Duality Revelation and Operator-Based Method in Viscoelastic Problems

Abstract

Viscoelastic materials are commonly used in civil engineering, biomedical sciences, and polymers, where understanding their creep and relaxation behaviors is essential for predicting long-term performance. This paper introduces an operator-based method for modeling viscoelastic materials, providing a unified framework to describe both creep and relaxation functions. The method utilizes stiffness and compliance operators, offering a systematic approach for analyzing viscoelastic problems. The operator-based method enhances the mathematical duality between the creep and relaxation functions, providing greater physical intuition and understanding of time-dependent material behavior. It directly reflects the intrinsic properties of materials, independent of input and output conditions. The method is extended to dynamic problems, with complex modulus and compliance derived through operator representations. The fractal tree model, with its constant loss factor across the frequency spectrum, demonstrates potential engineering applications. By incorporating a damage-based variable coefficient, the model now also accounts for the accelerated creep phase of rocks, capturing damage evolution under prolonged loading. While promising, the current method is limited to one-dimensional problems, and future research will aim to extend it to three-dimensional cases, integrate experimental validation, and explore broader applications.

1. Introduction

Viscoelastic materials are a class of functional materials with significant time-dependent mechanical properties, widely applied in civil engineering [1,2], biomedical fields [3,4], polymer materials [5,6], and energy [7] industries. For example, the creep behavior of rocks and concrete significantly affects the long-term safety of structures [8], while the relaxation properties of polymers directly relate to their energy storage and dissipation capabilities [9]. Accurately describing the mechanical response of viscoelastic materials in processes such as creep and relaxation is not only of great significance for engineering design but also serves as an important theoretical foundation for revealing the intrinsic properties of materials.

The development process, from the earliest mechanical models of fluids and solids by Newton and Hooke to the modern viscoelastic models, provides important insights. The expansion of research objects from simple models of Newtonian fluids and elastic solids to more complex viscoelastic models has mainly followed two important paths. The first path began with the exploration of physical models, where differential equations describing the stress–strain relationship were constructed by combining springs and dashpots, such as in the Maxwell and Kelvin–Voigt models in classical linear viscoelastic theory [10]. The second path focused on the innovation of mathematical models. Through the development of new mathematical forms for constitutive equations, the Scott–Blair model [11] first introduced fractional calculus, which effectively describes material behavior at different time scales. This is especially effective in the creep modeling of rocks, polymers, and composite materials [12]. Fractional calculus, as an extension of classical calculus, provides a powerful tool for viscoelastic material modeling [13,14,15], but fractional models face several challenges: their mathematical form is complex [16] and difficult to apply directly in engineering practice; additionally, these models often lack a direct connection with the intrinsic properties of materials, making it difficult to reveal the underlying physical mechanisms [17]. Despite these differences in approach, both methods reflect an organic combination of mathematics and mechanics, ultimately leading to significant and innovative outcomes.

From the perspective of rheology, an important advancement in the field is the discovery that constitutive relations with fractional derivatives can be described using fractal structures [18,19,20,21]. Unlike classical viscoelastic models, fractal structures are modeled using self-similar structures composed of countless combinations of springs and dashpots. The fractional-order constitutive relations embedded in these structures can be solved through operations on the complex modulus in the frequency domain [22] or derived via Heaviside operations in the time domain [18,19]. This work is not only of significant theoretical value but also provides a bridge between the fields of fractals and fractional derivatives [15,23,24].

To overcome the limitations of classical and fractional-order models, there is an urgent need for a new theoretical framework that can uniformly describe both the creep and relaxation behaviors of viscoelastic materials. This study presents an operator-based methodology to establish a novel framework for abstracting the mechanical responses of materials into a systematic operator representation. By defining the stiffness operator and compliance operator, it unifies the description of the creep and relaxation functions. The advantage of the operator-based method lies in its ability to not only directly reflect the intrinsic properties of materials but also to provide a clear physical intuition and a higher degree of mathematical duality, offering a fresh perspective for modeling viscoelastic problems.

The remainder of this paper is structured as follows: Section 2 reviews the fundamentals of classical linear viscoelastic theory; Section 3 introduces the operator-based method and defines the mathematical expressions for the stiffness and compliance operators; Section 4 extends the operator-based method to dynamic mechanical analysis; Section 5 explores the method for viscoelastic problems with variable-coefficient creep functions; and Section 6 summarizes the theoretical contributions of the operator-based method. Additionally, Appendix A compares the classical differential equation approach with the operator-based method. Appendix B presents the solution process for a fractional-order viscoelastic model via an operator-based method.

2. Classical Linear Viscoelastic Theory

2.1. Creep Function and Relaxation Function

Linear viscoelastic materials exhibit time-dependent mechanical behavior, combining characteristics of both elastic solids and viscous fluids. These materials obey the superposition principle, which allows for the linear mapping of stress and strain histories. The creep and relaxation functions are fundamental to understanding this time-dependent behavior under simple loading conditions.

Under the stress $\sigma \left(t\right)={\sigma }_{0}H\left(t\right)$ applied to a linear viscoelastic material, the time-dependent strain response is expressed as follows:

$$\epsilon (t)={\sigma }_{0}J(t),$$

(1)

where $J\left(t\right)$ is referred to as the creep function, also known as the creep compliance. $H\left(t\right)$ is the Heaviside function, also known as the unit step function, and is defined as follows:

$$H(t)=\left\{\begin{array}{lll}1\hfill & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em} t⩾0\hfill & \hfill \\ 0\hfill & \hspace{0.33em}\hspace{0.33em}\hspace{0.33em} t<0\hfill & \hfill \end{array}\right..$$

(2)

The creep function represents the strain value that changes over time under a unit stress, and is generally a non-decreasing function of time. When studying stress relaxation of materials, the stress response to a strain $\epsilon \left(t\right)={\epsilon }_{0}H\left(t\right)$ is expressed as follows:

$$\sigma (t)={\epsilon }_{0}E(t),$$

(3)

where $E\left(t\right)$ is referred to as the relaxation function, also known as the relaxation modulus. It represents the stress response under a unit strain and is generally a non-increasing function of time.

These material functions can be extended to arbitrary stress or strain histories using the Boltzmann superposition principle. For a general stress history, the resulting strain is given by the hereditary integral:

$$\epsilon (t)={\displaystyle {\int }_0^{t}J}(t-\tau ){\displaystyle \frac{d\sigma (\tau )}{d\tau }}d\tau .$$

(4)

Similarly, the stress response to an arbitrary strain history is expressed as follows:

$$\sigma (t)={\displaystyle {\int }_0^{t}E}(t-\tau ){\displaystyle \frac{d\epsilon (\tau )}{d\tau }}d\tau .$$

(5)

These integral formulations inherently capture the memory effects in viscoelastic materials, where the current state depends on the entire loading history.

2.2. General Differential Constitutive Models

By connecting any number of springs and dashpots in series or parallel, a general one-dimensional linear viscoelastic differential constitutive equation can be obtained:

$$\sum _{k=0}^m{p}_k{\displaystyle \frac{{\mathrm{d}}^k\sigma }{\mathrm{d}{t}^k}}=\sum _{k=0}^n{q}_k{\displaystyle \frac{{\mathrm{d}}^k\epsilon }{\mathrm{d}{t}^k}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} n,m\in {\mathrm{Z}}^{+},$$

(6)

where $p_k$ and $q_k$ are constants determined by the material properties, with $p_0=1$ typically assumed. The constitutive relations for the basic elements, simple models, and general models are special cases of Equation (6). For example: If only the first terms on both sides of Equation (6) are taken, it represents the stress–strain relationship of a spring; if the first two terms are taken on both sides, it corresponds to the constitutive equation of a three-parameter solid; if the first three terms are taken, with $p_0=1$ and $q_0=0$, it gives the constitutive equation of the Burgers fluid model.

To discuss the material functions of general constitutive models, the Laplace transform (LT) method can be applied. By performing the LT on Equation (6) and considering the initial conditions at $t=0^{-}$ where stress, strain, and their higher derivatives are zero (i.e., satisfying the smoothness assumption), the resulting algebraic equation is expressed as follows:

$$\sum _{k=0}^m{p}_k{s}^k\overline{\sigma }(s)=\sum _{k=0}^n{q}_k{s}^k\overline{\epsilon }(s)$$

(7)

or

$$\overline{P}(s)\overline{\sigma }(s)=\overline{Q}(s)\overline{\epsilon }(s),$$

(8)

where $s$ is the transform variable; $p_k$ and $q_k$ are the same as those in Equation (6), determined by the material properties and independent of the stress and strain values. $\overline{\mathbf{P}}\left(s\right)$ and $\overline{\mathbf{Q}}\left(s\right)$ are polynomials in s

$$\overline{P}(s)=\sum _{k=0}^m{p}_k{s}^k;\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} \overline{Q}(s)=\sum _{k=0}^n{q}_k{s}^k.$$

(9)

To obtain the creep function, substitute $\sigma \left(t\right)={\sigma }_{0}H\left(t\right)$ into Equation (8) and consider the definition of the creep function from Equation (1), yielding:

$$\overline{\epsilon }(s)={\displaystyle \frac{\overline{P}(s)}{\overline{Q}(s)}}{\displaystyle \frac{{\sigma }_0}{s}}=\overline{J}(s){\sigma }_0,$$

(10)

where $\overline{J}\left(s\right)$ is the LT of the creep function:

$$\overline{J}(s)={\displaystyle \frac{\overline{P}(s)}{s\overline{Q}(s)}}.$$

(11)

The creep function can be obtained by performing the inverse Laplace transform (ILT):

$$J(t)={\mathcal{L}}^{-1}\left[\overline{J}(s)\right]={\mathcal{L}}^{-1}{\displaystyle \frac{\overline{P}(s)}{s\overline{Q}(s)}}.$$

(12)

Similarly, to obtain the relaxation function when $\epsilon \left(t\right)={\epsilon }_{0}H\left(t\right)$, combining with the definition of the relaxation function from Equation (3), we obtain the following:

$$\overline{\sigma }(s)={\displaystyle \frac{\overline{Q}(s)}{\overline{P}(s)}}{\displaystyle \frac{{\epsilon }_0}{s}}=\overline{E}(s){\epsilon }_0,$$

(13)

where $\overline{E}\left(s\right)$ is the LT of the relaxation function:

$$\overline{E}(s)={\displaystyle \frac{\overline{Q}(s)}{s\overline{P}(s)}}.$$

(14)

The relaxation function is obtained by performing the ILT:

$$Y(t)={\mathcal{L}}^{-1}\left[\begin{array}{c}\overline{E}(s)\end{array}\right]={\mathcal{L}}^{-1}{\displaystyle \frac{\overline{Q}(s)}{s\overline{P}(s)}}.$$

(15)

From Equations (11) and (13), it is clear that the creep function and the relaxation function of the material have a well-defined mathematical relationship in the Laplace space:

$$\overline{J}(s)\overline{E}(s)={\displaystyle \frac{1}{s^2}}.$$

(16)

Although constant stress and constant strain are the two most critical input conditions in viscoelastic problems, the creep function and the relaxation function derived from these inputs are important material functions. These functions not only reflect the intrinsic properties of the material but are also influenced by external input conditions. It is worth noting that these two material functions are not strictly dual in a mathematical sense, and their relationship can only be accurately represented in the Laplace space. In the following, this paper will propose a new research method that relies solely on the intrinsic properties of the material—the stiffness operator and compliance operator method—where a more perfect mathematical duality exists between the two.

3. Operator-Based Method to Viscoelastic Quasi-Static Mechanical Analysis

3.1. Operator-Based Method Based on LT

Yu et al. [25] introduced the concepts of the stiffness operator and compliance operator, but their definitions were not clearly articulated, and the fundamental significance of the operator-based method itself was not fully recognized. This section begins by deriving the definitions of the stiffness operator and compliance operator from the LT of the general differential-type constitutive equations. By deforming Equation (8), we obtain the following:

$$\mathit{T}(s)={\displaystyle \frac{\overline{\sigma }(s)}{\overline{\epsilon }(s)}}={\displaystyle \frac{\overline{Q}(s)}{\overline{P}(s)}}$$

(17)

or

$${\mathit{T}}^{-1}(s)={\displaystyle \frac{\overline{\epsilon }(s)}{\overline{\sigma }(s)}}={\displaystyle \frac{\overline{P}(s)}{\overline{Q}(s)}},$$

(18)

where $\mathit{T}\left(s\right)$ is the stiffness operator of the material, and ${\mathit{T}}^{-1}\left(s\right)$ is the compliance operator of the material. Therefore, the stress–strain relationship in operator form is expressed as follows:

$$\overline{\sigma }(s)=\mathit{T}(s)\overline{\epsilon }(s)$$

(19)

or

$$\overline{\epsilon }(s)={\mathit{T}}^{-1}(s)\overline{\sigma }(s).$$

(20)

If we consider the stress response of the material under strain input, the ILT of Equation (19) yields:

$$\sigma (t)={\mathcal{L}}^{-1}[\mathit{T}(s)]\ast \epsilon (t),$$

(21)

where * denotes the convolution operation. Specifically, to obtain the relaxation function, we substitute $\epsilon \left(t\right)={\epsilon }_{0}H\left(t\right)$, yielding:

$$E(t)={\int }_{0+}^t{\mathcal{L}}^{-1}[\mathit{T}(s)](\tau )\mathrm{d}\tau .$$

(22)

Similarly, if we consider the strain response under stress input, the ILT of Equation (20) gives:

$$\epsilon (t)={\mathcal{L}}^{-1}[{\mathit{T}}^{-1}(s)]\ast \sigma (t).$$

(23)

In particular, to obtain the creep function, we substitute $\sigma \left(t\right)={\sigma }_{0}H\left(t\right)$, yielding:

$$J(t)={\int }_{0+}^t{\mathcal{L}}^{-1}[{\mathit{T}}^{-1}(s)](\tau )\mathrm{d}\tau .$$

(24)

For general stress or strain input conditions, the corresponding results can be directly calculated using Equations (21) and (23) (as shown in Appendix A). It is important to note that the stiffness operator and the compliance operator are entirely determined by the intrinsic properties of the material and are independent of the input and output conditions. Once the stiffness or compliance operator is obtained, the operator kernel function can be determined through the ILT, thus enabling the derivation of the material’s relaxation function or creep function. By understanding the operators reflecting the system’s intrinsic properties, one can directly derive the material functions under other input conditions.

Clearly, the stiffness operator and the compliance operator satisfy the inverse relationship:

$$\mathit{T}\cdot {\mathit{T}}^{-1}=\mathbf{1}.$$

(25)

It is precisely because the stiffness operator and the compliance operator reflect the intrinsic properties of the material that they form a pair of mutually inverse operators, i.e., they are dual operators. Furthermore, combining Equations (22) and (24), we obtain the following:

$$\dot{J}(t)*\dot{E}(t)=\delta (t),$$

(26)

where $\dot{J}\left(t\right)$ and $\dot{E}\left(t\right)$ are the derivatives of the creep function and the relaxation function, respectively, representing the stress rate and strain relaxation rate, and $\delta \left(t\right)$ is the Dirac delta function. Equation (26) indicates that $\dot{J}\left(t\right)$ and $\dot{E}\left(t\right)$ are a pair of dual functions. The duality between the stress rate $\dot{J}\left(t\right)$ and strain relaxation rate $\dot{E}\left(t\right)$ is a beautifully symmetrical property. Therefore, it can be said that, compared to the creep function $J\left(t\right)$ and the relaxation function $E\left(t\right)$, the stress rate $\dot{J}\left(t\right)$ and strain relaxation rate $\dot{E}\left(t\right)$ exhibit higher symmetry. Equations (25) and (26) form a dual transformation. In other words, the stiffness operator and the compliance operator have dual invariance, and both creep theory and relaxation theory possess dual invariance. Through the dual transformation, one theory can be formally transformed into another.

3.2. Consistency with Operational Calculus

Heaviside [26] introduced the differential operator $\mathit{p}=\mathrm{d}/\mathrm{d}t$ when solving the telegraph equation. In the mid-20th century, Minkusinski [27] laid the foundation for operational calculus by treating functions as algebraic expressions of operators, establishing a complete framework based on algebraic theory. In operational calculus, the differential operator $\mathit{p}$ is defined as follows [18,26]:

$$\mathit{p}f\left(t\right)={\displaystyle \frac{\mathrm{d}f(t)}{\mathrm{d}t}}+f(0).$$

(27)

Equation (27) exactly corresponds to the time-domain differentiation principle of LT analysis. Yu et al. [28] demonstrated through rigorous mathematical analysis that the operator algebra method maintains strict consistency with LT theory, establishing operational calculus as a symbolic system grounded in LT fundamentals and algebraic manipulation. Compared with conventional integral transform methods, operational calculus offers enhanced notational efficiency in two aspects: (1) the symbolic representation avoids convolution integrals through operator composition rules, and (2) the solution procedure emphasizes algebraic manipulations in the operator domain rather than handling coupled spacetime derivatives. Consequently, we will adopt the differential operator $\mathit{p}$ (Heaviside’s operational notation) in subsequent sections, replacing the complex frequency parameter $s$ previously used in Section 3.1.

Building upon the axiomatic framework of operational calculus, the constitutive relationship for arbitrary linear viscoelastic elements admits a unified representation through the canonical stiffness operator $\mathit{T}\left(\mathit{p}\right)$ as:

$$\sigma \left(t\right)=\mathit{T}\left(\mathit{p}\right)\epsilon \left(t\right).$$

(28)

This is structurally identical to Equation (17). For a spring, $\mathit{T}$ is written as:

$$\mathit{T}=E.$$

(29)

For a dashpot, $\mathit{T}$ is written as:

$$\mathit{T}=\eta \mathit{p}.$$

(30)

For more general differential-type viscoelastic constitutive relations, the stiffness operator has two equivalent representations: one is given by Equation (17), i.e., the LT, and the other is derived by combining Equation (27), i.e., operational calculus, with an force–electricity analogy using algebraic operations. Appendix A contrasts classical differential systems with operator-based methods, demonstrating the latter’s computational efficiency and physical clarity through the generalized Kelvin–Voigt (GKV) model.

The creep theory and relaxation theory represented by operators $\mathit{T}\left(\mathit{p}\right)$ and ${\mathit{T}}^{-1}\left(\mathit{p}\right)$ are fully equivalent to the theories represented by functions $J\left(t\right)$ and $E\left(t\right)$. However, for engineers, the operator-based method provides a new research pathway. The operator algebra approach, in line with operational calculus, provides a clear and consistent framework for solving viscoelastic problems. Unlike traditional methods, the operator representation simplifies the handling of complex material behaviors by directly connecting the system’s intrinsic properties to its mathematical formulation. The physical intuition behind this approach lies in its ability to express the material’s stress–strain relationship using operators that reflect the system’s fundamental behavior, independent of the specific loading conditions.

3.3. Operator-Based Method for Fractional Differential Constitutive Models

Fractional calculus is a generalization of classical integer-order differentiation and integration. Over the past two decades, fractional calculus has become an important tool for physical and mathematical modeling and has found wide applications in various fields. In the field of rheology, Scott–Blair [11] first introduced fractional-order elements and developed various fractional-order derivative models. The stress and strain of elastic solids are proportional to the zero-order derivative of strain, while the stress and strain of Newtonian fluids are proportional to the first-order derivative of strain. Therefore, for viscoelastic materials, stress is proportional to the fractional-order strain. The general constitutive equation for viscoelastic bodies can be expressed as follows:

$$\sigma (t)={\eta }^{\beta }{\displaystyle \frac{{\mathrm{d}}^{\beta }\epsilon (t)}{\mathrm{d}{t}^{\beta }}},$$

(31)

where $\eta$ is the material’s viscosity coefficient ($Pa·s$) and $\beta$ is the order, with $0\le \beta \le 1$. Equation (31) is referred to as the Scott–Blair fractional-order element (also known as the Abel dashpot). This element can be viewed as a viscoelastic material that lies between an ideal solid and an ideal fluid and has been widely applied to modeling rock creep problems [29,30,31,32]. Therefore, it is necessary to provide an operator representation of the Scott–Blair element.

Liu et al. [33] have rigorously proven that, for the Riemann–Liouville type [34]:

$${}^{\mathit{RL}}\mathbf{D}^{\beta }f(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\beta )}}{\displaystyle \frac{{\mathrm{d}}^n}{\mathrm{d}{t}^n}}{\int }_0^t{(t-\tau )}^{n-\beta -1}f(\tau )\mathrm{d}\tau ,$$

(32)

and the Caputo-type fractional derivative definition [35,36]:

$${}^{\mathit{C}}\mathbf{D}^{\beta }f(t)={\displaystyle \frac{1}{\mathsf{\Gamma }(n-\beta )}}{\int }_0^t{(t-\tau )}^{n-\beta -1}{f}^{(n)}(\tau )\mathrm{d}\tau ,$$

(33)

the order of the fractional derivative strictly corresponds to the fractional exponent of the differential operator $\mathit{p}$. That is:

$${}^{\mathit{RL}}\mathbf{D}^{\beta }f(t)={\mathit{p}}^{\beta }f(t)-{\mathit{p}}^{\beta }\sum _{i=1}^n{\left[{}^{\mathit{RL}}\mathbf{D}^{\beta -i}f(t)\right]}_{t=0}{\displaystyle \frac{t^{\beta -i}}{\mathsf{\Gamma }(\beta -i+1)}},$$

(34)

and

$${}^{\mathit{C}}\mathbf{D}^{\beta }f(t)={\mathit{p}}^{\beta }\cdot f(t)-\sum _{i=0}^{n-1}{\mathit{p}}^{\beta -i-1}{f}^{(i)}(0+).$$

(35)

The difference between the two lies in the correction term. When the zero initial conditions of both are satisfied, the correction term becomes zero, and we obtain the following:

$${}^{\mathit{C}}\mathbf{D}^{\beta }f(t){=}^{RL}{D}^{\beta }f(t)={\mathit{p}}^{\beta }\cdot f(t).$$

(36)

Typically, the initial value conditions for the Caputo-type fractional derivative are easier to satisfy in engineering problems. Under this assumption, the Caputo-type fractional derivative ${\mathrm{d}}^{\beta }/\mathrm{d}{t}^{\beta }$ can be directly represented as ${\mathit{p}}^{\beta }$. For a Caputo-type Scott–Blair element that satisfies the zero initial value condition, its stiffness operator $\mathit{T}$ can be expressed as follows:

$$\mathit{T}={\eta }^{\beta }{\mathit{p}}^{\beta }.$$

(37)

Appendix B validates the operator-based method for fractional viscoelasticity using the fractional-order Nishihara model. This model incorporates Caputo-type Scott–Blair elements, a well-recognized approach in rock mechanics, as proposed by Zhou et al. [37]. For Scott–Blair elements under other types of fractional derivative definitions, Liu et al. [33] have provided operator representations for commonly used definitions and general solution methods, which are not repeated here.

4. Operator-Based Method to Viscoelastic Dynamic Mechanical Analysis

4.1. Operator-Based Method to Complex Modulus and Complex Compliance

In engineering applications, numerous materials and structures are subjected to short-duration external loads or time-varying alternating forces [38,39]. Consequently, investigating the dynamic mechanical properties of materials becomes essential [40,41]. A commonly employed approach for analyzing dynamic performance is the use of Fourier transforms, which characterize the viscoelastic properties in the frequency domain through complex modulus and complex compliance functions. To illustrate this, consider a harmonic strain excitation applied to the material:

$$\epsilon (t)={\epsilon }_0{\mathrm{e}}^{i\omega t}={\epsilon }_0(\cos \omega t+\mathrm{i}\sin \omega t).$$

(38)

For linear viscoelastic systems, the resultant stress response takes the following form:

$$\sigma (t)=\tilde{\sigma }{\mathrm{e}}^{i\omega t}={\sigma }_0{\mathrm{e}}^{\mathrm{i}(\omega t+\delta )}.$$

(39)

The complex modulus $\tilde{E}$ and complex compliance $\tilde{J}$ are formally defined as:

$$\tilde{E}(\omega )={\displaystyle \frac{\sigma (t)}{\epsilon (t)}}={\tilde{E}}_{\Re }(\omega )+\mathrm{i}{\tilde{E}}_{\Im }(\omega ),$$

(40)

$$\tilde{J}(\omega )={\displaystyle \frac{\epsilon (t)}{\sigma (t)}}={\tilde{J}}_{\Re }(\omega )-\mathrm{i}{\tilde{J}}_{\Im }(\omega ).$$

(41)

In the extant literature, $\tilde{E}$ is occasionally termed the dynamic modulus, with $\widetilde{E_{\mathrm{R}\mathrm{e}}}$ designated as the storage modulus and $\widetilde{E_{\mathrm{I}\mathrm{m}}}$ as the loss modulus. Analogously, $\tilde{J}$ is referred to as the dynamic compliance, where $\widetilde{J_{\mathrm{R}\mathrm{e}}}$ represents the storage compliance and $\widetilde{J_{\mathrm{I}\mathrm{m}}}$ corresponds to the loss compliance.

The physical significance of the complex modulus lies in its characterization of the stress response of a material subjected to alternating strain, serving as a critical indicator of dynamic viscoelastic behavior. This frequency-dependent quantity, expressed as a function of loading frequency $\omega$, remains independent of the amplitude of external stress/strain. In Equations (40) and (41), the real components quantify the instantaneous elastic response analogous to solid-like energy storage, hence termed storage modulus and storage compliance. Conversely, the imaginary components capture the phase-shifted response (π/2 phase difference) associated with viscous energy dissipation, referred to as loss modulus and loss compliance. These components satisfy the following interrelations:

$$\tan \delta ={\displaystyle \frac{{\tilde{E}}_{\Im }(\omega )}{{\tilde{E}}_{\Re }(\omega )}}={\displaystyle \frac{{\tilde{J}}_{\Im }(\omega )}{{\tilde{J}}_{\Re }(\omega ).}}$$

(42)

The loss factor is formally defined as $\tan \delta$.

When determining the dynamic modulus of geometrically complex structures (e.g., fractal structures), direct computation using Equations (38)–(41) becomes computationally intensive. As rigorously demonstrated by Yu et al. [28], the operational calculus method exhibits mathematical equivalence to integral transform techniques. Specifically, under zero initial conditions, the following correspondence holds among Fourier transforms, LT, and operational calculus:

$$\mathcal{F}[f^{(\alpha )}(t)]={(i\omega )}^{\alpha }\mathcal{F}[f(t)]\iff \mathcal{L}[f^{(\alpha )}(t)]=s^{\alpha }\mathcal{L}[f(t)]\iff f^{(\alpha )}(t)={\mathit{p}}^{\alpha }f(t).$$

(43)

Leveraging this similarity property in Equation (43) and the operator representation of stiffness, we propose a streamlined methodology for determining dynamic mechanical characteristics:

• Compute the total stiffness operator $\mathit{T}\left(\mathit{p}\right)$ of the structure;
• Substitute the differential operator $\mathit{p}$ with the complex variable $\mathrm{i}\omega$ in $\mathit{T}\left(\mathit{p}\right)$, yielding the dynamic complex modulus. The resultant real and imaginary components correspond to the storage modulus and loss modulus, respectively;
• Calculate the loss factor as the ratio of imaginary-to-real components.

This approach provides a streamlined methodology for determining the dynamic mechanical characteristics of materials, reducing the need for complex integral transform techniques.

4.2. Dynamic Performance Analysis of Fractal Tree Structure

Building on the operator-based method to viscoelastic dynamic mechanical analysis discussed in Section 4.1, this section extends the method to the analysis of the dynamic response of a fractal tree structure. The fractal tree model, a self-similar structure, offers a unique opportunity to explore the dynamic properties of complex, hierarchical systems. Previous research by Hu et al. [18,19] demonstrated through operational calculus formalism that self-similar fractal network topologies, such as the fractal tree structure shown in Figure 1, include a constitutive relationship involving fractional-order derivatives.

This section begins with a review of the stiffness operator method applied to the fractal tree structure. As shown in Figure 1, the fractal tree (Figure 1a) is equivalent to a fractal cell (Figure 1b), which in turn is equivalent to the fractal component (Figure 1c). By leveraging these equivalencies, we can establish an algebraic equation for the stiffness operator of the fractal tree. Solving this equation yields the stiffness operator for the fractal tree structure.

For the fractal tree structure, based on the stiffness equivalency, we have:

$${\mathit{T}}_F={\displaystyle \frac{{\mathit{T}}_F{\mathit{T}}_a}{{\mathit{T}}_F+{\mathit{T}}_a}}+{\displaystyle \frac{{\mathit{T}}_F{\mathit{T}}_b}{{\mathit{T}}_F+{\mathit{T}}_b}},$$

(44)

where ${\mathit{T}}_{\mathrm{F}}$ is the stiffness operator for the fractal tree, ${\mathit{T}}_{\mathrm{a}}=E$, and ${\mathit{T}}_{\mathrm{b}}=\eta \mathit{p}$. The left side of Equation (44) represents the stiffness operator for the fractal component, while the right side represents the stiffness operator for the fractal cell. Equation (44) is a quadratic algebraic equation for the operator T_F. Solving Equation (44) yields the operator for the fractal tree structure [18,19]:

$${\mathit{T}}_F=\sqrt{E\eta \mathit{p}}.$$

(45)

Here, the fractal operator ${\mathit{T}}_{\mathrm{F}}$ is a fractional operator of order 1/2. This non-rational operator, and the fractional derivatives behind it, inject new insight into the study of viscoelastic problems.

By implementing the variable substitution methodology outlined above, the complex modulus of the structure is directly obtained as:

$$\begin{array}{ll}\tilde{E}(\omega )& =\sqrt{E\eta \omega \mathrm{i}}\\ & ={\displaystyle \frac{\sqrt{2E\eta \omega }}{2}}(1+\mathrm{i})\end{array}.$$

(46)

The complex compliance is:

$$\begin{array}{ll}\tilde{J}(\omega )& ={\displaystyle \frac{1}{\sqrt{E\eta \omega \mathrm{i}}}}\\ & ={\displaystyle \frac{1}{\sqrt{2E\eta \omega }}}(1-\mathrm{i})\end{array}.$$

(47)

The loss factor is:

$$\tan \delta ={\displaystyle \frac{{\tilde{E}}_{\Im }(\omega )}{{\tilde{E}}_{\Re }(\omega )}}=1.$$

(48)

Specifically, the storage modulus and loss modulus are:

$${\tilde{E}}_{\Re }={\tilde{E}}_{\Im }={\displaystyle \frac{\sqrt{2E\eta \omega }}{2}}.$$

(49)

The storage compliance and loss compliance are:

$${\tilde{J}}_{\Re }={\tilde{J}}_{\Im }={\displaystyle \frac{1}{\sqrt{2E\eta \omega }}}.$$

(50)

More generally, for models characterized by the $\alpha$-order fractional derivative, where the stiffness operator is $\mathit{T}=c{\mathit{p}}^{\mathsf{\alpha }}$ and the compliance operator is ${\mathit{T}}^{-1}=1/\left(c{\mathit{p}}^{\mathsf{\alpha }}\right)$. Substituting the differential operator $\mathit{p}$ with the complex variable $\mathrm{i}\omega$, the complex modulus and complex compliance become:

$$\tilde{E}(\omega )=c{\omega }^{\alpha }(\cos {\displaystyle \frac{\alpha \pi }{2}}+\mathrm{i}\sin {\displaystyle \frac{\alpha \pi }{2}}),$$

(51)

$$\tilde{J}(\omega )={\displaystyle \frac{1}{c{\omega }^{\alpha }}}(\cos {\displaystyle \frac{\alpha \pi }{2}}-\mathrm{i}\sin {\displaystyle \frac{\alpha \pi }{2}}).$$

(52)

The loss factor is:

$$\tan \delta =\tan {\displaystyle \frac{\alpha \pi }{2}}.$$

(53)

This subsection introduces a method to directly obtain the dynamic modulus or compliance from the structure itself, without relying on harmonic input. This approach reinforces the idea that the stiffness and compliance of linear viscoelastic structures are intrinsic properties of the system, and the most natural way to explore the system’s eigen characteristics does not require dependence on external loading forms.

The dynamic mechanical responses of the fractal tree model are compared with the time-domain responses of the classical Kelvin–Voigt and GKV models in Figure 2. Time-domain analysis results show that both the fractal tree model and the Kelvin–Voigt model exhibit persistent phase lag effects across low-, medium-, and high-frequency regimes. However, differences in dynamic responses between the models cannot be fully discerned from time-domain results alone, necessitating supplementary analytical methods.

In plots where the storage modulus and loss modulus are represented on the horizontal and vertical axes, respectively (as shown in Figure 3), the loss factor corresponds geometrically to the slope of the curve, with $\delta$ denoting the angle relative to the coordinate axes. Equation (53) indicates that the curve of the fractional derivative viscoelastic model forms an angle of $\alpha \pi /2$ with the horizontal axis. Comparative analysis reveals that as frequency increases, the dynamic modulus transitions from a low storage/loss modulus region (lower-left quadrant) to a high storage/loss modulus domain (upper-right quadrant). Specifically, when frequency approaches zero, the structural response converges to quasi-static loading behavior.

In low-frequency regimes, the storage moduli of the Kelvin–Voigt and GKV models intersect the real axis, demonstrating solid-like quasi-static characteristics with non-zero equilibrium creep effects under step loading. Conversely, the Maxwell and fractal tree models asymptotically dissipate all external energy (zero residual strain). Under high-frequency loading, the Maxwell and GKV models exhibit pronounced solid-like energy storage, while the Kelvin–Voigt and fractal tree models display enhanced viscous dissipation.

Notably, the fractal tree model maintains a constant loss factor of unity across the entire frequency spectrum, positioning it at the boundary between energy storage and dissipation. This characteristic has important engineering implications. For example, Lin et al. [42] showed that myocardial patches with a $\mathrm{t}\mathrm{a}\mathrm{n}\delta =1$ achieve optimal biomechanical compatibility.

5. Discussion

5.1. Operator-Based Method for Creep Functions with Variable Coefficients

To describe the complete three stages of rock creep, especially the accelerated creep stage, a common approach is to consider material parameter damage, i.e., the variable coefficient creep function. For instance, the variable coefficient viscosity can be expressed as follows:

$$\eta =\eta \left[1-D\left(t\right)\right],\hspace{0.33em}\hspace{0.33em}\hspace{0.33em} 0\le D<1.$$

(54)

The corresponding variable coefficient Newtonian viscosity model can be expressed as follows:

$$\sigma (t)=\eta \left[1-D\left(t\right)\right]{\displaystyle \frac{d\epsilon (t)}{dt}}.$$

(55)

Before investigating the operator-based solution of Equation (55), it is necessary to first discuss whether it satisfies the definition of a constitutive relationship. In fact, Equation (55) contains explicit time-dependent terms $D\left(t\right)$, which cause the stress–strain relationship to be affected by the reference frame, violating the objectivity requirement of a constitutive relationship. Therefore, strictly speaking, Equation (55) cannot be considered a valid constitutive relationship.

When further analyzing the LT of Equation (55), the effect of time on strain is manifested not only as differentiation but also as the product of a time function and strain. Since the LT of a product of functions depends on their specific forms, it is not possible to directly obtain the stiffness operator and compliance operator through the LT.

However, due to the importance of damage-type creep functions in describing accelerated creep problems, it remains necessary to investigate this function. In the case of constant stress, the function can be treated as the product of a constant and a time-dependent function. Under this condition, the narrow sense compliance operator of creep can be derived from Equation (17) via the LT—this operator is only applicable under constant stress input conditions.

5.2. Creep Function of Physical Fractal Structure with Variable Coefficient

When a step stress $\sigma \left(t\right)={\sigma }_{0}H\left(t\right)$ is applied to the fractal tree structure described in Section 4.2, the creep response is given by:

$$\begin{array}{ll}\epsilon \left(t\right)& ={\mathcal{L}}^{-1}[{\displaystyle \frac{1}{\sqrt{E\eta p}}})]\ast \left[{\sigma }_{0}H\left(t\right)\right]\\ & =\left({\displaystyle \frac{1}{\sqrt{E\eta \pi t}}}\right)\ast \left[{\sigma }_{0}H\left(t\right)\right]\\ & ={\displaystyle \frac{2{\sigma }_0}{\sqrt{\pi }}}\sqrt{{\displaystyle \frac{t}{\tau }}}\end{array},$$

(56)

where $\tau =E\eta$. Although the fractal tree structure is effective in characterizing the viscoelastic behavior of biological materials, it struggles to accurately describe rock creep behavior, especially the third stage of rock creep, i.e., the accelerated creep stage. As described in Section 5.1 and Appendix B, one commonly used method is to consider damage in the rock. For example, Zhou et al. [43], based on ultrasonic experimental results and the trend in damage evolution, defined the damage factor as a negative exponential function, i.e.,

$$D=1-e^{-\alpha t},$$

(57)

where $\alpha$ is a variable related to the material properties of the rock. Substituting Equation (57) into Equation (55) yields:

$$\sigma (t)=\eta e^{-\alpha t}{\displaystyle \frac{d\epsilon (t)}{dt}}.$$

(58)

From Equation (17), the stiffness operator for the variable coefficient Newtonian viscosity model under constant stress can be derived as:

$$\mathit{T}=\eta (\mathit{p}-\alpha ).$$

(59)

Correspondingly, the stiffness operator ${\mathit{T}}_{\mathrm{D}\mathrm{F}}$ for the damage–fractal tree structure is:

$${\mathit{T}}_{DF}=\sqrt{E\eta (\mathit{p}-\alpha ).}$$

(60)

When a step stress $\sigma \left(t\right)={\sigma }_{0}H\left(t\right)$ is applied to the damage–fractal tree structure, the creep response is:

$$\begin{array}{ll}\epsilon \left(t\right)& ={\mathcal{L}}^{-1}[{\displaystyle \frac{1}{\sqrt{E\eta (p-\alpha )}}}]\ast \left[{\sigma }_{0}H\left(t\right)\right]\\ & ={\displaystyle \frac{e^{\alpha \tau }}{\sqrt{E\eta \pi t}}}\ast \left[{\sigma }_{0}H\left(t\right)\right]\\ & ={\displaystyle \frac{erfi(\sqrt{\alpha t)}}{\sqrt{\alpha \tau }}}{\sigma }_0\end{array}$$

(61)

where $erfi$ is the imaginary error function. Thus, the creep function for the damage–fractal tree structure is:

$$J\left(t\right)={\displaystyle \frac{erfi(\sqrt{\alpha t)}}{\sqrt{\alpha \tau }}}.$$

(62)

Figure 4 presents the sensitivity analysis of the creep function for the damage–fractal tree structure across different values of $\alpha$. The results suggest that this model holds significant potential for describing the accelerated creep phase of rock. Specifically, as $\alpha$ increases, the strain rate growth rate also increases, thereby enhancing the manifestation of accelerated creep in rock behavior.

However, it is important to note that the derivative of Equation (56) exhibits a non-increasing trend, indicating that the pure fractal tree structure shows a linear increase after the stabilization phase. This behavior does not align well with the experimentally observed accelerated creep, where the strain rate typically increases more rapidly. In contrast, the damage–fractal tree structure, incorporating a damage factor, more accurately characterizes the three-stage rheological behavior of rock, particularly the accelerated creep phase. This model offers a promising new approach for predicting the long-term performance of rocks and complex materials, addressing the limitations of the pure fractal model and enhancing its applicability to real-world rock mechanics.

Experimental data for model fitting were not included in this work. However, the model’s potential to capture the accelerated creep phase is supported by its functional form, particularly the increasing creep rate. By refining the model’s details for specific rock types, it is expected that experimental fitting will be achievable. Experimental validation will be addressed in future work to further refine the model and ensure its applicability in practical scenarios.

6. Conclusions

This paper introduces an operator-based method for modeling viscoelastic materials, providing a unified framework to describe both creep and relaxation behaviors. By utilizing stiffness and compliance operators, the method offers a systematic approach for analyzing viscoelastic problems.

• The operator-based method enhances the mathematical duality between the creep and relaxation functions, offering greater physical intuition and an intuitive understanding of time-dependent material behavior. It directly reflects the intrinsic properties of the system, independent of input and output conditions.
• The method is extended to dynamic problems, where the complex modulus and complex compliance are derived through operator representations. The fractal tree model, with its constant loss factor across the entire frequency spectrum, demonstrates its potential engineering value.
• By introducing a damage-based variable coefficient to the fractal tree model, it now has the potential to describe the accelerated creep phase of rocks. This enhancement allows the model to account for damage evolution in rock materials under prolonged loading, particularly during the accelerated creep stage. However, further experimental validation is required to fully substantiate its practical application.

Although the proposed method shows promise, its current application is limited to one-dimensional viscoelastic problems. Future research will focus on extending the method to three-dimensional cases, incorporating experimental validation, and exploring its broader relevance in engineering applications.