime \prime \prime }-{\beta }^{\prime \prime \prime }+{\overline{q}}^{*}=0,$$

(146)

with boundary conditions

$${\beta }^{\prime \prime }(0)=0,{\beta }^{\prime \prime }(1)=0,{[{\overline{c}}_2{\beta }^{\prime \prime \prime \prime }-{\beta }^{\prime \prime }]}_{{\overline{x}}_1=1}=0,{[{\overline{c}}_2{\beta }^{\prime \prime \prime }-{\beta }^{\prime }]}_{{\overline{x}}_1=1}=0.$$

(147)

There is a constant term in the solution of $\beta$ , which can be set to zero without loss of generality.

The solution of equations (143)–(145) reads

$$\overline{w}({\overline{x}}_1)={\overline{k}}_1+{\overline{k}}_2{\overline{x}}_1+{\overline{k}}_3{\overline{x}}_1^2+{\overline{k}}_4{\overline{x}}_1^3+\frac{{\overline{q}}^{*}}{24}{\overline{x}}_1^4+{\overline{k}}_5{e}^{\frac{1}{\sqrt{{\overline{c}}_2}}{\overline{x}}_1}+{\overline{k}}_6{e}^{-\frac{1}{\sqrt{{\overline{c}}_2}}{\overline{x}}_1},$$

(148)

with constants ${\overline{k}}_1,\dots ,{\overline{k}}_6$ given in Appendix 2.

It is now not difficult to verify, that $\overline{w}({\overline{x}}_1)$ in equation (148) converges for ${\overline{c}}_1\to {\overline{c}}_2$ to the corresponding deflection curve of classical elasticity. Furthermore, in the limit ${\overline{c}}_1\to 0$ , the deflection curve in equation (148) converges to the one according to the KG-Model-NA, the latter being determined in Broese et al. [43]. Figure 3 illustrates these issues. In particular, it can be seen that all graphs of the deflection curves in equation (148) for various values ${\overline{c}}_1$ are bounded from above by the classical solution and from below by the solution obtained for the KG-Model-NA.

Figure 3.

Distributions of $\overline{w}({\overline{x}}_1)$ predicted by the 3-PG-Model for various values of ${\overline{c}}_1$ with ${\overline{q}}^{*}=0.01$ , ${\overline{c}}_2=0.25$ .

It is remarkable, that the corresponding distributions $\beta ({\overline{x}}_1)$ are independent from ${\overline{c}}_1$ . This can be recognized from equations (146) and (147) which are in fact independent from ${\overline{c}}_1$ . Figure 4 shows the graph of the curve $\beta ({\overline{x}}_1)$ , which is common for all curves $\overline{w}({\overline{x}}_1)$ in Figure 3. W e proved in section 4.5.2 that the stress distributions according to the 3-PG-Model and the classical elasticity coincide. In Broese et al. [43], it is shown that, besides the ${\overline{\mathrm{\Sigma }}}_{11}$ distributions according to the KG-Model-NA, all other stress distributions predicted by the KG-Model are different from those according to the classical elasticity. Therefore, stress distributions according to the 3-PG-Model generally do not converge to those according to the KG-Model for $c_1\to 0$ .

Figure 4.

Graph of the curves $\beta ({\overline{x}}_1)$ , which are common to the curves $\overline{w}({\overline{x}}_1)$ in Figure 3.

Finally, it is perhaps of interest to remark, that for ${\overline{c}}_2$ = const. and increasing values of ${\overline{c}}_1$ the graphs of the curves $\overline{w}({\overline{x}}_1)$ become closer to the classical one. In the case of ${\overline{c}}_1$ = const. and increasing values of ${\overline{c}}_2$ the trends are opposite (see Figure 5). Note that for ${\overline{c}}_1$ = const., the distributions of $\beta (\overline{x})$ are dependent from ${\overline{c}}_2$ (see Figure 6), in opposite to the case ${\overline{c}}_2$ = const. where the distributions were independent from ${\overline{c}}_1$ .

Figure 5.

Distributions of $\overline{w}({\overline{x}}_1)$ predicted by the 3-PG-Model for various values of ${\overline{c}}_2$ with ${\overline{q}}^{*}=0.01$ , ${\overline{c}}_1=0.1$ .

Figure 6.

Graph of the curves $\beta ({\overline{x}}_1)$ corresponding to the curves $\overline{w}({\overline{x}}_1)$ in Figure 5.

5. Conclusion

A consistent Euler–Bernoulli bending theory for gradient elastic beams has been formulated in the paper. The material behavior is described by a gradient elasticity model, called 3-PG-Model, including both the Laplacian of the stress and the Laplacian of the strain. It is the analogon of the Three-Parameter viscoelastic solid. Starting from a standard formulation of the virtual work principle, it is shown that the Euler–Bernoulli geometrical constraints require to consider only an effective part of this principle. Alternatively, this effective work principle can be derived from the field equations of the three-dimensional material body. Moreover, the governing equations of bending might be derived as consequences of the effective virtual work principle, or alternatively directly from the three-dimensional field equations by integration.