Time-space multiscale model for viscoelastic construction materials using the initial stress method

Summary

Viscoelastic materials such as polymer-modified asphalt, rubberized concrete, and structural adhesives are widely used in civil engineering for damping and deformation-recovery properties. However, their time-dependent and heterogeneous features hinder long-term performance prediction and structural integrity assessment, necessitating an efficient, accurate multiscale analysis. This study proposes a multiscale finite element method (MsFEM) for viscoelastic materials, integrating the generalized Maxwell model (GMM) and the initial stress method. It converts time-domain convolution constitutive relations into incremental elastic problems, retains a constant stiffness matrix to boost efficiency, and uses mesoscale heterogeneity-representing basis functions for coarse-fine mesh coupling. Validated via axial rod and cantilever beam examples, the method shows high accuracy against analytical solutions, serving as a robust tool for predicting material long-term performance with applications in structural health monitoring, life cycle assessment, and sustainable design.

Graphical abstract

Highlights

• •Multiscale FEM with generalized Maxwell model for viscoelastic materials
• •Initial stress method for incremental solution with a constant stiffness matrix
• •Structural numerical validation with high accuracy against analytical solutions
• •Potential in long-term performance prediction and sustainability assessment

Applied sciences; Materials science

Introduction

Viscoelastic materials are extensively used in civil engineering applications, including vibration-damping layers, asphalt pavements, and polymer-based repair systems.^1,2 Their responses depend on both the magnitude of stress and the time.³ Typical viscoelastic materials include rubber, plastics, polymers, and certain composites. Other materials, such as rock, asphalt, and concrete, behave nearly elastically under normal conditions but exhibit pronounced viscoelastic characteristics under high temperatures or high pressures.^4,5,6,7 Therefore, accurate modeling of these behaviors is essential for ensuring the durability and safety of built structures, especially under cyclic mechanical and environmental loads.^8,9 As composite propellants, these materials display strong rate dependence, temperature sensitivity, and nonlinear viscoelastic behavior.^10,11,12 Under combined cyclic mechanical and thermal loads, they are prone to damage accumulation, aging, and crack propagation, which can severely compromise the service life and safety of engines.^13,14,15 High-fidelity models that can accurately capture these multi-physics responses must, therefore, be developed for assessing structural integrity in propulsion systems.

Numerous studies have demonstrated that construction materials exhibit intrinsic features across multiple, nested scales, with macroscopic mechanical behavior and failure mechanisms often driven by micro- and meso-scale structural evolution, including interfacial debonding, particle fracture, and microcrack propagation. Conventional homogenization methods often fail to capture these effects, particularly when material morphology is non-periodic or when long-term viscoelastic response is of interest. Multiscale finite element methods (MsFEMs)^16,17 have emerged as a means to bridge micro-, meso-, and macro-scale responses, enabling high-fidelity simulations without prohibitive computational cost.^18,19,20 Previous work has employed representative volume elements (RVEs) combined with finite element simulations to accurately predict the nonlinear viscoelastic response of propellant grains and their damage evolution.^21,22 With existing computational capabilities, multiscale analysis is a practical and feasible approach for examining macroscopic failure mechanisms and guiding the development of new materials or structural designs.

Multiscale analysis of heterogeneous materials has been extensively developed for elastic and elasto-plastic regimes. Extending these frameworks to viscoelastic composites presents distinct challenges owing to their inherent time- and history-dependent constitutive behavior.^23,24,25 Current research endeavors can be broadly classified into two areas: analytical homogenization^26,27 and computational multiscale methods.^28,29 Analytical approaches commonly employ transform-domain techniques, such as the elastic-viscoelastic correspondence principle (EVCP), together with micromechanical models (e.g., self-consistent, Mori-Tanaka, and unit-cell methods) to derive closed-form expressions for effective relaxation moduli and creep compliance in architectured materials, including fiber-reinforced, layered, and periodic composites. Computational strategies rely on numerical homogenization methods, primarily asymptotic homogenization coupled with the finite element method, and stochastic frameworks to predict macroscopic viscoelastic properties. These simulations systematically quantify the influence of microstructural morphology, including fiber distribution, interface properties, particle shape, volume fraction, and stochastic defects, on the macroscopic time-dependent response. The integration of these analytical and numerical approaches is important for establishing robust structure-property linkages and predicting long-term performance,³⁰ such as creep and stress relaxation, in advanced composite systems across engineering applications.^31,32,33,34 Multiscale analyses of viscoelastic materials predominantly rely on asymptotic homogenization, in which effective elastic properties are first derived through homogenization and then extended to the viscoelastic domain through the elastic-viscoelastic correspondence principle (EVCP. In this procedure, elastic moduli are replaced with Laplace-transformed viscoelastic parameters to establish constitutive relations in the Laplace domain, thereby embedding mesostructural information. The inverse Laplace transform is then employed to recover the corresponding time-domain relations, which are coupled with analytical or finite element formulations for macroscale computation. However, asymptotic homogenization requires periodicity of the representative volume element (RVE), a condition not satisfied by many composite materials, while the inverse Laplace operation in EVCP often introduces difficulties when addressing complex engineering problems.

This study introduces a MsFEM framework for viscoelastic materials based on the initial stress method and the generalized Maxwell model (GMM). The proposed approach avoids the need for Laplace-domain transformations and periodic assumptions, making it suitable for non-uniform and heterogeneous construction materials. Benchmark examples verify the method and demonstrate its applicability to structural analysis, health monitoring, and life cycle assessment of built infrastructure.

Multiscale finite element theory for viscoelastic materials

MsFEM was first proposed by Babuška et al.³⁵ Subsequent contributions by Hou^36,37 and Efendiev et al.³⁸ expanded the method and established its core concepts and fundamental implementation procedures. The method was originally applied to the coupled consolidation analysis of saturated heterogeneous porous media,³⁹ through the formulation of independent basis functions for the solid displacement and pore pressure fields. It was later extended to multidimensional vector field problems in solid mechanics,⁴⁰ and the coupling terms were incorporated into the basis functions.

In conventional FEM, polynomial basis functions (shape functions) are employed to interpolate the displacement field within each element. These functions are typically predefined analytical expressions and are independent of the element’s material properties, requiring each element to be homogeneous. For problems with multiscale features, the element size in conventional FEM must be smaller than the characteristic size of the heterogeneous material to achieve accurate solutions, which leads to significant computational cost. MsFEM addresses this limitation by dividing the computational domain into coarse and fine meshes, as illustrated in Figure 1. Thick lines represent the coarse-scale mesh, and thin lines represent the fine-scale mesh. The fine-scale mesh within each coarse element is referred to as a subgrid. Elements of the coarse-scale mesh are called coarse elements (macroscopic elements), and elements of the fine-scale mesh are called fine elements (microscopic elements). A single coarse element may also be referred to as a unit cell or an RVE. To capture mesoscale heterogeneity, such as spatial variations in elastic modulus or Poisson’s ratio, multiscale basis functions are numerically constructed by solving the local equilibrium equations within each coarse element. This enables the global problem to be solved on a coarse mesh with high accuracy and reduced computational cost. After the macroscale solution is obtained, the mesoscale response is recovered using the conversion relationships between mesoscale and macroscale quantities.

Figure 1.

MsFEM for a two-dimensional problem with pre-determined fine-grid, coarse-grid, and subgrids

The computational procedure of MsFEM for elastic problems consists of three stages: mesoscale computation, macroscale computation, and downscaling computation. Mesoscale computation constructs multiscale basis functions for each coarse element on the subgrids and generates the corresponding equivalent stiffness matrix. Macroscale computation then solves the physical problem on the coarse-scale meshes using the equivalent stiffness matrices derived from the mesoscale analysis. Downscaling computation is performed after obtaining the macroscale results and uses the transformation matrices associated with the basis functions to recover the structural response at the mesoscale. The following example of a planar continuum illustrates the implementation process and key formulas of MsFEM. Subscripts s and c denote subgrid and coarse-element quantities, respectively.

Construction of multiscale basic functions

For a coarse element within the computational domain shown in Figure 1, let its internal domain be denoted as Ω^K, where Ω^K⊂Ω. The equilibrium equations are solved numerically within each coarse element under specified boundary conditions to obtain the corresponding multiscale basis functions:

$$L{N}_i=0\text{in}{\mathrm{\Omega }}^K,i=\begin{array}{cccc}1\text{,}& 2\text{,}& \cdots \text{,}& m\end{array}$$ (Equation 1)

where L represents the elastic operator, and $Lu=\mathit{div}\left(D:\frac{1}{2}\left(\nabla u+{\left(\nabla u\right)}^T\right)\right)$; D represents the 4th-order stiffness tensor that characterizes the properties of the materials; u represents the displacement vector; N_i represents the basis function corresponding to the i-th node of the coarse element, satisfying $N_i{|}_j=N_i\left(\begin{array}{cc}{x}_j,& y_j\end{array}\right)={\delta }_{ij}$ for $i,j=\begin{array}{cccc}1\text{,}& 2\text{,}& \cdots \text{,}& m\end{array}$); δ represents the Kronecker delta; and m represents the node numbers in the coarse element, taken as m = 4 here.

For a plane problem, N_i consists of N_ixx and N_iyy, where $i=\begin{array}{cccc}1\text{,}& 2\text{,}& 3\text{,}& 4\end{array}$. The displacement of any node within the fine-scale mesh (subgrid) of the element is expressed as follows:

$$u=\sum _{i=1}^4{N}_{ixx}{u}_i^{\prime }+\sum _{i=1}^4{N}_{ixy}{v}_i^{\prime }$$ (Equation 2)

$$v=\sum _{i=1}^4{N}_{iyy}{v}_i^{\prime }+\sum _{i=1}^4{N}_{iyx}{u}_i^{\prime }$$ (Equation 3)

where N_iyx is the coupling term added to the basis function, representing the displacement in the y-direction at each microscopic node when a unit displacement occurs in the x-direction at the coarse node i. Equations 2 and 3 can be expressed in unified vector form:

$$U_s=N{U}_c$$ (Equation 4)

where U_s represents the displacement vectors of the subgrid nodes; U_c denotes the displacement vectors of the coarse element nodes, respectively. These vectors are expressed as:

$$U_s={\left[\begin{array}{ccccccc}{u}_1& v_1& u_2& v_2& \cdots & u_n& v_n\end{array}\right]}^T$$ (Equation 5)

$$U_c={\left[\begin{array}{cccccccc}{u}_1^{\prime }& v_1^{\prime }& u_2^{\prime }& v_2^{\prime }& u_3^{\prime }& v_3^{\prime }& u_4^{\prime }& v_4^{\prime }\end{array}\right]}^T$$ (Equation 6)

$$N={\left[\begin{array}{cccc}{R}_1^T& R_2^T& \cdots & R_n^T\end{array}\right]}^T$$ (Equation 7)

where,

$$R_i=\left[\begin{array}{cccccccc}{N}_{1xx}\left(i\right)& N_{1xy}\left(i\right)& N_{2xx}\left(i\right)& N_{2xy}\left(i\right)& N_{3xx}\left(i\right)& N_{3xy}\left(i\right)& N_{4xx}\left(i\right)& N_{4xy}\left(i\right)\\ N_{1yx}\left(i\right)& N_{1yy}\left(i\right)& N_{2yx}\left(i\right)& N_{2yy}\left(i\right)& N_{3yx}\left(i\right)& N_{3yy}\left(i\right)& N_{4yx}\left(i\right)& N_{4yy}\left(i\right)\end{array}\right],i=\begin{array}{cccc}1\text{,}& 2\text{,}& \cdots \text{,}& n\end{array}$$ (Equation 8)

where n is defined as the total number of nodes within the fine-scale mesh of the coarse element. For a specific fine element e within the subgrid of the domain Ω^K shown in Figure 1, the following transformation can also be established:

$$u_e=G_e{U}_c,G_e={\left[\begin{array}{cccc}{R}_{e1}^T& R_{e2}^T& R_{e3}^T& R_{e4}^T\end{array}\right]}^T$$ (Equation 9)

where G_e is the transformation matrix of fine element e, mapping its nodal displacements to the corresponding coarse element. The nodal displacement vector of element u_e in the subgrid,

$$u_e={\left[\begin{array}{cccccccc}{u}_{e1}^{\prime }& v_{e1}^{\prime }& u_{e2}^{\prime }& v_{e2}^{\prime }& u_{e3}^{\prime }& v_{e3}^{\prime }& u_{e4}^{\prime }& v_{e4}^{\prime }\end{array}\right]}^T$$ (Equation 10)

where e1, e2, e3, and e4 represent the four nodes of element e in the subgrid.

For scalar field problems, the following boundary condition must be satisfied when constructing the basis functions by solving Equation 1:

$$N_i=N_i^0\text{on}\text{the}\text{boundary}\text{of}{\mathrm{\Omega }}^K$$ (Equation 11)

where $N_i^0$ represents the boundary condition applied when constructing the basis function N_i.

For vector field problems, basis functions must be generated independently for each coordinate direction. Studies have demonstrated that the prescribed values of $N_i^0$ influence accuracy. The following explanation illustrates the application of linear boundary conditions using the basis function N_1xx.

To construct N_1xx under linear boundary conditions, linear constraints in the x-direction are applied along edges 1–4 and 1–2. Specifically, N_1xx varies linearly from $N_{1xx}\left(\begin{array}{cc}{x}_1,& y_1\end{array}\right)=1$ to $N_{1xx}\left(\begin{array}{cc}{x}_2,& y_2\end{array}\right)=0$ and $N_{1xx}\left(\begin{array}{cc}{x}_4,& y_4\end{array}\right)=0$. Along edges 2–3 and 3–4, the displacement in the x-direction is set to zero, while all boundary nodes are fixed in the y-direction (Figure 2). Under these conditions, the equilibrium Equation 1 is solved on the subgrid Ω^K using standard numerical methods, producing the displacement vector $N_1^x=\left\{\begin{array}{cc}{N}_{1xx}\text{,}& N_{1yx}\end{array}\right\}$. Repeating this procedure for the remaining directions yields the complete set of basis functions N for the coarse element.

Figure 2.

Linear boundary conditions for the unit displacement condition

The basic functions obtained through this procedure satisfy the partition of unity within the element:

$$\{\begin{array}{l}\begin{array}{cc}\sum _{i=1}^4{N}_{ixx}=1& \sum _{i=1}^4{N}_{iyy}=1\end{array}\\ \begin{array}{cc}\sum _{i=1}^4{N}_{iyx}=0& \sum _{i=1}^4{N}_{ixy}=0\end{array}\end{array}$$ (Equation 12)

This ensures that the rigid-body displacement of a coarse element remains consistent with the displacement continuity between adjacent elements. When applying linear boundary conditions, the edges of the coarse element are forced to deform linearly. This constraint yields a pronounced boundary layer effect along the edges of heterogeneous elements, which tends to overestimate stiffness in multiscale computations. To reduce this error, Hou et al.³⁷ proposed the super-sampling technique, which enlarges the computational domain for solving the basis functions to account for oscillations near coarse-element boundaries and improve solution accuracy. Details of this method are available in the literature.

Macroscopic computation of viscoelastic materials

In the subgrid domain, the conventional finite element discretized equations for a linear elastic problem are expressed as follows:

$$K_s{U}_s=F_s$$ (Equation 13)

where F_s is the nodal force vector, and K_s is the stiffness matrix assembled from all fine elements within the subgrid of a coarse element. The dimensions of K_s and F_s are n_s×n_s and n_s×1, respectively, where n_s denotes the total number of DOFs in the subgrid. Generally, n_s≫8. Substituting Equation 4 into Equation 13 and left-multiplying both sides by N^T yivelds:

$$K_c{U}_c=F_c$$ (Equation 14)

where F_c and K_c denote the nodal force vector and equivalent stiffness matrix of the coarse element. For a four-node coarse element, their dimensions are 8 × 8 and 8 × 1, respectively. They are expressed as follows:

$$K_c=N^T{K}_{s}N$$ (Equation 15)

$$F_c=N^T{F}_s$$ (Equation 16)

These relations provide the means for deriving the equivalent stiffness matrix of a coarse element once the integrated stiffness matrix K_s of fine elements within the subgrid is known. In the subgrid, if the stiffness matrix K_e of a specific fine element is given; for instance, element e located within domain Ω^K in Figure 1. Using Equation 9, the equivalent stiffness matrix of the coarse element can also be obtained as follows:

$$K_c=\sum _{e=1}^p{K}_e^{\prime },K_e^{\prime }=G_e^T{K}_e{G}_e$$ (Equation 17)

where p is the number of fine elements in the subgrid of a single coarse element.

Once the nodal force vector and the stiffness matrix of each coarse element are obtained, they are assembled into the global equivalent stiffness matrix and global nodal force vector of the structure using the standard finite element assembly procedure:

$$K_C=A_{c=1}^{N_c}{K}_c,F_C=A_{c=1}^{N_c}{F}_c$$ (Equation 18)

where K_C and F_C denote the global equivalent stiffness matrix and global nodal force vector, respectively, and A is the standard finite element assembly operator. The total number of coarse elements is N_c.

Once the global equivalent stiffness matrix and nodal force vector are determined, the macroscopic displacement vector U_C is obtained by solving the subsequent equation:

$$K_C{U}_C=F_C$$ (Equation 19)

Downscaling computation for viscoelastic materials

Once the macroscopic analysis is completed, the displacement vector U_c of a coarse element c is obtained from U_C. Using the transformation relations in Equation 4 or Equation 9, the nodal displacement vectors U_s of the subgrid are then determined. From U_s, the stress σ_e and the strain ε_e, and other microscopic quantities of each fine element e within the subgrid are calculated. This procedure, which derives microscopic quantities from macroscopic fields, is referred to as downscaling computation.

Initial stress method for analyzing viscoelastic materials through the generalized Maxwell model

In the analytical treatments based on the elastic-viscoelastic correspondence principle (EVCP), the Laplace transform is applied to obtain the elastic solution in the Laplace domain, and the inverse Laplace transform then recovers the viscoelastic solution in the time domain. EVCP requires an analytical solution to the corresponding elastic problem under identical boundary conditions. By contrast, the numerical analysis primarily employs the finite element method, with two implementation strategies determined by the adopted constitutive relation. The first strategy is the variable stiffness method, which relies on integral constitutive relations.⁴¹ This approach provides high accuracy but requires substantial computational effort. To improve computational efficiency, a second strategy, the constant stiffness method, uses differential constitutive relations. The viscous component of strain or stress is treated as an initial strain or initial stress, leading to the initial strain or initial stress method.⁴² The stiffness matrix remains unchanged, and each time step is solved recursively, which significantly reduces computational cost. Because the GMM is widely used, the principles and formulas of the initial stress method, which is formulated for this model, are presented later in discussion.

Generalized Maxwell model

The relaxation modulus for the context of the GMM is expressed as:

$$E\left(t\right)=E_{\infty }+\sum _{r=1}^m{E}_r{e}^{-\frac{t}{{\tau }_r}}$$ (Equation 20)

where E(t) is the relaxation modulus; E_∞ is the long-term elastic modulus; m is the number of Maxwell elements; E_r is the elastic modulus of the r-th Maxwell element; and ${\tau }_r=\frac{{\eta }_r}{E_r}$ is the relaxation time, with η_r denoting its viscosity coefficient. The parameters E_∞, E_r, η_r, and m are obtained by fitting relaxation test data using the least-squares method. Using the integral form of the constitutive relation, the constitutive equation is:

$$\sigma \left(t\right)={\epsilon }_{0}E\left(t\right)+{\int }_0^{t}E\left(t-s\right)\frac{\partial \epsilon \left(s\right)}{\partial s}ds$$ (Equation 21)

where the first term represents the stress at time t due to ε₀, and the integral term represents the stress generated by the evolution of strain over time.

Decomposition of total stress into elastic and relaxation contributions at any instant

From Equation 21, at t = t_n:

$${\sigma }^n={\epsilon }_{0}E\left(t_n\right)+{\int }_0^{t_n}E\left(t_n-s\right)\frac{\partial \epsilon \left(s\right)}{\partial s}ds$$ (Equation 22)

Applying integration by parts to the second term, and rearranging, yields:

$${\sigma }^n=E_0{\epsilon }_n-{\int }_0^{t_n}\epsilon \left(s\right)dE\left(t_n-s\right)={\sigma }_e^n-{\sigma }_v^n$$ (Equation 23)

where $E_0=E_{\infty }+\sum _{r=1}^m{E}_r=E\left(t=0\right)$ represents the elastic modulus at t = 0; ε_n = ε(t = t_n) denotes the strain at t = t_n; σ_n = σ(t = t_n) expresses the stress at t = t_n, ${\sigma }_e^n=E_0{\epsilon }_n$ is the elastic stress within the total stress at t = t_n relating to the initial elastic constants of the system, ${\sigma }_v^n={\int }_0^{t_n}\epsilon \left(s\right)dE\left(t_n-s\right)$ gives out the stress relaxation when t = t_n induced by the material’s viscosity. Equation 23 indicates that the total stress consists of an elastic component, which is independent of loading history, and a viscous component, which reflects the complete strain history. When the stress generated by viscous relaxation is treated as an initial stress, the numerical computation proceeds through the initial stress method.

Principle of the initial stress method for viscoelastic materials

From Equation 23, the total stress can be expressed as follows:

$${\sigma }^n={\sigma }_e^n-{\sigma }_v^n$$ (Equation 24)

The internal force within an element associated with this stress follows the principle of virtual displacements:

$${}_e{F}_{\mathrm{int}}^n=\underset{v}{\int }{}_e{B}^T{\left\{\sigma \right\}}^{n}dv=\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_e\right\}}^{n}dv-\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv=\underset{v}{\int }{}_e{B}^T{E}_0\left[D_0\right]{\left\{{\epsilon }_e\right\}}^{n}dv-\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv=\underset{v}{\int }{}_e{B}^T·E_0\left[D_0\right]·{}_{e}Bdv{\left\{u\right\}}_e^n-\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv$$ (Equation 25)

where ${}_e{F}_{\mathrm{int}}^n$ is the internal force in element e at t = t_n induced by the stress at the time step, _eB is the strain matrix, [D₀] is the elastic matrix that depends only on Poisson’s ratio, and ${\left\{u\right\}}_e^n$ is the element nodal displacement vector at t = t_n. Under the GMM, the strain in the spring component at any instant equals the strain in the Maxwell elements, corresponding to the total strain of the system at that time.

According to the internal and external force equilibrium equation,

$${}_e{F}_{\mathrm{int}}^n={}_e{F}_{ext}^n$$ (Equation 26)

Thus,

$$\underset{v}{\int }{}_e{B}^T{E}_0\left[D_0\right]{}_{e}Bdv{\left\{u\right\}}_e^n-\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv={}_e{F}_{ext}^n$$ (Equation 27)

where,

$$\underset{v}{\int }{}_e{B}^T{E}_0{\left[D_0\right]}_{e}Bdv{\left\{u\right\}}_e^n={}_e{F}_{ext}+{\underset{v}{\int }}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv={{}_{e}F}_{ext}^n+{{}_{e}F}_v^n$$ (Equation 28)

For simplicity, this relationship can be expressed as follows:

$$\left[k_e\right]{\left\{u\right\}}_e^n={}_e{F}_{ext}^n+{}_e{F}_v^n$$ (Equation 29)

where [k_e] refers to the element’s initial stiffness matrix, ${}_e{F}_{ext}^n$ denotes the nodal force vector, and ${}_e{F}_v^n=\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv$ represents the equivalent load arising from viscous relaxation (the initial stress load). The global stiffness matrix is formed by assembling the element matrices in Equation 29:

$$\left[K\right]{\left\{u\right\}}^n=F_{ext}^n+F_v^n$$ (Equation 30)

This linear system represents the global formulation of the initial stress method. The matrix [K] depends only on the elastic constants at the initial time and remains constant throughout the analysis. The nodal force vector $F_{ext}^n$ contains the applied loads, which may be constant or time-dependent. The term $F_v^n$ accounts for the additional load due to stress relaxation. Solving Equation 30 yields the displacement response at time t = t_n. Thus, solving the global system requires an accurate computation of $F_v^n$, obtained by assembling the element contributions from the viscous stress ${}_e{F}_v^n$. Since ${}_e{F}_v^n=\underset{v}{\int }{}_e{B}^T{\left\{{\sigma }_v\right\}}^{n}dv$. For viscoelastic materials, determining this stress relaxation ${\left\{{\sigma }_v\right\}}^n$ is the central step of the initial stress method.

Formulation of viscous stress in viscoelastic materials

For convenience, the derivation is first presented for the one-dimensional case; the results extend directly to three dimensions. According to Equation 23:

$${\sigma }_v^n={\int }_0^{t_n}\epsilon \left(s\right)dE\left(t_n-s\right)$$ (Equation 31)

Discretizing the interval 0∼t_n into [0, t₁, t₂, …, t_n-1, t_n], and the expression can be rewritten as:

$${\sigma }_v^n={\int }_0^{t_1}\epsilon \left(s\right)dE\left(t_n-s\right)+{\int }_{t_1}^{t_2}\epsilon \left(s\right)dE\left(t_n-s\right)+{\int }_{t_2}^{t_3}\epsilon \left(s\right)dE\left(t_n-s\right)+\cdots +{\int }_{t_{n-1}}^{t_n}\epsilon \left(s\right)dE\left(t_n-s\right)$$ (Equation 32)

If the strain is assumed constant during each step and equal to its initial value at that step, the expression reduces to:

$${\sigma }_v^n={\epsilon }_0\left[E\left(t_n-t_1\right)-E\left(t_n\right)\right]+{\epsilon }_1\left[E\left(t_n-t_2\right)-E\left(t_n-t_1\right)\right]+\cdots +{\epsilon }_{n-1}\left[E_0-E\left(t_n-t_{n-1}\right)\right]$$ (Equation 33)

Using Equation 20, the exponential kernel can be expressed in terms of the relaxation modulus, giving:

$$\{\begin{array}{l}{E}_0-E\left(t_n-t_{n-1}\right)=\sum _{r=1}^m{E}_r\left(1-e^{-\frac{t_n-t_{n-1}}{{\tau }_r}}\right)\\ E\left(t_n-t_2\right)-E\left(t_n-t_1\right)=\sum _{r=1}^m{E}_r\left(e^{-\frac{t_n-t_2}{{\tau }_r}}-e^{-\frac{t_n-t_1}{{\tau }_r}}\right)\\ E\left(t_n-t_1\right)-E\left(t_n\right)=\sum _{r=1}^m{E}_r\left(e^{-\frac{t_n-t_1}{{\tau }_r}}-e^{-\frac{t_n}{{\tau }_r}}\right)\end{array}$$ (Equation 34)

Substituting Equation 34 into Equation 33 and reorganizing yields:

$${\sigma }_v^n=\sum _{r=1}^m{\sigma }_{vr}^n$$ (Equation 35)

where ${\sigma }_{vr}^n={\epsilon }_0{E}_r\left(e^{-\frac{t_n-t_1}{{\tau }_r}}-e^{-\frac{t_n}{{\tau }_r}}\right)+{\epsilon }_1{E}_r\left(e^{-\frac{t_n-t_2}{{\tau }_r}}-e^{-\frac{t_n-t_1}{{\tau }_r}}\right)+\cdots +{\epsilon }_{n-1}{E}_r\left(1-e^{-\frac{t_n-t_{n-1}}{{\tau }_r}}\right)$ represents the viscous stress relaxation at time t = t_n. Applying the same procedure outlined in (Equation 31), (Equation 32), (Equation 33), (Equation 34), (Equation 35) for the next time step (t = t_n+1), we have ${\sigma }_v^{n+1}=\sum _{r=1}^m{\sigma }_{vr}^{n+1}$, where ${\sigma }_{vr}^{n+1}$ is the stress relaxation at t = t_n+1 and can be obtained by:

$${\sigma }_{vr}^{n+1}={\epsilon }_0{E}_r\left(e^{-\frac{t_{n+1}-t_1}{{\tau }_r}}-e^{-\frac{t_{n+1}}{{\tau }_r}}\right)+{\epsilon }_1{E}_r\left(e^{-\frac{t_{n+1}-t_2}{{\tau }_r}}-e^{-\frac{t_{n+1}-t_1}{{\tau }_r}}\right)+\cdots +{\epsilon }_{n-1}{E}_r\left(e^{-\frac{t_{n+1}-t_n}{{\tau }_r}}-e^{-\frac{t_{n+1}-t_{n-1}}{{\tau }_r}}\right)+{\epsilon }_n{E}_r\left(1-e^{-\frac{t_{n+1}-t_n}{{\tau }_r}}\right)$$ (Equation 36)

Assuming t_n+1-t_n = Δt_n+1, $e^{-\frac{t_{n+1}}{{\tau }_r}}=e^{-\frac{t_n+\Delta t_{n+1}}{{\tau }_r}}=e^{-\frac{t_n}{{\tau }_r}}·e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}$, $e^{-\frac{t_{n+1}-t_1}{{\tau }_r}}=e^{-\frac{t_n+\Delta t_{n+1}-t_1}{{\tau }_r}}=e^{-\frac{t_n-t_1}{{\tau }_r}}·e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}$, $e^{-\frac{t_{n+1}-t_2}{{\tau }_r}}=e^{-\frac{t_n+\Delta t_{n+1}-t_2}{{\tau }_r}}=e^{-\frac{t_n-t_2}{{\tau }_r}}·e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}$, $e^{-\frac{t_{n+1}-t_{n-1}}{{\tau }_r}}=e^{-\frac{t_n+\Delta t_{n+1}-t_{n-1}}{{\tau }_r}}=e^{-\frac{t_n-t_{n-1}}{{\tau }_r}}·e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}$, and $e^{-\frac{t_{n+1}-t_n}{{\tau }_r}}=e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}$. Equation 36 can be expressed as follows:

$${\sigma }_{vr}^{n+1}=e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}·{\epsilon }_0{E}_r\left(e^{-\frac{t_n-t_1}{{\tau }_r}}-e^{-\frac{t_n}{{\tau }_r}}\right)+e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}·{\epsilon }_1{E}_r\left(e^{-\frac{t_n-t_2}{{\tau }_r}}-e^{-\frac{t_n-t_1}{{\tau }_r}}\right)+\cdots +e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}·{\epsilon }_{n-1}{E}_r\left(1-e^{-\frac{t_n-t_{n-1}}{{\tau }_r}}\right)+{\epsilon }_n{E}_r\left(1-e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}\right)=e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}·{\sigma }_{vr}^n+{\epsilon }_n{E}_r\left(1-e^{-\frac{\Delta t_{n+1}}{{\tau }_r}}\right)$$ (Equation 37)

Equation 37 provides a recursive formula for computing the stress relaxation. At the first time step,

$${\sigma }_{vr}^1={\epsilon }_0{E}_r\left(1-e^{-\frac{\Delta t_1}{{\tau }_r}}\right)$$ (Equation 38)

Once the viscous stress at each time step is determined, substituting it into Equation 35 yields the total viscous stress relaxation of the GMM. The corresponding relaxation load is then obtained using the procedure described previously. The equilibrium equations are then solved to determine the displacements at each time step, from which the strains for the next time step are updated. This iterative process is repeated iteratively until the viscoelastic analysis is completed.

Results

Time-space multiscale finite element method framework for viscoelastic materials through the initial stress method

Multiscale finite element method steps for viscoelastic materials using the initial stress method

By combining the basic principles of MsFEM for elastic problems introduced in Section multiscale finite element theory for viscoelastic materials with the initial stress method for solving viscoelastic analysis using the GMM described in Section initial stress method for analyzing viscoelastic materials through the generalized Maxwell model, the following steps outline the MsFEM framework for viscoelastic materials using the initial stress method.

S0: Preprocessing

S0.1 The macroscopic mesh and the subgrids within each macroscopic element are generated;

S0.2 The parameters for the viscoelastic material are input: E_∞, E_i, η_i, μ (i = 1, 2, …, m), where m denotes the number of Maxwell elements in the GMM;

S0.3 The time domain is discretized as t = [0, t₁, t₂, …, t_N], with time increments of Δt₁ = t₁-0 = t₁, Δt₂ = t₂-t₁, ⋯, Δt_n = t_n-t_n-1 (n = 1, 2, …, N);

S0.4 Input the load time history P(t). Using the discretized time sequence, the load at each time step is determined as $P\left(t\right)=\left[\begin{array}{ccccc}P\left(t=0\right)& P\left(t=t_1\right)& P\left(t=t_2\right)& \cdots & P\left(t=t_N\right)\end{array}\right]$, denoted as $P\left(t\right)=\left[\begin{array}{ccccc}{P}_0& P_1& P_2& \cdots & P_N\end{array}\right]$. For time-independent loads, P(t) = P₀.

S1: Multiscale finite element method analysis at the initial time

S1.1 The material parameters at the initial time are determined: elastic modulus $E_0=E_{\infty }+\sum _{i=1}^m{E}_i$, Poisson’s ratio μ, and applied load P₀.

S1.2 Analysis of mesoscopic structure.

S1.2.1 Element analysis: Using the initial-time material parameters for each fine element, compute the stiffness matrix about each fine element within the subgrid of a coarse element. These fine-element matrices are assembled to form the integrated stiffness matrix of the subgrid for the corresponding coarse element.

S1.2.2 Construction of multiscale basic functions: For each DOF at the nodes of a coarse-element node, a unit displacement is imposed while all other DOFs are constrained to zero. The homogeneous equilibrium equations are then solved using the integrated stiffness matrices to obtain the basis functions corresponding to each DOF. These basic functions are then assembled into the basis function matrix. For instance, for a four-node quadrilateral coarse element, the basis function matrix is arranged such that nodes I, II, III, and IV correspond to the four coarse-element nodes in order as $N=\left[\begin{array}{cccccccc}{N}_{Ixx}& N_{Iyy}& N_{IIxx}& N_{IIyy}& N_{IIIxx}& N_{IIIyy}& N_{IVxx}& N_{IVyy}\end{array}\right]$.

S1.2.3 Determination of the transformation matrix: The transformation matrices relate the displacement vector of the subgrid to the displacement vector of the coarse element

$$\{\begin{array}{c}{R}_i=\left[\begin{array}{cccccccc}{N}_{Ixx}\left(i\right)& N_{Ixy}\left(i\right)& N_{IIxx}\left(i\right)& N_{IIxy}\left(i\right)& N_{IIIxx}\left(i\right)& N_{IIIxy}\left(i\right)& N_{IVxx}\left(i\right)& N_{IVxy}\left(i\right)\\ N_{Iyx}\left(i\right)& N_{Iyy}\left(i\right)& N_{IIyx}\left(i\right)& N_{IIyy}\left(i\right)& N_{IIIyx}\left(i\right)& N_{IIIyy}\left(i\right)& N_{IVyx}\left(i\right)& N_{IVyy}\left(i\right)\end{array}\right]\\ G_e={\left[\begin{array}{cccc}{R}_{e1}^T& R_{e2}^T& R_{e3}^T& R_{e4}^T\end{array}\right]}^T\\ u=N{u}_E^{\prime }\end{array}$$ (Equation 39)

where e1, e2, e3, and e4 denote the four nodes of fine element e within the subgrid, u represents the fine-element nodal displacement vector in the subgrid, and $u_E^{\prime }$ represents the coarse-element nodal displacement vector.

S1.3 Analysis of macroscopic structure.

S1.3.1 Computation of the equivalent stiffness matrix for coarse elements: Using the transformation relation $K_E=\sum _{e=1}^n{G}_e^T{k}_e{G}_e$, the stiffness matrices K_E of all coarse elements are then assembled into the global stiffness matrix K₀ according to the macroscopic node numbering. Here, k_e is the stiffness matrix of element e within the subgrid, and K_E represents the corresponding equivalent stiffness matrix for the coarse element.

S1.3.2 Solving for macroscopic nodal displacements at the initial time: Given the global stiffness matrix K₀U(t = 0) = P₀, U(t = 0), and nodal displacements $u_E^0$ can be solved for each coarse element.

S1.4 Downscaling analysis.

S1.4.1 Calculating the nodal displacements of the fine elements: Using $u^0=N{u}_E^0$, the nodal displacements $u_e^0$ of each fine element are computed.

S1.4.2 From the nodal displacements of the fine elements, the initial strain distribution within each fine element ${\epsilon }_e^0=B_e{u}_e^0$ is calculated.

S2: Multiscale finite element method analysis for subsequent time steps t = t₁, t₂, …, t_N.

S2.1 Analysis of mesoscopic structure.

S2.1.1 The recursive formula is applied to calculate the stress relaxation ${\sigma }_v^n$ of each fine element in the current time step.

S2.1.2 The additional nodal forces $F_{ve}^n=\underset{v}{\int }{B}^T{\sigma }_{ve}^{n}dv$ is calculated for all fine elements in the substructure using the stress relaxation obtained above.

S2.2 Macroscopic analysis

S2.2.1 The additional macroscopic forces are calculated: $F_v^n=\sum _e{N}_e^T{F}_{ve}^n$.

S2.2.2 $K_{0}U\left(t=t_n\right)=P_1+F_v^n$ is solved to obtain the macroscopic nodal displacements U(t = t_n) at t = t_n. From these results, the nodal displacements $u_E^n$ for each coarse element is determined.

S2.3 Downscaling analysis

S2.3.1 Determine the nodal displacements of the substructure: the nodal displacements $u_e^n$ for each sub-element is calculated using the transformation formula $u^n=N{u}_E^n$.

S2.3.2 The strain ${\epsilon }_e^n=B_e{u}_e^n$ in each fine element is calculated. These strains serve as the input for computing stress relaxation in the next time step.

The condition n ≤ N is satisfied. If satisfied, the computation is terminated, and the viscoelastic displacement results for all time steps are output. Otherwise, the process is reiterated from Step S2.1. For viscoelastic materials, the relaxation modulus varies rapidly at the initial stage and then evolves progressively more slowly with time. Therefore, from a theoretical standpoint, it is preferable to adopt a smaller time step at the early stage of the analysis and to gradually increase the time step as time progresses.

Example and discussion: One-dimensional axial rod

Three examples are presented to illustrate the computational procedure of the proposed method. The accuracy of the method is then verified by comparing its results with those obtained using other approaches. As shown in Figure 3, consider an axial rod with a uniform cross-section and a total length of 3L. The rod consists of three viscoelastic segments, each made of a different material and each occupying one-third of the total length. The left end is fixed, and the right end is subjected to a constant load P. The cross-sectional area is A, and the viscoelastic behavior of each segment is modeled using the GMM.

Figure 3.

Axial rod with uniform cross-section consisting of three viscoelastic segments and its viscoelastic constitutive model

The material parameters for each segment are defined as follows: Segment AB: The spring element has an elastic modulus of 3E_e. The elasticity coefficient of the single Maxwell element is 3E₁, and its viscosity coefficient is 3η₁. At the initial time step, the effective elastic modulus of the material is 3E₀ = 3E_e+3E₁, and the relaxation time is ${\tau }_1=\frac{3{\eta }_1}{3E_1}$; Segment BC: The spring element has an elastic modulus of 2E_e. The elasticity coefficient of the single Maxwell element is 2E₁, and its viscosity coefficient is 2η₁. At the initial time step, the effective elastic modulus is 2E₀ = 2E_e+2E₁, and the relaxation time is ${\tau }_1=\frac{2{\eta }_1}{2E_1}$; Segment CD: The spring element has an elastic modulus of E_e. The elasticity coefficient of the single Maxwell element is E₁, and its viscosity coefficient is η₁. At the initial time step, the effective elastic modulus is E₀ = E_e+E₁, and the relaxation time is ${\tau }_1=\frac{{\eta }_1}{E_1}$. For this example, it is assumed that E_e = 0.65 MPa, E₁ = 3 MPa, η₁ = 10.8 MPa s, τ₁ = 3.6s, E₀ = 3.65 MPa, A = 10 mm × 10 mm = 100 mm², l = 200 mm, and P = 2N. The objective is to compute the displacement at the free end of the axial rod over a total duration of 36 s.

Based on mechanics principles, the elastic displacement at different positions along the three-segment axial rod can be expressed as, $u_D=\frac{11Pl}{6EA}$, $u_C=\frac{5Pl}{6EA}$, $u_B=\frac{Pl}{3EA}$. According to EVCP, the displacement at each point in the three-segment viscoelastic bar is then given by²² $u_D=\frac{11Pl}{6A}f\left(t\right)$, $u_C=\frac{5Pl}{6A}f\left(t\right)$, $u_B=\frac{Pl}{3A}f\left(t\right)$, where $f\left(t\right)=\frac{1}{E_e}-\frac{E_1}{E_e\left(E_1+E_e\right)}{e}^{-\frac{E_1{E}_e}{\left(E_1+E_e\right){\eta }_1}t}=1.53846-1.264489e^{-0.049467t}$.

According to multiscale finite element theory, the axial rod is discretized into two levels of grids: a macroscopic grid and a mesoscopic subgrid. In this example, the macroscopic grid consists of a single element with two nodes, I and II. Within this coarse element, three fine elements are defined according to the material properties of the three segments, forming a subgrid with nodes numbered 1, 2, 3, and 4, as shown in Figure 4. The configuration of the macroscopic element and its mesoscopic subgrid is shown in Figure 5.

Figure 4.

Multiscale finite element discretization

(A) Macroscopic element; (B) mesoscopic subgrid.

Figure 5.

MsFEM results for the viscoelastic axial rod using the initial stress method, including the analytical solution and multiscale finite element solutions

The time step is uniform, Δt = 1 s, with t = [0, 1, 2, …, 36]. At t = 0, the material parameters are 3E₀ for Segment AB; 2E₀ for Segment BC; and E₀ for Segment CD.

Construction of basic functions

The stiffness matrices of the sub-elements are $k_1=\frac{3A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, $k_2=\frac{2A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, and $k_3=\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$. The assembled stiffness matrix of the subgrid is:

$$K=\left[\begin{array}{cccc}\frac{3A{E}_0}{l}& -\frac{3A{E}_0}{l}& 0& 0\\ -\frac{3A{E}_0}{l}& \frac{5A{E}_0}{l}& -\frac{2A{E}_0}{l}& 0\\ 0& -\frac{2A{E}_0}{l}& \frac{3A{E}_0}{l}& -\frac{A{E}_0}{l}\\ 0& 0& -\frac{A{E}_0}{l}& \frac{A{E}_0}{l}\end{array}\right]=\frac{A{E}_0}{l}\left[\begin{array}{cccc}3& -3& 0& 0\\ -3& 5& -2& 0\\ 0& -2& 3& -1\\ 0& 0& -1& 1\end{array}\right]$$ (Equation 40)

The multiscale basic functions, which incorporate the material properties of the mesoscopic structure, are then obtained by solving the homogeneous equations on the subgrid.

Let N_I(1) = 1 and N_I(4) = 0, solve for N_I(2) and N_I(3) as:

$$\frac{A{E}_0}{l}\left[\begin{array}{cccc}3& -3& 0& 0\\ -3& 5& -2& 0\\ 0& -2& 3& -1\\ 0& 0& -1& 1\end{array}\right]\left\{\begin{array}{l}1\\ N_I\left(2\right)\\ N_I\left(3\right)\\ 0\end{array}\right\}=\left\{\begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\right\}$$ (Equation 41)

In N_i(j), index i denotes macroscopic nodes, and index j denotes the number of microscopic nodes within the subgrid. From the second and third rows, we obtain $N_I\left(2\right)=\frac{9}{11}$ and $N_I\left(3\right)=\frac{6}{11}$. Thus, $N_I={\left[\begin{array}{cccc}{N}_I\left(1\right)& N_I\left(2\right)& N_I\left(3\right)& N_I\left(4\right)\end{array}\right]}^T={\left[\begin{array}{cccc}1& 9/11& 6/11& 0\end{array}\right]}^T$. Next, assuming N_II(4) = 1 and N_II(1) = 0, for the following equation is solved for N_II(2) and N_II(3):

$$\frac{A{E}_0}{l}\left[\begin{array}{cccc}3& -3& 0& 0\\ -3& 5& -2& 0\\ 0& -2& 3& -1\\ 0& 0& -1& 1\end{array}\right]\left\{\begin{array}{l}0\\ N_{II}\left(2\right)\\ N_{II}\left(3\right)\\ 1\end{array}\right\}=\left\{\begin{array}{l}0\\ 0\\ 0\\ 0\end{array}\right\}$$ (Equation 42)

From the second and third rows, we have: $N_{II}\left(2\right)=\frac{2}{11}$ and $N_{II}\left(3\right)=\frac{5}{11}$. Thus, $N_{II}={\left[\begin{array}{cccc}{N}_{II}\left(1\right)& N_{II}\left(2\right)& N_{II}\left(3\right)& N_{II}\left(4\right)\end{array}\right]}^T={\left[\begin{array}{cccc}0& 2/11& 5/11& 1\end{array}\right]}^T$. The matrix of basic functions is:

$$N=\left[\begin{array}{cc}{N}_I& N_{II}\end{array}\right]=\left[\begin{array}{cc}1& 0\\ 9/11& 2/11\\ 6/11& 5/11\\ 9& 1\end{array}\right]={\left[\begin{array}{cccc}{R}_1^T& R_2^T& R_3^T& R_4^T\end{array}\right]}^T$$ (Equation 43)

where $R_1=\left[\begin{array}{cc}1& 0\end{array}\right]$, $R_2=\left[\begin{array}{cc}9/11& 2/11\end{array}\right]$, $R_3=\left[\begin{array}{cc}6/11& 5/11\end{array}\right]$, $R_4=\left[\begin{array}{cc}0& 1\end{array}\right]$, and $u=N{u}_E^{\prime }\Rightarrow \left\{\begin{array}{l}{u}_1\\ u_2\\ u_3\\ u_4\end{array}\right\}=\left[\begin{array}{cc}1& 0\\ 9/11& 2/11\\ 6/11& 5/11\\ 0& 1\end{array}\right]\left\{\begin{array}{l}{u}_I\\ u_{II}\end{array}\right\}\Rightarrow \{\begin{array}{l}{u}_1=u_I\\ u_2=\frac{9}{11}{u}_I+\frac{2}{11}{u}_{II}\\ u_3=\frac{6}{11}{u}_I+\frac{5}{11}{u}_{II}\\ u_4=u_{II}\end{array}$. Here, u is the nodal displacement vector of the subgrid, and $u_E^{\prime }$ is the nodal displacement vector of the coarse element.

Macroscopic analysis

The equivalent stiffness matrices are computed using $K_E=\sum _{e=1}^3{k}_e^{\prime }$, where $k_e^{\prime }=G_e^T{k}_e{G}_e$ (e = 1, 2, 3). For Element 1, $k_1=\frac{3A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, $G_1=\left[\begin{array}{cc}1& 0\\ 9/11& 2/11\end{array}\right]$, $k_1^{\prime }=\frac{12}{121}\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$; for Element 2: $k_2=\frac{2A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, $G_2=\left[\begin{array}{cc}9/11& 2/11\\ 6/11& 5/11\end{array}\right]$, $k_2^{\prime }=\frac{18}{121}\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$; for Element 3: $k_3=\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$, $G_3=\left[\begin{array}{cc}6/11& 5/11\\ 0& 1\end{array}\right]$, $k_3^{\prime }=\frac{36}{121}\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$. Thus, $K_E=\sum _{e=1}^3{k}_e^{\prime }$ = $\frac{6}{11}\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]$. Since the example contains only one coarse element, the equivalent stiffness matrix also represents the stiffness matrix of the full structure. Given the obtained equivalent stiffness matrix, the equation K_Eu_E = F can be solved as $\frac{6}{11}\frac{A{E}_0}{l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]\left\{\begin{array}{l}{u}_I\\ u_{II}\end{array}\right\}=\left\{\begin{array}{l}0\\ P\end{array}\right\}$. Applying the boundary condition u_I = 0 yields $u_{II}=\frac{11}{6}\frac{Pl}{A{E}_0}=u_D$. Thus, the macroscopic displacement at the initial time step is: $u_E=\left\{\begin{array}{l}0\\ \frac{11Pl}{6A{E}_0}\end{array}\right\}$.

Subsequent calculations for downscaling analysis

Using the basic functions, the nodal displacements within the subgrid are obtained from the macroscopic displacements: u₁ = u_I = 0, $u_2=\frac{9}{11}{u}_I+\frac{2}{11}{u}_{II}=\frac{Pl}{3E_{0}A}$, $u_3=\frac{6}{11}{u}_I+\frac{5}{11}{u}_{II}=\frac{5}{6}\frac{Pl}{E_{0}A}$, and $u_4=\frac{11}{6}\frac{Pl}{E_{0}A}$. The strain matrix for each element is $B=\left[\begin{array}{cc}-\frac{1}{l}& \frac{1}{l}\end{array}\right]$, thereby resulting in ${\epsilon }_1^0=\frac{P}{3E_{0}A}$, ${\epsilon }_2^0=\frac{P}{2E_{0}A}$, and ${\epsilon }_3^0=\frac{P}{E_{0}A}$. Using the initial stress method, for t = 1s, ${\sigma }_{v1}^1={\epsilon }_1^{0}3E_1\left(1-e^{-\frac{1}{{\tau }_1}}\right)$, ${\sigma }_{v2}^1={\epsilon }_2^{0}2E_1\left(1-e^{-\frac{1}{{\tau }_1}}\right)$, and ${\sigma }_{v3}^1={\epsilon }_3^0{E}_1\left(1-e^{-\frac{1}{{\tau }_1}}\right)$. Therefore, ${\sigma }_{v1}^1={\sigma }_{v2}^1={\sigma }_{v3}^1=\frac{E_1}{E_0}\frac{P}{A}\left(1-e^{-\frac{1}{{\tau }_1}}\right)={\sigma }_v^1$. The reaction force components can be obtained as follows: $F_{v1}^1={\int }_0^l{B}^{T}A{\sigma }_{v1}^{1}dl={\int }_0^l\left[\begin{array}{l}-1/l\\ 1/l\end{array}\right]A{\sigma }_{v1}^{1}dl=\left[\begin{array}{l}-\frac{E_1}{E_0}P\left(1-e^{-\frac{1}{{\tau }_1}}\right)\\ \frac{E_1}{E_0}P\left(1-e^{-\frac{1}{{\tau }_1}}\right)\end{array}\right]=F_{v2}^1=F_{v3}^1$, thereby $F_v^1=\sum _{e=1}^3{F}_{ve}^{\prime }$ and $F_{ve}^{\prime }=N_e^T{F}_{ve}^1$ . Considering $N_1=\left[\begin{array}{cc}1& 0\\ 9/11& 2/11\end{array}\right]$ and $N_1^T=\left[\begin{array}{cc}1& 9/11\\ 0& 2/11\end{array}\right]$, $F_{ve}^{\prime }=N_e^T{F}_{ve}^1$ can be updated as $F_{v1}^{\prime }=N_1^T{F}_{v1}^1=\left[\begin{array}{cc}1& 9/11\\ 0& 2/11\end{array}\right]\left[\begin{array}{l}-\frac{E_1}{E_0}P\left(1-e^{-\frac{1}{{\tau }_1}}\right)\\ \frac{E_1}{E_0}P\left(1-e^{-\frac{1}{{\tau }_1}}\right)\end{array}\right]$. Introducing $E=\frac{E_1}{E_0}\left(1-e^{-\frac{1}{{\tau }_1}}\right)$ yields $F_{v1}^1=\left\{\begin{array}{l}-EP+\frac{9}{11}EP\\ \frac{2}{11}EP\end{array}\right\}=\left[\begin{array}{l}-\frac{2}{11}EP\\ \frac{2}{11}EP\end{array}\right]$, $F_{v2}^{\prime }=N_2^T{F}_{v2}^1=\left[\begin{array}{l}-\frac{3}{11}EP\\ \frac{3}{11}EP\end{array}\right]$, and $F_{v3}^{\prime }=N_3^T{F}_{v3}^3=\left[\begin{array}{l}-\frac{6}{11}EP\\ \frac{6}{11}EP\end{array}\right]$, finally resulting in the expression as $F_v^1=\sum _{e=1}^3{F}_{ve}^{\prime }=\left[\begin{array}{l}-EP\\ EP\end{array}\right]$.

At t = 1 s, the total load is $F^1=\left[\begin{array}{l}0\\ P\end{array}\right]+\left[\begin{array}{l}-EP\\ EP\end{array}\right]=\left[\begin{array}{l}-EP\\ P+EP\end{array}\right]$. Solve the equation: $K_0{\left\{u_E\right\}}_{t=1}=F^1$ and $\frac{6A{E}_0}{11l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]{\left\{\begin{array}{l}{u}_I\\ u_{II}\end{array}\right\}}_{t=1}=\left[\begin{array}{l}-\\ P_1\end{array}\right]$, where P₁ = P + EP. Under the boundary condition u_I = 0, $u_{II}^1=\frac{11P_{1}l}{6A{E}_0}$ and ${\left\{u_E\right\}}_{t=1}=\left\{\begin{array}{l}0\\ \frac{11P_{1}l}{6A{E}_0}\end{array}\right\}$. Downscaling again yields u₁ = u_I = 0, $u_2=\frac{P_{1}l}{3E_{0}A}$, $u_3=\frac{5}{6}\frac{P_{1}l}{E_{0}A}$, and $u_4=\frac{11}{6}\frac{P_{1}l}{E_{0}A}$; ${\epsilon }_1^1=\frac{P_1}{3E_{0}A}$, ${\epsilon }_2^1=\frac{P_1}{2E_{0}A}$, and ${\epsilon }_3^1=\frac{P_1}{E_{0}A}$.

For t = 2s, ${\sigma }_{v1}^2={\sigma }_{v2}^2={\sigma }_{v3}^2={\sigma }_v^2$, $F_v^2=\left[\begin{array}{l}-E\left(P{e}^{-\frac{1}{{\tau }_1}}+P_1\right)\\ E\left(P{e}^{-\frac{1}{{\tau }_1}}+P_1\right)\end{array}\right]$, and $F^2=\left[\begin{array}{l}0\\ P\end{array}\right]+F_v^2=\left\{\begin{array}{l}-\\ P_2\end{array}\right\}$ where $P_2=P+E\left(P{e}^{-\frac{1}{{\tau }_1}}+P_1\right)$. Solve the equation: $K_0{\left\{u_E\right\}}_{t=2}=F^2$, we obtain $\frac{6A{E}_0}{11l}\left[\begin{array}{cc}1& -1\\ -1& 1\end{array}\right]{\left\{\begin{array}{l}{u}_I\\ u_{II}\end{array}\right\}}_{t=2}=\left[\begin{array}{l}-\\ P_2\end{array}\right]$. Applying the boundary condition u_I = 0 yields $u_{II}^2=\frac{11P_{2}l}{6A{E}_0}$, and ${\left\{u_E\right\}}_{t=2}=\left\{\begin{array}{l}0\\ \frac{11P_{2}l}{6A{E}_0}\end{array}\right\}$. From downscaling analysis, u₁ = u_I = 0, $u_2=\frac{P_{2}l}{3E_{0}A}$, $u_3=\frac{5}{6}\frac{P_{2}l}{E_{0}A}$, $u_4=\frac{11}{6}\frac{P_{2}l}{E_{0}A}$; ${\epsilon }_1^2=\frac{P_2}{3E_{0}A}$, ${\epsilon }_2^2=\frac{P_2}{2E_{0}A}$, and ${\epsilon }_3^2=\frac{P_2}{E_{0}A}$. By repeating this process, the displacements for all subsequent time steps are obtained. The MsFEM results computed using the initial stress method are shown in Figure 5, together with the analytical EVCP solution. Figure 6 presents the corresponding relative error of the MsFEM analysis. The analysis flowchart is presented in supplemental information.

Figure 6.

Relative errors of the MsFEM results in the time domain

Result analysis

The results indicate that, even with only a single coarse element, the proposed method reproduces the same overall trend of the analytical solution and accurately captures the creep behavior of viscoelastic materials under constant loading. From the error curve in Figure 6, the error is initially very small, then increases over time, reaching a maximum of approximately 5% around 10 s, and gradually decreases thereafter. The error remains negative throughout, indicating that the numerical solution consistently underestimates the displacement relative to the analytical solution. The pattern of an initial increase followed by a decrease in error is attributed to the characteristics of the GMM. The relaxation modulus indicates that the curve is initially steep, with rapid changes in slope. After the relaxation time, the curve flattens, and the relaxation modulus varies only minimally. In this example, the relaxation time of τ₁ = 3.6s is assumed to be τ₁ = 3.6s, meaning that at t = 3.6s, the stress decays to $\frac{1}{e}$ of its initial value (a 63.2% reduction), indicating very rapid stress relaxation. Consequently, near the relaxation time (or within a similar time range), the material parameters change rapidly, and uniform time discretization leads to increased numerical error. Beyond this range, the error decreases. To improve accuracy, smaller time steps may be used near the relaxation time, while larger steps are appropriate farther from it.

The computations also reveal that, for an axial rod composed of three different materials, the conventional finite element method requires at least three elements. At each time step, the global stiffness matrix is of size 4 × 4. By contrast, the proposed MsFEM requires only a single coarse element, and the stiffness matrix for each time step is 2 × 2. For larger-scale applications, the size of the MsFEM system of equations is, therefore, considerably smaller than that of the conventional finite element method. A key advantage of the initial stress method is that the stiffness matrix remains unchanged throughout the analysis, and the basic functions are computed only once. At each time step, evaluating stress relaxation requires only the stress relaxation and strain from the previous step, without the need to store values from all historical data. As a result, computational efficiency increases and memory usage decreases.

Example and discussion: Homogeneous cantilever beam

Consider a cantilever beam with its left end fixed and its right end free (Figure 7). The beam has a length L = 192 cm, a height h = 32 cm, and a thickness b = 1 cm, with a moment of inertia $I=\frac{b{h}^3}{12}=\frac{{32}^3}{12}{\text{cm}}^4$. The free end is subjected to a tangential force P = 1000N, and the material has a Poisson’s ratio of μ = 0.3. The material’s viscoelastic behavior is modeled using the GMM with a single Maxwell element, as shown in Figure 7. The material parameters are: E_e = 2 × 10⁴MPa, E₁ = 3 × 10⁴MPa, η₁ = 1.08 × 10⁵ MPa s, and ${\tau }_1=\frac{{\eta }_1}{E_1}=3.6\mathrm{s}$, with an initial elastic modulus of E₀ = E_e+E₁ = 5 × 10⁴MPa. The objective is to compute the viscoelastic displacement of the cantilever beam under the applied load over 36 s. It should be emphasized that the proposed method is developed for viscoelastic materials of the generalized Maxwell type. As long as the relaxation modulus can be represented in the form of Equation 20, the proposed formulation and solution strategy remain applicable.

Figure 7.

Homogeneous cantilever beam under a tangential force at the free end and its generalized Maxwell model (GMM) with a single Maxwell element

This cantilever beam can be treated as a plane stress problem. According to elasticity theory,²³ the elastic displacement at the free end is $\Delta =\frac{P{L}^3}{3EI}=\frac{1000\times {1920}^3\times 12}{3\times 5\times {10}^4\times 10\times {320}^3}=\frac{216\times 4}{500}=1.728\text{mm}$. According to EVCP, the viscoelastic displacement at the free end can be computed as $\Delta \left(t\right)=\frac{P{L}^3}{3I}f\left(t\right)$, where $f\left(t\right)=\frac{1}{E_e}-\frac{E_1}{E_e\left(E_1+E_e\right)}{e}^{-\frac{E_1{E}_e}{\left(E_1+E_e\right){\eta }_1}t}=\left(5-3e^{-0.111111t}\right)\times {10}^{-5}$. When t = 0, $f\left(t\right)=2\times {10}^{-5}=\frac{1}{E_0}$ and Δ(t = 0) = 1.728 mm. When t = +∞, $f\left(t\right)=5\times {10}^{-5}=\frac{1}{E_e}$, yielding $\Delta \left(t=+\infty \right)=1.728\times \frac{5\times {10}^4}{2\times {10}^4}=1.728\times 2.5=4.32\text{mm}$.

For this plane stress problem, four-node rectangular finite elements are used. According to MsFEM, the beam is discretized into coarse and fine meshes. At the macroscale, the mesh consists of 12 coarse-element columns along the span and 2 element layers along the height, giving a total of 24 coarse elements. Each coarse element measures 16 cm × 16 cm. The total number of macroscopic meshes contains 39 nodes and 78 DOFs. The coarse-scale mesh layout is shown in Figure 8. Each coarse element is further subdivided into subgrid elements. Two fine-mesh schemes are adopted for comparison: 2 × 2 and 4 × 4. In the 2 × 2 scheme, each coarse element contains 4 fine elements, 9 internal nodes, and 18 internal DOFs. Each fine element measures 8 cm × 8 cm. In the 4 × 4 scheme, each coarse element contains 16 fine elements, 25 internal nodes, and 50 internal DOFs. Each fine element measures 4 cm × 4 cm. The two fine-mesh schemes are shown in Figures 8A and 8B. The geometric details of both discretization schemes are summarized in Table 1.

Figure 8.

Discretization of coarse-scale and fine-scale meshes

(A) Coarse-scale mesh, (B) 2 × 2 fine-mesh subdivision, and (C) 4 × 4 fine-mesh subdivision.

Table 1.

Mesh discretization for the cantilever beam

Scheme	Number of coarse elements	Dimension of coarse elements	Number of macroscopic nodes	Number of macroscopic DOFs	Number of fine elements	Dimension of fine elements	Number of internal nodes	Number of internal DOFs
2 × 2	24	16 × 16	39	78	4	8 × 8	9	18
4 × 4	24	16 × 16	39	78	16	4 × 4	25	50

Note: Element dimensions are in cm. Fine-mesh data refer to the subgrid elements within a single coarse element.

To analyze the cantilever beam using the proposed method, a MsFEM program was implemented in Julia following the procedures described in Section time-space multiscale finite element method framework for viscoelastic materials through the initial stress method. The vertical viscoelastic solutions at the midspan node pertaining to the free end, computed under different fine-mesh discretizations, are presented in Figure 9, together with the results from the analytical solution. Figure 10 shows the corresponding relative errors. Because the beam represents a periodic structure, the geometry and material properties of each coarse element are identical. Therefore, only a single coarse element is required for computing the basic functions and its equivalent stiffness matrix for a typical coarse element. The assembled global stiffness matrix remains constant throughout the subsequent displacement calculations, which significantly improves computational efficiency. In addition, because stress relaxation is computed recursively at each time step, the complete strain and stress history does not need to be stored. Only the strain and stress relaxation from the previous step are required, thus reducing memory usage.

Figure 9.

MsFEM results for viscoelasticity under different mesh discretizations, including the analytical solution and the numerical solutions

Figure 10.

Relative errors for different mesh discretizations in the time domain

Figure 9 shows the time-dependent vertical displacement solution. The figure includes the MsFEM results for the two fine-mesh discretizations and the analytical solution obtained via EVCP. For the applied load and material parameters, the initial vertical displacement at the free end is approximately 1.7 mm. Owing to the viscoelasticity of the material, the displacement increases over time under constant loading, reaching approximately 2.5 times the initial value at 36 s. This duration corresponds to ten times the relaxation time (τ = 3.6 s) of the viscous element in the GMM, a point at which the viscoelastic displacement is nearly stabilized. The results indicate that, for the same coarse-scale mesh, increasing the number of sub-elements within each coarse element improves computational accuracy. The relative errors shown in Figure 10 follow a trend similar to that observed in the one-dimensional axial rod example: the error initially increases, reaches a peak at approximately 10 s, and then decreases. As explained in Section 5.1.4, this behavior primarily results from the uniform time discretization scheme. In practice, a refined time discretization near the relaxation time can reduce this error. The analytical viscoelastic displacement was derived from the elastic solution using EVCP. The elastic solution was obtained under the constraints (u)_x=0,y=0=(v)_x=0,y=0 = 0 and ${\left(\frac{\partial v}{\partial x}\right)}_{x=0,y=0}=0$. Changing the constraints to (u)_x=0,y=0=(v)_x=0,y=0 = 0 and ${\left(\frac{\partial u}{\partial y}\right)}_{x=0,y=0}=0$ alters the form of the elastic displacement. The first constraint assumes that a differential segment parallel to the x axis at the fixed end remains parallel to the x axis after deformation, while a differential segment parallel to the y axis rotates slightly clockwise. By contrast, the second constraint assumes that a differential segment parallel to the y axis remains parallel to the y axis, while a differential segment parallel to the x axis rotates slightly counterclockwise. Neither constraint can be strictly realized in practice. In finite element analysis, the fixed boundary is typically approximated by constraining both horizontal and vertical displacements of all nodes at the fixed-end cross section to zero. This approximation differs slightly from the assumptions used in the analytical solution and partially contributes to the relative errors shown in Figure 10.

Example of a non-homogeneous cantilever beam

The geometric dimensions, constraint conditions, and loading scenario of the cantilever beam are the same as in the example from Section example of a non-homogeneous cantilever beam. However, the material is no longer homogeneous; instead, it is assumed that its elastic modulus varies randomly within a certain range. The algorithm proposed in this article can still be used for multiscale finite element analysis in this case. However, since the material parameters differ within each coarse element, the basis functions and equivalent stiffness matrices need to be computed individually for each coarse element before assembling the global stiffness matrix for solution.

As there is no analytical solution for the non-homogeneous cantilever beam, a fine-mesh discretization is employed for comparison. The beam is discretized into a 48 × 8 mesh–48 elements along the length and 8 layers along the height—resulting in an element size of 4 cm × 4 cm. This yields 384 elements, 441 nodes, and 882 degrees of freedom. The generalized Maxwell model shown in Figure 7 is still used to represent the material’s viscoelastic behavior. The elastic spring constant E_e for different elements is assumed to vary randomly between 2000 MPa and 40000 MPa, while the other viscoelastic parameters remain unchanged. Consequently, the initial elastic modulus E₀ of different elements randomly varies between 3.2 × 10⁴ MPa and 7 × 10⁴ MPa, forming a non-periodic structure. A viscoelastic analysis is performed based on this fine mesh using the initial stress method from reference,⁴² and the results serve as the benchmark for comparison.

When applying the proposed algorithm for multiscale analysis, the macroscopic mesh division is identical to that in the example from Section example and discussion: homogeneous cantilever beam: 12 elements along the length and 2 layers along the height. The resulting coarse mesh is shown in Figure 8A. The fine-scale mesh within each coarse element follows the 4 × 4 sub-mesh scheme depicted in Figure 8C, meaning the sub-element size is 4 cm × 4 cm. The mesh information for both the multiscale finite element method and the traditional finite element method is summarized in Table 2.

Table 2.

Mesh information for the non-homogeneous cantilever beam

Method	Number of Macroscopic Elements	Macroscopic Element Size	Number of Macroscopic Nodes	Macroscopic DOF	Number of Fine-scale Elements	Fine-scale Element Size	Number of Fine-scale Nodes
Traditional FEM	384	4 cm × 4 cm	441	882	–	–	–
FEM in this study	24	16 cm × 16 cm	39	78	16	4 cm × 4 cm	25

The multiscale finite element method adopted in this study relaxes the periodicity assumption required by asymptotic homogenization, making it applicable to non-periodic structures with random material variations or irregular geometries. Although the need to compute basis functions and equivalent stiffness matrices separately for each coarse element reduces computational efficiency compared to periodic cases, the independence of each sub-scale analysis allows for effective parallelization in large-scale problems. In this example, the proposed method reduces the degrees of freedom from 882 to 78, drastically cutting the cost of solving linear equations at each time step, while maintaining an error of only about 5–6% compared to traditional refined-mesh solutions—a level acceptable for engineering purposes, see Figure 11. Further improvements in accuracy could involve oversampling techniques or higher-order elements in future studies.

Figure 11.

Viscoelastic displacement at the midpoint of the free end of the non-homogeneous cantilever beam

Discussion

This study presents a time-space multiscale finite element framework for viscoelastic materials, utilizing the initial stress method and the generalized Maxwell model. The method efficiently captures rate-dependent and history-dependent behaviors while maintaining computational tractability through a constant stiffness matrix and a recursive stress-update scheme.

The framework constructs multiscale basis functions and equivalent stiffness matrices in a single initial stage. By exploiting the time-invariant stiffness matrix produced by the initial stress approach, it circumvents the repeated assembly of the global stiffness matrix at each time step, a routine requirement in traditional variable-stiffness methods. This design improves computational efficiency and is well-suited for long-term creep and stress-relaxation analysis of civil engineering materials and components.

Stress relaxation is updated using a recursive formulation that requires only the stress and strain states from the immediately preceding time step. This elegant scheme eliminates the need to store the complete history of field variables, thereby eliminating a major data storage bottleneck in conventional viscoelastic analysis. The significant reduction in memory requirements and enabling footprint paves the way for large-scale numerical simulations of complex structural systems containing viscoelastic components.

In contrast to asymptotic homogenization methods that require periodic microstructures, the proposed approach accommodates truly non-uniform material phases without scale-separation constraints. It accurately captures realistic microstructural topology in composites under general boundary conditions.

This framework provides a basis for modeling rate-dependent damage, aging effects, and thermo-mechanical coupling in building materials. It offers strong potential for durability evaluation and structural integrity assessment, offering an effective numerical tool for sustainable structural design.

Limitations of the study

Although the proposed method shows promising results, some limitations exist. The generalized Maxwell model may not capture complex nonlinear or aging behaviors under varying environmental conditions. Numerical errors can arise from time discretization, especially near rapid relaxation stages. For highly heterogeneous materials, the basic functions might not fully resolve local stress concentrations. The framework has yet to be validated under dynamic or coupled thermo-mechanical loads typical in real civil structures. Additionally, the preprocessing overhead increases for large non-periodic systems. Future work will extend the model to include damage and aging, improve time integration, and validate against full-scale experiments.

Resource availability

Lead contact

Further information and requests for resources should be directed to and will be fulfilled by the lead contact, Yanchao Wang (wang_yanchao2020@163.com).

Materials availability

This study did not generate unique reagents.

Data and code availability

• •All data reported in this article will be shared by the lead contact upon request.
• •This article does not report original code.
• •Any additional information required to reanalyze the data reported in this article is available from the lead contact upon request.

Acknowledgments

This research was supported by the Stable Support Project of the National Key Laboratory of Solid Rocket Propulsion, China (Grant No. SPL2025JJ002, SY41YYF202309043), the National Natural Science Foundation of China (Grant No. U22B20131, 52108387), and the Natural Science Basic Research Program of Shaanxi (Grant No. 2025JC-YBMS-508). All the support is gratefully acknowledged.

Author contributions

Conceptualization, X.W.; methodology, Y.W., H.Q., J.B., X.W., X.H., and Z.Z.; software, Y.W., J.B., and Z.Z.; writing – original draft preparation, X.W., Y.W., H.Q., J.B., Z.Z., and X.H.; writing – review and editing, X.W., Y.W., H.Q., J.B., Z.Z., and X.H.; visualization, Z.Z. and Y.W.; supervision, H.Q., X.H., and Z.Z.; funding acquisition, X.W. and Y.W. All authors have read and agreed to the published version of the article.

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have influenced the work reported in this article.

STAR★Methods

Key resources table

REAGENT or RESOURCE	SOURCE	IDENTIFIER
Software and algorithms
Python 3.11	Commercially Available Software	N/A
ABAQUS 2022	Commercially Available Software	N/A
Origin 2021	Commercially Available Software	N/A
Microsoft Office PowerPoint	Commercially Available Software	N/A

Experimental model and study participant details

This study does not use experimental model and subject details typical in the life sciences.

Method details

The core of the time-space multiscale analysis method for viscoelastic construction materials proposed in this study lies in integrating the Generalized Maxwell Model (GMM), the initial stress method, and the Multiscale Finite Element Method (MsFEM). By discretizing the computational domain into macroscopic coarse meshes and mesoscopic subgrids, it numerically constructs multiscale basis functions embedding mesoscopic heterogeneity and derives the macroscopic equivalent stiffness matrix. The initial stress method is utilized to convert the viscoelastic constitutive relation into a series of incremental elastic problems, and the viscous relaxation stress is updated via a recursive formula (requiring only stress and strain information from the immediately preceding time step). This method avoids Laplace transforms and periodicity assumptions, balancing computational efficiency and memory usage. Validated through cases of axial rods, homogeneous and heterogeneous cantilever beams, it can accurately capture the long-term creep and stress relaxation behaviors of materials, providing an efficient numerical tool for structural durability assessment.

Quantification and statistical analysis

This study does not include quantification and statistical analysis.

Published: February 21, 2026