Effective Mechanical Properties Research Articles

Effect of ferromagnetic particles on the elastic and magnetostriction parameters of bulk superconductors

Based on the magnetic dipole theory and the Eshelby's equivalent inclusion theory, a theoretical model of ferromagnetic particle inclusion-superconducting matrix is established in this paper, intrinsic strain is introduced and uniform stress boundary conditions are used to calculate the mechanical properties of the composite material. The effects of changes in parameters such as the elastic modulus, volume fraction ratio, and magnetic field of ferromagnetic particle inclusions on the effective mechanical properties of bulk superconductors such as the effective elastic modulus, effective Poisson's ratio, and effective magnetostriction were compared in detail, providing a theoretical reference for the preparation of superconducting materials containing ferromagnetic particle inclusions.

Geometric void-multiscale model for evaluating the effect of bead width and layer height on voids in FDM parts

PurposeThis paper aims to present a geometrical void model in conjunction with a multiscale method to evaluate the effect of interraster distance, bead (raster) width and layer height, on the voids concentration (volume) and subsequently calculate the final mechanical properties of the fused deposition modeling parts at constant infill.Design/methodology/approachA geometric model of the voids inside the representative volume element (RVE) is combined with a two-scale asymptotic homogenization method. The RVEs are subjected to periodic boundary conditions solved by finite element (FE) to calculate the effective mechanical properties of the corresponding RVEs. The results are validated with literature and experiments.FindingsBead width from 0.2 to 0.3 mm, reported a decrease of 25% and 24% void volume for a constant layer height (0.1 and 0.2 mm – 75% infill). It is reported that the void’s volume increased up to 14%, 32% and 36% for 75%, 50% and 25% infill by varying layer height (0.1–0.2 and 0.3 mm), respectively. For elastic modulus, 14%, 9% and 10% increase is reported when the void’s volume is decreased from 0.3 to 0.1 mm at a constant 75% infill density. The bead width and layer height have an inverse effect on voids volume.Originality/valueThis work brings values: a multiscale-geometric model capable of predicting the voids controllability by varying interraster distance, layer height and bead width. The idealized RVE generation slicer software and Solidworks save time and cost (<10 min, $0). The proposed model can effectively compute the mechanical properties together with the voids analysis.

Macroscopic mechanical properties of fluid-saturated sandstone at variable temperatures

Rocks can be viewed as composites of solid minerals with pores or cracks filled with softer material such as pore fluids, kerogen, bitumen, and other organic matter. The mechanical properties of highly viscous soft phases are highly sensitive to ambient temperatures and lead to temperature-dependent static and dynamic observations with the composite rock. However, the constituents operate by forming some effective (averaged) mechanical properties of the composite rock, and yet these averaged properties are still little known. To reveal such macroscopic temperature-dependent mechanical properties and measure their values in rock samples, a double-porosity model of porous rock with nonlinear viscosity is developed. The model is based on rigorous continuum mechanics with physically meaningful, real-valued, and time- and frequency-independent material properties and elegantly unifies the existing frequency-dependent microscopic squirt flow and mesoscopic wave-induced fluid flow models. The approach is used to accurately model the broad attenuation peaks and Young’s modulus dispersion observed in previously published laboratory experiments with glycerol-saturated Berea sandstone and invert for its mechanical properties. The observations are explained as mainly due to the temperature-dependent elastic coupling caused by non-Newtonian fluid within microcracks. Several hitherto unknown mechanical properties of the rock are constrained quantitatively: the average porosity of the microcracks, the effective high-pressure bulk modulus of the drained frame, the internal stiffness defect within the rock frame, the solid viscosities associated with bulk and shear deformations, and an exponent of nonlinearity for viscosity. These parameters constitute a Biot-consistent mechanical model of the rock, which can be used to simulate its behavior in arbitrary experimental environments. The rigorous first-principle model can be used in many applications: detailed and physically accurate interpretations of laboratory experiments, numerical wavefield simulations and seismic data inversion, reservoir characterization, geothermal exploration, thermal-enhanced oil recovery, and exploration for deep oil and gas resources in high-temperature environments.

Topology optimization of UAV structure based on homogenization of honeycomb core

In this paper, a new optimization design strategy is explored to achieve a rapid design of lightweight structures for unmanned aerial vehicles (UAVs) using honeycomb sandwich panels. Based on the existing performance equivalent method of honeycomb core material, an effective mechanical property homogenization model of honeycomb material is proposed and applied to finite element analysis. A finite element model is established to analyze the mechanical response of honeycomb sandwich panels under different bending loads, and the rationality of the model is verified by mechanical tests. The error between the simulation results and the experimental results is about 10%. Based on the variable density method and rational approximation of material properties interpolation method, a topology optimization model with the minimum structural flexibility as the objective is constructed. By comparing with the experiment-based design scheme, it is found that the optimized design scheme has a more reasonable structure and better performance. The optimized design can effectively reduce the structural weight and improve the load transfer path while ensuring structural stiffness. In this way, the influence of stress concentration and large deformation on the structure is avoided, which is of great significance for the design of UAV structures.

On-demand contactless programming of nonlinear elastic moduli in hard magnetic soft beam based broadband active lattice materials

Engineered honeycomb lattice materials with high specific strength and stiffness along with the advantage of programmable direction-dependent mechanical tailorability are being increasingly adopted for various advanced multifunctional applications. To use these artificial microstructures with unprecedented mechanical properties in the design of different application-specific structures, it is essential to investigate the effective elastic moduli and their dependence on the microstructural geometry and the physics of deformation at the elementary level. While it is possible to have a wide range of effective mechanical properties based on their designed microstructural geometry, most of the recent advancements in this field lead to passive mechanical properties, meaning it is not possible to actively modulate the lattice-level properties after they are manufactured. Thus the on-demand control of mechanical properties is lacking, which is crucial for a range of multi-functional applications in advanced structural systems. To address this issue, we propose a new class of lattice materials wherein the beam-level multi-physical deformation behavior can be exploited as a function of external stimuli like magnetic field by considering hard magnetic soft beams. More interestingly, effective property modulation at the lattice level would be contactless without the necessity of having a complex network of electrical circuits embedded within the microstructure. We have developed a semi-analytical model for the nonlinear effective elastic properties of such programmable lattice materials under large deformation, wherein the mechanical properties can be modulated in an expanded design space of microstructural geometry and magnetic field. The numerical results show that the effective properties can be actively modulated as a function of the magnetic field covering a wide range (including programmable state transition with on-demand positive and negative values), leading to the behavior of soft polymer to stiff metals in a single lattice microstructure according to operational demands.

Towards the design of nature-inspired materials: Impact of complex pore morphologies via higher-order homogenization

Even though the development of novel materials that mimic nature is widely used in a variety of engineering and scientific fields, the relationship between effective material properties and underlying, often complex pore morphology is still not fully understood. To address this knowledge gap and accelerate the development of novel nature-inspired materials, this paper adopts a higher-order asymptotic homogenization method to numerically investigate the effect of complex micropore morphology on the effective mechanical properties of a porous system. Specifically, we create unique pore morphologies with varying levels of complexity that serve as a more realistic representation of natural materials. We then use the second-order homogenization method to capture the role of pore size, shape, orientation, and distribution on effective properties. By creating different pore morphologies, we systematically studied the relationship between morphology and effective mechanical properties. The results highlight the necessity of higher-order parameters to fully capture the role of realistic pore morphologies on effective mechanical properties and provide a path forward in the design of nature-inspired materials.

Cage‐shaped octaphenyl silsesquioxane with micro‐nano dispersibility for strengthening intumescent flame retardancy in polypropylene composites

AbstractIn this work, a silicon‐, phosphorus‐, nitrogen‐containing flame retardant system was constructed via the combination of the caged organic octaphenyl silsesquioxane (OPS) and an intumescent flame retardant (IFR) composed of piperazine pyrophosphate and melamine polyphosphate. Compared with the IFR mixture alone, the designed OPS/IFR system simultaneously endowed polypropylene (PP) with a higher charring flame‐retardant effect and better mechanical properties under the same loading amount. With only 0.5 wt% OPS and 19.5 wt% IFR in PP, both 3.2 mm‐ and 1.6 mm‐thick flame‐retardant composite reached UL 94 V‐0 level, while the sample with IFR only passed the 3.2 mm‐thick UL 94 V‐0 rating. Moreover, compared with IFR, OPS/IFR with a specific mass ratio also performed higher limited oxygen indexes, improved anti‐dripping behavior in the glow‐wire test, and suppressed the heat and smoke release to a larger extent of PP composite. The main mechanism was that a phosphorus/silicon synergistic charring effect occurred in PP composites during the combustion, thus forming more compact char layers with rich phosphorus/silicon compounds. The investigation of the multielement OPS/IFR system also disclosed a more effective flame retardant working way in PP through synergistic charring effect. Meanwhile, due to the mico‐nanoscale dispersibility of OPS in PP, the constructed composites kept higher mechanical performances compared with the sample loaded with IFR alone.

Formulation and Evaluation of Film Forming Solution of Tavaborole for Treatment of Skin Infections

Film forming solutions (FFS) as a novel approach attracted the interest of pharmaceutical scientist for skin drug delivery system development and characterization concerning with their mechanical properties and water vapor permeability. FFS were developed by varying concentration of the film forming polymers also nature and content of the plasticizer. The formulations were prepared using 32 full factorial design. They were tested for drying time, drug release, antifungal activity, skin irritation and stability studies. The FFS was characterised for pH, viscosity, drug content, effective dosage volume and mechanical properties of the film formed after application; bioadhesion and water vapour permeability were also tested. All the formulations showed results within acceptable range for various tests. The optimized formulation showed drug release of 97.2% and antifungal activity in terms of efficacy as 99.2%. Such a formulation can be claimed to decrease duration of therapy, will be more accepted by the patients and be a breakthrough in treating skin infections.

INFLUENCES OF P AND Sr ON MICROSTRUCTURE AND ITS EFFECTS ON MACHINABILITY AND MECHANICAL PROPERTIES OF HYPEREUTECTIC Al-20%Si

Hypereutectic Al-Si alloys are widely utilized in the automotive industry due to their high strength-to-weight ratio, minimal thermal expansion, and superior castability. The downside of hypereutectic Al-Si alloys is the formation of coarse primary silicon particles. The primary Si phases were controlled by the growth-hindering agents, phosphorus (refiner), and strontium (modifier) by using a conventional stir casting technique at room temperature. The microstructural changes were observed through an optical microscope and SEM analysis. The additions of 0.08% P have ensured the formation of the uniformly distributed fine-grained particles in the alloy. The primary silicon particle size was reduced from 220[Formula: see text][Formula: see text]m to 150[Formula: see text][Formula: see text]m as compared to the untreated alloy. The tensile and yield properties of the treated alloy were increased by 12.7% when compared to the untreated alloy with a hardness of 131 BHN. The treated alloy imparted better impact toughness, ensuring a ductile mode of failure through the fractography studies. The influence of the microstructure on the machinability of the alloy was investigated in a dry environment with uncoated and coated inserts (code: CCGT 09T304 FL K10) by varying the process parameters, i.e. speed, feed rate, and depth of cut (DoC). Modification of the primary and eutectic Si phases of an Al-20Si hypereutectic alloy increases machinability with coated inserts as well as its mechanical properties.

Can Different Parameter Sets Lead to Equivalent Optima between Geometric Accuracy and Mechanical Properties in Arburg Plastic Freeforming?

Technological advances have led to the increased use of plastic-based additive manufacturing processes for the production of consumer goods and spare parts. For this reason, the need for the best possible mechanical properties while maintaining geometric accuracy is becoming increasingly important. One of these additive manufacturing processes is the Arburg Plastic Freeforming process, which differs from the widely used Fused Filament Fabrication process in the way that droplets are discharged along a track instead of continuous extruded tracks. As with all other plastic-based additive manufacturing processes, due to the round shape of the tracks, voids occur between the individual tracks during manufacturing, which effects mechanical properties. In contrast to previous work, which mainly focused on how the mechanical properties change with a change in a single printing parameter, this work focused more closely on the interaction of three relevant printing parameters considered as a parameter set. Their influence on the mechanical properties was investigated by tensile tests, the influence on the residual porosity by density measurements and the influence on the geometric accuracy by surface roughness measurements. It was shown that by considering the parameters as a parameter set, states of high density and therefore high mechanical properties while reaching minimal surface roughness can be achieved for significantly more combinations than previously assumed. However, for these states the residual porosity was slightly different. This difference was explained by a parameter-dependent deformation factor of the droplets, which influences the maximal possible degree of filling during manufacturing. For the optimization of arbitrary parameter sets, an analytical model was derived.

Enhanced energy absorption performance of 3D printed 2D auxetic lattices

Auxetic lattices have attracted increasing attention due to their unusual mechanical behavior and potential for an array of applications. However, a narrow window of stiffness realizable for a given cell topology limits their applications. In this study, a pair of novel 2D re-entrant auxetic lattices capable of exhibiting enhanced stiffness and energy absorption is proposed by introducing vertical ligaments into conventional re-entrant structures. These modified re-entrant auxetic lattices were realized via fused deposition additive manufacturing. The deformation patterns and the energy absorption characteristics of 3D printed auxetic lattices under quasi-static compression were investigated both via Finite Element (FE) simulations and experiments. The effective elastic stiffness of the proposed lattices was theoretically estimated. The FE results corroborated by experiments, elucidate the role of different sub-cells on the effective mechanical properties of the proposed auxetic lattices. The results indicate that the proposed structures — Type A and B variants, exhibit enhanced stiffness (+355%) and superior energy absorption (+165%) in comparison to conventional 2D re-entrant lattices of the same mass. Furthermore, the findings of the study suggest that the strength, stiffness, energy absorption capacity and Poisson’s ratio of 2D auxetic lattices can be tailored by tuning the sub-cell properties and cell wall thickness.

Experiments on the dynamic Poisson's ratio of PUR foams

AbstractPolyurethane (PUR) foams are used in various industrial applications, including vibration damping and noise reduction. Because the effective mechanical properties are needed for the engineering design process, this work presents a method to measure the dynamic Poisson's ratio for the typical range of sound frequencies. A hydraulic pulse testing machine is used to apply the harmonic displacement excitation of the cylindrical specimens for 1 − 100 Hz. The analysis of the time‐dependent deformation data is based on video recordings of a high‐speed camera and a specially derived image processing algorithm.For the Poisson's ratios of the measured foams no significant frequency dependence was found. For PUR foam with an open‐cell or mixed‐cell microstructure the dynamic Poisson's ratio is approximately 0.3. Poisson's ratios of about 0.4 were measured for PUR foams with a high density and a closed‐cell microstructure.

New preparation method of microstructurally and mechanically standardized PETG specimens by material extrusion additive manufacturing and machining

Additive manufacturing processes are recent and lack standardized methods to characterize the mechanical properties of the printed material. Sample preparations in the literature consist in the direct printing of specimens, which induces uncontrolled interactions between printing conditions of the specimens and material properties. This study aims to develop a method for the specimen preparation to characterize the behavior of materials manufactured by material extrusion additive manufacturing. The proposed method consists in preparing printed plates from polyethylene terephthalate glycol-modified filament into a unique design before machining out specimens at different orientation angles. A comparison to directly printed specimens at the same orientation angles is performed to confront the proposed method to a commonly used one in the literature, and to understand the cause of discrepancies and the lack of predictiveness in additive manufacturing. The prepared specimens are characterized by X-ray tomography and tensile tests to assess process-structure-property relationships. The analysis highlights the structure heterogeneity of the as-printed specimen compared to the machined one, which influences the mechanical properties of the specimen. The new hybrid additive-subtractive preparation method for the specimen limits the superimposed thermo-kinetic effects caused by changes in deposition strategy, including coalescence and flow instability, and allows to measure effective mechanical properties based on a homogenized multiscale structure that is more representative of large parts. In addition, machining smooths the surface without thermal alterations, which leads to more accurate mechanical testing regarding cross-sectional area measurement, crack initiation and fracture mode. The proposed method is thought to standardize the preparation of additively manufactured materials based on extrusion to ease comparison between studies, to help the qualification of technical and industrial parts, and to measure intrinsic material’s properties based on a controlled microstructure for understanding of the process-related behavior and properties of material.

Average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity: Effective elastic properties of composites

The effective mechanical properties of composites are strongly influenced by the morphology, volume fraction, and properties of the inclusion phase. Extensive theoretical models for the evaluation of effective properties are mainly based on the continuum micromechanics derived from the well-known Eshelby’s solution of an ellipsoidal inclusion problem, which are subjected to the limitations on the Eshelby tensor for a non-ellipsoidal inclusion. This work devises a three-step method to derive the average Eshelby tensor of an arbitrarily shaped inclusion from convexity to non-convexity in an elastic field. The proposed three-step method is user-friendly and avoids the singularity that arises from solving the complicated integrals for an arbitrarily shaped inclusion. The calculated average Eshelby tensors of inclusions with different morphologies, including Platonic solids, superballs, superellipsoids, doughnuts, helices, and non-convex particles, are thereafter compared against the existing data reported in the literature, in order to verify the reliability of the present method. Moreover, we also incorporate the proposed method into a micromechanical model to predict the effective elastic moduli of two-phase composites, in which congruent non-ellipsoidal inclusions of a specific volume fraction are randomly distributed in a homogeneous matrix. Comparisons with the direct finite element method (FEM) simulations indicate that the developed micromechanical model is a reliable means to evaluate the elastic properties of composites. The proposed average Eshelby tensor of an arbitrarily shaped inclusion can also be generalized to the theoretical predictions of the effective transport properties of composites with inclusions of diverse types and shapes, such as particles, rocks, granules, pores, fibers, and cracks.

A Study on Effective Compressive and Shear Properties of Tetrahedral Lattice Materials

Tetrahedral lattice materials can be used as the core of a sandwich structure. The properties of tetrahedral lattice materials can be controlled by modifying their geometrical parameters and relative density. In this paper, a tetrahedral lattice structure deformation mechanism-based theoretical analysis model is established to predict the effective mechanical properties of the structure under compressive and shear loadings. The analytical solutions are subsequently verified by finite element analysis of a large-scale lattice material model. Based on the obtained results, the effects of the geometrical parameters, relative density, and shear deformation are discussed. At a specific relative density, as strut inclination angle increases: (1) the effective compressive modulus in the z-direction increases; (2) the effective compressive modulus in x- and y-directions, and the effective shear modulus in xy-, xz-, and yz-directions firstly increases but then decreases; (3) the effective Poisson’s ratios [Formula: see text] and [Formula: see text] increase, whereas, [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text] decrease. With an increase in relative density, the effective compressive and shearing modulus increase, the effective Poisson’s ratios remain constant, [Formula: see text] and [Formula: see text] are always equal to 0 when the strut inclination angle is [Formula: see text]. The effect of shear deformation on the effective mechanical properties increases as the slenderness ratio increases. The predicted effective properties enable the tetrahedral lattice unit cells to be treated as “material” in the design and analysis process.

Thermo-Electro-Chemo-Mechanical Coupled Modeling of Solid Oxide Fuel Cell with LSCF-GDC Composite Cathode

Intricate relationships between transport phenomena, reaction mechanisms, and mechanical aspects likely affect the durability of solid oxide fuel cell (SOFC) stack. This study presents a modeling framework that combines thermo-electro-chemo models (including the methanol conversion process and the electrochemical reactions of the carbon monoxide as well as the hydrogen) and a contact thermo-mechanical model that considers the effective mechanical properties of composite electrode material. Detailed parametric studies are performed focusing on the inlet fuel species (hydrogen, methanol syngas) and flow arrangements (co-flow, counter-flow) under typical operating conditions (operating voltage 0.7 V), and performance indicators of the cell, such as the high-temperature zone, current density, and maximum thermal stress were discussed for parameter optimization. The simulated results show that the high temperature zone of the hydrogen-fueled SOFC is located at the central part of units 5, 6, and 7, and the maximum value is about 40 K higher than that of methanol syngas-fueled SOFC. The charge transfer reactions can occur throughout the cathode layer. The counter-flow improves the trend of the current density distribution of hydrogen-fueled SOFC, while the effect on the current density distribution of methanol syngas-fueled SOFC is small. The distribution characteristics of the stress field within SOFC are extremely complex, and the inhomogeneity of the stress field distribution can be effectively improved by feeding methanol syngas. The counter-flow improves the stress distribution state of the electrolyte layer of methanol syngas-fueled SOFC, and the maximum tensile stress value is reduced by about 37.7%.

On the best volume fraction distributions for functionally graded cylinders, spheres and disks – A pseudospectral approach

The paper is devoted to the optimization of axisymmetric structures made of functionally graded materials and subject to mechanical and thermal loads. The novelty of the results is that the volume fraction distribution is not limited to a power-law variation, as in most of the works available in the literature, but can be any (piecewise continuous) function. This approach leads to an intrinsic tailoring approach, in the sense it occurs without prefixing the spatial distribution of effective mechanical properties a priori, and therefore exploiting at best the inhomogeneity of functionally graded material. After recalling the governing equations and showing some recent results concerning candidate solutions for the optimal volume fraction distribution in some particular cases, several instances of the optimization problems aiming at minimizing occurring maximum stresses are formulated. We show that all these formulations can be treated within the same numerical approach based on the so-called pseudospectral methods. In the last part of the paper we describe how these methods have been effectively applied to the considered problems and we discuss the yielded solutions comparing them, where possible, with power-law solutions.

A flexible design framework to design graded porous bone scaffolds with adjustable anisotropic properties

Since the success of bone regenerative medicine depends on scaffold morphological and mechanical properties, numerous scaffolds designs have been proposed in the last decade, including graded structures that are suited to enhance tissue ingrowth. Most of these structures are based either on foams with a random pore definition, or on the periodic repetition of a unit cell (UC). These approaches are limited by the range of target porosities and obtained effective mechanical properties, and do not permit to easily generate a pore size gradient from the core to the periphery of the scaffold. In opposition, the objective of the present contribution is to propose a flexible design framework to generate various three-dimensional (3D) scaffolds structures including cylindrical graded scaffolds from the definition of a UC by making use of a non-periodic mapping. Conformal mappings are firstly used to generate graded circular cross-sections, while 3D structures are then obtained by stacking the cross-sections with or without a twist between different scaffold layers. The effective mechanical properties of different scaffold configurations are presented and compared using an energy-based efficient numerical method, pointing out the versatility of the design procedure to separately govern longitudinal and transverse anisotropic scaffold properties. Among these configurations, a helical structure exhibiting couplings between transverse and longitudinal properties is proposed and permits to extend the adaptability of the proposed framework. In order to investigate the capacity of common additive manufacturing techniques to fabricate the proposed structures, a subset of these configurations is elaborated using a standard SLA setup, and subjected to experimental mechanical testing. Despite observed geometric differences between the initial design and the actual obtained structures, the effective properties are satisfyingly predicted by the proposed computational method. Promising perspectives are offered concerning the design of self-fitting scaffolds with on-demand properties depending on the clinical application.

An integrated approach to evaluate mechanical performance of date palm filler reinforced polypropylene

In this study, the mechanical properties of newly developed bio-composites comprising both untreated and chemically treated date palm agro-residue fillers and polypropylene matrix are experimentally evaluated. Then, the experimental results are integrated into a numerical technique using a spring network model to estimate the tensile properties of the date palm filler in a powder form. Indeed, experimentally measuring these properties is a tedious task and time consuming. Besides, the spring network is employed to predict the effective mechanical properties of the bio-composite and simulate progression of fracture including crack onset and evolution in a two-dimensional model of the material. Next, the elastic modulus and strength as obtained numerically are fed into proper analytical models after adjusting relevant input parameters to calculate both stiffness and strength of the bio-composites. The best analytical models along with their adjusted empirical constants that closely fit the experimental outputs are identified. Both numerical and analytical modeling techniques exhibited excellent agreement with the experimental counterparts and as such could act as efficient tools in producing natural fiber polymeric composites with tailored mechanical properties. Besides, the derived bio-composites serve as eco-friendly, renewable, economically competitive material that can be used to design a broad range of products in industries including, construction, automotive, sport and others.

A new multi-scale modeling method for needled C/C composites

Needled composites have stronger interlaminar properties compared to two-dimensional fiber composite structures due to the introduction of z-directional fibers in the fiber composite layup. An effective multi-scale finite element model is developed in this paper for the complex structure of needle-punched composites. Three scales, namely fRVE, pRVE, and lRVE, are modeled from the fiber scale, the delamination scale, and the laminate scale. using ABAQUS, periodic boundary conditions are applied to the RVEs at different scales to obtain the effective mechanical properties of various RVEs. Meanwhile, the effect of needling density on the needled composites is predicted by applying the above multi-scale model. The increase of needling density can enhance the out-of-plane effective mechanical properties of the composites, but it also weakens the in-plane effective mechanical properties. The maximum error of the elastic modulus calculation is 17.68% with that of the reference, thus verifying the rationality of the multi-scale model.