Variable-order approach to nonlocal elasticity: theoretical formulation, order identification via deep learning, and applications

Abstract

This study presents the formulation of the variable-order continuum mechanics theory and its application to the analysis of nonlocal heterogeneous solids. The variable-order continuum theory enables a unique approach to model the response of solids exhibiting position-dependent nonlocal behavior. The formulation also guarantees frame-invariance provided that proper constraints on the functional definition of the variable-order are imposed. The study also presents a deep learning approach to identify the variable-order distribution describing the behavior of the medium. This methodology presents a very promising route for the practical application of the variable-order theory to real-world problems, especially when the microstructure is not known a priori and must be inferred from the physical response of the medium. The capabilities of the variable-order theory are illustrated by numerically simulating the static response of nonlocal beams having either a porous or a functionally graded core. The reduced-order variable fractional model shows excellent accuracy and significant computational efficiency when compared with a reference solution produced by a 3D finite element model that fully resolves the beam geometry.

Appendix

1.1 A: Definitions of variable-order fractional derivatives

Variable-order fractional operators were first conceptualized by Samko et al.  [29] in 1993 as a natural extension of CO fractional operators. Over time, researchers have presented several definitions of VO fractional derivatives. The most significant discriminant factor, between the different definitions, consisted in the memory behavior of the operator. A detailed review of the different definitions, properties, and their applications can be found in [37, 72]. Here below, we report the definitions of the type-I and type-II VO Caputo operators that have been used in this work.

Type-I: If f(x) and $\alpha (x)$ are continuous real-valued functions on (a, b), the left- and right-handed VO Caputo derivative to the order $\alpha (x)>0$ with no order-memory are defined as:

$$\begin{aligned}&\text {Left-handed derivative:}~ {}^C_a D^{\alpha (x)}_x f(x) \nonumber \\&\quad = \frac{1}{\Gamma (n-\alpha (x))} \int _{a}^{x} \frac{ D^n_{x^\prime } f(x^\prime )}{(x - x^\prime )^{1+\alpha (x)-n}} \mathrm {d}x^\prime \end{aligned}$$

(30a)

$$\begin{aligned}&\text {Right-handed derivative:}~ {}^C_x D^{\alpha (x)}_b f(x) \nonumber \\&= \frac{(-1)^n}{\Gamma (n-\alpha (x))} \int _{x}^{b} \frac{ D^n_{x^\prime } f(x^\prime )}{(x^\prime - x)^{1+\alpha (x)-n}} \mathrm {d}x^\prime \end{aligned}$$

(30b)

where $n=\lceil {\alpha (x)}\rceil$ is the upper integer bound on $\alpha (x)$ at the spatial location x, $\Gamma (\cdot )$ is the Gamma function, and $x^\prime$ is a dummy spatial variable of integration. As discussed in Sect. 2$\alpha (x)\in (0,1)$ throughout this work. Under this latter condition, the expressions in Eq. (30) can be simplified as:

$$\begin{aligned}&\text {Left-handed derivative:}~ {}^C_a D^{\alpha (x)}_x f(x) \nonumber \\&\quad = \frac{1}{\Gamma (1-\alpha (x))} \int _{a}^{x} \frac{ D^1_{x^\prime } f(x^\prime )}{(x - x^\prime )^{\alpha (x)}} \mathrm {d}x^\prime \end{aligned}$$

(31a)

$$\begin{aligned}&\text {Right-handed derivative:}~ {}^C_x D^{\alpha (x)}_b f(x) \nonumber \\&\quad = \frac{-1}{\Gamma (1-\alpha (x))} \int _{x}^{b} \frac{ D^1_{x^\prime } f(x^\prime )}{(x^\prime - x)^{\alpha (x)}} \mathrm {d}x^\prime \end{aligned}$$

(31b)

Analogous expressions can be obtained for the Type-II and Type-III operators presented in the following when $\alpha (x)\in (0,1)$. We do not provide them here for the sake of brevity.

Type-II: If f(x) and $\alpha (x)$ are continuous real-valued functions on (a, b), the left- and right-handed VO Caputo derivative to the order $\alpha (x)>0$ with weak order-memory are defined as:

$$\begin{aligned}&\text {Left-handed derivative:}~ {}^C_a D^{\alpha (x^\prime )}_x f(x)\nonumber \\&\quad = \int _{a}^{x} \frac{1}{\Gamma (n-\alpha (x^\prime ))} \left[ \frac{D^n_{x^\prime } f(x^\prime )}{(x - x^\prime )^{1+\alpha (x^\prime )-n}} \right] \mathrm {d}x^\prime \end{aligned}$$

(32a)

$$\begin{aligned}&\text {Right-handed derivative:}~ {}^C_x D^{\alpha (x^\prime )}_b f(x) \nonumber \\&\quad = \int _{x}^{b} \frac{(-1)^n}{\Gamma (n-\alpha (x^\prime ))} \left[ \frac{ D^n_{x^\prime } f(x^\prime )}{(x^\prime - x)^{1+\alpha (x^\prime )-n}} \right] \mathrm {d}x^\prime \end{aligned}$$

(32b)

Type-III: If f(x) and $\alpha (x)$ are continuous real-valued functions on (a, b), the left- and right-handed VO Caputo derivative to the order $\alpha (x)>0$ with weak order-memory are defined as:

$$\begin{aligned}&\text {Left-handed derivative:}~ {}^C_a D^{\alpha (x-x^\prime )}_x f(x)\nonumber \\&\quad = \int _{a}^{x} \frac{1}{\Gamma (n-\alpha (x-x^\prime ))} \left[ \frac{D^n_{x^\prime } f(x^\prime )}{(x - x^\prime )^{1+\alpha (x-x^\prime )-n}} \right] \mathrm {d}x^\prime \nonumber \\ \end{aligned}$$

(33a)

$$\begin{aligned}&\text {Right-handed derivative:}~ {}^C_x D^{\alpha (x-x^\prime )}_b f(x)\nonumber \\&\quad = \int _{x}^{b} \frac{(-1)^n}{\Gamma (n-\alpha (x-x^\prime ))} \left[ \frac{ D^n_{x^\prime } f(x^\prime )}{(x^\prime - x)^{1+\alpha (x-x^\prime )-n}} \right] \mathrm {d}x^\prime \nonumber \\ \end{aligned}$$

(33b)

Note the differences within the definitions of the different VO derivatives. While the strength of the power-law kernel, that is the exponent of the denominator, is fixed for the point x for the type-I derivative, the strength of the type-II kernel is a function of the dummy variable $x^\prime$ and the strength of the type-III kernel a function of the relative separation between x and $x^\prime$. This is exactly the reason which leads to difference in the memory characteristics of the different definitions. Detailed discussions on the properties of these derivatives, including linearity, time invariance, memory characteristics (both operator- and order-memory), Laplace transforms, and physical realization using switches can be found in [30].

1.2 B: Frame-invariance of the VO continuum model - Type-I v/s Type-II v/s Type-III operators

In the following, we will analyze the frame-invariance of the variable-order continuum formulation resulting from the use of either type-I or type-II or type-III VO Caputo derivatives. Consider a rigid-body motion superimposed on a general point $\varvec{X}$ (see Fig. 1a) of the reference configuration of the body as:

$$\begin{aligned} \varvec{\Psi }(\varvec{X},t)=\mathbf{c} (t)+\mathbf{Q} (t)\varvec{X}, \end{aligned}$$

(34)

where $\mathbf{Q} (t)$ is a proper orthogonal tensor denoting a rotation and $\mathbf{c} (t)$ is a spatially constant term representing a translation. Under this rigid-body motion, the fractional deformation gradient tensor denoted as $\tilde{\mathbf{F }}^{\Psi }_X$ should be an orthogonal tensor such that $\tilde{\mathbf{F }}^{\Psi T}_X\tilde{\mathbf{F }}^{\Psi }_X = \mathbf{I}$. More specifically, the fractional deformation gradient tensor should transform as $\tilde{\mathbf{F }}^{\Psi }_X = \mathbf{Q}$ (similar to the classical continuum case where $\mathbf{F} ^{\Psi } = \mathbf{Q}$) such that the strain measures are null. As discussed in detail in [10], the fractional deformation gradient tensor $\tilde{\mathbf{F }}^{\Psi }_X$ is defined as the fractional-order derivative of the motion $\Psi$ with respect to the reference coordinates, that is, $\tilde{\mathbf{F }}^{\Psi }_X = D^{\alpha (\varvec{X},\varvec{X}^\prime )}_{\varvec{X}} \varvec{\Psi }$.

Type-I derivative: Consider the formulation involving type-I VO derivatives. Recalling the definition of the type-I VO-RC derivative from Eqs. (2,30) it follows that:

$$\begin{aligned} \tilde{\mathbf{F }}_{X_{ij}}^{\Psi }= & {} \frac{1}{2} \Gamma (2-\alpha (\varvec{X})) \left[ \frac{L_{-_{j}}^{\alpha (\varvec{X})-1}}{\Gamma (1-\alpha (\varvec{X}))} \int _{X_{-_{j}}}^{X_j} \frac{D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t)}{(X_j-{X}^\prime _j)^{\alpha (\varvec{X})}} \mathrm {d}{X}^\prime _j \right. \nonumber \\&\left. + \frac{L_{+_{j}}^{\alpha (\varvec{X})-1}}{\Gamma (1-\alpha (\varvec{X}))} \int _{X_{j}}^{X_{+_j}} \frac{D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t)}{({X}^\prime _j-X_j)^{\alpha (\varvec{X})}} \mathrm {d}{X}^\prime _j \right] \end{aligned}$$

(35)

where $\varvec{X}^\prime$ is the dummy vector representing the spatial variable, and $L_{-_j}$ and $L_{+_j}$ are the length scales corresponding to the horizon of nonlocality in the reference configuration. $D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t)$ simplifies as:

$$\begin{aligned} D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t) = \frac{\mathrm {d} \Psi _{i} (\varvec{X}^\prime ,t) }{\mathrm {d}{X}^\prime _j} = \frac{\mathrm {d}}{\mathrm {d}{X}^\prime _j}(c_i + Q_{ik}{X}^\prime _{k}) \end{aligned}$$

(36)

Noting that $\frac{\mathrm {d}c_i(t)}{\mathrm {d}{X}^\prime _j}=0$ and $\mathbf{Q} =\mathbf{Q} (t)$ it follows that:

$$\begin{aligned} D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t)=Q_{ik}{X}^\prime _{k,j}=Q_{ik}\delta _{kj}=Q_{ij} \end{aligned}$$

(37)

Thus, under the rigid body motion $\varvec{\Psi }$:

$$\begin{aligned} \tilde{\mathbf{F }}_{X_{ij}}^{\Psi }= & {} \frac{1}{2} \Gamma (2-\alpha (\varvec{X})) Q_{ij} \left[ \frac{L_{-_{j}}^{\alpha (\varvec{X})-1}}{\Gamma (1-\alpha (\varvec{X}))}\right. \nonumber \\&\int _{X_{-_{j}}}^{X_j} \frac{ 1 }{(X_j-{X}^\prime _j)^{\alpha (\varvec{X})}} \mathrm {d}{X}^\prime _j \nonumber \\&\left. + \frac{L_{+_{j}}^{\alpha (\varvec{X})-1}}{\Gamma (1-\alpha (\varvec{X}))} \int _{X_{j}}^{X_{+_j}} \frac{ 1 }{({X}^\prime _j-X_j)^{\alpha (\varvec{X})}} \mathrm {d}{X}^\prime _j \right] \end{aligned}$$

(38)

Since the exponent of the power-law kernel $\alpha (\varvec{X})$ is independent of the integrating variable $\varvec{X}^\prime$, the above expression can be easily simplified (by treating $\varvec{X}$ as a constant within the integration) as:

$$\begin{aligned} \tilde{\mathbf{F }}_{X_{ij}}^{\Psi }= & {} \frac{1}{2} \left[ L_{-_{j}}^{\alpha (\varvec{X})-1} (X_j-X_{-_{j}})^{1-\alpha (\varvec{X})}\right. \nonumber \\&\left. + L_{+_{j}}^{\alpha (\varvec{X})-1} (X_{+_{j}}-X_j)^{1-\alpha (\varvec{X})} \right] Q_{ij} \end{aligned}$$

(39)

In the above simplifications we have used the following property of the $\Gamma (\cdot )$ function: $\Gamma (2-\alpha ) = (1-\alpha )\Gamma (1-\alpha )$. As highlighted in §2, the length scales $L_{-_j}$ and $L_{+_j}$ are taken such that: $L_{-_{j}}=X_j-X_{-_{j}}$ and $L_{+_{j}}=X_{+_{j}}-X_j$. This has also been illustrated schematically in Fig. 1b. By substituting these relations in Eq. (39), it follows that $\tilde{\mathbf{F }}^{\Psi }_X = \mathbf{Q}$ at all times. We also emphasize that the nonlocal formulation allows for an exact treatment of frame invariance in the presence of asymmetric horizons which occur at points close to material boundaries and interfaces. The different horizon lengths $L_{-_j}$ and $L_{+_j}$ enables the truncation of the horizon at points close to or on the boundary in order to exactly satisfy frame-invariance.

Type-II derivative: We consider a general formulation involving the left- and right-handed type-II VO Caputo derivatives. More specifically, we replace the length scale factors introduced in Eq. (2) with general multiplying factors $c_1$ and $c_2$, and then find expressions for $c_1$ and $c_2$ such that the resulting formulation is frame-invariant. Using the definition of the type-II VO Caputo derivatives from Eq. (32) it follows that:

$$\begin{aligned} \begin{aligned} \tilde{\mathbf{F }}_{X_{ij}}^{\Psi }&= c_1 \int _{X_{-_{j}}}^{X_j} \frac{1}{\Gamma (1-\alpha (\varvec{X}^\prime ))} \left[ \frac{D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t)}{(X_j-{X}^\prime _j)^{\alpha (\varvec{X}^\prime )}} \right] \mathrm {d}{X}^\prime _j \\&\quad + c_2 \int _{X_{j}}^{X_{+_j}} \frac{1}{\Gamma (1-\alpha (\varvec{X}^\prime ))} \left[ \frac{D^1_{{X}^\prime _j} \Psi _{i}(\varvec{X}^\prime ,t)}{({X}^\prime _j-X_j)^{\alpha (\varvec{X}^\prime )}} \right] \mathrm {d}{X}^\prime _j \end{aligned}\nonumber \\ \end{aligned}$$

(40)

Retracing the steps in Eqs. (36,37), under the rigid body motion $\varvec{\Psi }$, we obtain:

$$\begin{aligned} \tilde{\mathbf{F }}_{X_{ij}}^{\Psi }= & {} Q_{ij} \left[ c_1 \int _{X_{-_{j}}}^{X_j} \frac{(X_j-{X}^\prime _j)^{-\alpha (\varvec{X}^\prime )}}{\Gamma (1-\alpha (\varvec{X}^\prime ))} \mathrm {d}{X}^\prime _j \right. \nonumber \\&\left. + c_2 \int _{X_{j}}^{X_{+_j}} \frac{({X}^\prime _j-X_j)^{-\alpha (\varvec{X}^\prime )}}{\Gamma (1-\alpha (\varvec{X}^\prime ))} \mathrm {d}{X}^\prime _j \right] \end{aligned}$$

(41)

For frame-invariance, it is necessary that $\tilde{\mathbf{F }}_{X_{ij}}^{\Psi }=Q_{ij}$ at all times and for all points within the nonlocal solid. More specifically, we obtain the following equation:

$$\begin{aligned}&c_1 \underbrace{\int _{X_{-_{j}}}^{X_j} \frac{(X_j-{X}^\prime _j)^{-\alpha (\varvec{X}^\prime )}}{\Gamma (1-\alpha (\varvec{X}^\prime ))} \mathrm {d}{X}^\prime _j}_{\text {Cannot be simplified further}}\nonumber \\&\quad + c_2 \underbrace{\int _{X_{j}}^{X_{+_j}} \frac{({X}^\prime _j-X_j)^{-\alpha (\varvec{X}^\prime )}}{\Gamma (1-\alpha (\varvec{X}^\prime ))} \mathrm {d}{X}^\prime _j}_{\text {Cannot be simplified further}} = 1 \end{aligned}$$

(42)

To ensure that the above relation holds true for all points $\varvec{X}$, at all time instants, and for every order distribution, it is essential to evaluate the integrals highlighted in the equation above. However, given the functional variation of the VO, it is not straightforward to analytically evaluate the integrals. In fact, the possibility to obtain an analytical expression of the solution depends on the specific functional variation of the fractional-order. For the specific cases, where the analytical solution does not exist, it might not be possible to achieve a frame-invariant formulation. While the integrals could certainly be numerically evaluated, the numerical route poses additional computational challenges. The latter comment stems from the fact that the values of the specific integrals highlighted above depend on the position $\varvec{X}$ as well as the specific functional variation of the order. Further, contrary to the type-I case where these factors turn out as the dimensions of the horizon of nonlocality, no physical interpretation can be conclusively drawn for the factors when using type-II derivatives.

Type-III derivative: By using the definition of the type-III derivatives and the arguments presented for the type-II derivative, Eq. (41) can be modified for a formulation using type-III derivatives as:

$$\begin{aligned} \tilde{\mathbf{F }}_{X_{ij}}^{\Psi }= & {} Q_{ij} \Bigg [ c_1 \underbrace{\int _{X_{-_{j}}}^{X_j} \frac{(X_j-{X}^\prime _j)^{-\alpha (\varvec{X}-\varvec{X}^\prime )}}{\Gamma (1-\alpha (\varvec{X}-\varvec{X}^\prime ))} \mathrm {d}{X}^\prime _j}_{\text {Cannot be simplified further}}\nonumber \\&+ c_2 \underbrace{\int _{X_{j}}^{X_{+_j}} \frac{({X}^\prime _j-X_j)^{-\alpha (\varvec{X}-\varvec{X}^\prime )}}{\Gamma (1-\alpha (\varvec{X}-\varvec{X}^\prime ))} \mathrm {d}{X}^\prime _j}_{\text {Cannot be simplified further}} \Bigg ] \end{aligned}$$

(43)

Given the specific form of the VO, similar to type-II derivatives, it is not always possible to obtain a closed form expression for the factors $c_1$ and $c_2$. It immediately follows that the remarks made above for type-II also hold true for type-III derivatives.

To summarize, the use of type-II and type-III VO derivatives is more likely to lead to non frame-invariant formulations. For cases, where a frame-invariant formulation can be achieved, the procedure to obtain the different factors is not general because frame-invariance must be re-valuated for every VO and at every point in the nonlocal solid. This makes the formulation computationally intensive and additionally, the obtained factors do not admit clear physical interpretations, unlike the length scale factors used in type-I derivatives.

1.3 C: Derivation of the governing equations

Theorem: The displacement field $\varvec{u}(\varvec{x})\in \psi$, a class of all kinematically admissible displacement fields, which solves the Eqs. (7, 8) minimizes the total potential energy functional given in Eq. (6) in the class $\psi$. Conversely, the displacement field minimizing the total potential energy functional in Eq. (6) solves the fractional-order nonlocal beam governing Eqs.  (7, 8).

Proof

Let $\varvec{u}^*\in \psi$ be the unique solution to the system of Eqs. (7,8). Next, we assume $\varvec{u} = \varvec{u}^* + \delta \varvec{u}$ is another kinematically admissible field such that $\delta \varvec{u}\in \psi ^*$. The class $\psi ^*$ is similar to the class $\psi$ except for the boundary points $x\in \{0,L\}$, where the displacement degrees of freedom $\{u_0,w_0,D^1_x w_0\}=0$ in context of Eqs. (7,8). In the following, all quantities with the superscript $\square ^*$ correspond to the displacement field $\varvec{u}^*$. Under the above conditions, following the principles of variational calculus, Eq. (6) can be expressed as:

$$\begin{aligned} \Pi [\varvec{u}] = \Pi [\varvec{u}^*] + \delta \Pi + \frac{1}{2}\delta ^2\Pi \end{aligned}$$

(44)

where $\delta \Pi$ and $\delta ^2\Pi$ are the first and second variations of $\Pi$ from $\varvec{u}^*$. Using the Eqs. (5, 6, 9), the first variation $\delta \Pi$ is obtained as:

$$\begin{aligned} \delta \Pi= & {} \underbrace{\int _{0}^{L} N_{xx}^* \left[ D_{x}^{\alpha (x)}\delta u_0 \right] \mathrm {d}x}_{{\mathcal {I}}_1} - \underbrace{\int _{0}^{L} M^*_{xx} \left[ D_{x}^{\alpha (x)} \left( D^1_x \delta w_0 \right) \right] \mathrm {d}x}_{{\mathcal {I}}_2} \nonumber \\&- \int _{0}^{L} F_a \delta u_0 \mathrm {d}x - \int _{0}^{L} F_t \delta w_0 \mathrm {d}x \end{aligned}$$

(45)

$\square$

${Simplification\, of\, {\mathcal {I}}_1}$: We first simplify the term indicated as ${\mathcal {I}}_1$. Using the definition of the VO-RC fractional derivative given in Eq. (2), we obtain:

$$\begin{aligned} {\mathcal {I}}_1= & {} \frac{1}{2} \Bigg [ \underbrace{ \int _0^L N_{xx}^* \left[ l_-^{\alpha (x)-1} \Gamma (2-\alpha (x)) {}^{C}_{x-l_-} D^{\alpha (x)}_{x} \delta u_0 \right] \mathrm {d}x}_{{\mathcal {I}}_{11}} \nonumber \\&- \underbrace{\int _0^L N_{xx}^* \left[ l_+^{\alpha (x)-1} \Gamma (2-\alpha (x)) {}^{C}_{x}D^{\alpha (x)}_{x + l_+} \delta u_0 \right] \mathrm {d}x}_{{\mathcal {I}}_{12}} \Bigg ]\nonumber \\ \end{aligned}$$

(46)

From the definitions for the VO left- and right-handed Caputo derivatives given in Eq. (30) we obtain:

$$\begin{aligned} {\mathcal {I}}_{11}= & {} \int _0^L N^*_{xx} ~ l_-^{\alpha (x)-1} \frac{\Gamma (2-\alpha (x))}{\Gamma (1-\alpha (x))}\nonumber \\&\left[ \int _{x - l_-}^{x} \left( x - x^\prime \right) ^{-\alpha (x)} \left( D^1_{x^\prime } \delta u_0 \right) \mathrm {d}x^\prime \right] \mathrm {d}x \end{aligned}$$

(47a)

$$\begin{aligned} {\mathcal {I}}_{12}= & {} - \int _0^L N^*_{xx} ~ l_+^{\alpha (x)-1} \frac{\Gamma (2-\alpha (x))}{\Gamma (1-\alpha (x))} \nonumber \\&\left[ \int _{x}^{x + l_+} \left( x^\prime - x \right) ^{-\alpha (x)} \left( D^1_{x^\prime } \delta u_0 \right) \mathrm {d}x^\prime \right] \mathrm {d}x \end{aligned}$$

(47b)

By changing the order of integration in the above equation, we obtain:

$$\begin{aligned} {\mathcal {I}}_{11}= & {} \int _0^L D^1_{x^\prime } \delta u_0 \nonumber \\&\left[ \int _{x^\prime }^{x^\prime + l_-} l_-^{\alpha (x)-1} \frac{\Gamma (2-\alpha (x))}{\Gamma (1-\alpha (x))} \left( x - x^\prime \right) ^{-\alpha (x)} N^*_{xx} \mathrm {d}x \right] \mathrm {d}x^\prime \nonumber \\ \end{aligned}$$

(48a)

$$\begin{aligned} {\mathcal {I}}_{12}= & {} - \int _0^L D^1_{x^\prime } \delta u_0\nonumber \\&\left[ \int _{x^\prime - l_+}^{x^\prime } l_+^{\alpha (x)-1} \frac{\Gamma (2-\alpha (x))}{\Gamma (1-\alpha (x))} \left( x^\prime - x \right) ^{-\alpha (x)} N^*_{xx} \mathrm {d}x \right] \mathrm {d}x^\prime \nonumber \\ \end{aligned}$$

(48b)

Substituting the above expressions within Eq. (46) and using the definition of the VO Riesz integral given in Eq. (10), we obtain:

$$\begin{aligned} {\mathcal {I}}_1= & {} \int _0^L \left[ D^1_{x^\prime } \delta u_0 \right] \left[ I^{1-\alpha (x)}_{x^\prime } N^*_{xx} \right] \mathrm {d}x^\prime \nonumber \\\equiv & {} \int _0^L \left[ D^1_{x} \delta u_0 \right] \left[ I^{1-\alpha (x^\prime )}_{x} N^*_{xx} \right] \mathrm {d}x \end{aligned}$$

(49)

The above integrals are further evaluated using integration by parts in order to transfer the derivative from the independent the variable (axial displacement) to the secondary variable (stress resultant):

$$\begin{aligned} {\mathcal {I}}_1 = \delta u_0 \left[ I^{1-\alpha (x^\prime )}_x N^*_{xx} \right] \bigg \vert _{0}^{L} - \int _0^L \delta u_0 D^1_x\left[ I^{1-\alpha (x^\prime )}_x N^*_{xx} \right] \mathrm {d}x\nonumber \\ \end{aligned}$$

(50)

Now by using the definition of the VO R-RL derivative given in Eq. (11), we obtain:

$$\begin{aligned} {\mathcal {I}}_1 = \delta u_0 \left[ I^{1-\alpha (x^\prime )}_x N^*_{xx} \right] \bigg \vert _{0}^{L} - \int _0^L \delta u_0 \left[ {\mathfrak {D}}^{\alpha (x^\prime )}_x N^*_{xx} \right] \mathrm {d}x \end{aligned}$$

(51)

${Simplification\, of\, {\mathcal {I}}_2}$: By retracing the steps through Eqs. (46-51), it can be similarly shown that:

$$\begin{aligned} {\mathcal {I}}_2= & {} \left( \delta D^1_x w_0\right) \left[ I^{1-\alpha (x^\prime )}_x M^*_{xx} \right] \bigg \vert _{0}^{L} - \delta w_0 \left[ {\mathfrak {D}}^{\alpha (x^\prime )}_x M^*_{xx} \right] \bigg \vert _{0}^{L} \nonumber \\&+ \int _0^L \delta w_0 D^1_x \left[ {\mathfrak {D}}^{\alpha (x^\prime )}_x N^*_{xx} \right] \mathrm {d}x \end{aligned}$$

(52)

Now by using the variational simplifications in Eqs. (45,51, 52) and the governing equations in Eq. (7,8) it can be shown that $\delta \Pi =0$. Additionally, the second variation $\delta ^2\Pi$ is given as:

$$\begin{aligned} \delta ^2\Pi = \int _\Omega E_0 \delta [{\varepsilon }_{xx}] \delta [{\varepsilon }_{xx}] \mathrm {d}V \end{aligned}$$

(53)

For any nontrivial $\delta \varvec{u}\in \psi ^*$ we have from the above equation $\delta ^2\Pi >0$, which leads us to the inequality:

$$\begin{aligned} \Pi [\varvec{u}] = \Pi [\varvec{u}^*] + \frac{1}{2} \delta ^2\Pi \ge \Pi [\varvec{u}^*] ~~\forall ~~\varvec{u}\in \psi \end{aligned}$$

(54)

It follows that the equality holds iff $\varvec{u}=\varvec{u}^*$ $\forall ~\varvec{x}\in \Omega$. It follows immediately that, as claimed in the Theorem above, the displacement field $\varvec{u}^*$ which solves the system of equations in Eqs. (7, 8) minimizes the functional $\Pi$ in the class $\psi$.

Conversely, let $\varvec{u}^*$ be the unique solution to the minimization problem: $\varvec{\min }(\Pi [\varvec{u}])$ such that $\varvec{u}\in \psi$. The minimization implies that for any variation $\delta \varvec{u}\in \psi ^*$, $\delta \Pi$ evaluated at $\varvec{u}^*$ must be identically zero. The $\delta \Pi$ is evaluated through Eqs. (45,51, 52) where $\varvec{u}^*$, minimizes the functional $\Pi$. It follows that the stress field corresponding to the displacement field $\varvec{u}^*$ uniquely satisfies the equilibrium Eqs. (7,8), and thus the set of fields $\{\varvec{u}^*,{\varvec{\varepsilon }}^*,{\varvec{\sigma }}^*\}$ solve the variable fractional-order Euler-Bernoulli beam equations.