Elastic-viscoelastic Correspondence Principle Research Articles

Dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading

This paper conducts the dynamic analysis of fractional poroviscoelastic reinforced subgrade under moving loading. Based on the Biot theory and transversely isotropic (TI) parameter expression of the geogrid reinforced subgrade, the governing equations of the poroelastic reinforced subgrade are established in the wavenumber domain by the double Fourier transform. Considering the viscosity of the soil skeleton and the flow-dependent viscosity between the soil skeleton and pore water, the governing equations are extended to the fractional poroviscoelastic medium by introducing the Zener viscoelastic model, fractional calculus theory and the dynamic elastic-viscoelastic correspondence principle. Combining boundary conditions and interlayer continuity conditions, the extended precise integration method (PIM) and double Fourier integral transform are employed to obtain the solution of fractional poroviscoelastic reinforced subgrade in the spatial domain. After the numerical validation, a sensitivity analysis of the relaxation time, permeability, reinforcement ratio and the load velocity are conducted.

Time-domain expression and mechanical response of the viscoelastic Poisson's ratio in asphalt pavement

Poisson's ratio (PR) is a fundamental mechanical parameter for materials. Unlike the constant Elastic Poisson's Ratio (EPR), the Viscoelastic Poisson's Ratio (VEPR) is significantly influenced by external factors such as environmental temperature, stress level, and loading frequency. Using the VEPR to characterize the stress state of asphalt pavements provides a more accurate reflection of their actual service conditions. Moreover, the VEPR plays a more critical role than the EPR in predicting fatigue life, analyzing temperature sensitivity, and optimizing pavement structure design. In this study, the Elastic-Viscoelastic Correspondence Principle (EVCP) and the Zener mechanical model were combined to derive a time-domain expression for the VEPR of asphalt pavement. Fiber Bragg Grating (FBG) sensors were embedded within various structural layers of an in-service road to collect mechanical response and temperature data under traffic loads. Based on the measured temperature data, Finite Element Models (FEMs) were developed for both the VEPR and EPR conditions. The mechanical responses were validated using these FEMs, and the results were compared with the strain data collected by the FBG sensors. The results indicated that both VEPR and EPR increase most rapidly between 20°C and 35°C, with VEPR resulting in higher transverse and longitudinal strains and stresses than EPR, while vertical strain remained constant. Furthermore, the VEPR was better suited to actual driving conditions, with a peak error of only 3.97 % and an average error of 10.04 %.

Analysis of reflective cracking in asphalt overlaid jointed concrete airfield pavements using a 3D generalized finite element approach

ABSTRACT Asphalt Concrete (AC) overlays are a common strategy for maintaining PCC airfield pavements. However, the movement of joints in the underlying pavement may lead to reflective cracks propagating to the overlay. The problem is highly three-dimensional resulting in mixed mode of cracks due to the aircraft gear loading configurations and underlying concrete pavement joint spacing and design. The development of a 3D model of airfield pavements to predict reflective cracking using Finite Element Method (FEM) and Generalized Finite Element Method (GFEM), is presented in this study. GFEM enrichment strategy and global-local approach are used to achieve efficient framework for developing the models from both computational and user time perspective. Viscoelastic fracture analysis was performed using elastic-viscoelastic correspondence principle. Sensitivity of the domain size, order of approximation and boundary conditions are investigated in the numerical simulations. It is shown that the crack initiation and propagation is a combination of Mode-I (tensile) and Mode-II (shearing) crack opening. The presented models contributes to the mechanistic empirical reflective cracking design algorithm for the FAA pavement design program.

Study on Stability of Elastic Compression Bending Bar in Viscoelastic Medium

In the southeastern coastal regions of China, thick layers of marine soft soil are widely distributed, exhibiting characteristics such as high compressibility, high porosity, low strength, high sensitivity, and easy thixotropy, and these viscoelastic behaviors of foundation soil have significant implications for elastic compression bending bar, as evidenced by issues such as post-construction settlement of roadbeds and long-term operation deviation of bridge pile foundations. In this study, a mechanical model of an elastic bar embedded in an elastic and viscoelastic medium, fixed at the base and free at the top, is established based on the Winkler foundation assumption. The deflection function of a bar subjected to both axial force and locally distributed horizontal load is derived using the Rayleigh-Ritz method. Utilizing the elastic-viscoelastic correspondence principle, the viscoelastic medium surrounding the bar is modeled as an elastic medium in which the ground reaction coefficient varies within phase space formulation. This study provides a robust theoretical foundation for soft soil foundation engineering projects and fills a significant gap in the literature by offering a comprehensive framework for understanding displacement in elastic bars within viscoelastic media. Drawing upon the derivation of the deformation function for elastic rods within a viscoelastic medium, the findings of this research hold significant applicability across a range of domains. These include, but are not limited to, the expansion of roadways in regions characterized by coastal soft soil, as well as the monitoring of deformation and lifespan in bridge pile foundations.

Analysis of fractures in linear viscoelastic media using a generalized finite element method and the elastic–viscoelastic correspondence principle

A methodology for an efficient analysis of 3-D fracture mechanics problems in linear viscoelastic media is presented. It combines the Elastic–Viscoelastic Correspondence Principle (EVCP), a Generalized Finite Element Method (GFEM), and an inverse Laplace transform (ILT) method to compute Energy Release Rate (ERR) and Crack Mouth Opening Displacement (CMOD) for viscoelastic fracture problems in the time domain. The method is denoted EVCP-GFEM-ILT and consists of solving a reference elastic problem in the Laplace domain using a quadratic p-hierarchical GFEM, computing quantities of interest in this domain, and transforming them to the time domain with the aid of a numerical ILT method. One of the key features of the method is that a single elastic solution is required, regardless of the time interval of the target viscoelastic problem. A detailed verification of the methodology against analytical and reference numerical solutions is performed. Problems subjected to either non-homogeneous Neumann or Dirichlet boundary conditions of increasing complexity are considered. Two ILT methods are studied and the one based on Fourier series is shown to deliver accurate results for all tested boundary conditions, as well as non-trivial viscoelastic material models. The computational cost of the method applied to a 3-D problem is compared with the one for Abaqus which solves the problem directly in the time domain using a FEM mesh with quarter-point elements. A fully 3-D mixed-mode fracture problem with a non-trivial viscoelastic material model is solved to highlight the applicability of the proposed methodology. Finally, a problem of practical interest is analyzed in order to show how the methodology proposed in this work can be applied to efficiently handle large-scale simulations.

Viscoelastic analysis of surface deformation above salt caverns for gas storage based on semi-infinite space

The construction of salt caverns for energy storage is of strategic significance, but the operation of underground gas storages in salt formations will cause surface deformation. The research and analysis of surface deformation is an important part of the stability evaluation of salt caverns for gas storage. From the perspective of classical mechanics theory, the analytical solution of surface deformation above salt caverns for gas storage is deduced. The surface displacement caused by the contraction of a salt cavern is regarded as the boundary displacement caused by the surface force acting on a spherical cavern in a semi-infinite space. The equivalent conversion of load form is carried out based on the mirror principle. The vertical subsidence and horizontal displacement of the surface of the elastic semi-infinite space is obtained. According to the elastic-viscoelastic correspondence principle, the viscoelastic solution of the surface displacement of the semi-infinite body, regarded as Maxwell body, can be obtained. A numerical model of a salt cavern for gas storage is established based on the engineering practice. The results of the analytical solution and the numerical simulation are compared and analyzed to verify the accuracy of the analytical solution. The analytical solution was applied to predict the surface displacement above a group of caverns, and the influence of the layout of salt caverns and cavern center spacing on the surface displacement is analyzed. The research results show that the regular triangular layout is slightly better than the square layout. The larger the cavern center spacing, the smaller the maximum surface subsidence. The proposed theoretical prediction model can provide a theoretical basis for controlling surface deformation for the construction of gas storage caverns.

Microbuckling behavior of unidirectional fiber-reinforced shape memory polymer composite undergoing compressive deformation

The microbuckling mechanics of unidirectional fiber-reinforced shape memory polymer composite (SMPC) with low fiber volume fraction were investigated, and the mechanical models were formulated considering the attenuation of shear strain in the resin matrix near the buckled fibers. We deduced the analytical expression of the attenuation function and the key parameters during the microbuckling process of SMPC, including the critical buckling wavelength and strain. The values determined by finite element analysis verified the accuracy of the above theoretical predictions. The nonlinear stress–strain relationship during the post-buckling process was also investigated. Additionally, the classical elastic–viscoelastic correspondence principle was established to determine the dynamic buckling behaviors of SMPC at different temperatures. The viscoelastic parameters of shape memory polymer (SMP) were obtained from the isothermal stress relaxation experiments, and then the variation of buckling wavelength with time at different temperatures was evaluated.

Thermal buckling and free vibration of viscoelastic functionally graded sandwich shells with tunable auxetic honeycomb core

In this paper, thermal buckling and free vibration of viscoelastic functionally graded sandwich shell with tunable auxetic honeycomb core is investigated. The sandwich shell has functionally graded material (FGM) skins at top and bottom and tunable auxetic honeycomb core at the middle. The viscoelastic behavior of the skins and core are modeled according to complex modulus approach in combination with the elastic–viscoelastic correspondence principle. Poisson's ratio of the auxetic honeycomb core is tunable. It has positive or negative value, which depends on an angle parameter of the unit cell of the auxetic honeycomb core. The equations of motion for the viscoelastic FGM sandwich shell with tunable auxetic honeycomb core are derived using Hamilton's principle. Analytical solution for simply supported shell is obtained using Navier's solution procedure. The effects of the geometry parameters, temperature rise, parameters of the tunable auxetic honeycomb core are studied.

Finite volume based asymptotic homogenization of viscoelastic unidirectional composites

The previously developed finite volume based asymptotic homogenization theory designed for unidirectional composites with finite-sized microstructures is further extended to non-ageing linear viscoelastic region via correspondence principle. The theory is a unified framework able to accommodate general loadings, both out-of-plane and in-plane. It first transforms the problems in time domain to Laplace domain via elastic–viscoelastic correspondence principle, solves the problems in transformed region, and transforms the solutions back to time domain through numerical inversion technique. In Laplace domain, the microscale problems are solved up to the third term of expansion, and the homogenized stiffness tensors of all orders are passed to the macroscale, leading to the solutions of macroscale displacements. The method’s capability is assessed through comparison with direct numerical solution, showing increasing accuracy with increasing truncation order and decreasing scale ratio, and further enhancement by reconstructing boundary layer. The robustness of theory is verified through parametric studies. The possibility of reducing the highest order of spatial derivative of homogenized displacement is also investigated. The finite volume based methodology marks the theory’s uniqueness among the existing higher order asymptotic homogenization methods, providing an efficient alternative to solve the viscoelastic composite structural problems.

A viscoelastic constitutive model for shape memory polymer composites: Micromechanical modeling, numerical implementation and application in 4D printing

In this work, a novel micromechanics-based thermo-viscoelastic constitutive model for shape memory polymer composites (SMPCs) is proposed and applied to four-dimensional (4D) printed SMPCs. The multi-branch constitutive model is used to simulate the time- and temperature-dependent mechanical behavior of the shape memory polymer matrix. According to the elastic–viscoelastic correspondence principle, the equivalent viscoelastic stiffness tensor of the composite is obtained in the micromechanics framework of energy-based effective strain theory and the Mori-Tanaka homogenization scheme, in which a two-parameter interfacial damage model is adopted to consider the displacement discontinuity and traction continuity conditions at the interface between the inclusion and the matrix. The numerical integration scheme for the three-dimensional (3D) viscoelastic constitutive model of SMPCs is presented, and the finite element application is implemented by the user material subroutine UMAT of ABAQUS. After identifying the model parameters with experimental data, the tensile stress-strain curves and stress relaxation phenomena of 4D printed unidirectional SMPCs are successfully described by theoretical simulations. Besides, theoretical simulations also adequately predict the shape memory behavior of 4D printed composites and complex members.

Stroh formalism for various types of materials and deformations

Abstract The Stroh formalism is a complex variable formulation developed originally for solving the problems of two-dimensional linear anisotropic elasticity. By separation of the third variable for the linear variation of displacements along the thickness direction, it was proved to be applicable for the problems with coupled stretching-bending deformation. By the Radon transform which maps a three-dimensional solid to a two-dimensional plane, it can be applied to the three-dimensional deformation. By the elastic-viscoelastic correspondence principle, it is also valid for the viscoelastic materials in the Laplace domain. By expansion of the matrix dimension, it can be generalized to the coupled-field materials such as piezoelectric, piezomagnetic and magneto-electro-elastic materials. By introducing a small perturbation on the material constants, it can also be applied to the degenerate materials such as isotropic materials. Thus, in this paper, the Stroh formalism for several different types of materials (anisotropic elastic, piezoelectric, piezomagnetic, magneto-electro-elastic, viscoelastic) and deformations (two-dimensional, coupled stretching-bending, three-dimensional) are organized together and presented in the same mathematical form.

An analytical model for 3D consolidation and creep process of layered fractional viscoelastic soils considering temperature effect

The behavior of long-term deformation of saturated soil under external loads has been one of the research focuses in geological engineering and geotechnical engineering. The temperature change, especially in the fields of geological radioactive waste disposal and geothermal system, usually significantly influence the consolidation and creep process of soils. This paper presents an analytical three-dimensional (3D) model for investigating the consolidation and creep behavior of layered soils considering temperature effect. Based on the analytical layer element elastic solution, fractional calculus theory and elastic-viscoelastic correspondence principle, an analytical 3D model is built by introducing the coefficients of thermal expansion and viscosity varying with temperature into the fractional Merchant model (FMM). Verifications against published analytical and numerical results are provided, followed by extensive parametric studies to investigate the effects of type of viscoelastic model, temperature change, fractional order and viscosity of FMM, layered characteristic of soils on consolidation and creep process. It’s evident that the present model predicts long-term behavior of layered viscoelastic soils well and can be used to evaluate the effect of temperature change on consolidation and creep process in geological engineering and geotechnical engineering.

Time-domain asymptotic homogenization for linear-viscoelastic composites: mathematical formulae and finite element implementation

Asymptotic homogenization (AH) is a rigorous method used for modeling multiscale mechanical behavior in periodic composites. Unlike conventional AH methods, which predict the effective viscoelasticity using the elastic–viscoelastic correspondence principle, an AH method was proposed to directly calculate the effective material properties in the time domain, thereby enabling the complex transform to be omitted between the time and Laplace domains. This time-domain AH method is reformulated based on the integral form rather than the differential form of the Kelvin–Voigt viscoelastic model. Thus, multiple time-domain-based characteristic elasticity and viscosity displacement tensors are replaced by one integral form characteristic displacement tensor in the polymer matrix composite. Using mathematical formulae, a numerical implementation method was then developed by establishing location- and time-related stress loads. An in-house Python subroutine was also programmed to compute the effective viscoelastic properties from the finite element implementation results. Unidirectional and woven fabric composite homogenization results were comparable to those obtained using the conventional representative volume element method, indicating high effectiveness and accuracy. The results suggest that the proposed time-domain AH method is a powerful tool for modeling the two-scale mechanical behavior of viscoelastic composites and is compatible with the existing commercial software.

Viscoelastic adhesive contact between a sphere and a two-dimensional nano-wavy surface

For contact involving viscoelastic materials with nanoscale roughness surfaces, both the viscoelasticity and the adhesive interaction play significant roles, and an in-depth understanding of this viscoelastic adhesive contact behavior is essential. This paper establishes a numerical viscoelastic adhesive contact model between a smooth sphere and a two-dimensional nano-wavy surface based on the Derjaguin approximation and elastic–viscoelastic correspondence principle, and the wavy surface is considered by introducing the height distribution into the surface gap equation. Influences of approach speed, contact position and wavy surface parameters on the viscoelastic adhesive contact behavior are quantitatively analyzed. It is shown that the viscoelasticity can be ignored at large approach speed, while when the speed is small, viscoelasticity significantly increases the maximum contact area and pull-off force. Variations of maximum contact area and pull-off force with contact position or wavy surface parameters are not completely consistent, where the contact positions with relatively large maximum contact area may exhibit small pull-off force. The pull-off force changes monotonically with increasing wavy parameter, while the maximum contact area changes nonmonotonically. This study can offer guidance for the application of wavy surface in viscoelastic adhesion regulation, and provide basis for the surface design and parameter determination of viscoelastic materials.

Constitutive Model of N15 Propellant considering the Confining Pressure Effect

To describe the effect of confining pressure on the mechanical responses of N15 propellant, a constitutive model considering the confining pressure effect was first established for N15 propellant based on the elastic-viscoelastic correspondence principle. Then, the mechanical properties of N15 solid propellant under different confining pressures were obtained using confining pressure test system, and the obtained results indicate that the initial modulus of propellant did not change with confining pressure, but the maximum tensile strength, rupture strength, the maximum elongation, and elongation at break increased with increasing confining pressure. In conjunction with propellants’ mesoscopic structure and cross-section analysis, the mechanical mechanism of confining pressure effect on propellant was initially disclosed. Due to confining pressure, the particle dewetting inside the propellant was reduced, the hole propagation was delayed, and crack extension inhibited germination, proving that confining pressure has a strengthening impact on the propellant. Finally, assuming that the model parameters were dependent on pressure, the model parameters acquisition and validation were conducted. The results demonstrated that constitutive model can describe confining pressure influence on the mechanical properties of N15 propellant accurately.

Theoretical prediction of elastic modulus at different states and squirt-flow-related attenuation: extension of Cracks-Pores Effective Medium model

SUMMARYSquirt flow plays an essential role in elastic modulus dispersion and attenuation for fluid-saturated cracked porous rocks. The Mavko–Jizba model and relevant modified models can describe the squirt flow well based on the related elastic moduli, such as dry/drained bulk modulus. However, when these elastic moduli are challenging to attain, it is impossible to model the squirt-flow-related elastic moduli and attenuations with the models. On the other hand, the effective medium theory (EMT) model can estimate these elastic moduli, but cannot predict the undrained/relaxed and partially relaxed saturated elastic moduli and the squirt-flow-related attenuations. This paper extended an EMT model—Cracks–Pores Effective Medium (CPEM) model—to cover the undrained/relaxed and partially relaxed states following the elastic–viscoelastic correspondence principle. The proposed model [i.e. frequency-dependent CPEM (CPEMF) model] can thus estimate the elastic moduli over the different states (dry/drained, undrained/relaxed, partially relaxed and unrelaxed) and associated attenuations. It agrees well with the prediction of the modified Mavko–Jizba–Gurevich model (MJGZ-HF) at unrelaxed state and is precisely consistent with the prediction of Gassmann at undrained/relaxed state. Also, it analytically shows good consistency with the modified Mavko–Jizba–Gurevich model (MJGZ-MF) at partially relaxed state. The numerical simulations of CPEM/CPEMF models and MJGZ-HF/MJGZ-MF models show good agreement at the different states. Furthermore, we interpreted the experimental data on a basaltic sample and a sandstone sample with the CPEM/CPEMF models. The CPEMF model's predictions of elastic modulus at different states and associated modulus dispersion/attenuation are in good agreement with the corresponding measured ones, suggesting that the proposed CPEMF model can efficiently predict the elastic moduli at different states (dry/drained, undrained/relaxed, partially relaxed and unrelaxed) and quantify the squirt-flow-related elastic modulus dispersion and attenuation among different states well.

Transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids

Consider the transient thermal stress analysis for a material that is anisotropic (direction-dependent), viscoelastic (time-dependent), and thermorheologically simple (temperature-dependent). To solve this problem, three fundamental theories are employed: (1) complex variable Stroh formalism for anisotropic elasticity, (2) elastic-viscoelastic correspondence principle (EVCP) or time-stepping method (TSM) for viscoelasticity, and (3) time-temperature superposition principle (TTSP) for thermorheologically simple materials. By using the Stroh formalism, all different kinds of anisotropic elastic materials can be analyzed by using the same formulae. With EVCP and TSM, the problem of viscoelasticity can be solved via its corresponding problem of elasticity. By TTSP, the temperature-dependent constitutive relation can be expressed by using the reference temperature and the reduced time defined through shift factor. Thus, through the combined use of these theories, the complicated problems of transient anisotropic thermo-viscoelasticity can be solved by using the available solution tool for anisotropic elasticity such as the boundary element method (BEM). Based upon this concept, two novel BEMs are developed for the transient thermal stress analysis of temperature-dependent anisotropic viscoelastic solids. One is BEM-EVCP, and the other is BEM-TSM. The final results show that both of them can provide accurate solutions, and BEM-TSM is much more efficient than BEM-EVCP.

On the consolidation and creep behaviour of layered viscoelastic gassy sediments

Gassy sediments are frequently encountered in engineering geology and geotechnical engineering. Gas bubbles contained in these sediments significantly change the engineering properties, and further influence the consolidation and creep process. The long term behaviour of gassy sediment is drastically different from that of saturated and unsaturated soils. This study presents an analytical solution for exploring such behaviour within layered viscoelastic gassy sediments under different type of load. An Analytical layer element method together with a Laplace-Hankel transform are employed to solve the governing equations, and the viscoelastic solution is then captured with the aid of the elastic-viscoelastic correspondence principle. Comparisons of the present solution with published semianalytical solutions and experimental results, for a single and double layered soils, are provided. Extensive parametric analyses were carried out to explore the effects of the soil skeleton model, fractional order, type of load, saturation degree and stratification characteristics on consolidation and creep processes. In the present model, the permeability is saturation degree- and porosity-dependent and the modulus of elasticity is time-dependent, which can simulate the long term behaviours of gassy sediments reasonably.

Metoda VECD jako narzędzie do oceny trwałości zmęczeniowej konstrukcji nawierzchni

Postęp technologiczny i wprowadzenie do stosowania zaawansowanych materiałów o ponadstandardowych właściwościach, np. mieszanek mineralno-asfaltowych z lepiszczami wysokomodyfikowanymi polimerami HiMA (ang. highly modified asphalt), wymaga poszukiwania metody umożliwiającej kompleksowe uwzględnienie cech tych materiałów w obliczeniach. Jedną z takich metod może być metoda wykorzystująca model rozproszonego niszczenia w ośrodku lepkosprężystym (ang. viscoelastic continuum damage model, VECD). W artykule zaprezentowano metodologię opartą na modelu VECD oraz możliwości, jakie daje przy wymiarowaniu nawierzchni drogowych. Opisano poszczególne komponenty metody: analogię sprężysto-lepkosprężystą, mechanikę rozproszonego niszczenia i zasadę równoważności temperaturowo-czasowej z przyrostowym niszczeniem. Przedstawiono rów¬nież koncepcję uproszczonych obliczeń w metodzie S-VECD (ang. simplified viscoelastic continuum damage model). Artykuł zawiera opisy opracowanych kryteriów zniszczenia. Przybliżono również metody badań, które dostarczają danych wejściowych. W końcowej części artykułu przedstawiono przykładową analizę porównawczą z wykorzystaniem metody S-VECD dla konstrukcji z mieszankami mineralno-asfaltowymi z HiMA w warstwie podbudowy i mieszankami mineralno-asfaltowymi z HiMA we wszystkich warstwach.

Analytical solutions and boundary element analysis for holes and cracks in anisotropic viscoelastic solids via time-stepping method

In this study the time-stepping method, which can convert the problem of anisotropic viscoelasticity into two sets of elastic-like system, is employed for both analytical solutions and boundary element method (BEM) to solve the two-dimensional problems of holes and cracks in anisotropic viscoelastic solids. Based upon this method and the available solutions for the corresponding elastic problems, some analytical solutions for the two-dimensional problems of holes and cracks in anisotropic viscoelastic solids are presented, such as the one with an elliptical hole, a straight crack, two collinear cracks, or collinear periodic cracks under uniform load at infinity. We also present the fundamental solution for the problems with an elliptical hole, and use this solution to construct a special boundary element method (SBEM). Further extension of SBEM is made to develop the boundary-based finite element method (BFEM) to deal with the problems with multiple holes and cracks. The advantage of SBEM and BFEM is that no meshes are required on the boundaries of holes and cracks. To show the accuracy and efficiency of the proposed method, several representative examples are implemented and compared with those obtained by the finite element method and the elastic-viscoelastic correspondence principle. The results show that the solutions obtained by different methods are in well agreement with each other, and the ones obtained by the time-stepping method are around 3 to 36 times faster than the others depending on the problems and methods.