Representative Unit Cell Research Articles

Multi-scale modelling of the damage behavior for 3D woven composites with variant structure

In the process of fabricating a three-dimensional (3 D) woven perform, the variant structures introduced during adding and reducing tows, lead to changes of local tow orientation and woven pattern, which affects the mechanical response of 3 D woven composite. In this study, specimens with variant structure were manufactured by adding and reducing tow techniques, and representative unit cells of normal and variant structure were established by topological method. A multi-scale damage model was proposed to analyze the damage behaviors and predict the strength of 3 D woven composites with the user subroutine UMAT of ABAQUS/Standard. The results show that the variation of tensile modulus and strength between the experimental and simulation value were less than 4%. In addition, it is proved that the damage propagation of the macro-scale model with variant structure can be successfully captured.

A numerical study about the effects of the metal volume fraction on the effective properties of a porous piezoelectric composite with metalized pore boundaries

This work investigates a developed porous piezoelectric composite with metal layers introduced at the interfaces between the piezoelectric and vacuum phases. These metal layers can be added to improve the mechanical and electromechanical properties of the composite. The ordinary porous piezocomposite, which contains only vacuum pores, and the piezocomposite with metal inclusions, whose pores are filled with metal, were explored as specific cases. We constructed the finite element model for all composites understudy and determined their macro properties using the effective moduli theory, which depends on the well-known Hill-Mandel principle. We considered a simple representative unit cell cubic volume, which consists of a piezoelectric matrix with a spherical pore at its center. We compared the properties of all systems and explored the effects of the metal fraction increase on the effective moduli. There are drastic effects on the dielectric permittivity and piezoelectric properties of the considered composite due to the presence of metal inclusion. With an appropriate design of the porous piezocomposite with a metalized pore surface, better effective piezoelectric strain coefficients and transverse transduction coefficient can be achieved. As a result, it is expected that piezoelectric transducers made from this piezocomposite would perform well in piezoelectric motors, transverse piezoelectric sensors, and transverse piezoelectric energy harvesters.

A modified series-parallel electrical resistivity model of saturated sand/clay mixture

Electrical resistivity tomography, ERT, is a geophysical method that is used to image subsurface geomaterials for various geological, environmental and geotechnical engineering problems. The success of ERT depends on the availability of a robust electrical resistivity model of soils that is able to characterise the different type of soils using its electrical resistivity measurements. The aim of this paper is to present an electrical resistivity/conductivity model for isotropic saturated sand/clay mixture that considers the effects of sand content, and surface conduction of clay content. The proposed model is formulated using a modified series-parallel mixing model theory which uses phase-volumetric and surface conduction model parameters. The phase-volumetric parmeters describe the configuration of the different soil/water phases in a representative soil unit cell which includes solid soil particles, free water, diffuse double layer (DDL) water. The surface conduction model parameters are used to describe the volume fraction of DDL water in the unit cell, and its electrical conductivity compared to the electrical conductivity of free water. Simple test methods are used in this study to determine the proposed model parameters. Laboratory experimental programme was also conducted in this study to validate the proposed model. Two types of clays, namely, kaolin and bentonite, and one type of clean sand were used to constitute five different sand/clay and clay/clay mixtures. Dynamic compaction method using Standard Proctor tools was used to constitute the testing samples. The electrical conductivities of different reconstituted soil samples were measured and compared with the proposed model predictions, and a good agreement was achieved. The small differences between the model predictions and the experimental measurements could be attributed to the slight deviation of the laboratory test samples from the model assumptions, which require the soil sample to fulfil fully saturated and electrical isotropy conditions.

Design of Kinematic Connectors for Microstructured Materials Produced by Additive Manufacturing.

The main characteristic of materials with a functional gradient is the progressive composition or the structure variation across its geometry. This results in the properties variation in one or more specific directions, according to the functional application requirements. Cellular structure flexibility in tailoring properties is employed frequently to design functionally-graded materials. Topology optimisation methods are powerful tools to functionally graded materials design with cellular structure geometry, although continuity between adjacent unit-cells in gradient directions remains a restriction. It is mandatory to attain a manufacturable part to guarantee the connectedness between adjoining microstructures, namely by ensuring that the solid regions on the microstructure’s borders i.e., kinematic connectors) match the neighboring cells that share the same boundary. This study assesses the kinematic connectors generated by imposing local density restrictions in the initial design domain (i.e., nucleation) between topologically optimised representative unit-cells. Several kinematic connector examples are presented for two representatives unit-cells topology optimised for maximum bulk and shear moduli with different volume fractions restrictions and graduated Young’s modulus. Experimental mechanical tests (compression) were performed, and comparison studies were carried out between experimental and numerical Young’s modulus. The results for the single maximum bulk for the mean values for experimental compressive Young’s modulus () with 60 show a deviation of . The single maximum shear for the experimental compressive Young’s modulus mean values () with 60, exhibit a deviation of . For graded structures, the experimental mean values of compressive Young’s moduli (), compared with predicted total Young’s moduli (), show a deviation of for the bulk graded structure. The main results show that the single type representative unit-cell experimental Young’s modulus with higher volume fraction presents a minor deviation compared with homogenized data. Both (i.e., bulk and shear moduli) graded microstructures show continuity between adjacent cells. The proposed method proved to be suitable for generating kinematic connections for the design of shear and bulk graduated microstructured materials.

Temperature Dependent Modelling of Fibre-Reinforced Thermoplastic Organo-Sheet Material for Forming and Joining Process Simulations

Joining and local forming processes for fibre-reinforced thermoplastics (FRTP) like hole-forming or variations of the clinching process require an in-depth understanding of the process induced effects on meso-scale. For numerical modelling with a geometrical description of a woven fabric, adequate material models for a representative unit cell are identified. Model calibration is achieved employing a mesoscopic finite-element-approach using the embedded element method based on tensile tests of the consolidated organo-sheets and a phenomenological evaluation of photomicrographs. The model takes temperature dependent stiffness and fibre tension failure into account.

Mechanical properties of a hybrid auxetic metamaterial and metastructure system

We propose in this work an innovative hybrid auxetic metamaterial with a centersymmetric unit cell and tessellation topology similar to the one provided by the missing rib configuration. The tessellation proposed is applied to different core unit cells (star shape, cross-chiral shape with same dimensions, and reentrant). The effects of the geometric parameters of the cells on the in-plane mechanical properties of this hybrid auxetic metamaterial system are investigated via finite elements (FEMs). Representative unit cells (RUCs) with optimal mechanical behaviors are identified; those configurations exhibit the larger negative Poisson’s ratios and enhanced specific moduli. Designs related to two groups of auxetic metastructures with cylindric and cubic shapes are then developed based on the optimized RUCs along x and y directions. The equivalent mechanical performance of these metastructures under internal pressure is evaluated from a numerical standpoint. Auxetic cylindrical metastructures can be tailored by adjusting the number of the optimized RUCs along the circumferential and longitudinal directions, together with the geometric parameters of the optimized RUC itself. These hybrid auxetic metamaterials and metastructures provide the potential for multifunctional applications in biomechanics, flexible electronics, and aerospace.

Representative unit cell model of elastic spherical particle composite with interphase and/or general imperfect interface

We consider the representative unit cell (RUC) model of the elastic spherical particle composite with (a) interphase layer and (b) general imperfect interface in the form proposed by Bövik (1994). The rigorous analytical solution to both RUC model problems is obtained by the multipole expansion method in a unified manner. Fulfillment of the matrix-to-inhomogeneity elastic contact conditions reduces the boundary value problem to the linear algebraic system for multipole strengths and provides a highly efficient algorithm for numerical study. The proposed approach applies to all physically consistent models of imperfect interface including the nonlinear ones. In so doing, use of the vector spherical harmonics makes the solving procedure simple and straightforward. Comparison of the accurate numerical data obtained from the interphase and interface models confirms correctness of the Bövik model and illustrates its applicability limits in the multi-particle environment. The developed RUC model enables consideration of composite with the spherical particles of diverse size, properties, interphases and interface bonding, with an adequate account of interaction between the inhomogeneities. Comparison with the laboratory test data for syntactic foam shows that a careful modeling of microstructure is a prerequisite for reliable prediction of the effective elastic properties of composite.

Broadband, frequency-independent acoustic metasurfaces through inverse design

The recent development of acoustic metasurfaces has broadened the capabilities of wave engineering devices with a compact profile. However, the application of acoustic metasurfaces in many scenarios has been limited by their bandwidths. In this work, we report the realization of broadband acoustic metasurfaces that deliver frequency-independent functionalities. Three examples are demonstrated, namely beam steering, focusing and acoustic levitation with fixed refraction angle, constant focal depth and stable levitation. This is achieved by analyzing the requirement of each functionality and engineering the dispersion of the unit cells. The architecture of the unit cells is obtained through inverse design and each unit cell displays a unique dispersive response outlined by the theoretical requirements. The designed metasurface is experimentally verified and we demonstrate a fractional bandwidth of around 100% for all three functionalities. We also attempt to reveal the mechanism for such broadband by characterizing the behavior of representative unit cells.

A micro-mechanical constitutive model for cracked plies of laminated composites considering the constraint effect of the adjacent plies

Three types of representative unit cell (RUC) of [±θ/904]S composite laminates and cracked plies were created to obtain the constitutive model of the cracked plies and to compare their accuracy of the calculation. The RUCs represent the damaged laminates, cracked plies, and cracked plies with additional constraints. A correction factor was defined to improve the accuracy of transverse modulus estimation using the RUC of cracked plies with constraints. The correction factor and numerical results were discussed to illustrate the constraint effect of the adjacent plies on the cracked plies in terms of stiffness, orientation, and thickness.

Flexural edge waves in a Kirchhoff plate carrying periodic edge resonators and resting on a Winkler foundation

Flexural edge waves in a Kirchhoff plate, carrying periodically spaced spring–mass​ resonators at its edge and laid on a Winkler foundation, are considered. A spectral element method is applied on a representative unit cell to develop the dynamic stiffness matrix of the structure. In light of structural periodicity, Bloch wave theorem is used to derive a quadratic eigenvalue problem of the wave propagation constants, which are numerically solved for. Frequency bands are analyzed using the attenuation and phase constants. Two types of bandgaps are identified: one is attributed to the existence of attached resonators and the other is due to the fact that these resonators are arranged periodically. It is found that these bandgaps are influenced by the value of the resonator’s natural frequency and the stiffness of the elastic foundation. The analytically realized bandgap structures agree very well with the finite-element obtained dispersion curves by COMSOL.

Multiscale analysis of out-of-plane masonry elements using different structural models at macro and microscale

A novel two-scale modeling approach, linking different structural models at macro and microscale, is proposed to describe response of masonry walls with periodic texture. At the higher macroscopic scale, the real heterogeneous material is modeled as a homogenized medium, considering the classical Mindlin-Reissner theory for flat shells. At the lower microscopic scale, a representative masonry Unit Cell (UC), accounting for the actual geometry, arrangement and nonlinear behavior of constituent materials, is analyzed in detail by resorting to a three-dimensional Cauchy model. The UC is modeled as the assembly of elastic bricks and nonlinear zero-thickness interfaces, in which the sliding frictional and damaging mechanisms are concentrated. To perform the macro–micro information transition a proper kinematic map is defined, whereas the upscaling process is performed via a homogenization procedure based on the Transformation Field Analysis (TFA), properly extended to the case of interfaces. The developed homogenization procedure invokes a generalized Hill-Mandel principle and requires to satisfy ‘non-standard’ constraints at the microlevel, for which the perturbed Lagrangian method is employed.Numerical applications are performed to prove the model efficiency in describing the response of a running bond UC subjected to in-plane and out-of-plane loads. Special attention is devoted to the analysis of shell bending and shear behavior, comparing the results obtained with the proposed model with those recovered by detailed micromechanical analyses.

Flow and thermal transport characteristics of Triply-Periodic Minimal Surface (TPMS)-based gyroid and Schwarz-P cellular materials

High-porosity “cellular materials” are desirable for manufacturing multifunctional heat exchangers. While stochastic metal foams and regular strut-based topologies have been the focus of many prior investigations, there is very limited research on mathematically defined, triply-periodic minimal surfaces (TPMS). To this end, convective heat transport in sheet-based TPMS topologies, viz. gyroid and Schwarz P are numerically investigated and compared against “Tetrakaidecahedron,” which is a representative unit cell of stochastic metal foams. The candidacy of sheet-based topologies over strut-based in conjugate thermal transport applications was evaluated by plotting the product against the pumping power, where gyroid outperformed the other two topologies.

Thermo-Viscoelastic Response of 3D Braided Composites Based on a Novel FsMsFE Method.

A homogenization-based five-step multi-scale finite element (FsMsFE) simulation framework is developed to describe the time-temperature-dependent viscoelastic behavior of 3D braided four-directional composites. The current analysis was performed via three-scale finite element models, the fiber/matrix (microscopic) representative unit cell (RUC) model, the yarn/matrix (mesoscopic) representative unit cell model, and the macroscopic solid model with homogeneous property. Coupling the time-temperature equivalence principle, multi-phase finite element approach, Laplace transformation and Prony series fitting technology, the character of the stress relaxation behaviors at three scales subject to variation in temperature is investigated, and the equivalent time-dependent thermal expansion coefficients (TTEC), the equivalent time-dependent thermal relaxation modulus (TTRM) under micro-scale and meso-scale were predicted. Furthermore, the impacts of temperature, structural parameters and relaxation time on the time-dependent thermo-viscoelastic properties of 3D braided four-directional composites were studied.

Anisotropic behaviors of helically corrugated cylindrical shells: Homogenized in-plane stiffness

Thin-walled corrugated structures have been widely used in engineering applications for centuries, because corrugation enables engineers to tailor directional dependent properties despite the structures being made of isotropic materials. However, the vast majority of research has paid attention to corrugated plates and relatively scarce studies are on corrugated shells. This paper, therefore, aims to investigate elastic behaviors of thin cylindrical shells that possess sinusoidal corrugating patterns in the helical direction. Elastic responses of undulated shells under three types of deformation, namely, axial elongation, axial torsion, and radial expansion are thoroughly investigated by using the finite-element method (FEM) via ABAQUS™. The effective stiffness of the corrugated cylinders is determined by a homogenization technique in a representative unit cell element. A variety of FEM models are created according to four groups of dimensionless parameters, i.e. radius-to-thickness ratio, radius-to-corrugating depth ratio, helical angle and the number of waveform (thread) in a cylinder cross section. The results show that the degree of anisotropy induced by the corrugated shells is different from that of a fiber-reinforced composite material, significantly sensitive to their configurations, and can be extraordinarily higher than the perfect round shell made of the same parent material. Unorthodox coupling quantities of torsional-axial, torsional-radial, and axial-radial coupling deformations are encountered at some specific geometries. Anisotropy design space in the form of spherical maps is given as a guideline to effectively design or select a proper helically corrugated cylindrical shell with the desired stiffness properties.

Homogenization towards chiral Cosserat continua and applications to enhanced Timoshenko beam theories

A homogenization methodology for the construction of effective Cosserat substitution media for heterogeneous materials is proposed, combining a variational principle in linear elasticity with the extended Hill-Mandel lemma accounting for the introduced generalized kinematics. The proposed method is general and can be applied to a wide class of architected materials and composites prone to such micropolar effects. The microscopic displacement field of the initially heterogeneous continuum splits into a homogeneous part polynomial in the generalized kinematic measures and a fluctuation involving localization operators. The tensors of effective micropolar moduli are formulated as integrals over a representative unit cell utilizing of the displacement localizators, solution of classical, and higher-order unit cell problems. The proposed method has the chief advantage of delivering size-independent higher-order effective moduli and to remedy the deficiencies of most of existing higher order homogenization schemes towards generalized continua in the literature. Based on the developed homogenization method, the effective micropolar moduli of the tetrachiral lattice and composites made of a tetrachiral lattice reinforcement are computed to elaborate an enhanced Timoshenko microstructured beam model exhibiting couplings between different deformation modes induced by the response of its underlying tetrachiral microstructure.

A simplified metaelastic model for coated sphere-filled random composites

A simplified metaelastic model is presented to study long-wavelength dynamics of random composites filled with coated rigid spheres under the condition that the characteristic wavelength of the displacement field is much larger than the average distance between adjacent coated rigid spheres. The model is characterized by a simple differential relation between the displacement field of the composite and the displacement field of the mass center of a representative unit cell. The validity and accuracy of the model are demonstrated by comparing its predicted bandgap frequencies with known numerical and experimental data. The efficiency and merits of the model are demonstrated by applying it to study vibration isolation of coated rigid sphere-filled composite rods and (periodic or non-periodic) free vibration caused by initial displacement or velocity disturbance of the embedded rigid spheres inside an otherwise static composite rod. The proposed model could offer a simple method to study various long-wavelength metaelastic dynamic problems of coated rigid sphere-filled random composites.

High-order three-scale computational method for elastic behavior analysis and strength prediction of axisymmetric composite structures with multiple spatial scales

A novel high-order three-scale (HOTS) computational method for elastic behavior analysis and strength prediction of axisymmetric composite structures with multiple spatial scales is developed in this paper. The multiple heterogeneities of axisymmetric composite structures we investigated are taken into account by periodic distributions of representative unit cells on the mesoscale and microscale. First, the new micro–meso–macro coupled HOTS computational model for elastic problems of axisymmetric composite structures is established based on multiscale asymptotic analysis, which breaks through the traditional multiscale assumptions and includes three main components. Two classes of mesoscopic and microscopic auxiliary cell functions are constructed on the mesoscale and microscale, respectively. The macroscopic homogenization problems are defined on global axisymmetric structures by twice up-scaling procedures from microscale to mesoscale and then from mesoscale to macroscale. Moreover, the asymptotic HOTS solutions are constructed for approximating multiscale elastic problems of axisymmetric structures and the numerical accuracy analysis of the HOTS solutions is given in detail. Then, the strength prediction formulas for axisymmetric composite structures with multiple spatial scales are presented based on the high-accuracy elastic behavior analysis from the proposed HOTS computational model. Furthermore, the corresponding HOTS numerical algorithm based on the finite element method (FEM) is presented for analyzing the mechanical behaviors and predicting the strength of axisymmetric composite structures with multiple spatial scales in detail. Finally, some numerical examples are reported to verify the feasibility and effectiveness of the proposed HOTS computational method. In this study, a unified three-scale computational framework is offered, which enables the simulation of mechanical behaviors of axisymmetric composite structures with multiple spatial scales.

Influence of non-uniformly periodic distribution of fibers in composites on the stress field and effective shear modulus under anti-plane shear

Based on the complex variable techniques combined with the boundary collocation method, a semi-analytical procedure is proposed to explore the anti-plane shear behavior of non-uniformly periodic fibrous composites under uniform remote shear loadings. Specific series with unknown coefficients are introduced to describe the complex potentials of the representative unit cell (RUC) of the composite. The unknown coefficients are determined from the continuity conditions on the interface, the periodic boundary conditions imposed on the edge of the unit cell and the remote loading conditions. Once the complex potentials are determined, the effective moduli of the composite are obtained according to the average-field theory. Extensive numerical examples are provided to investigate the influence of the periodic distribution of the inclusions, the moduli of each component, and the volume fraction of the inclusions on the local stress fields and effective shear moduli of the composite. Numerical results show that the inhomogeneous distribution of the inclusions may induce a continuous area with high equivalent stress in the matrix under certain loadings and material properties of each component. However, we could overcome this disadvantage by changing the components or the periodic arrangement of the inclusions. For the case of a RUC with double inclusions, the two inclusions with the same shear moduli arranged symmetrically in the direction of 45 degrees may induce the transverse isotropy of the composite. Nevertheless, when the two inclusions are horizontally arranged and the volume fraction of the inclusions is large enough (more than 10%), the composite generally shows orthotropic characteristics.

Homogenous asymptotic analysis on vibration energy dissipation characteristics of periodical honeycomb reinforced composite laminate filled with viscoelastic damping material

In this paper, free transverse vibration energy dissipation properties of honeycomb reinforced composite laminated structures are studied. By applying the homogenous asymptotic method (HAM) and representative unit cell (RUC) approach, considering the frequency- and temperature-dependent properties of the viscoelastic damping material (VDM), the constitutive equations of the laminated composite structure is obtained. Based on the transverse wave energy propagation and transmission theory in solid structures, the structural loss factor (SLF) of the composite structure can be ultimately determined, which is a frequency- and temperature-dependent variable according to the same properties of VDM’s physical parameters that were applied. Several geometry factors including the thickness of the VDM layer, the width and the inner angle of the honeycomb reinforcements, influence on the vibration energy dissipation characteristics of the composite structure are investigated qualitatively and quantitatively. The results are validated by the Galerkin method based asymptotic approach, which shows consistency with each other, especially when the external excitation frequency larger than 500 Hz in the considered frequency ranges.

Micromechanics-based constitutive modeling of magnetostrictive 1–3 and 0–3 composites

This study presents micromechanics models for the analysis of magneto-thermo-mechanical coupled hysteresis responses for magnetostrictive composites, comprising of an elastic polymer reinforced by magnetostrictive continuous fibers and particles, respectively, subjected to a cyclic magnetic field loading under various circumstances. Constitutive relation for the magnetostrictive reinforcements that can describe nonlinear strain and magnetic flux density in a variety of thermal environments is employed, while the polymeric matrix is assumed to be a linear elastic material. A simplified unit-cell micromechanics model is developed for the magnetostrictive composites whose microstructures are idealized with periodically distributed arrays of cubic representative unit cells. To obtain the overall nonlinear responses of the magnetostrictive composites, linearized micromechanical relations are first used to provide trial solutions followed by iterative schemes in order to correct errors from linearizing the nonlinear responses. The micromechanical estimations are validated by experimental data available in literature. For comparison, the Mori-Tanaka micromechanics model is further reformulated for the magnetostrictive composites. The presented micromechanical formulations can reveal the effects of constituent volume fraction, applied prestress, temperature, the geometry of the magnetostrictive reinforcement, and the direction of applied magnetic field on the overall performance of the magnetostrictive composites under periodic magnetic field loadings.