Generalized Kelvin–Voigt Creep Model in Fractal Space–Time

Abstract

In this paper, we study the creep phenomena for self-similar models of viscoelastic materials and derive a generalization of the Kelvin–Voigt model in the framework of fractal continuum calculus. Creep compliance for the Kelvin–Voigt model is extended to fractal manifolds through local fractal-continuum differential operators. Generalized fractal creep compliance is obtained, taking into account the intrinsic time $\tau$ and the fractal dimension of time-scale $\beta$. The model obtained is validated with experimental data obtained for resin samples with the fractal structure of a Sierpinski carpet and experimental data on rock salt. Comparisons of the model predictions with the experimental data are presented as the curves of slow continuous deformations.

1. Introduction

Fractal geometry deals with bodies whose geometrical shapes are very complicated [1], where their zigzag trajectories or discontinuities are difficult to describe, as they do not conform to the notions of integrability and differentiability in the conventional sense [2]: for example, coastlines, mountains, clouds, broccoli, and so on.

These bodies were defined by Mandelbrot [3] as fractals, which almost always have a fractional Hausdorff dimension that exceeds its topological dimension [4].

The Hausdorff dimension can be defined as a measure of the complexity or roughness of these kind of shapes and can be treated as the degree to which a set embedded in the Euclidean space $E^n$ fills it. The mathematical definition of a Hausdorff dimension was introduced by Besicovitch based on the works of Felix Hausdorff and Carathódory [5]; however, it is difficult to determine in real-life fractals. Therefore, in practice, the box-counting dimension [4], which is a numerical approximation of the Hausdorff dimension, is used. Some practical implications of the Hausdorff dimension of fractals in civil, hydrology and biomechanical engineering can be found in [6,7,8,9].

However, to characterize the fractal topology of a set and take into account its ramification, connectivity, lacunarity, and dynamic properties, it is necessary to consider some other fractal dimensional numbers [10]. It has been argued that the main topological features can be quantified by the following six independent dimensional numbers [11]: the topological dimension $d_t$, the Hausdorff dimension $d_H$, the topological Hausdorff dimension $d_{tH}$ [12], the chemical dimension $d_{ch}$ [13], the spectral dimension $d_s$ [14], the Hausdorff dimension of random walk $d_W$ [15], the shortest (minimal) path dimension $d_{min}$ between two randomly chosen points on a fractal [16], and the spatial degree of freedom dimension $\nu$ [17].

Different physical phenomena were studied in fractal materials, including the diffusion processes in the work of Golmankhaneh and Ochoa-Ontiveros [18] and fluid flow in [19] by Yang et al.; Konas et al. [20] investigated the correlation between the fractal dimension of the wood anatomy structure and impact energy; and Pan et al. [21] developed a fractal prediction model of dry friction of rough surfaces, to mention a few. In these studies, different fractional operators have been proposed [22,23,24] and efficiently used [25] to describe in detail many physical phenomena.

In this paper, we study the creep behavior, which is defined as the slow continuous deformation of a material under constant stress under the assumption that it can be well described when the set of dimensional numbers of the domain under study is determined.

Creep is a special issue of the time-dependent behavior of materials [26], which occurs under a constant stress [27]. It has become of great importance, especially for polymers. One of the most popular models to describe it is the conventional Kelvin–Voigt creep equation defined as [28]

$$\begin{array}{c}\hfill {\nabla }_t\epsilon +{\eta }^{-1}E\epsilon ={\eta }^{-1}\sigma ,\end{array}$$

(1)

where $\epsilon$ is the strain, $\sigma$ is the magnitude of applied stress, E represents the elastic modulus of the material and $\eta$ is the coefficient of viscosity. Then, the creep strain can be obtained as

$$\begin{array}{c}\hfill \epsilon \left(t\right)={\displaystyle \frac{\sigma }{E}}\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left(-{\displaystyle \frac{Et}{\eta }}\right)\right].\end{array}$$

(2)

Different experimental studies (Rieth [29], Yang et al. [30], Farfan-Cabrera and Pascual-Francisco [31]) and numerical works [32,33] have been carried out on Euclidean samples to validate the Kelvin–Voigt model.

Moreover, the creep properties of a viscoelastic 3D-printed sample with a Sierpinski carpet structure have been experimentally studied by Pascual-Francisco et al. [34] utilizing standard calculus on samples with fractal geometry. It was found that the creep Poisson’s ratio decreases in the fractal samples, and their stiffness is reduced.

On the other hand, Zhang et al. [35], Ribeiro et al. [36] and Bouras and Vrcelj [37] proposed different extended creep models using fractional calculus. Likewise, Garra et al. [38] developed a new generic Lomnitz logarithmic creep law via Hadamard fractional calculus. Lastly, fractal approaches to characterize creep modulus and compliance were introduced by Cai et al. [39], Wang et al. [40] and Yin et al. [41] from the models of Hausdorff derivative proposed by Chen [42].

In the early years of this century, Carpinteri et al. [43,44] suggested the well-celebrated Carpinteri’s fractal consisting of the Cartesian product of a Cantor set and the Sierpinski carpet. It is an alternative model to describe the mechanical behavior in a virtual fractal continuum. After that, the concept of a fractal continuum using the approximation of non-differentiable functions defined on fractals by differentiable analytic envelopes was introduced by Tarasov [45]. Moreover, local and non-local methodologies of fractional calculus in continuum mechanics have been formulated by several authors: Carpinteri [46,47], Drapaca [48], Ostoja-Starzewski [49,50], Balankin [51,52,53], Sumelka [54], Tarasov [55,56], and Lazopoulos [57], to name a few. In particular, in this paper, the model introduced by Balankin and Elizarraraz [51,52], called fractal continuum calculus, $F^{\gamma }$-CC is applied. $F^{\gamma }$-CC takes into account the fractal geometry and the fractal topology through the different numbers of fractal dimensions or specific relations between them, which provide additional details in the description of physical phenomena that occur in the fractal under study.

Methodologies that use only the Hausdorff dimension sometimes can lead to very different solutions of the same problem for physical fractal domains. Considering additional fractal dimensional numbers (as in $F^{\gamma }$-CC), it is possible to avoid this risk, since in this case, the topological, metrological, morphological and topographic properties are taken into account. In Figure 1, an example of fractals having the same value of Hausdorff dimension with different connectivity, ramification and degrees of freedom is presented.

A generalized formulation of the Kelvin–Voigt creep equation from ordinary to fractal calculus to depict the creep behavior in viscoelastic materials is developed in this work. In the analysis, six basic fractal dimensional numbers of fractal domain are determined so that the developed formulation includes its fractal topology.

The goodness and effectiveness of the suggested fractal model are demonstrated through a series of experimental creep tests carried out on samples with a fractal structure similar to the Sierpinski carpet, whose fractal dimensional numbers are known.

The manuscript is organized as follows: An overview of fractal continuum calculus is provided in Section 2. A fractal creep model developed using fractal continuum calculus is presented in Section 3. Section 4 is devoted to validating the fractal formulation obtained, comparing it with the experimental results. Section 5 presents the analysis and discussion of results. The paper finished with the conclusions in Section 6.

2. Fractal Continuum Calculus

A fractal set is not completely characterized by its Hausdorff (or box-counting) dimension. A more complete description of a fractal can be obtained using additional fractal dimensional numbers (see

Table 1. Fractal dimensional numbers and scaling exponents.

	Parameter and Symbol		Definition
$\mathrm{D}\mathrm{i}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{a}\mathrm{l} \mathrm{n}\mathrm{u}\mathrm{m}\mathrm{b}\mathrm{e}\mathrm{r}\mathrm{s}$	Box-counting dimension	$d_B$	Box-counting dimension is defined via the scaling relation $N_{\ell }$∼${\ell }^{d_B}$, where $N_{\ell }$ is the number of n-dimensional boxes of size ℓ needed to cover the fractal object [4]
	Topological Hausdorff dimension	$d_{tH}$	$d_{tH}=\mathrm{m}\mathrm{i}\mathrm{n}\{s:\exists X\subseteq \mathrm{\Omega } \mathrm{s}\mathrm{u}\mathrm{c}\mathrm{h} \mathrm{t}\mathrm{h}\mathrm{a}\mathrm{t} d_B\left(X\right)\le s-1 \mathrm{a}\mathrm{n}\mathrm{d}d(\mathrm{\Omega }\setminus X)\le 0\}$. This dimension defines the ramification of a fractal [12].
	Chemical dimension	$d_{ch}$	$d_{ch}={lim}_{\ell \to 0}ln{N}_{\ell }/ln\ell$, where $N_{\ell }(\ell )$ is the number of points connected with an arbitrary point inside of the $d_{\ell }$-dimensional ball of radius ℓ around this point. This parameter indicate the fractal connectivity [13].
	Hausdorff dimension of the minimal path	$d_{min}$	The Hausdorff dimension of the minimum path is defined via the scaling relation $\langle {\ell }_{min}\rangle \propto r^{d_{min}}$, where ${\ell }_{min}$ is the shortest distance between two randomly chosen points on the network, while r is the Euclidean distance between these points and $\langle \dots \rangle$ denotes the ensemble average [13].
	Number of spatial degrees of freedom	$n_{\nu }$	The number of independent directions in which a walker can move without violating any constraint imposed on it, which represents a topological feature [17].
	Number of dynamical degrees of freedom	$d_s$	The number of effective dynamical degrees of freedom is equal to the spectral dimension, defined as $\mathrm{\Omega }\left(\omega \right)\sim {\omega }^{d_s-1}$, where $\mathrm{\Omega }\left(\omega \right)$ is the density of vibrations modes of the fractal and $\omega$ is the frequency. This implies a harmonic metric [11].

) that characterize the ramification, connectivity, harmonical and topological properties of matter.

Although the standard and fractional calculus remain suitable in many cases, fractal calculus provides a specialized mathematical framework developed for equations with fractal solutions.

In this regard, the fractal continuum calculus is an approach designed to describe the behavior of heterogeneous materials with fractal properties, which consists of the mapping of a fractal body from integer coordinates $x_k\in E^3$ to the fractional coordinates ${\zeta }_k\in F^{\gamma }$ as it is geometrically illustrated in Figure 2. If the fractal domain is path connected, it implies that $d_{ch}=d_H<3$, and the number of mutually orthogonal fractional coordinates is determined by the integer part of $d_{ch}$ in the same way as the topological dimension d determines the number of mutually orthogonal Cartesian coordinates in a d-dimensional domain.

In the $F^{\gamma }$-CC, the infinitesimal volume element is defined as $d{V}_{d_H}=d{\zeta }_k\left(x_k\right)d{A}^{\left(k\right)}=c_1^{\left(k\right)}\left(x_k\right)c_2^{\left(k\right)}d{x}_{k}d{A}_2^{\left(k\right)}=c_3\left(x_k\right)d{V}_3=c_{3}d{x}_{1}d{x}_{2}d{x}_3$, where $d{A}^{\left(k\right)}=d{x}_i·d{x}_j$ and $d{A}^{\left(k\right)}$ are the infinitesimal area elements on the intersection between $F^{\gamma }$ and a two-dimensional plane normal to the k-axis and $F^{\gamma }\in E^3$, respectively, $c_2^{\left(k\right)}\left(x_{j\ne k}\right)$ is the density of admissible states in the plane of this intersection, and $c_3=c_1^{\left(k\right)}{c}_2^{\left(k\right)}$.

Then, the measure in $F^{\gamma }$ is given by $\int d{V}_3{c}_3\sim {\ell }^{3-d_H}{L}^{d_H}$, $\int d{A}_2^{\left(k\right)}{c}_2^{\left(k\right)}\sim {\ell }^{1-d_A^{\left(k\right)}}{L}^{d_A^{\left(k\right)}}$, and $\int d{\zeta }_k=\int d{x}_k{c}_1^{\left(k\right)}\sim {\ell }^{1-{\alpha }_k}{L}^{{\alpha }_k}$ [58].

The fractal continuum norm is given by $\Vert A\Vert ={\left[{\sum }_k^3{\zeta }_k^{2\gamma }\right]}^{1/2\gamma }$, where $\gamma =d_{ch}/3\le 1$, and the mapping of the integer coordinates $x_k\in E^3$ to the fractal coordinates ${\zeta }_k\in F^{\gamma }$ is defined as

$$\begin{array}{c}\hfill \zeta =\ell {\left({\displaystyle \frac{x}{\ell }}+1\right)}^{\alpha },\end{array}$$

(3)

here, ℓ is the lower cutoff of fractality, $\alpha =d_H-d_A$ denotes the Hausdorff dimension in each fractal direction ${\zeta }_k\in E^3$ of order $0<\alpha <1$, and $d_A$ denotes the Hausdorff dimension of a cross-section of the fractal body (see the integer and fractal configurations sketched in Figure 2). The distance between two points $A,B\in F^{\gamma }$ is given by $\mathrm{\Delta }(A,B)={\left[{\sum }_x^3{\mathrm{\Delta }}_x^{2\gamma }\right]}^{1/2\gamma }$, where ${\mathrm{\Delta }}_x=\Vert {\zeta }_{ax}-{\zeta }_{bx}\Vert$, and the fractal continuum gradient is defined as ${\nabla }^{\alpha }f={\mathbf{e}}_x{\nabla }_x^{\alpha }f$, where ${\mathbf{e}}_x$ are basis vectors and

$$\begin{array}{c}\hfill {\nabla }_x^{\alpha }f=\underset{\zeta \to {\zeta }^{\prime }}{lim}{\displaystyle \frac{f\left({\zeta }^{\prime }\right)-f\left(\zeta \right)}{{\zeta }^{\prime }-\zeta }}=\underset{x\to x^{\prime }}{lim}{\displaystyle \frac{f\left(x^{\prime }\right)-f\left(x\right)}{\mathrm{\Delta }\left(x^{\prime },x\right)}}={\displaystyle \frac{1}{\alpha }}{\left({\displaystyle \frac{x}{\ell }}+1\right)}^{1-\alpha }{\nabla }_{x}f,\end{array}$$

(4)

is the spatial fractal continuum derivative. The divergence operator is ${\nabla }^{\alpha }·\overrightarrow{F}$ and the fractal continuum Laplacian in the integer space is defined as ${\mathrm{\Delta }}^{\alpha }f={\nabla }^{\alpha }·{\nabla }^{\alpha }f={\sum }_x^3{\left(c_1^{\left(x\right)}\right)}^{-2}$$\left[{\nabla }_x^2+{\displaystyle \frac{\gamma -\alpha }{x}}{\nabla }_x\right]f$ with $c_1^{\left(x\right)}=\alpha {\ell }^{1-\alpha }{x}^{\alpha -1}$.

On the other hand, the temporal fractal continuum derivative defined in the $F^{\gamma }$-CC approach is suggested by Balankin and Elizarraraz [52] as

$$\begin{array}{c}\hfill {\nabla }_t^{\beta }f=\underset{{\zeta }_{\beta }\to {\zeta }_{\beta }^{\prime }}{lim}{\displaystyle \frac{f\left({\zeta }_{\beta }^{\prime }\right)-f\left({\zeta }_{\beta }\right)}{{\zeta }_{\beta }^{\prime }-{\zeta }_{\beta }}}=\underset{t\to t^{\prime }}{lim}{\displaystyle \frac{f\left(t^{\prime }\right)-f\left(t\right)}{\mathrm{\Delta }\left(t^{\prime },t\right)}}={\left({\displaystyle \frac{t}{\tau }}+1\right)}^{1-\beta }{\nabla }_{t}f,\end{array}$$

(5)

where $\tau$ is the intrinsic time and $\beta$ is the intrinsic time exponent. In reference [17], the following relation was derived:

$$\begin{array}{c}\hfill \beta =d_s/n_{\nu },\end{array}$$

(6)

where $n_{\nu }=2d_{ch}-d_s$ is the number of effective spatial degrees of freedom of a random walker on the studied fractal.

The derivatives defined above can be calculated in the conventional sense and employed to generalize equations and formulations of material scale invariants from conventional to fractal calculus. This methodology was successfully utilized in several areas of science and technology by Balankin et al. [59], Balankin [60], Samayoa et al. [61,62,63] and Damián-Adame et al. [64].

However, there are several methodologies of fractal continua with different fractional differential operators whose functions are differentiable in a standard sense. So, the non-standard local derivatives in the fractal continuum framework can be expressed in terms of standard derivatives. In

Table 2. Local fractional derivatives in fractal continuum approach.

Definition	Equation	Derivative
$D^{\alpha }f=\underset{x\to x^{\prime }}{lim}{D}_x^{\alpha }\left[f\left(x^{\prime }\right)-f\left(x\right)\right]$	$D^{\alpha }f={\displaystyle \frac{d^{n}f}{d{x}^n}}$	Kolwankar–Gangal [65]
${\displaystyle \frac{d^{H}f}{d{x}^{\alpha }}}=\underset{x\to x^{\prime }}{lim}{\displaystyle \frac{f\left(x\right)-f\left(x^{\prime }\right)}{{\left(x\right)}^{\alpha }-{\left(x^{\prime }\right)}^{\alpha }}}$	${\displaystyle \frac{d^{H}f}{d{x}^{\alpha }}}={\displaystyle \frac{x^{1-\alpha }}{\alpha }}{\displaystyle \frac{df}{dx}}$	Hausdorff [42]
${\nabla }^{H}f={\displaystyle \frac{df}{d\zeta }}=\underset{x\to x^{\prime }}{lim}{\displaystyle \frac{f\left(x\right)-f\left(x^{\prime }\right)}{\zeta \left(x\right)-\zeta \left(x^{\prime }\right)}}$	${\displaystyle \frac{df}{d\zeta }}={\displaystyle \frac{1}{\alpha }}{\left(1+{\displaystyle \frac{x}{\ell }}\right)}^{1-\alpha }{\displaystyle \frac{df}{dx}}$	Balankin (Equations (4) and (5)) [51]
${\displaystyle \frac{df}{d_{S}x}}=\underset{x\to x^{\prime }}{lim}{\displaystyle \frac{f\left(x\right)-f\left(x^{\prime }\right)}{k\left(x\right)-k\left(x^{\prime }\right)}}$	${\displaystyle \frac{df}{d_{S}x}}={\left({\displaystyle \frac{dk}{dx}}\right)}^{-1}{\displaystyle \frac{df}{dx}}$	Structural [66]
$D^{\alpha }\left[f\left(x\right)\right]=\underset{ϵ\to 0}{lim}{\displaystyle \frac{f\left(x{e}^{ϵt^{-\alpha }}\right)-f\left(x\right)}{ϵ}}$	$D^{\alpha }f=x^{1-\alpha }{\displaystyle \frac{df}{dx}}$	Deformed [67]

, some examples of alternative definitions of local fractional derivatives within the fractal continuum approach are shown.

3. Fractal Continuum Kelvin–Voigt Creep Model

Constitutive equation of Kelvin–Voigt creep in the mechanics of fractal continuum are obtained under the analogous assumptions like those obtained in conventional calculus (for details see [28]), where the fractal Kelvin–Voigt model can be expressed in terms of the temporal fractal continuum derivative using Equation (5), so we have

$$\begin{array}{c}\hfill {\nabla }_t^{\beta }\epsilon +{\eta }^{-1}E\epsilon ={\eta }^{-1}\sigma ,\end{array}$$

(7)

and the fractal Kelvin–Voigt model can be written according to the standard time derivatives using the proportionality constant ${(t/\tau +1)}^{1-\beta }$ as follows:

$$\begin{array}{c}\hfill {\left({\displaystyle \frac{t}{\tau }}+1\right)}^{1-\beta }{\nabla }_t\epsilon +{\eta }^{-1}E\epsilon ={\eta }^{-1}\sigma ,\end{array}$$

(8)

so, the creep strain is given by

$$\begin{array}{c}\hfill \epsilon \left(t\right)={\displaystyle \frac{\sigma }{E}}\left[1-C\left({\displaystyle \frac{E\tau }{\eta \beta }}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left\{-\left({\displaystyle \frac{E\tau }{\eta \beta }}\right){\left({\displaystyle \frac{t}{\tau }}+1\right)}^{\beta }\right\}\right],\end{array}$$

(9)

where C is a real constant.

Note that Equation (7) reduces to the conventional Kelvin–Voigt creep equation when $\beta =1$. Moreover, Equation (8) coincides with the standard creep strain described by Equation (2) when the constant C is defined as $C=(\eta \beta /E\tau )\mathrm{e}\mathrm{x}\mathrm{p}(E\tau /\eta \beta )$. Therefore, from Equation (8) and the above-mentioned observations, the fractal continuum creep strain can be obtained as

$$\begin{array}{c}\hfill \epsilon \left(t\right)={\displaystyle \frac{\sigma }{E}}\left\{1-\mathrm{e}\mathrm{x}\mathrm{p}\left\{-{\displaystyle \frac{E\tau }{\eta \beta }}\left[{\left({\displaystyle \frac{t}{\tau }}+1\right)}^{\beta }-1\right]\right\}\right\},\end{array}$$

(10)

and the creep compliance of the fractal continuum Kelvin–Voigt model can be obtained as

$$\begin{array}{c}\hfill J\left(t\right)={\displaystyle \frac{\epsilon \left(t\right)}{\sigma }}.\end{array}$$

(11)

A mechanical picture of this relationship is presented in Figure 3, which predicts its behavior for different values of $0<\beta \le 1$.

The main result of this work is Equation (8), which is a generalized equation of the Kelvin–Voigt creep model in the fractal space–time continuum that considers the fractal topology of viscoelastic materials through the $\tau$ (intrinsic time) and $\beta$ (intrinsic time exponent) as well as the fractal continuum creep strain and fractal continuum creep compliance described by Equations (10) and (11), respectively.

4. Experimental Validation of Fractal Continuum Creep Model

The objective of this section is to experimentally validate the fractal continuum Kelvin–Voigt creep equations obtained in the previous section, for samples with the fractal structure of a classical Sierpinski carpet.

4.1. Sierpinski Carpet

The Sierpinski carpet ${\mathcal{S}}^{\ell }$ is a self-similar fractal, and it is a two-dimensional version of the Cantor middle-ℓ set ${\mathcal{C}}^{\ell }\subset [0,1]=L\in \Re$. Then, ${\mathcal{S}}^{\ell }$ can be constructed by an iterative process from unit square ${[0,1]}^2$. The initial square is divided into $\ell \times \ell$ sub-squares of equal size, and the interior of $B^2$ sub-squares are removed. Iterating this process ad infinitum, ${\mathcal{S}}^{\ell }\in {\Re }^2$ is obtained, and its Hausdorff dimension is defined as [68]

$$\begin{array}{c}\hfill d_H={\displaystyle \frac{log\left[N{(\ell )}^2-B^2\right]}{logN(\ell )}},\end{array}$$

(12)

where $N(\ell )$ is the number of boxes covering the fractal mass of the Sierpinski carpet, and $B^2$ is the number of deleted boxes of the fractal mass. Note that $\ell \times \ell$ is the size of the sub-squares, which are eliminated in the k-th iteration of the corresponding Sierpinski carpet.

The connectivity of a Sierpinki carpet is determined by its chemical dimension, which can be obtained by the scaling law $N(L/{\ell }_{ch})\sim {(L/{\ell }_{ch})}^{d_{ch}}$. In the above-mentioned scaling law, $N(L/{\ell }_{ch})$ denotes the number of $d_{ch}$-dimensional boxes of size ${\ell }_{ch}$ needed to cover the fractal according to the scale invariance principle, where ${\ell }_{ch}$ is measured with respect to the geodesic metric over the Sierpinski carpet. In this context, Cristea [69] proved that for the Sierpinski carpet, the chemical and Hausdorff dimensions are equal ($d_H=d_{ch}$), as its Euclidean and geodesic metrics are equivalent.

On the other hand, the density of Sierpinski carpet vibration modes scales with frequency $\omega$ as $\mathrm{\Omega }\left(\omega \right)\sim {\omega }^{d_s-1}$, where $d_s$ represents the spectral dimension determining scaling properties of the eigenvalues of the Laplacian defined on $S^{\ell }$. Moreover, the spectral dimension of Sierpinki carpet can be computed as [70]

$$\begin{array}{c}\hfill d_s=2{\displaystyle \frac{log\left[N{(\ell )}^2-B^2\right]}{log\left[N{(\ell )}^2+B^2\right]}}.\end{array}$$

(13)

It is well known that for the Sierpinski carpet, its Hausdorff dimension is equal to its chemical dimension ($d_H=d_{ch}$), which implies that its fractal dimension of the shortest path is equal to one as $d_{min}=d_H/d_{ch}$ [15].

In

Table 3. Fractal dimensions characterizing Sierpinski carpet ${\mathcal{S}}^{\ell }$.

Classical (ℓ)	${\mathit{d}}_{\mathit{t}}$	${\mathit{d}}_{\mathit{H}}$	${\mathit{d}}_{\mathbf{ch}}$	${\mathit{d}}_{\mathbf{min}}$	${\mathit{d}}_{\mathit{s}}$	${\mathit{n}}_{\mathit{\nu }}$	$\mathit{\beta }$
${\displaystyle \frac{1}{3}}$	1	${\displaystyle \frac{log8}{log3}}$	${\displaystyle \frac{log8}{log3}}$	1	$1.806$	$1.98$	$0.91$

, the fractal dimensional numbers for a classical Sierpinski carpet are presented; meanwhile, in Figure 4, the experimental set up of samples with Sierpinski carpet configurations is presented.

4.2. Experimental Details

Resione F69 flexible resin [34] samples used for the validation of proposed model were fabricated with fractal geometry extracted from the Sierpinski carpet of the fifth iteration $k=5$ with the mechanical properties presented in

Table 4. Mechanical properties of fractal material.

Tensile Strength	Percent Elongation at Break	E	Nominal Strain at Break
4.87 ± 0.6 MPa	$175\pm 9$	23.53 MPa	$182\pm 8$

. At least 10 samples were used in the experimental test.

All samples analyzed had the shape of rectangular cuboids of thickness $w=1\pm 0.12$ mm with length $L=100\pm 0.12$ mm along the $x_1$-axis and a height of $h=15\pm 0.12$ mm in the $x_3$-axis direction. In Figure 4b, it can be seen that the designed sample is part of the fifth iteration of the basic Sierpinski carpet, constructed with 5 squares, and each one of them is of the third iteration. By the self-similarity properties, the fractal samples have the same Hausdorff dimension as the entire Sierpinski carpet.

Figure 5a presents the graph of the experimental values of $\epsilon \left(t\right)$ for the period of time up to 7200 s.

5. Theoretical and Experimental Results

Figure 3 shows curves for fractal continuum creep compliance with several values of $\beta$. It is a straightforward matter to see that the value of $J\left(t\right)$ is becoming lower as the fractal dimension of the time scale ($\beta$) decreases; however, all fractal creep compliances in Figure 5b converge for a large period of time ($t>15\times {10}^5$ s). Note that when $\beta =1$, the creep compliance of the fractal continuum Kelvin–Voigt model reduces to standard form. The graph inset in Figure 3 displays a close up of the creep behavior for values of $\beta$ whose curves match with the results reported in Figure 3 in [39].

In Figure 5a, the fractal creep model prediction obtained from Equation (10) with $\beta =d_s/\nu =0.91$ (see Table 3) is presented, and it is in a very good agreement with the experimentally obtained data from the samples with the classic Sierpinski carpet structure.

In addition, theoretical analysis with the model suggested in this work was carried out for the experimental data of creep behavior in rock salt published by Cristescu [71] in order to show its efficiency and precision with alternative data to our study, which can be observed in Figure 5b. It can also be observed that the parameters p and $\beta$ for both the fractal [39] and fractal continuum Kelvin–Voigt models are very similar. It can be concluded that the parameter $\beta$ is an inherent property related to the fractal features of the material under study, as the order of the fractal continuum derivative is linked to Hausdorff, spectral and chemical dimensions as well as to the degrees of freedom of the fractal object.

On the other hand, it was found that the fractal continuum creep strain rate

$$\begin{array}{c}\hfill {\epsilon }^{\prime }\left(t\right)={\displaystyle \frac{\sigma }{\eta }}{\left({\displaystyle \frac{t}{\tau }}+1\right)}^{\beta -1}\mathrm{e}\mathrm{x}\mathrm{p}\left\{-{\displaystyle \frac{E\tau }{\eta \beta }}\left[{\left({\displaystyle \frac{t}{\tau }}+1\right)}^{\beta }-1\right]\right\},\end{array}$$

(14)

for different applied stresses has the same slope. A linear regression analysis employing the least squares method shows that ${\epsilon }^{\prime }\left(t\right)\sim e^{mt}$ with $m\approx -1/2$. Meanwhile, that ${\epsilon }^{\prime }\left(t\right)$ increases linearly as $\sigma$ increases (see Figure 6 for $\beta =0.953$ data). This behavior is consistent with the experimental and numerical results reported in Rieth [29] and [72], respectively.

In Figure 7, a graph of ${\epsilon }^{\prime }\left(t\right)$ as a function of $\beta$ is presented to study the influence of fractality on the fractal continuum creep strain rate along specific times. There is the initial instantaneous strain in Figure 7 that implies an increment in ${\epsilon }^{\prime }\left(t\right)$; this phenomenon was extensively studied by Zhang et al. [35]. However, for $t>2$ s, the creep strain rate decreases as the order of $\beta$ increases. The last means that when the fractal material under study has more mass characterized by its $d_H$, then its ${\epsilon }^{\prime }\left(t\right)$ decreases, as it is observed in Figure 7.

The obtained model has a resemblance to the results acquired in models that incorporate fractional geometry. However, it includes the topology and fractal geometry through the different Hausdorff dimensions of the fractal materials under study. Moreover, it captures the material’s heterogeneity using the alpha parameter (which depends on its fractal geometry and topology) and its dynamical characteristics through the spectral dimension.

Consequently, the engineering implications of a generalized fractal continuum Kelvin–Voigt creep equation that extends the standard equation to fractal manifolds is able to describe the creep phenomena in viscoelastic materials that have non-classical characteristics, possess scale invariance and cannot be described using conventional calculus. It is worth noting that the proposed model contains the fractal parameter $\beta$, which is an additional parameter not included in the preceding fractional models.

6. Conclusions

A fractal generalization of the standard Kelvin–Voigt creep model is obtained using the fractal continuum approach.

The concept of a fractal continuum Kelvin–Voigt creep solution is established, where the discontinuities in fractal domains are mapped to the continua domains in the fractal continuum space–time using local fractional differential operators and proportionality constants given in Equations (3) and (5) such that the creep phenomena can be described in complex and heterogeneous domains.

The slow continuous deformation given by Equation (10) is linked with the topology and fractal geometry of viscoelastic fractal materials with the scaling exponent $\beta \in (0,1]$, which depends on the Hausdorff dimension of the self-similar model of viscoelastic material and its chemical dimension. In the special case of $\beta =1$, Equation (10) reduces to Equation (2), as it is shown in Figure 3.

The creep strain rate for the fractal continuum case is presented in Figure 6, where its behavior is shown. It was observed that if the order of $\beta$ increases, the ${\epsilon }^{\prime }\left(t\right)$ decreases. The effects of the fractal geometry and topology of domain under study are obtained introducing the parameter $\beta$, which depends on the Hausdorff, chemical and spectral dimensions.

The developed fractal creep model provides a good prediction for the creep strain behavior in viscoelastic materials with a self-similar structure. The model obtained is a generalized version of the standard Kelvin–Voigt creep model, which represents a particular case when $\beta =1$.