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Micromechanics: Overall Properties of Heterogeneous Materials
Abstract
Part 1 Overall properties of heterogeneous solids: aggregate properties and averaging methods aggregate properties, averaging methods elastic solids with microcavities and microcracks linearly elastic solids, elastic solids with traction-free defects, elastic solids with micrcavities, elastic solids with microcracks elastic solids with micro-inclusions overall elastic modulus and compliance tensors, examples o elastic solids with elastic micro-inclusions, upper and lower bounds for overall elastic moduli, self-consistent differential and related averaging methods, Eshelby's tensor and related topics solids with periodic microstructure general properties and field equations, overall properties of solids with periodic microstructure, mirror-image decomposition of periodic fields. Part 2 Introduction to basic elements of elasticity theory: foundations geometric foundations, kinematic foundations, dynamic foundations, constitutive relations elastostatic problems of linear elasticity boundary-value problems and extremum principles three-dimensional problems solution of singular problems. Appendix: references.
... Here, we investigate the equivalent properties of the PZT-Ni and the OPS as functions of the inclusion's volume proportion while taking polarization field homogeneity or non-uniformity into account. A simple periodic RVE with only one spherical inclusion was created, and the boundary value problem was solved using the finite element method (FEM) while taking into account linear essential boundary conditions, which are identical to periodic boundary conditions [20][21][22][23]. Finally, the effective moduli were calculated from the FEM findings using the Hill-Mendel principle. ...
... Using the conventional approach of the efficient moduli theory [15,[21][22][23], we applied the linear essential displacement and electric potential boundary conditions as follows: ...
... Here <...> = (1/| |) (...) d , δ ij is the Kronecker delta, S 0 , E 0 are arbitrary nonzero constants, characterizing macroscopic values of deformations and E-field strength, respectively. As can be seen from (18)- (22), in total, from the solutions of problems AQ4 I-V, a complete set of 10 effective moduli of the piezoceramic composite is determined, and the same piezoelectric moduli are found twice and, up to the error of numerical methods, should have the same values [23]. ...
This study compares the equivalent properties of a PZT-Ni composite to those of a standard PZT-Air porous piezocomposite. The effective properties of both composites were estimated principally numerically using the finite element method and the Hill-Mendel principle, with fully homogeneous and highly inhomogeneous polarized piezoceramic matrix models. The numerical findings concerning the fully polarized model were verified analytically using Mori-Tanaka homogenization technique. The elastic moduli of PZT-Ni and PZT-Air composites and the dielectric permittivity of PZT-Air composite exhibited no significant dependence on polarization inhomogeneity; however, the piezoelectric coefficients showed a considerable dependence on the chosen polarizations model. The incorporation of metal inclusions into the piezoceramic matrix increases the polarization field at the interface, which improves the composite’s homogenized dielectric permittivity moduli. The PZT-Ni composite’s improved dielectric permittivity boosts its efficiency in lighting electronics, voltage controllers, and multilayer small-volume high-performance capacitors.
... Consider a volume V in a three-dimensional Euclidean space E 3 bounded by a regular surface ∂V . The conservation of linear momentum is (Landau & Lifshitz 1959b;Nemat-Nasser & Hori 2013) ...
... where ∇ ∇ ∇ is the del operator, ⊗ is the tensor product, the superscript " T " denotes transpose, ε ε ε is the strain tensor (ε ε ε is the strain-rate tensor), v is the velocity field. The del operator, ∇ ∇ ∇, is a vectorial differential operator, denoted by (Li & Wang 2008;Nemat-Nasser & Hori 2013) ...
Understanding the underlying mechanisms of seismic attenuation and moduli dispersion in fluid-saturated cracked porous rocks is of great importance for the development of non-invasive methods to characterize the subsurface. Wave-induced fluid flow at the pore scale, so-called squirt flow, is responsible for seismic attenuation and moduli dispersion at sonic and ultra-sonic frequencies and may be relevant at seismic frequencies. The squirt flow associated attenuation is usually quantified using analytical models. However, numerical experiments suggest that the squirt flow related dissipation is sensitive to fine details of the pore geometry, which can only be modeled numerically. Most of the existing numerical studies explore this phenomenon using simplified models, and there is a lack of numerical studies that model the physics in realistic pore geometries with sufficient numerical resolution. As a result, the impact of wave-induced fluid flow on the effective static and time-dependent mechanical characteristics in realistic settings is still poorly understood. I address these issues by developing a numerical method to model the effective mechanical properties of a hydro-mechanically coupled system at the pore scale suitable for graphical processing units (GPUs). A numerical evaluation of attenuation and modulus dispersion due to squirt flow in models based on 3D micro-tomography images of cracked Carrara marble is presented. It is shown that the local hydraulic conductivity can be quantitatively estimated from the numerically evaluated effective properties. The accuracy of the numerical results is carefully analyzed. This study improves the understanding of the underlying mechanisms of attenuation and moduli dispersion in fluid-saturated cracked rocks. The new method can be applied to model squirt flow for entire laboratory samples in the centimeter scale which was not possible a decade ago.
... Among the physical parameters of materials, Young's modulus and Poisson ratio are the most important. However, Young's modulus parameter of different single-layer materials is different; the equivalent Young's modulus of multilayer materials is different in different bending directions and needs theoretical calculation [11]. ...
... I z and W of hollow circular cross section are respectively calculated by Eqs. (11) and (12). ...
High-temperature superconducting (HTS) cable, with massive current carrying capability and low electric power loss, is always at the cutting edge of the strong electric fields. The concentric three-phase HTS cable usually subjects to the impact of electromagnetic and mechanical forces. The forces will lead to the shape change of the cable, which may damage the cable and cause the degeneration of the critical current (Ic). In this paper, an analysis model of stress-strain and bending properties of the 10 kV/1 KA cable based on laminated theory is built. Laminated beam theory can simplify REBCO superconducting tape and concentric three-phase HTS cable to analyze stress-strain distribution. A finite element method (FEM) simulation model is established to analyze the critical bending radius of the HTS cable. Meanwhile, the normalized of the critical current of the cable is obtained at different bending radii. The analysis results will provide the theoretical basis for cable pipelaying and line relay protection in grid.
... Among those well-established methods are the Mori-Tanaka (MT) model, the self-consistent (SC) and generalised self-consistent schemes, or differential and incremental schemes; cf. [12]. More elaborate models allow for taking into account the secondary factors influencing the effective properties, namely packing and size effects. ...
... It can also be seen that the degree of anisotropy increases with the increase of the Poisson's ratio of the matrix material, and for a given porosity and pore distribution, the greatest anisotropy is obtained when the matrix material is incompressible, i.e., for m = 0.5. Formulas (12) and specifically (14)-(16) provided by the mean-field cluster scheme enable straightforward and immediate finding of the effective properties required, e.g., in large-scale simulations of structures made of composites or porous materials. However, their safe use should be preceded by model validation. ...
Although the elastic properties of porous materials depend mainly on the volume fraction of pores, the details of pore distribution within the material representative volume are also important and may be the subject of optimisation. To study their effect, experimental analyses were performed on samples made of a polymer material with a predefined distribution of spherical voids, but with various porosities due to different pore sizes. Three types of pore distribution with cubic symmetry were considered and the results of experimental analyses were confronted with mean-field estimates and numerical calculations. The mean-field ‘cluster’ model is used in which the mutual interactions between each of the two pores in the predefined volume are considered. As a result, the geometry of pore distribution is reflected in the anisotropic effective properties. The samples were produced using a 3D printing technique and tested in the regime of small strain to assess the elastic stiffness. The digital image correlation method was used to measure material response under compression. As a reference, the solid samples were also 3D printed and tested to evaluate the polymer matrix stiffness. The anisotropy of the elastic response of porous samples related to the arrangement of voids was assessed. Young’s moduli measured for the additively manufactured samples complied satisfactorily with modelling predictions for low and moderate pore sizes, while only qualitatively for larger porosities. Thus, the low-cost additive manufacturing techniques may be considered rather as preliminary tools to prototype porous materials and test mean-field approaches, while for the quantitative and detailed model validation, more accurate additive printing techniques should be considered. Research paves the way for using these computationally efficient models in optimising the microstructure of heterogeneous materials and composites.
... Another important scheme is the well-known two-scale computational homogenization (FE 2 ) which determines the effective properties by two nested BVPs (boundary-value-problems) along with the corresponding scale transition law, see for instance [49][50][51][52][53][54][55][56][57][58][59][60][61] In this case, the material behavior at the microscopic level is analyzed by employing the representative volume element concept, whereas a homogenization technique is considered to compute the macroscopic response. We refer to [62][63][64] for fundamental homogenization principles of local mechanical responses. The above-introduced analytical and physically motivated mathematical models lead to pronounced computational costs. ...
... The goal is to return this constitutive information from a finer scale to the macro level. The RVE acts as a statistically representative portion of the heterogeneous microstructure (grains separated by grain boundary, voids, inclusion, crack, and other similar defects), see [64]. Its size must be chosen such that it is large enough to be representative or rather such that it sufficiently accounts for the character and distribution of heterogeneities. ...
Material modeling using modern numerical methods accelerates the design process and reduces the costs of developing new products. However, for multiscale modeling of heterogeneous materials, the well-established homogenization techniques remain computationally expensive for high accuracy levels. In this contribution, a machine learning approach, convolutional neural networks (CNNs), is proposed as a computationally efficient solution method that is capable of providing a high level of accuracy. In this work, the data-set used for the training process, as well as the numerical tests, consists of artificial/real microstructural images (“input”). Whereas, the output is the homogenized stress of a given representative volume element RVE\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {RVE}$$\end{document}. The model performance is demonstrated by means of examples and compared with traditional homogenization methods. As the examples illustrate, high accuracy in predicting the homogenized stresses, along with a significant reduction in the computation time, were achieved using the developed CNN model.
... There are various kinds of microstructure distribution in composites, and periodic microstructure is one of the typical distribution forms. The periodic structure has basic cells with repeated distribution, which indicates that the arrangement of inclusions in the material is from completely disordered to strictly ordered [1][2][3][4]. Many natural materials have periodic or nearly periodic microstructures. ...
... Modern composite material design, especially computer-aided material design, usually refers to the design of cells with periodic distribution, which can be directly developed by microstructure engineering to develop various new materials. For the research on mechanical properties of periodic microstructures, the analysis methods include micromechanics methods [3,4] (e.g., dilute method, self-consistent method, differential method, Mori-Tanaka method, generalized self-consistent method), Fourier series expansion method [6], asymptotic homogenization method [7], interaction direct derivation [8], periodic finite element method [9], boundary element method [10], eigenfunction-variational method [11], hybrid element method [12], eigenelement method [13], analytic function boundary value problem theory [14], etc. ...
The effective elastic properties of composites with double periodic nanofibers are studied theoretically. Based on the Gurtin–Murdoch surface elasticity theory, the elastic field in the nanocomposite can be expanded by applying a functional variational method to a unit cell. The analytical solution of the effective anti-plane shear modulus of the periodic nanocomposites is presented. The convergence of the analytical results is discussed. The comparisons of the obtained macroscopic and nanoscale solutions with the existing results show the effectiveness and accuracy of the proposed method. Based on the analytical solution obtained, the size effect of the effective properties of the periodic nanocomposites is discussed. The effects of the period ratio of microstructure, the fiber/matrix stiffness matching and the nanoporous volume fraction on the effective shear modulus of the nanocomposites are discussed in detail.
... The equivalent permeability of this inclusion is determined using numerical results obtained from the BEM. Lastly, the macroscopic permeability is analytically derived by applying various homogenization methods discussed in micro-mechanical books such as [28][29][30]. These methods include the dilute distribution and Mori-Tanaka schemes. ...
The primary objective of this work is to determine the effective permeability of porous media consisting of an isotropic permeable solid matrix containing pores of arbitrary shapes. Fluid flow through the matrix phase is modeled by Darcy’s law, while the flow inside the pores follows the Stokes equations. The interfaces between the matrix phase and inclusions are defined by the general form of the Beavers-Joseph-Saffman conditions. To achieve this objective, the Boundary Element Method (BEM) is first developed to solve the coupled Darcy and Stokes problem related to fluid flow through an infinite solid phase containing an arbitrarily shaped pore under a uniform prescribed pressure gradient at infinity. In contrast to the classical BEM where integration equations are often singular, our method, incorporating both finite difference and analytical integration schemes, overcomes this inconvenience. Additionally, compared to the commonly used numerical method based on the finite element method, our approach, which only requires discretization of the solid/fluid interface, significantly enhances computational speed and efficiency. Subsequently, each pore is substituted with an equivalent permeable inclusion, and its permeability is determined. Finally, employing classical micromechanical schemes, the macroscopic permeabilities of the porous material under consideration are estimated. These macroscopic permeability estimates are then compared with the relevant data available in the literature, as well as several numerical results provided by the finite element method.
... Multiscale homogenization can be traced back to the early work of Eshelby [35], who derived the analytical stiffness of a composite RVE. Extensions of Eshelby's model can be found in Willis [36], Nemat-Nasser and Hori [37], Ponte Castaneda and Suquet [38], and Drugan and Willis [39]. However, these models are derived analytically or semi-analytically, and thus are limited to RVEs with simplified geometry or linear material model. ...
... where Ω is the domain of the RVE and ‖ ‖ Ω ‖ ‖ is its volume. Adopting the Hill-Mandel condition [44], ⟨ ⟩ ∶⟨ ⟩ = ⟨ ∶ ⟩, the internal virtual work of the macro element will be equal to ...
The rapid development of additive manufacturing (AM) technology and the consequential microstructural variations have brought the simulation aspects into focus. To capture an accurate mechanical response, multiscale methods are used to determine the macroscopic constitutive behaviour. Among these methods, the recently introduced Direct FE² (DFE²) technique has shown some success in predicting material behaviour via the direct incorporation of representative microstructure volumes into the macroscopic structure. The present study highlights the predictive modelling capabilities of DFE² linked with an element elimination technique (EET) in investigating the performance of fused filament fabrication (FFF) samples under tension and compression. Parts were manufactured by FFF with various raster angles to vary the failure characteristics. Their mechanical responses and fracture behaviours were simulated by DFE² and benchmarked against conventional FE, highlighting that the results of DFE² were competitive against FE. Most importantly, the combination of EET with DFE² resulted in predictive insights regarding changes in the failure surface for samples with varying raster angles.
... Although the experimental approaches have been used in the literature to investigate the effects of void, fibre volume fraction, fibre aspect ratio, types of fibre reinforcements and cross-sectional geometry of fibre on the mechanical properties of 3D printed composite parts [25][26][27][28][29], making the prediction of the properties of short fibre composites is more challenging compared to isotropic materials. To deal with these critical issues, a micromechanical approach based on homogenization technique [30] is used for investigating the multi-axial properties of 3D-printed composite materials/ parts. It makes a positive contribution to predict the properties of composite parts based on the physical properties of individual phases, arrangements of fibres in matrix materials, as well as the geometry used to build the composite part. ...
Mechanical behaviour of 3D-printed composite parts is affected by the volume fraction, aspect ratio and type of fibre reinforcement. Although in the literature experimental approaches have been used to characterise the effects of the above factors on the mechanical properties of 3D printed parts, time and cost of the manufacturing process as well as the uncertainty associated with a large number of experimental techniques are the key issues. This study aims to address these challenges by developing a methodology based on a multi-scale Finite Element (FE) analysis of representative volume element (RVE) of 3D printed composite parts to predict the effective orthotropic properties. To account for the effects of fibre features, RVEs were modelled considering variables of volume fraction, aspect ratios and type of short fibres. To study the main and interaction effects of the above variables on the mechanical properties of 3D printed composite parts, a structured approach based on the Design of Experiments is used. The FE stress analysis of the RVE provides an understanding about the potential failure modes such as interfacial debonding between fibres and matrix, interlayer and intralayer delamination that may occur in load-bearing 3D printed composite parts. The FE computed mechanical properties are validated against experimental data through a series of mechanical testing of flexure, Iosipescu, and short beam shear which were conducted in conjunction with the Digital Image Correlation technique. As a result, certainty is obtained in using the proposed approach for a fast iterative design of 3D printed composite parts prior to industrial applications.
... The unit cell of random two difference size circle inclusions In the case of circle inclusion, the function I α (ξ ) is given by Nemat-Nasser[21] ...
This work studies solutions to determine the macro-elastic moduli of three-phase composite materials with imperfect interfaces in 2D. Which is based on the coated circular assemblage model of Hashin-Strikman and the polarization approximation method (PA) to develop the formulae for elastic moduli of circle inclusions with spring-layer and surface-stress imperfect interfaces. From that, explicit algebraic expressions were obtained to estimate the elastic moduli of three-phase composites with imperfect interfaces, in which two phases are different with circular inclusions distributed randomly in the matrix. Besides, the FFT algorithm and the differential approximation (DA) are also developed to determine the elastic moduli of the three-phase composite with imperfect interfaces. The results of the FFT numerical methods will be compared with the DA and PA results with different material cases to show the effectiveness of the applied methods.
... This suggests that a significant minority of particles participate in the strong network. Mori-Tanaka has been shown to be robust with higher inclusion volume fractions because this homogenization scheme implicitly takes into account interactions between inclusions [45,64,65]. Because the strong network permeates throughout the assembly, Mori-Tanaka is the ideal homogenization scheme to implicitly capture the complex strong network structure. ...
Granular media is ubiquitous, playing a vital role in a diverse set of applications. The complex microstructure of granular media results from assorted particle shapes, morphologies, and packings, make it difficult to predict its macroscopic behavior. Under compression, these complex microstructures enable highly anisotropic and heterogenous behaviors, including creation of highly-loaded particles (i.e. force chains) supported by clusters of minimally-loaded particles. While many existing constitutive models relate state variables describing microscale behavior to continuum properties, these models do not generally consider the mesoscale interactions between the force chain network and minimally-loaded particles. Here, we develop a micromechanics model that connects micro-scale force chain mechanics to macro-scale mechanical behavior through explicit consideration of the interaction between force chains and minimally-loaded particles. We first examine the elastic behavior of a force chain using a spring model, explicitly considering the mesoscale interactions between the force-chains and surrounding regions. We then construct an equivalent inclusion problem to calculate macroscopic mechanical response as analytical functions of microscopic properties, with proper consideration of mesoscale interactions. We present our calibration and validation approaches, showing the model’s predictive abilities. Finally, we examine the effect of relevant microscopic quantities on macroscopic response, demonstrating the importance of these mesoscale interactions on bulk deviatoric behavior.
... A r = (K r −K 0 ) −1 ; K r , which K r means the thermal conductivity of r-phase. Further elaboration on the derivation of the formula and the con guration of the micromechanical models can be found in the referenced literatures [32][33][34]. These comparative analyses substantiate the theoretical models against measured data, providing a nuanced understanding of the in uence of HGM inclusion on the conductive properties of the composites. ...
The rapid evolution and expansion of digital infrastructures has underscored the signi cance of electromagnetic interference (EMI) shielding composites. However, there has been a notable dearth of efforts to explore EMI shielding performance in the context of elevated temperatures. This study introduces hollow glass microspheres (HGM) to enhance the EMI shielding performance of carbon ber (CF)-embedded conductive cement under elevated temperatures. Experimental outcomes delineate the impact of HGM inclusion on compressive strength, electrical and thermal conductivities, and EMI shielding performance. HGM incorporation is observed to reduce thermal conductivity, consequently improving EMI shielding performance at elevated temperatures by decreasing re ection and increasing absorption properties. The investigation incorporates comprehensive analyses, including XRD, TG, MIP, and micro-CT, to systematically examine the EMI shielding test outcomes at elevated temperatures. In conclusion, the utilization of HGM has the potential to yield super lightweight EMI shielding composites with enhanced EMI shielding performance at elevated temperatures.
... Nach den Theorien der porösen Medien und Mehrphasen-Materialien, lautet im Falle von einem elastischen Material und kleinen Verzerrungen, das allgemeine Helmholtz-Energiepotential [Ostwald et al., 2014;Nemat-Nasser and Hori, 2013]: ...
Materialmodelling of CMC Materials, User Subroutine UMAT for Abaqus
... Also, the self-consistent model was not used because it is more suitable for polycrystals and aggregates. Finally, they used a differential scheme (Nemat-Nasser et al., 1996) because of its advantage where the volume fraction of the crystalline phase (the reinforcing phase) is high. In the differential scheme, the crystalline phase was introduced using small increments where the crystalline fraction was added step-by-step under the assumption of dilute-distribution. ...
Highly heterogeneous and complex micro-structure of semi-crystalline polymers challenges accurate prediction of their macroscopic behavior. Micro-mechanical models establish a relationship between the micro-structure and macroscopic properties (structure–property relationship), and are able of predicting not only the macro-scale behavior, but also the evolution of the micro-structure. Therefore, micro-mechanical modeling can be used as a virtual experiment to predict the overall behavior of semi-crystalline polymers, where the effect of any single micro-structural parameter can be investigated. These parameters include morphological information about distribution of amorphous and crystalline phases, and constitutive properties of both phases. In this review paper, two main categories of micro-mechanical models, including mean-field and full-field models, are reviewed in detail. Three different groups of mean-field models, namely single-phase, two-phase, and three-phase models are discussed. Besides, the morphology of semi-crystalline polymers together with different deformation mechanisms, involved in different deformation regimes, are illustrated.
... These investigations have looked into various aspects, including the inter-laminar hybridization of Kevlar-based laminates with E-glass [8], carbon fibrebased laminates with Kevlar [9], and the intra-laminar hybridisation of carbon fibre-based laminae with E-glass [10,11]. These studies have predominantly employed computational micromechanics of composites through the use of representative volume element (RVE) models [12] with periodic boundary conditions [13]. However, there is currently a lack of computational studies focusing on the intra-laminar hybridization of carbon/epoxy laminates incorporating hollow glass fibres. ...
In this work, unidirectional composite laminae with intra-laminar fibre hybridisation (i.e. two fibre types within a matrix) are studied to understand the influence of solid and hollow glass fibre content on the homogenised specific lamina properties of and the matrix micro-stress fields in carbon/solid-E-glass/epoxy and carbon/hollow-E-glass/epoxy laminae. A 3D representative volume element (RVE) model is developed for the micromechanical analysis of unidirectional composite laminae with intra-laminar fibre hybridisation by considering random fibre distribution. The random sequential expansion (RSE) algorithm is modified to generate fibre hybrid microstructures. The RVE model is validated with analytical models. Using the RVE model, carbon/solid-E-glass/epoxy and carbon/hollow-E-glass/epoxy fibre hybrid laminae are studied. The results show that the intra-laminar hybridisation of carbon/epoxy laminae with hollow glass fibre content can affect the density and significantly alter the homogenised specific transverse lamina properties, while the reduction in the specific longitudinal elastic properties is negligible. Moreover, the RVE models with random fibre microstructures show that the presence of hollow glass fibres can considerably alter the matrix micro-stress fields in carbon/hollow-E-glass/epoxy laminae when compared to the micro-stress fields in carbon/solid-E-glass/epoxy laminae.
... The RVE contains a finite number of constituents, each represented by a constitutive model. An homogenization method is used to find the macro constitutive response of RVE, where continuum mechanics is applied at the macro scale with macro constitutive equations [32]. This two-scale approach allows the consideration of microstructural effects on the macroscopic response of the solid body, without the need to solve the problem at the micro-scale for the whole solid body, which would be computationally expensive. ...
Large Format Additive Manufacturing (LFAM) has gained prominence in the aerospace and automotive industries, where topology optimization has become crucial. LFAM facilitates the layer-by-layer production of sizeable industrial components in carbon fiber (CF) reinforced polymers, however 3D printing at large scales results in warpage generation. Printed components are deformed as residual stresses generated due to thermal gradients between adjacent layers. This paper tackles the problem at two different scales: the micro and macroscale. Initially, the microstructure characterization of the thermoplastic ABS matrix composite material enriched with 20% short CF is used in the development of numerical models to understand the mechanical behavior of the studied material. Numerical modeling is performed simultaneously by means of Mean-Field (MF) homogenization methods and Finite Element Analysis (FEA). Outcomes validated with corrected experimental mechanical testing results show a discrepancy in the elastic modulus of 7.8% with respect to FE multi-layer analysis. Micro-level results are coupled with the a macroscopic approach to reproduce the LFAM process, demonstrating the feasibility of the tool in the development of a Digital Twin (DT).
... The material is a C/C composite, and the CTE of the orthotropic fiber and the isotropic matrix are listed in Table 4. According to [52], the current RVE is not a proper hexagonal-packed RVE but is considered a hexagonal bundle enforced in a square cross-section shape. The void content ranges from 2 to 8%, and the homogenized CTE is compared with [39], where 3D Abaqus finite elements equal to 43725 for the 57% RVE and 43680 for the 80% case were used. ...
This work presents results of numerical simulations to investigate the effect of different void percentages on composite materials’ Coefficient of Thermal Expansion (CTE) and local stress fields. A random distribution of voids is considered within the Representative Volume Element (RVE) matrix, and different types of microstructures are considered, including square-packed and randomly distributed fibers. The use of a higher-order beam model within the framework of Carrera Unified Formulation (CUF) leads to a Component-Wise (CW) approach, resulting in an accurate, 3D description of the cross-section although using a 1D formulation. Numerical results for different fiber volume fractions and void concentration percentages demonstrate the agreement of the computed effective coefficients of thermal expansion of the present micromechanical thermoelastic model with references from the literature. The local stress fields are affected by voids, with higher effects over the matrix. Furthermore, higher void fractions lead to higher variability of stress peaks.
... is the Fourier transform of the characteristic function I 1 (x), which is sometimes referred to as the "shape function" or "shape coefficient" (see for instance Nemat-Nasser [16]). Interestingly, the shape function is explicitly known in certain special cases, such as microstructures with non-overlapping cylindrical and spherical inclusions (for more details, refer to [4]). ...
This paper investigates, with the help of micromechanics, the overall nonlinear filtration law of a porous solid containing impervious inclusions. At the microscopic scale, we assume that the fluid flow in the porous region is described by the Forchheimer law. More specifically, the nonlinear variational homogenization approach is considered to derive closed-form bounds for the macroscopic filtration law, which is based on velocity or pressure gradient-dependent potentials applied to a unit cell with uniform boundary conditions. The approach is developed in the case of a porous solid with impervious cylinders or spheres, for which we derive explicit analytic upper and lower bounds, considering appropriate simple trial fields. The closed-form expressions of the non-linear bounds are provided and compared to FFT solutions for various microstructures.
... Understanding the relationship between microstructures and macro-properties (e.g., fracture, toughness, strength, etc.) has been a remarkable interest in recent decades in order to control the macroscopic properties of materials by changing their microstructures [7][8][9][10][11][12]. ...
Statistical continuum mechanics theory was used to simulate the inelastic stress of polycrystalline materials using two-point statistics. For the experimental part, the Electron beam melting (EBM) technique (Arcam EBM Q10 additive machine) was used to fabricate cylindrical rods of Ti-6Al-4V both in horizontal and vertical directions. Electron backscatter diffraction (EBSD) technique was employed to achieve statistically reliable orientation maps of vertically and horizontally printed samples. In this study, high strain rate compression tests at six different strain rates were performed, and the stress-strain curves were generated. This work is amongst the first attempts to model the microstructure of additively manufactured hexagonal alloys under compressive loadings using the statistical continuum mechanics theory. The model is capable of simulating reasonably large microstructures (statistically representative) with a practical computational cost and accuracy, unlike numerical models that require a high computational cost. It should be noted that in additive manufacturing, due to large grains and high anisotropy, microstructures used in the simulations should be large enough to include sufficient information from the material’s structure. Therefore, using finite element models would be very challenging here. On the other hand, the statistical continuum mechanics theory uses the statistical representation of the material’s characteristics for solving the governing equations with Green’s function that enables this methodology to use more microstructure characteristic information without having a noticeable change to the computational cost. The proposed model in this study uses different microstructure characteristics such as crystal grain orientation, total slip systems, active slip systems, gain morphology, and chemical phases that are obtained from EBSD images for simulating the inelastic mechanical behavior of polycrystalline materials. Although this model simulates polycrystalline materials by considering various crystal and grain information, unlike numerical methods, it doesn’t simulate the grain interactions well and we cannot study local deformation and crack nucleation sites. This model works very well for simulating the overall behavior of material instead of each individual grain and failure analysis. This model has shown a good combination of computational cost and accuracy in which the error between the simulated and experimental strength for vertical and horizontal samples was 6.21% and 8.07%, respectively.
... Nonetheless, a more comprehensive topdown view of MPSE reveals selected other key aspects of bringing SERVE methods into MPSE design, and these are addressed in the remainder of the paper. The fundamental assumption behind the ATDBC and UTBC is that stresses and strains immediately outside the simulated RVE are constant [56,77,[146][147][148]. These boundary conditions ignore the presence of heterogeneities exterior to the RVE and their interaction with those in the interior. ...
Mechanical properties of materials and associated engineered components are controlled by the material structure at various lengths and time scales. As materials are being further utilised to the maximum extent of their capabilities, tails on property distributions become significant. These tails are often driven by the extremities of microstructural feature distributions, suggesting the need for a statistically relevant description of the microstructure and a reciprocity relationship with the range of property measurement capabilities and the models that represent this information.
Representative volume elements (RVE) and statistically equivalent representative volume elements (SERVE) have emerged as frameworks for such microstructural characterisation and quantification.
This review covers the evolution of quantitative microstructure description for use in material behaviour predictions from homogenised representations, large volume statistical representation, to the determination of the minimum spatial size to statistically represent a microstructure based on features of interest and properties of interest.
... In addition, the presence of abundant clay minerals (ellipsoidal) with preferred orientation makes the sediment inherently anisotropic (Bennett et al., 1991). There are several approaches that can be applied to predict the effective elastic properties of a micro-heterogeneous medium (Berryman, 1992;Dai et al., 2012;Frisch, 1968;Ghosh et al., 2010a,b;Gubernatis and Krumhansl, 1975;Hornby et al., 1994;Hudson, 1991;Jakobsen et al., 2000;Sheng, 1992;Watt et al., 1976;Marín-Moreno et al., 2017;Nemat-Nasser and Hori, 1993;Pan et al., 2022). The combination of the self-consistent approximation (SCA) and differential effective medium (DEM) theory is argued to be the most useful to model the gas hydrate-bearing sediments. ...
The importance of gas hydrate as an alternative energy resource is ascending because of its worldwide distribution along the continental margins, in the permafrost regions and deep-water lakes, its richness, and high fuel productivity. Seismic survey is used predominantly to identify gas hydrate deposits at the regional scale, whereas drilling and coring are used for ground-truthing at the local scale. However, reliable quantification is the utmost priority to evaluate its resource potential commercially and understand its effect on the environment. Pressure core analysis is a direct way to know the morphology and amount of gas hydrate in a reservoir, while rock physics modelling is used widely to estimate gas hydrate saturation from remotely sensed geophysical data. Elevated velocity and resistivity are two parameters mostly used in rock physics theory to estimate the gas hydrate concentration in sediments, whereas, pore-water salinity is one of the important proxy methods used to estimate gas hydrate saturation. Several rock physics models based on empirical, semi-empirical, and laws of physics exist in the literature that relate the observed anomalous physical properties like velocity and resistivity to reservoir properties like porosity, permeability, saturation etc. However, both direct and indirect methods have certain limitations, which are important of accurately quantifying gas hydrate. We review in detail the existing rock physics theories with possible uncertainties to estimate gas hydrate saturation using velocity, which provides reliable estimates with less uncertainty than that from resistivity.
... Moreover, several computational studies have investigated the mechanical behaviour of synthetic/synthetic fibre hybrid composite laminates, such as the inter-laminar hybridisation of Kevlar fibre-based laminates with E-glass [21], carbon fibre-based woven laminates with Dyneema [22], carbon fibre-based laminates with Kevlar [23], and on the intralaminar hybridisation of carbon fibre-based laminae with E-glass [24,25]. These studies have mostly been based on the computational micromechanics of composites using representative volume element (RVE) models [26] with periodic boundary conditions [27]. However, no computational studies have been reported on the intra-laminar hybridisation of natural fibre-based laminates. ...
This study investigates the effect of intra-laminar fibre hybridisation on the homogenised lamina properties and micro-stress fields, with an emphasis on natural and synthetic fibre combinations. A representative volume element (RVE) approach, considering random fibre distributions, is employed to study unidirectional flax/E-glass/epoxy composite laminae. The results indicate that intra-laminar hybridisation with natural and synthetic fibre combinations can offer opportunities to tailor the homogenised lamina properties, especially to vary the effective density and thus achieve high specific properties. In addition, it is shown that the intra-laminar hybridisation can significantly alter the micro-stress and strain fields, potentially affecting intra-laminar damage mechanisms.
... The fourth-order Hill tensor of an ellipsoid can be expressed using the second-order derivative of the Green's function, as has been demonstrated in classical works on micromechanics (Mura, 1987;Nemat-Nasser et al., 1996). ...
This paper introduced a novel microstructure-based constitutive model designed to comprehensively characterize the intricate mechanical behavior of anisotropic clay rocks under the influence of water saturation. The proposed model encompasses elastoplastic deformation, time-dependent behavior, and induced damage. A two-step homogenization process incorporates mineral compositions and porosity to determine the macroscopic elastic tensor and plastic yield criterion. The model also considers interfacial debonding between the matrix and inclusions to capture rock damage. The application of the proposed model is demonstrated through an analysis of Callovo-Oxfordian clayey rocks, specifically in the context of radioactive waste disposal in France. Model parameters are determined, followed by numerical simulations of various laboratory tests including lateral decompression tests with constant mean stress, triaxial compression tests under different water saturation conditions, and creep tests. The numerical results are compared with corresponding experimental data to assess the efficacy of the proposed model.
... The RVE is considered the smallest representative volume and corresponds to a periodic fiber packing sequence [17,18,19]. Micromechanics is used to evaluate the behavior of multiphase materials [1,20,21], and various approaches are available for obtaining homogenized properties, and local stress and strain fields [22]. ...
... 17 Nemat-Nasser and Hori (1993) attribute the result to three sources: (Hill, 1967) cited in the main text, another paper of Hill's from 1963 and a paper by Mandel in 1980. Hill's two papers do not contain Eq. (2), let alone the general proof. ...
The paper discusses from first principles all aspects relevant to the plasticity of porous materials. Emphasis is laid on unhomogeneous yielding, defined as the process of yielding and plastic flow under gradient-free macroscopically nonuniform deformation. The nonuniformity is represented by strain localization in one or more bands of finite thickness. A universal feature of all intrinsic yield criteria is their dependence upon the normal and shear tractions resolved on the band. When specialized to isotropy, a Mohr–Coulomb criterion and a Rankine–Tresca criterion emerge as two extremes. The latter is an ideal that typifies the yield behavior of porous materials under arbitrary loading. The general theory stands for a finite number of bands or yield systems. Its overall structure bears some features of crystal plasticity, but with dependence upon the resolved normal stress. The evolution of microstructural parameters can be given in general terms, being solely based on the kinematic constraints of unhomogeneous yielding and matrix incompressibility. Throughout the paper, the competition with homogeneous yielding, heretofore taken for granted, is analyzed with or without strain and strain-rate hardening effects. We close by discussing the thermodynamic consistency of this new class of constitutive relations and a link to strain-gradient theories.
... As the simplest and computationally most efficient method, Mean-Field Homogenization computes homogenized stress-and strain fields as volume averages over composite constituents. More complex homogenization methods involve numerical and semi-analytical schemes, as well as extensions to elasto-visco-plastic materials (see ref. [17][18][19][20][21] and references therein). ...
Inhomogeneously swollen elastomers are an emergent class of materials, comprising elastic matrices with inclusion phases in the form of microgel particles or osmolytes. Inclusion phases can undergo osmotically driven swelling and deswelling over orders of magnitude. In the swollen state, the inclusions typically have negligible Young's modulus, and the matrix is strongly deformed. In that regime, the effective mechanical properties of the composite are governed by the matrix. Laying the groundwork for a generic analysis of inhomogeneously swollen elastomers, we develop a model based on incremental mean-field homogenization of a hyperelastic matrix. The framework allows for the computation of the macroscopic effective stiffness for arbitrary hyperelastic matrix materials. For an in-depth quantification of the local effective stiffness, we extend the concept of elastic stiffness maps to incompressible materials. For strain-stiffening materials, stiffness maps in the swollen state highlight pronounced radial stiffening with a non-monotonic change in stiffness in the hoop direction. Stiffening characteristics are sensitive to the form of constitutive models, which may be exploited in the design of hydrated actuators, soft composites and metamaterials. For validation, we apply this framework to a Yeoh material, and compare to recently published data. Model predictions agree well with experimental data on elastomers with highly swollen embedded microgel particles. We identify three distinct regimes related to an increasing degree of particle swelling: first, an initial decrease in composite stiffness is attributed to particle softening upon liquid intake. Second, dilute particle swelling leads to matrix stiffening dominating over particle softening, resulting in an increase in composite stiffness. Third, for swelling degrees beyond the dilute limit, particle interactions dominate further matrix stiffening.
... In this geometrical micro-scale model, each phase, constituent or singularity is explicitly represented as demonstrated in Figure 3.1. Several authors have been proposing their definition of RVE (Hashin, 1983;Nemat-Nasser and Hori, 1993;Gitman et al., 2007) but the main idea is that the RVE should be large compared to the microstructural singularities, being statistically representative of them, but it must be small enough so that it can be considered an infinitesimal point at the macro-scale, allowing the application of the homogenisation procedure. This concept leads to the so-called scales separation principle (Hashin, 1983) and is better understood through the following equation: ...
A large strains multi-scale computational framework is formulated to model the behaviour of polycrystalline materials under slip plasticity, mechanically-induced martensitic transformation and intergranular fracture. For that purpose, a constitutive model is proposed to simulate crystal-plasticity-like phenomena by combining thermodynamically consistent constitutive equations for crystallographic slip and martensitic transformation. The model includes an anisotropic hyperelastic law with self- and latent-hardening for describing slip plasticity evolution and a crystallographic theory of martensitic transformations for mechanically induced martensite formation. The complete set of highly coupled equations is expressed in a single system of equations and solved with a monolithic solution procedure based on the Newton-Raphson method, where the complete linearisation of the residual equations at both local and global equilibrium levels leads to asymptotic quadratic rates of convergence. The quasi-static discretised evolution equations are integrated with a fully implicit scheme, except for the critical resolved slip stresses, which employ the generalised midpoint rule. The plastic flow is integrated with an implicit exponential integrator to exactly preserve plastic incompressibility.
Viscous regularisations for both deformation mechanisms are pursued to overcome numerical difficulties and model the behaviour over a wide range of strain-rates. Three numerical techniques are introduced to improve the reliability and efficiency of the model in addressing the rate-independent limit case of viscous rate-dependent formulations. These dramatically improve the model's performance by enabling more significant incremental steps to be used within the monolithic solution procedure. In addition, an efficient strategy is suggested to complete the transformation process when the full martensitic transformation is approached at a given integration point. Numerical examples are provided to demonstrate the model's efficiency and predictive capabilities, and the impact of each numerical technique is evaluated through a series of ablation studies.
In this thesis, the cohesive zone model is used to investigate intergranular fracture in polycrystalline materials. The Park-Paulino-Roesler (PPR) cohesive model, which is based on a potential and has distinct fracture properties for both normal and tangential directions, is utilised. The conditions that can lead to unstable solution branches when simulating fracture problems in an implicit and quasi-static manner are described, and two strategies for addressing these numerical issues are discussed, including the extension of a quasi-static formulation to a dynamic one. A multi-scale model that relates discontinuous microstructures to a continuous macroscopic domain is also developed using the method of multi-scale virtual power (MMVP). This model includes micro-cracks and inertial effects in the micro-scale equilibrium problem.
A large strain fracture-based computational homogenisation procedure is introduced to homogenise traction-separation laws and fracture properties from microstructural responses obtained using the aforementioned formulation. It is derived from a crack-based Hill-Mandel principle, described in the reference configuration, and uses an energetic-based damage variable, proposed in this work for the PPR cohesive model, to accurately define the crack domain and compute the crack homogenised quantities. A method for accurately computing the homogenised unit normal vector of the equivalent macroscopic crack is also proposed. Numerical studies on the impact of slip plasticity and martensitic transformation on the intergranular crack propagation mechanism illustrate the features of the developed modelling framework.
... The solid phase is described as a linear isotropic elastic material. The conservation of linear momentum is (Landau and Lifshitz, 1959b;Nemat-Nasser and Hori, 2013) ∇ · σ ¼ 0; ...
Gassmann’s equations were derived several decades ago and continue to be widely used in applied geophysics. Gassmann’s equations allow us to calculate the elastic moduli of a fully saturated rock from dry rock moduli knowing the porosity, fluid bulk modulus, and bulk modulus of the solid grains. These equations are treated as exact in the scientific community, but there is a lack of comprehensive numerical validation. Furthermore, recently several publications appeared in the literature postulating a logical error in the derivation of Gassmann’s equations. Therefore, I develop a numerical validation of Gassmann’s equations. For that, I use a 3D finite-element approach to resolve the conservation of linear momentum that is coupled with the stress-strain relations for the solid phase and the quasistatic linearized compressible Navier-Stokes momentum equation for the fluid phase. Finally, a convergence study validating the correctness of Gassmann’s equations for a particular yet arbitrarily chosen “generic” pore geometry is presented. The arbitrary model geometry is simple as compared with real rocks; however, it is sufficiently complex with elements resembling wider pore bodies and narrower pore throats to, in general, validate Gassmann’s equations. MATLAB routines to reproduce the presented results are provided.
... The literature presents numerous analytical models that can estimate the elastic property of polymer matrix composites (PMCs) from their constituent properties. For instance, classical theories such as rule-of-mixture (ROM), modified rule-of-mixture (MROM) and enhanced models such as the continuous periodic fiber model (CPFM), the concentric cylindrical assembly (CCA), Mori-Tanaka (MT), and the Bridging Model (BM), have been widely employed to predict composite elastic response [5][6][7][8][9][10][11][12][13][14][15][16]. However, there are only a few micromechanical strength theories that can accurately predict the composite strength from its constituent properties [4,8,10,[17][18][19]. Huang et al. [8] developed the BM to estimate the stress field inside a single-fiber repeating unit cell (RUC) with square packing. ...
The transverse strength of fiber-reinforced composites is a matrix-dominated property whose accurate prediction is crucial to designing and optimizing efficient, lightweight structures. State-of-the-art analytical models for composite strength predictions do not account for fiber distribution, orientation, and curing-induced residual stress that greatly influence damage initiation and failure propagation at the microscale. This work presents a novel methodology to develop an analytical solution for transverse composite strength based on computational micromechanics that enables the modeling of stress concentration due to representative volume elements (RVE) morphology and residual stress. Finite element simulations are used to model statistical samples of composite microstructures, generate stress-strain curves, and correlate statistical descriptors of the microscale to stress concentration factors to predict transverse strength as a function of fiber volume fraction. Tensile tests of thin plies validated this approach for carbon- and glass-reinforced composites showing promise to obtain a generalized analytical model for transverse composite strength prediction.
... To model the plastic deformation of metals, microstructure-property correlation needs to be defined mathematically. For this, the theory of homogenization assumes prime importance, according to which, the heterogeneities in a microstructure are assumed to be at continuum length scale [8]. Homogenization is applied to evaluate the plastic response of polycrystalline materials through two approaches. ...
The present study is an attempt to model dynamic recrystallization (DRX) in a single phase metal using a mean field crystal plasticity (MFCP) based approach. A new empirical equation is proposed to model nucleation, in which the nucleation rate is a function of
microstructure and plasticity descriptors that are known to affect DRX behavior, such as the temperature, strain rate, grain fineness and stored energy. Grains undergo nucleation when their dislocation density exceeds a threshold value. Subsequently, new grains grow based on the difference in stored deformation energy with respect to the average value over all grains. The MFCP-DRX model is able to correctly predict trends for the flow stress, dislocation density evolution, grain size evolution and kinetics across a range of temperatures and strain
rates for uniaxial compression. Transition of the flow stress from single to multiple peaks is observed with increasing temperature and decreasing strain rate, thus comparing well against known DRX trends. The evolutions of crystallographic texture during DRX in unaxial
compression and plane strain compression are compared against experimental observations. A sensitivity analysis is conducted to understand the effect of variables on nucleation and growth. The competition between nucleation and growth dominated deformation in different strain regimes is analyzed in detail across various temperatures and strain rates.
... For solids and geomaterials, mechanical constitutive models (material laws) express a macroscopic view of force and displacement correspondence, resulting from multiscale mechanisms such as molecular deformation, granular displacements, or mesoscale deformation localization, in terms of homogenized (averaged) quantities, i.e., stresses and strains, in a representative elementary volume (REV) (Nemat-Nasser and Hori, 1999). They build the foundation to correlate conservation laws, e.g., conservation of momentum, with the kinematics of the system, e.g., displacements. ...
We examine a method to compute the unknown fracture toughness of a heterogeneous material by the means of a time dependent boundary condition. This boundary condition imposes a steadily propagating crack on an exactly defined heterogeneous body, from which it is possible to derive the corresponding fracture toughness of the underlying material. The goal is to identify toughening mechanisms and to compute the toughness parameter for the particular material definition, without restricting the model to randomness or small contrast.
复合材料及其三明治板具有良好的刚度和较高的阻尼，因而在工程中受到广泛的应用，准确预测其振动阻尼具有重要意义。本文遵循从材料到结构，从微观到宏观的研究思路，系统研究了复合材料及其三明治板振动阻尼特性，揭示其耗能机理，主要内容包括材料动态特性表征、纤维复模量、复合材料复模量敏感性、三明治板均质化方法、点阵结构振动阻尼。本文主要研究内容及创新性成果如下：
（1）在材料动态特性表征方面，建立了一种基于广义麦克斯韦模型Prony级数曲线拟合的自动平移方法，以获得材料的动态特性主曲面。该方法通过最小化两条拟合曲线之间垂直距离的平方和来实现自动平移，综合考虑储能模量和损耗模量的权重。结果表明，综合考虑储能模量和损耗模量得到的主曲面与实验结果吻合更好。
（2）基于通用单胞模型和相应性原理，提出了一种通过拟合计算获得纤维复模量的方法，解决了以往研究中因缺乏纤维动态属性而假设纤维无阻尼或各向同性阻尼的问题。通过复合材料和基体的动态力学属性计算得到了英国产和中国产T700碳纤维的各向异性复模量，利用有限元法验证了该方法计算碳纤维的轴向复模量和横向复模量的准确性。
（3）采用基于方差的敏感性分析方法对碳纤维复合材料的复模量进行了全局敏感性分析，研究了纤维体积含量、纤维复模量、基体复模量对单向碳纤维复合材料和3种层合板的复模量的影响大小，揭示了复合材料的耗能机理。结果表明，碳纤维复合材料的损耗因子主要受基体损耗因子和纤维轴向损耗因子影响，在较小程度上受纤维横向损耗因子影响，几乎不受纤维体积含量和纤维轴向剪切损耗因子影响。
（4）建立了一种非平截面法预测三明治板振动阻尼的代表体积元均质化方法，解决了平截面法过度约束截面的问题。平截面法利用平截面假设来施加激励，非平截面法通过预先计算四点弯曲和四点扭转得到的真实截面位移来施加激励，计算代表体积元在8个正弦激励下的复响应得到板壳复刚度矩阵，结合等效单层模型、一阶剪切变形理论、有限元模态应变能法计算模态阻尼。分别以含阻尼层的点阵三明治板和波纹三明治板为研究对象，通过实验和实体模型验证了2种代表体积元均质化方法的准确性。结果表明，2种方法对损耗因子的预测准确性相差不大，但非平截面法预测的自然频率比平截面法更加准确。
（5）发展了一种基于经典层合板理论、矩阵位移法、应变能法解析计算复合材料正交点阵桁架三明治板的板壳复刚度的方法，以及一种新的制备工艺。利用连续铺丝模压成型工艺制备了连续平面桁架，通过内桁架插入外桁架进行装配得到了连续正交点阵桁架三明治板。实验结合理论研究了环氧树脂、乙烯-醋酸乙烯酯共聚物（EVA）、丁腈橡胶3种胶粘剂对点阵三明治板振动阻尼的影响规律。实体模型、平截面法壳模型、非平截面法壳模型、解析法壳模型4种理论方法均能准确预测环氧树脂和EVA三明治板的振动阻尼，但对橡胶三明治板振动阻尼的预测准确性有待提高。EVA胶粘剂以较小的刚度和强度损失，提高了三明治板的模态阻尼。
Composite materials and sandwich panels have excellent stiffness and high damping, making them widely used in engineering applications. Accurately predicting their vibration damping is of great significance. In this paper, following the research idea from materials to structures, from micro to macro, the vibration damping characteristics of composite materials and sandwich panels are systematically studied, and the energy dissipation mechanism is revealed. The main contents include the characterization of material dynamic properties, fiber complex moduli, sensitivity of complex moduli of composite materials, homogenization methods of sandwich panels, and vibration damping of lattice structures. The main research contents and innovative achievements of this paper are as follows:
(1) In terms of material dynamic characterization, an automatic shift method based on the generalized Maxwell model Prony series curve fitting is developed to obtain the master surfaces of material dynamic characteristics. The method achieves automatic shift by minimizing the sum of squares of the vertical distances between two fitting curves, and weights the storage modulus and loss modulus together. The results indicate that the master surfaces obtained by considering both the storage modulus and the loss modulus agree better with experimental results.
(2) A method for obtaining the complex moduli of fibers through fitting calculations is proposed based on generalized method of cells and corresponding principle. This method solves the problem of assuming that fibers have no damping or isotropic damping due to the lack of dynamic properties of fibers in previous studies. The anisotropic complex moduli of T700 carbon fibers produced in the the United Kingdom and China are calculated according to the dynamic mechanical properties of composites and matrix, and the accuracy of the axial and transverse complex moduli of carbon fibers is verified using the finite element method.
(3) A global sensitivity analysis of the complex moduli of carbon fiber composites is performed using variance-based sensitivity analysis. The effects of fiber volume fraction, fiber complex moduli, and matrix complex moduli on the complex moduli of unidirectional carbon fiber composites and three types of laminates are studied to reveal the energy dissipation mechanism of composite materials. The results showed that the loss factors of carbon fiber composites are mainly affected by the matrix loss factor and the fiber axial loss factor, slightly affected by the fiber transverse loss factor, and almost unaffected by the fiber volume fraction and the fiber axial shear loss factor.
(4) A representative volume element (RVE) homogenization method based on non-planar cross-sections is developed to predict the vibration damping of sandwich panels, addressing the over-constraining issue of the planar cross-section method. While the planar cross-section method uses the assumption of a planar cross-section to apply excitation, the non-planar cross-section method applies excitation based on the actual cross-section displacements obtained from four-point bending and four-point torsion simulations. The complex stiffness matrix of the sandwich panel is computed by calculating the complex response of the RVE under eight sinusoidal excitations. The modal damping is calculated by combining the equivalent single-layer model, first-order shear deformation theory, and finite element-modal strain energy method. Two sandwich panel types, lattice and corrugated sandwich panels with damping layers, are used to validate the accuracy of the two RVE homogenization methods via experiments and solid models. Results show that the two methods have similar accuracy in predicting the loss factors, but the natural frequencies predicted by the non-planar cross-section method are more accurate than those predicted by the planar cross-section method.
(5) A method based on classical lamination theory, matrix displacement method, and strain energy method is developed to analytically calculate the complex stiffness of composite orthogonal lattice truss sandwich panels. A new manufacturing process is also developed using continuous fiber laying and compression molding to produce continuous plane trusses, which are assembled by inserting inner trusses into outer trusses to create continuous orthogonal lattice truss sandwich panels. The effects of three types of adhesives, epoxy resin, ethylene-vinyl acetate copolymer (EVA), and nitrile butadiene rubber (NBR), on the vibration damping of lattice sandwich panels are experimentally studied in combination with theoretical analysis. Four theoretical methods, including solid model, shell model with planar cross-section method, shell model with non-planar cross-sections method, and analytical model, accurately predicted the vibration damping of epoxy resin and EVA sandwich panels, but the accuracy of predicting the vibration damping of rubber sandwich panels needs to be improved. EVA adhesive improved the modal damping of sandwich panels with a small loss in stiffness and strength.
The appropriate understanding and formulation of rock discontinuities via FEM is still challenging for rock engineering, as continuous algorithms cannot handle the discontinuities in rock mass. Also, different failure modes of rock samples, containing tensile and shear failure, need to be computed separately. In this study, a novel double-phase field damage model was introduced with two independent phase field damage variables. The construction of the proposed model follows the thermodynamics framework from the overall Helmholtz free energy, with elastic, plastic and surface damage components. The proposed model is calibrated via traditional damage variables, based on ultrasonic wave velocity measurement and acoustic emission monitoring, and both show great consistency between simulation results and laboratory observations. Then the double-phase field damage model is applied to COMSOL software to simulate microcrack propagation in a pre-fractured rock sample. Both lateral and wing cracks are observed in this study, manifested as shear- and tensile-dominated cracks. We also observed different microcracking mechanisms in the proposed numerical models, such as tensile and shear cracking, the influence of plastic strain and the percolation between tensile and shear microcracks. Overall, this study provides valuable insights into the mechanics of microcracking in rocks, and the proposed model shows promising results in simulating crack propagation.
This article focuses on the computational analysis of sandwich composite materials based on polypropylene, polyester, glass, and cotton fibers. In the automotive components prepared from these fiber materials, the various components are used in different proportions. Through the manufacturing process, isotropic materials become somewhat anisotropic. Part of this article is aimed at obtaining input values of material characteristics for calculations using finite element analysis (FEM) and the comparison of experimental results with FEM-based material models created using the Digimat 2023.1 software. This article analyzes the modeling of two-phase as well as multiphase composite materials. This work focuses on calculations using FEM according to the test defined in the PR375 standard for loading the finished product in the luggage compartment of a car. The defined methodology enables the application of the FEM-based calculation directly to the product design in the initial phase of research. The construction and production of expensive prototypes and the subsequent production of automotive parts is replaced by computer-based simulation. This procedure makes it possible to simulate several optimization cycles over a relatively shorter time. From the results of computational simulations, it is clear that materials based on PP/PET/glass fibers show a much higher modulus of elasticity than materials created using cotton, i.e., materials of the PP/PET/cotton type. In order to achieve a high strength and stiffness, it is, therefore, appropriate to use glass fibers in the composite materials used for such applications.
Description
STP 1309 gives you access to the latest research data, technical advances, and novel experimental techniques for the evaluation of continuous-fiber ceramic composites (CFCCs).
This new volume also introduces the first published reports of applications of recently approved ASTM test methods for CFCCs, and contains significant information for many engineering applications where materials may be exposed to service cycles in various aggressive environments.Types of CFCCs examined include those processed with chemically infiltrated, polymer-impregnated, sintered, melt-infiltrated, or viscous glass-infiltrated matrices.
19 peer-reviewed papers focus on the following five categories:
• Room-Temperature Test Results/Methods;
• High-Temperature Test Results/Methods;
• Nondestructive Characterization;
• Modeling and Processing;
• and Testing of Tubes.
In orthopedic and dental surgery, the implantation of biomaterials within the bone to restore the integrity of the treated organ has become a standard procedure. Their long-term stability relies on the osseointegration phenomena, where bone grows onto and around metallic implants, creating a bone-implant interface. Bone is a highly hierarchical material that evolves spatially and temporally during this healing phase. A deeper understanding of its biomechanical characteristics is needed, as they are determinants for surgical success. In this context, we propose a multiscale homogenization model to evaluate the effective elastic properties of bone as a function of the distance from the implant, based on the tissue’s structure and composition at lower scales. The model considers three scales: hydroxyapatite foam (nanoscale), ultrastructure (microscale), and tissue (mesoscale). The elastic properties and the volume fraction of the elementary constituents of bone matrix (mineral, collagen, and water), the orientation of the collagen fibril relative to the implant surface, and the mesoscale porosity constitute the input data of the model. The effect of a spatiotemporal variation in the collagen fibrils’ orientation on the bone anisotropic properties in the proximity of the implant was investigated. The findings revealed a strong variation of the components of the effective elasticity tensor of the bone as a function of the distance from the implant. The effective elasticity appears to be primarily sensitive to the porosity (mesoscale) rather than to the collagen fibrils’ orientation (sub-micro scale). However, the orientation of the fibrils has a significant influence on the isotropy of the bone. When analyzing the symmetry properties of the effective elasticity tensor, the ratio between the isotropic and hexagonal components is determined by a combination of the porosity and the fibrils’ orientation. A decrease in porosity leads to a decrease in bone isotropy and, in turn, an increase in the impact of the fibrils’ orientation. These results demonstrate that the collagen fibril orientation should be taken into account to properly describe the effective elastic anisotropy of bone at the organ scale.
High-velocity impact events are ubiquitous in our modern world, from aerospace applications to ballistic impacts, and understanding how materials respond to these extreme conditions is of paramount importance. This chapter explores the intricate domain of high-velocity impact modeling in materials science, offering a multiscale perspective that encompasses macro, micro, and nanoscale phenomena. We begin by elucidating the historical evolution of this field, addressing existing research, and highlighting the pressing gaps and challenges. Multiscale modeling, a pivotal approach to comprehending the intricacies of high-velocity impact, is introduced, along with an exploration of its methodologies and tools. As materials undergo dramatic changes when subjected to high velocities, we delve into the various responses, such as shock waves, deformation, fracture, and phase transitions, emphasizing the role of material properties in these processes. Furthermore, we survey a spectrum of modeling techniques, including finite element analysis, molecular dynamics, and continuum mechanics, delineating their advantages and limitations and exemplifying how they can be integrated across scales. Real-world case studies underscore the practical implications of high-velocity impact modeling, demonstrating how multiscale perspectives contribute to solving complex materials science challenges. This chapter also considers the broader implications and future directions of the research related to high-velocity impact modeling, especially in fields like aerospace, automotive, and defense.
The skin is the largest organ in the human body and serves various functions, including mechanical protection and mechanosensation. Yet, even though skin’s biomechanics are attributed to two main layers—epidermis and dermis—computational models have often treated this tissue as a thin homogeneous material or, when considering multiple layers, have ignored the most prominent heterogeneities of skin seen at the mesoscale. Here, we create finite element models of representative volume elements (RVEs) of skin, including the three-dimensional variation of the interface between the epidermis and dermis as well as considering the presence of hair follicles. The sinusoidal interface, which approximates the anatomical features known as Rete ridges, does not affect the homogenized mechanical response of the RVE but contributes to stress concentration, particularly at the valleys of the Rete ridges. The stress profile is three-dimensional due to the skin’s anisotropy, leading to high-stress bands connecting the valleys of the Rete ridges through one type of saddle point. The peaks of the Rete ridges and the other class of saddle points of the sinusoidal surface form a second set of low-stress bands under equi-biaxial loading. Another prominent feature of the heterogeneous stress pattern is a switch in the stress jump across the interface, which becomes lower with respect to the flat interface at increasing deformations. These features are seen in both tension and shear loading. The RVE with the hair follicle showed strains concentrating at the epidermis adjacent to the hair follicle, the epithelial tissue surrounding the hair right below the epidermis, and the bulb or base region of the hair follicle. The regions of strain concentration near the hair follicle in equi-biaxial and shear loading align with the presence of distinct mechanoreceptors in the skin, except for the bulb or base region. This study highlights the importance of skin heterogeneities, particularly its potential mechanophysiological role in the sense of touch and the prevention of skin delamination.
Intergranular ductile fracture is a failure mode that may arise in many metallic alloys used in industrial applications. It manifests as the successive nucleation, growth, and coalescence of cavities at grain boundaries. Thus, simulation of intergranular ductile fracture in polycrystals requires modeling those three different stages at the scale of grain boundaries, i.e. at the interface between two different crystals. In this study, a yield criterion for the coalescence of cavities at the interface between two isotropic materials obeying Mises plasticity is first developed by limit analysis in order to provide some insights into that phenomenon. This criterion is checked against numerical limit analysis under combined tension and shear and is found to agree with unit-cell simulations quantitatively. The model is then extended to crystals so as to account for the complex coupling between loading state, crystallographic orientations, and void microstructure in intergranular coalescence. This second criterion is also assessed through comparisons to numerical limit analysis for an FCC crystal lattice. The agreement is very good in the case of coalescence by internal necking and the trends displayed by coalescence under combined tension and shear are captured correctly. Some implications of the model on the competition between transgranular and intergranular ductile fracture are discussed. Finally, by combining this model with an existing criterion for void growth at grain boundaries, a multi-surface yield function relevant to intergranular ductile fracture is obtained and compared to unit-cell simulations.
Confining pressure, strain rate and anisotropy are three essential factors that affect the mechanical behaviors of shale. In this study, a damage constitutive model including the effects of all these three factors is developed to predict the deformation and strength of sedimentary shale. The inherent elastic tensor and the damage of shale are considered to be transversely isotropic. The statistical trans-scale theory is adopted to establish the formulation of the damage variable which connects the microscopic element damage to macroscopic rock failure. A generalized Lade criterion combined with the fabric tensor is proposed to take into account the anisotropic strength and tension–compression asymmetry of shale, which successfully predicts the different properties under tension and compression. The strain rate and confining pressure change shale’s stiffness and strength by influencing anisotropic elastic moduli and damage evolution rates. The theoretical predictions of the established model are in good agreement with the experimental results in the literature for various cases, demonstrating that the model can predict the stress–strain relations of transversely isotropic shale under complicated loading conditions.
In this study, micromechanical modeling will be performed for composite materials containing fillers oriented randomly in the matrix. The purpose of this study is to derive more general and explicit solutions for the effective thermal and electromagnetic properties of such composite materials without restricting the properties and shapes of the fillers. For this purpose, it is assumed that the physical properties of the filler are the same anisotropic properties as orthorhombic materials, and the shape of the fillers is ellipsoidal. This model is analyzed by micromechanics combining the Eshelby's equivalent inclusion method with the self-consistent method or the Mori-Tanaka's theory. Solutions of the effective thermal and electromagnetic properties both for composite materials containing many kinds of fillers with different shapes and physical properties and for polycrystalline materials can be also derived. Using the obtained solutions, the effect of the shape, the anisotropy, and the volume fraction of the filler on the effective thermal conductivity is examined for the carbon filler / polyethylene and the two types of quartz particles (and voids) / polyethylene. As a result, for the carbon filler / polyethylene, it is found that the effective thermal conductivity of the material when the shape of the filler is flat is about 20% higher than that when the shape of the filler is fibrous. Furthermore, when the shape of the carbon filler is flat, the result when the carbon filler is assumed to be isotropic is significantly different from that when the filler is assumed to be anisotropic. From the above, when the filler is oriented randomly in the material, it is found that simultaneously considering not only the shape of the filler but also its anisotropic properties is important to accurately evaluate the effective physical properties of the composite material. For two types of quartz particles (and voids) / polyethylene materials, the experimental result agrees better with the result of the Mori-Tanaka's theory than that of the self-consistent method, even if the volume fraction of the filler is more than 50%. From the above results, it is found that the analytical solutions of this study can generally explain the experimental results and can be applied to actual materials.
Anisotropy is an important property that is widely present in crustal rocks. Efforts have been devoted to providing a constitutive model that can describe both inherent and stress-induced anisotropy in rock. Different from classic models, that are based on stress invariants or strain tensors, we propose here an anisotropic damage microplane model to capture the characteristics of rock properties in different orientations (i.e., their anisotropy). The basic idea is to couple continuum damage mechanics with the classic microplane model. The stress tensor in the model is dependent on the integration of microplane stresses in all orientations. The damage state of any element in the model is determined by the microplane that satisfies the maximum tensile stress criterion or Mohr–Coulomb criterion. An ellipsoidal function was used to characterize the failure strength, where the orientation of the failure plane changes with the preferred orientation of defects in the rock. The proposed model is validated against laboratory experiments performed on brittle material with orientated cracks and granite under true triaxial compression. The fracture pattern and the effect of the intermediate principal stress are numerically predicted by our anisotropic damage model and we discuss relationships between the damage evolution and the anisotropy of the rock under true triaxial compression. The proposed numerical model, based on microplane theory, offers a new approach to analyzing the effect of crack orientation on the deformation and fracture of brittle rock.
Ductile fracture in metal alloys occurs by growth and coalescence of cavities. Growth, also referred to as homogeneous yielding, refers to rather diffuse plasticity around cavities, while coalescence, also termed as inhomogenous yielding, corresponds to the localization of plasticity along some planes or directions. Coalescence can develop in various patterns; three coalescence modes have been observed experimentally: internal necking, coalescence in columns and void sheeting. Plastic anisotropy of the material is known to have a significant effect on both homogeneous and inhomogeneous yielding. Therefore, in the present study, yield criteria accounting for the transition from homogeneous yielding to inhomogeneous yielding modes in anisotropic porous materials are obtained using kinematic limit analysis on a cylindrical unit-cell with a coaxial cylindrical cavity. Two types of plastic anisotropy are considered: Hill (1948) plasticity and crystal plasticity. The proposed analytical yield criteria are compared to numerical limit analysis computations and are found to qualitatively agree with simulations. In particular, plastic anisotropy, void shape effects and their coupling are well captured, especially regarding yield stresses and deformation modes. Finally, an homogenized model for Hill porous materials is obtained by supplementing evolution laws for microstructural parameters (void aspect ratio and ligament size ratios) derived from sequential limit analysis. Proposed evolution laws are then discussed in the light of numerical results and experimental evidence.
Evaluation of the elastic properties of self-compacting fibre-reinforced concrete is one of the primary concerns in civil and structural engineering. This paper investigates the elastic properties of self-compacting fibre-reinforced concrete with varying coarse aggregate and steel fibre content. Traditionally, the elastic properties of concrete are measured experimentally which incurs sig- nificant cost and time overhead. In this paper, a two-step homogenisation approach is proposed for predicting the elastic properties of self-compacting fibre-reinforced concrete. In the first step, the mortar, air voids and aggregates are homogenised based on mean-field homogenisation using the Mori-Tanaka model. X-ray computed tomography (CT) scanning technique was employed to analyse and determine the volume fractions, shapes, numbers of pores for validation purposes. In the second step, a finite element model of representative volume elements is generated with steel fibre inclusions and homogenised concrete to determine the overall macroscale elastic modulus of SCFRC. The results show that the content of aggregates, steel fibres and porosity in the mix has a substantial effect on the elastic modulus. The influence of fibre orientation on the elastic modulus SCFRC is also investigated. The results obtained from the homogenisation method were compared with those obtained from an experimental study and it was found that the maximum error in the elastic modulus prediction using the proposed multiscale homogenisation approach was less than 4%. This agreement between multiscale homogenisation results and experimental data highlights the feasibility of using the two-step homogenisation approach in the development of SCFRC. It has been demonstrated that the proposed homogenisation method can efficiently replace time- consuming laboratory tests, saving both resources and time.
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