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A New Approach to the Application of Mori–Tanaka's Theory in Composite Materials
Abstract
This paper is a reconsideration and reformulation of the Mori-Tanaka's theory in its application to the computation of the effective properties of composites. Previous applications of the theory in this context continued to be linked with eigenstrain, equivalent inclusion, and back stress concepts, and many times involved energy considerations. In this paper we adopt the ‘direct approach’ of defining and computing effective moduli. By elucidating the nature of the approximation in applying Mori—Tanaka's theory to composites insofar as the ‘concentration-factor’ tensors are concerned, we achieve a straightforward exposition and interpretation of the method which are different than those existing in previous formulations. The analysis is given for two-phase composites with anisotropic elastic constituents and an inclusion phase consisting of aligned or randomly oriented ellipsoidal particles. The derived simple expressions for the predicted stiffness and compliance tensors permit a proof of the self-consistency of the method, a discussion of the predictions' relation to the Hashin-Shtrikman bounds in the case of isotropic constituents and randomly oriented ellipsoidal particles, and finally a derivation of some new results in randomly cracked bodies with penny-shaped cracks.
... Notably, when the volume fraction of reinforcements in a composite is low (less than 20%), accounting for their influence can effectively addressed via a mean-field strategy. The method assumes the surrounding area of reinforcements as if it's part of the matrix affected by a strain similar to the matrix's overall average strain (Mori-Tanaka method) [5] or alternatively, by using a medium mirroring the composite's effective stiffness (self-consistent method) [6]. Upon the mean-field approximation, the composite's effective properties, considering multiple reinforcements, can be determined by utilizing solutions of single inhomogeneity problems. ...
... Consequently, when the volume fraction of reinforcement is relatively low, usually less than 20%, it becomes feasible to approximate the average strain field within the reinforcements by embedding a single inhomogeneity within a matrix under the influence of the average strain field of the multi-component composites, a method commonly referred to as the mean-field approximation. In the Mori-Tanaka mean-field homogenization scheme, the relation between the average strain fields in the matrix, inhomogeneity, and composite (represented as 0 , 1 , and (= A , applied strain), respectively) are related as follows [5]: ...
... As the volume fraction c 1 of inhomogeneity tends towards very small values (i.e., the matrix volume fraction becomes very large, c 0 ≈ 1 ), the tensor A converges to T , which is the strain concentration tensor for single inhomogeneity problem. By considering the volume-averaged stress and strain fields within both the matrix and reinforcement ( = c 0 0 + c 1 1 = c 0 L 0 ∶ 0 + c 1 L 1 ∶ 1 and = c 0 0 +c 1 1 , respectively), it becomes possible to predict the effective stiffness ( L eff ) in the Mori-Tanaka scheme for composites as follows [5]: ...
In the realm of technologically important short fiber and particulate-reinforced composites, homogenization approaches based on micromechanics are extensively explored for estimating inherent effective properties. This review provides a comprehensive overview of the core principles underpinning micromechanics-based homogenization, as well as its advancements and applications encompassing: (i) predicting nonlinear reactions under complex and cyclic loading conditions, (ii) accounting for interfacial imperfections, and (iii) estimating various effective physical properties. We also delve into the integration of (iv) data-driven strategies, aiming to augment the accuracy of predictions. We conclude the article by discussing a seminal challenge, (v) the prediction of localized failure.
... where u and σ n , respectively, represent displacement and external force vectors, S represents the outer surface of the composite material, ε 0 and σ 0 represent constant strain and stress tensors, respectively, x is the directional tensor, n represents the external normal of surface S. This set of boundary conditions can be used to define the stiffness and flexibility tensors of effective composite media [22]. According to the above equation, ...
... where L i and M i (I = 1, 2) are the stiffness and flexibility tensors of each phase, respectively, and the direction-dependent tensor (also known as the "concentration factor" [22]), A and B are defined as ε (2) ...
With the increasing implementation of sustainable development strategies, recycled concrete (RC) has garnered attention in research circles due to its substantial environmental and economic advantages. The presence and properties of various interface transition zones (ITZs) in RC play a vital role in its mechanical properties. This research uses a combination of multiphase inclusion theory and finite element numerical simulation to investigate and compare the impact of ITZs on concrete’s mechanical properties. The multiphase inclusion theory offers a theoretical framework for understanding ITZ behavior in concrete, categorizing it into new mortar, old mortar, new ITZ, old ITZ, and natural aggregate based on meso-structure. With simplified RC at the mesoscale, the study accurately predicts the mechanical properties of RC by adjusting the elastic modulus, Poisson’s ratio, and thickness of new and old ITZ models. Through finite element simulation and theoretical validation, the study achieves a minimal error of 6.24% in predicting the elastic modulus and 1.75% in predicting Poisson’s ratio. These results highlight the effectiveness of multiphase inclusion theory in capturing the meso-structure characteristics of RC and forecasting its macroscopic mechanical behavior while comprehensively considering the complexity of ITZs.
... (3) Perfect bonding is assumed between two phases. Homogeneous and isotropic assumptions of cement mortar are common in the study of concretelike materials [10], and perfect bonding between two phases is a consensus in the derivation of elastic theory [11]. When voids are not fully saturated, their elasticity is approximated to zero [12]; thus, the stiffness of voids is negligible when compared to cement mortar. ...
... The coefficient in elasticity matrix of the two-phase composite with aligned ellipsoidal inclusion phase can be given as 11 ...
Foamed concrete has gained increasing attention for its excellent engineering performance. Endeavors have been made to experimentally and numerically elucidate the mechanism of foamed concrete, while few theoretical methods are used to analytically calculate mechanical properties. This paper proposes a theoretical framework to describe the elastic behavior of foamed concrete by combining Eshelby’s equivalent inclusion theory with the Mori–Tanaka method. In the analytical model, foamed concrete is regarded as a two-phase composite comprising a homogeneous cement matrix and randomly oriented ellipsoidal air void. During derivation, the elastic modulus, strength, bulk modulus, shear modulus, and Poisson ratio are measured. A good agreement is established between the proposed method and experimental data.
... 。面板材料主要是金属或复合材料。传统的夹芯材料主要是 泡沫 [29,30] 、蜂窝 [31] 、聚合物 [32,33] 。随着对低质量、高强度、高刚度、多功能的不断追 求，众多新型夹芯结构材料被提出 [34][35][36][37][38] ，其 包括新型蜂窝夹芯 [39,40] 、 负泊松比夹芯 [41] 、 波纹夹芯 [42] 、格栅夹芯 [43][44][45] 、Y 形夹芯 [46] 、自折叠夹芯 [47] 、截头圆顶夹芯 [48] 、点阵 夹芯 [49][50][51][52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68] [76][77][78][79][80] 和 Arrhenius 方程 [81][82][83] [105] 。Hashin-Shtrikman 边界基于最小势能和变分原 理得到了复合材料弹性模量的上界和下界 [106] 。同心圆柱装配模型采用变分法得到了 圆形纤维增强复合材料的弹性模量 [107,108] 。自洽法假设纤维被嵌入一个无穷大的基体 之中，因此该方法也被称为嵌入模型 [109][110][111] 。Christensen 和 Lo [112] 在自洽法的基础上 进行改进得到广义自洽法，该模型包含三个相，分别是中心的夹杂物、外环的基体材 料、最外部的无穷大等效均质材料，推导了三维球形夹杂复合材料和二维横观各向同 性纤维增强复合材料的弹性模量。Mori-Tanaka 法最早由 Mori 和 Tanaka 提出 [113] ，其 认为在基体中任意位置的应力等于基体的平均应力，与进行平均处理的基体区域位置 无关，之后又有许多学者对该方法进行了改进和应用 [114][115][116][117][118][119] 。微分模型最初是被用来 分析刚性球体悬浮液的粘度 [120] ，而后该方法被用来计算两相复合材料的有效弹性模 量 [121,122] 。Halpin-Tsai 方程是一种具有经验系数的半经验方程，其经验系数是通过拟 合实验数据得到的，相比于混合定律可以更准确地预测复合材料的弹性模量 [123][124][125] 。 任意分层同心圆柱模型，假设中心的圆柱体纤维被外面的环状层包裹，环状层的数量 可以是任意层 [126][127][128] 。Saravanos 和 Chamis [129,130] 提出一种基于能量法的综合细观力学 方法，并 应用于方形基体内嵌单纤维代表体积元阵列。Aboudi 等人假设复合材料是具 有重复单胞的周期性结构，先后提出了单胞模型 [131,132] 、通用单胞模型 [133,134] 、高精 度通用单胞模型 [135,136] 。周期性微观结构微观力学采用傅里叶级数，假设均匀应变为 分段常数，考虑掺杂物的几何形状，计算得到具有周期性微观结构的复合材料的弹性 模量，并应用于时域粘弹性力学 [137][138][139] 。胞元横向模型将一个正方形的胞元沿着纤维 ...
... Mori-Tanaka 法(MT) Mori 和 Tanaka 等人 [113][114][115][116][117][118][119] 1973 ...
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.
... In the second step, the effective properties of the LCC are calculated using an elastic (linear) homogenization model. The most common linear homogenization is performed by the Mori-Tanaka model [7,8]. ...
... The elasto-plastic properties of metal-matrix composites (MMCs), in particular, those with an aluminum alloy matrix, have been studied the most. The matrix material obeys the power law of hardening according to Eq. (8). When solving the inverse problem, the step of changing the secant shear modulus in Eq. (16) was μ 0.25 GPa. ...
Based on the first-order Mori–Tanaka secant model, a numerical-analytical model of elasto-plastic deformation of matrix composites is developed. The calculation results of the analytical model are "fitted" to the results of numerical simulation. The fitting parameter of the model is the ratio of the constrained equivalent strain of the matrix in the numerical model to the constrained equivalent strain of the matrix in the Mori–Tanaka model. Identification of the fitting parameter is performed using a reference stress–strain curve calculated numerically for a basic composite composition. For an arbitrary composition composite, a linear dependence of the fitting parameter on the inclusion volume fraction is used. A good agreement was obtained between the results of the calculation by the numerical-analytical model and the numerical data for isotropic and transversely isotropic composites with different contents, morphology, and contrast properties of the phases.
... These inclusions are assumed to be embedded www.nature.com/scientificreports/ in a cementitious material (without capillary pores). The capillary pores and the surrounding cementitious material together can be homogenized using the Mori-Tanaka scheme 36 to obtain the Young's modulus of the homogeneous material at a larger scale of observation (REV II), i.e., hardened cement paste. At this observation scale (REV II), the sand particles and air pores, also assumed to be spherical in shape, are embedded in the homogeneous hardened cement paste matrix (whose properties were just computed using homogenization). ...
Precisely estimating material parameters for cement-based materials is crucial for assessing the structural integrity of buildings. Both destructive (e.g., compression test) and non-destructive methods (e.g., ultrasound, computed tomography) are used to estimate Young’s modulus. Since ultrasound estimates the dynamic Young’s modulus, a formula is required to adapt it to the static modulus. For this formulas from the literature are compared. The investigated specimens are cylindrical mortar specimens with four different sand-to-cement mass fractions of 20%, 35%, 50%, and 65%. The ultrasound signals are analyzed in two distinct ways: manual onset picking and full-waveform inversion. Full-waveform inversion involves comparing the measured signal with a simulated one and iteratively adjusting the ultrasound velocities in a numerical model until the measured signal closely matches the simulated one. Using computed tomography measurements, Young’s moduli are semi-analytically determined based on sand distribution in cement images. The reconstructed volume is segmented into sand, cement, and pores. Young’s moduli, as determined by compression tests, were better represented by full-waveform inversions (best RMSE = 0.34 GPa) than by manual onset picking (best RMSE = 0.87 GPa). Moreover, material parameters from full-waveform inversion showed less deviation than those manually picked. The maximal standard deviation of a Young’s modulus determined with FWI was 0.36, while that determined with manual picking was 1.11. Young’s moduli from computed tomography scans match those from compression tests the closest, with an RMSE of 0.13 GPa.
... In such a case, the elastic stiffness tensor of a composite with misaligned fibers may be determined in terms of the elastic constants of a unidirectional composite weighted by the fiber orientation distribution. In this scenario, the elastic constants of the unidirectional composite can be determined by using well-known micromechanical models such as Mori-Tanaka (Benveniste 1987;Pierard et al. 2004) or numerical homogenization (Böhm and Rasool 2016;Burczyński et al. 2010;Kouznetsova et al. 2001;Segurado and Llorca 2002). Advani and Tucker (1987) proposed to use the orientation tensors a ij and a ijkl to describe the orientation distribution of fibers and provided closed-form solution for the stiffness tensor of composite with misaligned fibers: ...
The orientation distribution of fibers in discontinuous fiber composite materials is influenced by various factors associated with the manufacturing process. Predicting fiber orientation distribution can be achieved through software simulation of the process or experimental methods, such as X-ray computed tomography. As uncertainties related to the reconstruction of fiber orientation distribution may be unavoidable in practical cases, this paper investigates the impact of some of these uncertainties on the effective elastic constants of composites through Monte Carlo simulations. The primary objective of this study is to address how the ratio of fiber detection and the measurement error of individual fiber orientation influence the orientation tensors and effective elastic constants. To predict the elastic properties of composites under various scenarios of fiber orientation distributions, a micromechanical model incorporating an orientation averaging procedure has been used. Three cases of different fiber orientation distributions have been analyzed. The conducted Monte Carlo simulations enabled the presentation of a quantitative description of the uncertainty associated with the reconstruction of the orientation distribution of fibers, including the effective elastic constants. The resulting distributions of orientation tensors and elastic constants have been analyzed and discussed.
... The volume fraction of the inclusions, their elastic properties (e.g., Young's modulus and Poisson's ratio), and the elastic properties of the matrix phase are key inputs for the calculations. The Mori-Tanaka method employs an averaging technique to estimate the effective properties of the composite [63,64]. ...
This work develops a polarization-dependent analytical model using terahertz time-domain spectroscopy (THz-TDS) for computing strain in materials. The model establishes a correlation between volumetric strain and the change in time of arrival for a THz pulse by using the dielectrostrictive properties, variations in doping particle density, and changes in the thickness of the sample resulting from Poisson’s effects. The analytical model is validated through strain mapping of polydimethylsiloxane (PDMS) doped with passive highly dielectrostrictive strontium titanate (STO). Two experiments, using an open-hole tensile and a circular edge-notch specimen are conducted to show the efficacy of the proposed. The stress relaxation behavior of the composite is measured and accounted for to prevent changes in strain during the measurement window. The THz strain mapping results are compared with the finite element model (FEM) and surface strain measurements using the digital image correlation (DIC) method. The experimental findings exhibit sensitivity to material features such as particle clumping and edge effects. The THz strain map shows a strong agreement with FEM and DIC results, thus demonstrating the applicability of this technique for surface and sub-surface strain mapping in polymeric composites.
... Furthermore, the inclusions need to have the same shape and aspect ratio. [20] The studies in this article are based on the two-step homogenization (Mori-Tanaka/Voigt model) approach. ...
This article presents an approach regarding the interlinking of simulation programs to determine mechanical properties of foamed polymers. The software Moldex3D is used to simulate the heterogeneous foam structure, which serves as a database for material modeling. This material modeling is performed with the software Digimat and evaluated using various modeling approaches. The material models are characterized on one side by a homogeneous structure and on the other by the simulated heterogeneous foam structure. In addition, different failure indicators are used and evaluated, which were originally developed for fiber‐reinforced materials. The simulation of the tensile test is carried out with the software Marc‐Mentat. Herein, local stresses and strains are calculated, which reach a maximum value due to the influence of the failure indicators. The results are used to show the extent to which the integration of the heterogeneous foam structure proves to be beneficial. It is additionally demonstrated that the applied failure indicators cannot be used for the prediction of maximum stress and strain simultaneously. Thus, for the time of this study, it is formulated to include both indicators to calculate the maximum stress and strain or to evaluate alternative approaches regarding failure prognosis.
... (e.g., Taya and Arsenault (2016)). Benveniste (1987) improved this approach and encompassed a significant portion of this relevant research (Fig. 8). ...
... The Mori-Tanaka approach provides a framework for estimating the internal stress within a matrix that incorporates inclusions by using eigenstrains. Benveniste [6] reformulated this approach, elucidating the assumptions made within the theory. Their method employs the Mori-Tanaka tensor to relate the stresses and strains in the matrix and fiber, which are linked through a concentration tensor. ...
This study aims to critically assess different micromechanical analysis models applied to carbon-fiber-reinforced plastic (CFRP) composites, employing micromechanics-based homogenization to accurately predict their effective properties. The paper begins with the simplest Voigt and Reuss models and progresses to more sophisticated micromechanics-based models, including the Mori–Tanaka and Method of Cells (MOC) models. It provides a critical review of the areas in which these micromechanics-based models are effective and analyses of their limitations. The numerical analysis results were confirmed through finite element simulations of the periodic representative volume element (RVE). Furthermore, the effective properties predicted by these micromechanics-based models were validated via experiments conducted on IM7/5320-1 composite material with a fiber volume fraction of 0.62.
... Therefore, the need for a fast and accurate method for solving the micromechanical problem over different varying spatial microstructures emerges naturally. Computational homogenization methods operate on a particular spatially resolved microstructure and determine the macroscopic properties in a direct manner, alleviating the restrictive geometric assumptions of mean field or analytical homogenization approaches [10][11][12][13] The combination of Fast Fourier Transform (FFT) approaches [14][15][16][17] and model-order reduction (MOR) frameworks [18][19][20] yields promising results. However, the MOR-based approach is restricted to the models in the micro-scale that it is trained on. ...
A key challenge for the virtual characterization of components manufactured using short fiber-reinforced thermoplastics (SFRTs) is the inherent anisotropy which stems from the manufacturing process. To address this, a multi-scale approach is necessary, leveraging deep material networks (DMNs) as a micromechanical surrogate, for a one-stop solution when simulating SFRTs under highly nonlinear long-term load cases like creep and fatigue. Therefore, we extend the a priori fiber orientation tensor interpolation for quasi-static loading (Liu et al. in Intelligent multi-scale simulation based on process-guided composite database. arXiv:2003.09491, 2020; Gajek et al. in Comput Methods Appl Mech Eng 384:113,952, 2021; Meyer et al. in Compos Part B Eng 110,380, 2022) using DMNs with a posteriori approach. We also use the trained DMN framework to simulate the stiffness degradation under fatigue loading with a linear fatigue-damage law for the matrix. We evaluate the effectiveness of the interpolation approach for a variety of load classes using a dedicated fully coupled plasticity and creep model for the polymer matrix. The proposed methodology is validated through comparison with composite experiments, revealing the limitations of the linear fatigue-damage law. Therefore, we introduce a new power-law fatigue-damage model for the matrix in the micro-scale, leveraging the quasi-model-free nature of the DMN, i.e., it models the microstructure independent of the material models attached to the constituents of the microstructure. The DMN framework is shown to effectively extend material models and inversely identify model parameters based on composite experiments for all possible orientation states and variety of material models.
... In the Mori-Tanaka scheme [12] as reformulated by Benveniste [26,13], the field in the matrix at a sufficient distance from an inclusion is approximated by the constant value of its mean. As a result, the existence of further inclusions is encoded in the mean field of the matrix, and thus the method takes into account the particle interactions. ...
Several analytical mean-field homogenization methods, which take into account the particle volume fraction, shape and orientation are readily available to estimate the effective properties of particulate composites. Models have also been proposed to account for the spa
tial distribution of the particles. The classical Ponte-Castañeda and Willis (PCW) model is based on a parametrization of the statistical distribution law, while the Interaction Direct Derivative (IDD) model associates a matrix cell to each inclusion, representative of close
interactions. In the literature, the use of the IDD is commonly reduced to the particular case of the classical Mori and Tanaka (MT) scheme or to the aforementioned PCW model. The present study introduces an original approach to calibrate the IDD model, for 2D
linear conductivity, based on representative images of the microstructure. The links between the models and the range of validity of the IDD model are discussed. Besides, an “IDD-based” PCW model and a two-step scheme are proposed for situations where the IDD estimate is inconsistent (lack of major symmetry). Finally, an image analysis method using Voronoı̈ diagrams is implemented to define the cells associated to each inclusion and supply the models. The method is validated by comparisons between the obtained IDD and PCW estimates, the Mori-Tanaka (MT) model and benchmark full-field numerical simulations.
Accounting for the inclusion distribution is seen to lead to better estimates, both qualitatively (by capturing anisotropic behaviors due to the sole distribution) and quantitatively. Possible extensions to elastic composites are discussed.
... For the Mori-Tanaka scheme the strain localisation tensor A MT , is formulated according to Benveniste et al. 44,45 , following the theory by Eshelby 37 , that the strain for a homogeneous and ellipsoidal inclusion in an infinite matrix is constant. It depends on the fourth order identity tensor I, the stiffness tensor for the matrix C m , the stiffness tensor for the fibre C f and Eshelby's tensor P. The strain localisation tensor is then a fourth-order tensor and can be expressed as ...
Among micro-scale imaging technologies of materials, X-ray micro-computed tomography has evolved as most popular choice, even though it is restricted to limited field-of-views and long acquisition times. With recent progress in small-angle X-ray scattering these downsides of conventional absorption-based computed tomography have been overcome, allowing complete analysis of the micro-architecture for samples in the dimension of centimetres in a matter of minutes.
These advances have been triggered through improved X-ray optical elements and acquisition methods. However, it has not yet been shown how to effectively transfer this small-angle X-ray scattering data into a numerical model capable of accurately predicting the actual material properties. Here, a method is presented to numerically predict mechanical properties of a carbon fibre-reinforced polymer based on imaging data with a voxel-size of 100 μm corresponding to approximately fifteen
times the fibre diameter. This extremely low resolution requires a completely new way of constructing the material’s constitutive law based on the fibre orientation, the X-ray scattering anisotropy, and the X-ray scattering intensity. The proposed method combining the advances in X-ray imaging and the presented material model opens for an accurate tensile modulus prediction for volumes of interest between three to six orders of magnitude larger than those conventional carbon fibre orientation image-based models can cover.
... The study mainly focuses on S-glass/polypropylene/epoxy laminae under transverse tension and transverse shear loading to understand the matrix-dominant behaviour of such fibre-hybrid composite laminates. The predicted effective transverse elastic moduli are compared against Mori-Tanaka [28,29] and Chamis [30,31] analytical models for conventional unidirectional composite laminae. The validated RUC models are used to conduct parametric studies and the results are presented. ...
This paper investigates the effect of intra-laminar fibre hybridisation, i.e., primary and secondary fibres within a matrix, on the homogenised properties and micro-stress fields in uni-directional polymer composite laminae. The study is focused on S-glass/epoxy laminae which are hybridised with secondary fibres (e.g., polypropylene). Two-dimensional repeating unit cells (2D RUCs) with periodic microstructures are developed to conduct the micro-mechanical analyses under transverse tensile and transverse shear loading conditions. Uni-directional fibre-hybrid S-glass/epoxy laminae with different secondary fibres are studied by varying (a) the periodic microstructure and (b) the material properties of the constituent fibres to assess the effect of such geometric and material variations on the homogenised elastic lamina properties and intra-lamina micro-stress fields. The results show that intra-laminar fibre hybridisation significantly affects the elastic lamina properties and micro-stress fields. Notably, the presence of the secondary fibres significantly increases or reduces the stress fields in the matrix and at the fibre-matrix interfaces (i.e. normal and shears stress components)–depending on the microstructure and the stiffness of the secondary fibres–which could be explored to manipulate the damage modes and thus energy dissipation mechanisms.
... Research focusing on the characteristics of composites through techniques that analyze the representative volume element (RVE) is known as research using a homogenization technique [17][18][19][20]. The Eshelby model served as the primary tool for the development of widely employed analytical and homogenization methods [21][22][23][24]. For the purpose of modeling the investigated composite material, these methods are insufficient due to significant simplification of the RVE geometry. ...
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.
... In such a case, we are dealing with an anisotropic PD layer v l i (52) that presents no additional difficulties because all equations in Sections 3-5 also hold for a general case of anisotropy of both the layers and matrix. At the 50-year history, the original MTM (see [13,66,69]) was applied and developed to a very wide class of local micromechanical problems. All these applications are easily generalized to the currently considered PD multilayer CM. ...
The basic feature of the peridynamics (PD) is a continuum description of material behavior as the integrated nonlocal force interactions between material points. Besides the conventional local theory, the PD equation of motion introduced by Silling (J Mech Phys Solids 48: 175-209, 2000) has no spatial derivatives of displacement. A linearized bond-based PD model is used for the analysis of random structure CMs subjected to the remote volumetric homogeneous boundary conditions. Effective properties are expressed through the local stress polarization tensor averaged over the external interaction interface of inclusions rather than in an entire space. Any spatial derivatives of displacement fields are not required. Inclusions are considered as identical aligned layers with a statistically homogeneous distribution in the space. For one infinite layer in the infinite homogeneous matrix, 3D PD equilibrium equation is reduced to the 1D integral equation with 1D micromodulus obtained by integrations of the original 3D micromodulus over the cross-sections (parallel to layers) of a horizon region. One estimates the average strain and stress fields in the extended layer by the use of averaging displacement and traction over the external interaction interface. Effective moduli for PD multilayered CM are estimated in the matrix form representations usually used in locally elastic multilayered CM and based on the consideration of the normal and tangential parts of the effective moduli matrix.
... For numerical methods, the difference method (FDM) (Abushaikha & Terekhov, 2020), finite volume method (FVM) , finite element method (FEM) (Wu et al., 2022), dissipative particle dynamics (DPD) (Phan-Thien et al., 2018), direct simulation Monte Carlo (Hong & Morris, 2022), smoothed-particle hydrodynamics (Islam et al., 2022), and lattice Boltzmann method (LBM) (Ju et al., 2020;Li et al., 2023aLi et al., , 2024 are utilized commonly. On the other hand, the relevant theoretical models include Katz-Thompson model (Katz & Thompson, 1986), generalized effective medium theory (Mclachlan, 1987), Mori-Tanaka method (Benveniste, 1987;Gong et al., 2011;Xu, Zhang, et al., 2019), differential effective medium approximation (D-EMA) (Norris et al., 1985;Xu et al., 2018), double/ multiple-inclusion model (Hori & Nemat-Nasser, 1993;Xu et al., 2017), and Kozeny-Carman (KeC) equation (Carman, 1937;Chen & Yao, 2017;Kozeny, 1927;Zhang & Schaap, 2019). Among them, one of the most commonly used semi-empirical formulas is the Kozeny-Carman (KeC) equation, as shown in Eq. (1). ...
The microstructure of granular media, including grain’s shape- and size-polydispersities, orientation, and area fraction can potentially affect its permeability. However, few studies consider the coupling effects of these features. This work employs geometrical probability and stereology to establish quantitative relationships between the above microstructural features and the geometric tortuosity of the two-dimensional granular media containing superellipse, superoval, and polygon grains. Then the lattice Boltzmann method (LBM) is used to determine the permeabilities of these granular media. By combining the tortuosity model and the LBM-derived permeabilities, modified K-C equations are formulated to predict the permeability and the shape factor, considering the grain’s shape- and size-polydispersities, orientation, and area fraction. The reliability of these methods can be verified
by comparing with both our simulations and available experimental, theoretical, and numerical data reported in the literature. The findings implicate that the tortuosity and permeability of the granular media are strongly correlated with the grain’s shape, orientation, and area fraction but unaffected by the size polydispersity and spatial arrangement of grains. Only circularity is not enough to derive a unified formula for considering the impact of grain shape on tortuosity and permeability, other shape parameters need to be explored in the future.
... The authors chose the representation (41) to point out the linearity of the orientation average in F. It should be noted, that the orientation average of the unidirectional stiffness tensor mathematically corresponds to a Voigt-like averaging [57,58]. Different approaches such as performing orientation average on strain-localization tensor approximations or effective unidirectional compliance tensors (Reuss-like) exist [44,59]. ...
Fiber orientation tensors (FOT) are widely used to approximate statistical orientation distributions of fibers within fiber-reinforced polymers. The design process of components made of such fiber-reinforced composites is usually accompanied by a virtual process chain. In this virtual process chain, process-induced FOT are computed in a flow simulation and transferred to the structural simulation. Within the structural simulation, effective macroscopic properties are identified based on the averaged information contained in the FOT. Solving the field equations in flow simulations as well as homogenization of effective stiffnesses necessitates the application of a closure scheme, computing higher-order statistical moments based on assumptions. Additionally, non-congruent spatial discretizations require an intermediate mapping operation. This mapping operation is required, if the discretization, i.e., mesh, of the flow simulation differs from the discretization of the structural simulation. The main objective of this work is to give an answer to the question: Does the sequence of closure and mapping influence the achieved results? It will turn out, that the order influences the result, raising the consecutive question: Which order is beneficial? Both questions are addressed by deriving a quantification of the closure-related uncertainty. The two possible sequences, mapping followed by closure and closure followed by mapping, yield strongly different results, with the magnitude of the deviation even exceeding the magnitude of a reference result. Graphical consideration reveals that for both transversely isotropic and planar FOT-input, invalid results occur if the mapping takes place prior to closure. This issue is retrieved by orientation averaging stiffness tensors. As a by-product, we explicitly define for the first time the admissible parameter space of orthotropic fourth-order fiber orientation tensors and define a distance measure in this parameter space.
Reduced order models (ROMs) are often coupled with concurrent multiscale simulations to mitigate the computational cost of nonlinear computational homogenization methods. Construction (or training) of ROMs typically requires evaluation of a series of linear or nonlinear equilibrium problems, which itself could be a computationally very expensive process. In the eigenstrain‐based reduced order homogenization method (EHM), a series of linear elastic microscale equilibrium problems are solved to compute the localization and interaction tensors that are in turn used in the evaluation of the reduced order multiscale system. These microscale equilibrium problems are typically solved using either the finite element method or semi‐analytical methods. In the present study, a reduced order variational spectral method is developed for efficient computation of the localization and interaction tensors. The proposed method leads to a small stiffness matrix that scales with the order of the reduced basis rather than the number of degrees of freedom in the finite element mesh. The reduced order variational spectral method maintains high accuracy in the computed response fields. A speedup higher than an order of magnitude can be achieved compared to the finite element method in polycrystalline microstructures. The accuracy and scalability of the method for large polycrystals and increasing phase property contrast are investigated.
The compressive strength of concrete with low strength aggregate volume fractions of 10%, 20%, and 30% was calculated using both the graphical analysis method and mesomechanics method. In the calculation method based on graphical analysis, three-dimensional random spherical aggregate models of concrete with different volume contents of low strength aggregate were established and sliced. The results show that the graphical analysis method can effectively calculate the compressive strength of concrete with different volume contents of low strength aggregate. In the graphical analysis method, the relative errors of the calculated compressive strength of concrete with low strength aggregate volume fractions of 10%, 20%, and 30% were 4.84%, 4.84%, and 6.43%, respectively. Three-phase concrete models composed of mortar, aggregate, and interfacial transition zone were analyzed through the method of mesomechanics. The calculation results of the mesomechanics method show that the compressive strength of concrete was controlled by low strength aggregate, and the calculated compressive strength of concrete decreased with the increase in low strength aggregate volume content. In the mesomechanics method, the relative errors of the calculated compressive strength of concrete with low strength aggregate volume fractions of 10%, 20%, and 30% were 1.36%, 1.74%, and 3.7%, respectively. It can be found that the calculation results of the mesomechanics method are closer to the test values and have smaller relative errors, which indicate that the calculation method of mesomechanics theory is superior to the method of graphical analysis.
Mean field homogenization methods like the Mori–Tanaka formulation are used to determine the effective response of heterogeneous materials. Typically, these methods are deployed for inclusions within the isotropic matrix. This is due to the availability of closed-form solutions for Eshelby’s tensor when the surrounding medium is isotropic. However, in real life, the matrix can often be transversely isotropic or even orthotropic. This paper proposes a model for the implementation of the Mori–Tanaka formulation for all types of matrices. The proposed method is implemented and validated against full finite element models for three length scales: effective properties (RVE level), phase average stresses (constituent level) and interface stresses. The proposed models are seen to deliver reasonable results at the different length scales for transversely isotropic as well as orthotropic matrices.
Velocities of low-frequency seismic waves and, in most rocks, sonic logging waves depend on the compressibility of the undrained rock, which is conventionally computed from the drained rock compressibility using Gassmann's equation. Although more comprehensive and accurate alternatives exist, the simplicity of the equation has made it the preferred fluid substitution model for geoscience applications. In line with recent publications, we show that Gassmann's equation strictly applies only to rocks with a microhomogeneous void space microstructure that is devoid of cracks and microcracks. We use a rock physics model that separates the respective compliance contributions of pores and cracks on dry (drained) moduli and show that Gassmann's model does not apply to rocks with measurable crack density. A fourth independent bulk modulus (in addition to the bulk moduli of the mineral matrix, dry frame, and saturating fluid) is required to take the effect of cracks into account and perform fluid substitution modeling for rocks with pores and cracks more accurately than prescribed by Gassmann's equation. Therefore, we propose combining the Vernik-Kachanov model with Brown-Korringa's equation for more reliable modeling of undrained bulk compressibility for reservoir rocks with measurable crack density. To conclude, a practical quantification of the applicability of Gassmann's equation based on the combined effects of crack density and stress sensitivity is proposed.
Carbon nanotube (CNT) has fostered research as a promising nanomaterial for a variety of applications due to its exceptional mechanical, optical, and electrical characteristics. The present article proposes a novel and comprehensive micromechanical framework to assess the viscoelastic properties of a multiscale CNT-reinforced two-dimensional (2D) woven hybrid composite. It also focuses on demonstrating the utilisation of the proposed micromechanics in the dynamic analysis of shell structure. First, the detailed constructional attributes of the proposed trans-scale composite material system are described in detail. Then, according to the nature of the constructional feature, mathematical modelling of each constituent phase or building block’s material properties is established to evaluate the homogenised viscoelastic properties of the proposed composite material system. To highlight the novelty of this study, the viscoelastic characteristics of the modified matrix are developed using the micromechanics method of Mori–Tanaka (MT) in combination with the weak viscoelastic interphase (WI) theory. In the entire micromechanical framework, the CNTs are considered to be randomly oriented. The strength of the material (SOM) approach is used to establish mathematical frameworks for the viscoelastic characteristics of yarns, whereas the unit cell method (UCM) is used to determine the viscoelastic properties of the representative unit cell (RUC). Different numerical results have been obtained by varying the CNT composition, interface conditions, agglomeration, carbon fibre volume percentage, excitation frequency, and temperature. The influences of geometrical parameters like yarn thickness, width, and the gap length to yarn width ratio on the viscoelasticity of such composite material systems are also explored. The current study also addresses the issue of resultant anisotropic viscoelastic properties due to the use of dissimilar yarn thickness. The results of this micromechanical analysis provide valuable insights into the viscoelastic properties of the proposed composite material system and suggest its potential applications in vibration damping. To demonstrate the application of developed novel micromechanics in vibration analysis, as one of the main contributions, comprehensive numerical experiments are conducted on a shell panel. The results show a significant reduction in vibration amplitudes compared to traditional composite materials in the frequency response and transient response analyses. To focus on the aspect of micromechanical behaviour on dynamic response and for the purpose of brevity, only linear strain displacement relationships are considered for dynamic analysis. These insights could inform future research and development in the field of composite materials.
The use of nano-fillers as reinforcement in natural fibers-based hybrid composites has gained prominence in multiple sectors in recent years because of their virtuous mechanical and physical characteristics. The impeccable properties of nano-fillers like their high aspect ratio and larger surface area have made them to be used in areas for instance, sectors like aviation, automotive, and biotechnology fields. This study focuses on examining how various weight percentages of nano-calcium carbonate (NCaCO3) fillers (2%, 5%, 7%) impact the elastic properties of innovative hybrid composites blended with banana and kenaf fibers, combined with epoxy. The elastic characteristics of the suggested composite, including longitudinal elastic modulus (LEM), transverse elastic modulus (TEM), longitudinal Poisson’s ratio (LPR), and longitudinal shear modulus (LSM), are analyzed through micromechanical models such as the Mori–Tanaka (M–TA) model, generalized self-consistent (GS-C) model, and modified Halpin–Tsai (M-HT) model. The composite consisting of a solitary banana fiber sheet, a solitary NCaCO3 mix epoxy sheet, and another solitary kenaf fiber sheet is modeled in ANSYS APDL simulation software. The composite’s layers are organized in a specific order: starting with banana fiber at 90° orientations, followed by a layer of NCaCO3 and epoxy at 0° orientations, and concluding with kenaf fiber at 90° orientations. The ANSYS software is employed to analyze the total sum deformation and strength of the suggested composite. The outcomes obtained from this research are contrasted and confirmed through comparison with existing literature. The inclusion of 7 wt% of NCaCO3 in the suggested hybrid composite is found to have the highest elasticity and ductility in comparison with 2 wt% and 5 wt% of NCaCO3. The composite containing 7 wt% NCaCO3 demonstrates the greatest load-bearing capability. Additionally, while calculating the elastic characteristics of the proposed composite, both the modified Halpin–Tsai (M-HT) model and the generalized self-consistent model (GS-C) outperform the Mori–Tanaka model (M–TA). Furthermore, the hybrid impact is computed for the suggested composite to analyze the tensile strain rates at which failure occurs for banana and kenaf fibers within the composite hybrid structure. The computed hybrid value of 0.5 indicates that the failure rate of a non-hybridized composite is 50% more than the hybridized composite. This signifies that the hybrid composites have high load-bearing strength, high elasticity, and stiffness.
This article aims at proposing a new mean-field homogenization framework for the study of composites undergoing fully coupled thermomechanical processes. Strongly dissipative phenomena during high or moderate cyclic loading conditions in a structural component made of a composite material cause significant interplay between mechanical and thermal fields. The proposed framework attempts to address such effect by combining the Mori–Tanaka scheme and the Transformation Field Analysis (TFA) theory and by developing a multiscale framework capable of taking into account thermomechanically coupled processes. The numerical simulations performed in the examples section and validations with computations using periodic homogenization and full-structure analysis demonstrate the proposed strategy’s accuracy and robustness. The numerical simulation of a tube shows the model’s ability to simulate cyclic loading conditions with significantly less computational cost than the alternative FE
computation strategies. This drastic computational time reduction is due to the semi-analytical formalism of the micromechanics methodology.
Decomposing the representative volume element (RVE) of short fiber-reinforced plastics (SFRPs) into several pseudograins (PGs) is essential for understanding its effective mechanical behavior. However, conventional PG decomposition methodologies are limited by their high computational costs due to iteration-based algorithms. To address this, we propose a machine learning-assisted PG decomposition procedure that utilizes a series–parallel artificial neural network (ANN) system to facilitate the time-consuming decomposition process. To validate the effectiveness of our proposal, we implemented a two-step homogenization framework of SFRP that consists of the series–parallel ANN system, Mori-Tanaka model, and Voigt model into ABAQUS user material subroutine (UMAT). The elastic modulus values predicted by the UMAT are found to be in good agreement with both DIGIMAT-MF and experimental values, while also maintaining low computational time.
Short fiber-reinforced thermoplastics have complex mechanical behavior due to local variations in the fiber orientation and concentration distribution. In particular, the mechanical properties of infrared-welded components are difficult to predict due to reorientation and accumulation of fibers in the weld line. In this work, a model is developed that is able to consider locally varying fiber orientation and local variation of the fiber concentration to calculate the elastic material properties. The model is applied to infrared-welded components and the results are compared with experimental investigations. Two materials are studied, PA6 GF50 and PPA GF35. The material properties are characterized in order to obtain the necessary information for the simulation. The validation of the model shows a good agreement between the simulation and the experiments.
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.
Description
This collection of work represents the tremendous progress that has recently been made toward the characterization and modeling of Titanium Matrix Composites (TMCs), including microstructural aspects, stress analysis, and life models. 29 papers divided into five sections cover:
• Interfacial Properties and Microstructure
• Fiber Bridging Behavior
• Inelastic Material Behavior and Modeling
• Fatigue
• and Life Predictions.
The paper introduces a novel data-driven homogenization model developed for the computation of stress-strain curves for composites featuring arbitrary orientation distributions, aspect ratios, and volume fractions of fibres. The objective of the proposed model is to deliver both time-efficient and accurate results while taking into account a relatively small database. The database is generated through numerical solutions of the equivalent inclusion problem, encompassing results for predefined orientations and aspect ratios of the single inclusion (fibre). The database created for a specific matrix-fibre system can be employed by the surrogate model to predict outcomes for various cases involving different fibre configurations. The surrogate model comprises four key modules: aspect ratio interpolation, a discrete orientation averaging procedure involving a genetic algorithm, a spatial transformation procedure, and an iterative Mori-Tanaka scheme. The validity of the proposed data-driven model has been confirmed by comparing the obtained results with those acquired through analysis of the complex representative volume element (RVE) using Fast Fourier Transform (FFT)-based homogenization, as well as through comparisons with experimental results from the existing literature.
This study investigated the microscale behaviours and damage evolutions of graphene/epoxy nanocomposites under uniaxial tensile loadings. Computational molecular dynamics (MD) simulation was applied to model the microstructural constituents of graphene/epoxy composites and the evaluation of their mechanical and the damage responses at different temperatures. The computed results revealed the influence of the proportions of graphene and temperatures on the mechanical properties, the damage evolution and the change in strain energy associated with failure of the nanocomposites. The study has added to the understanding of the damage development and failure of the graphene nanoplatelets (GNP)/epoxy composites.
This study aims to develop an interpretable ensemble machine learning (EML) method for predicting the homogenized elastic properties of unidirectional fiber-reinforced composite. Three machine learning models—Random Forest (RF), eXtreme Gradient Boosting Machine (XGBoost), and Light Gradient Boosting Machine (LGBM) are selected for constructing the EML model. The ground-truth dataset is created via knowledge-based finite element (FE) simulations with representative volume element (RVE) models of composites considering randomly dispersed fibers. We thoroughly evaluate the EML model’s performance using the metrics of accuracy, efficiency, interpretability, and generalization. The obtained results indicate that: (1) the EML model outperforms the base ML model with high precision (R2 = 0.962, MSE = 5.41); (2) the SHapley Additive explanations (SHAP) based on cooperative game theory is used to interpret the predictions, with the global interpretations identifying fiber volume content as the most influential variable on the composite and the local interpretations examining the key influencing mechanisms of each feature; (3) the EML model demonstrates good generalization ability on experimental data, and it can accurately forecast the homogenized properties of the composite with genuine fibers.
For an elastic body containing periodically distributed voids, several effective techniques are presented which can be used to obtain the effective elastic moduli with any desired degree of accuracy. The results include the effects of void geometry as well as void interactions. For a body containing spherical voids, numerical results are presented and compared with those obtained by other methods.
Internal inconsistencies and limitations of typical failure criteria are pointed out. A judicious linear approximation seems preferable to the complicated nonlinear analyses in predicting the first ply failure. Curing stresses induced during fabrication should be included in the analysis. When the applied strain exceeds the failure strains of weaker plies, cracks begin to appear in these plies. The authors' preliminary results show this formation of cracks to be predictable. More work is needed in the effects of the first ply failure on the subsequent structural behavior under various loading and environmental conditions. A simple comparison is shown among the methods for the prediction of notched strength - the inherent flat model, the point and average stress critieria, and Barenblatt-Dugdale model. A possible application of the resistance method is also suggested. Delamination, which is unique to composite laminates, should be recognized as an independent failure mode and treated accordingly.
The basis for the estimates is given in the form of an auxiliary solution characterizing the behavior of a single embedded site. Both classical self-consistent methodologies (SCM) and rationalized differential formulations (DCM) are developed, with emphasis on physical realism, applications to polycrystalline and various composite materials (including porous media and plate bending) are compared favorably with many experimental results. Recent attempts to generate effective plasticity/creep models plus estimates of thermal/electrical/pore-fluid transmissivity, are examined critically. A general conclusion is that DCM provides remarkably accurate descriptions but that even more detailed characterization of microstructure methodologies is essential for extreme circumstances, such as discrete site interaction or dominance of nicrogeometric connectivity. Fully numerical simulation may be avoided by specializations of exact integral equations: among these, only the self-consistent approximation and a few isolated second-order estimates have been explored.
Description
This publication, Recent Advances in Composites in the United States and Japan, contains papers presented at the United States/Japan Symposium on Composite Materials which was held in Hampton, Virginia, 6-8 June 1983. The symposium was sponsored by ASTM Committees D30 on High Modulus Fibers and Their Composites and E09 on Fatigue in cooperation with the National Aeronautic and Space Administration. Jack R. Vinson and Minoru Taya, University of Delaware, served as symposium chairman and secretary, respectively. Jack R. Vinson and Minoru Taya are editors of this publication.
Van der Poel's method for calculating the shear modulus of a particulate composite agrees well with experimental data, but its validity has been questioned, and it was applicable only to composites in which the matrix material is incompressible. These limitations are removed in this paper in which an error in the original derivation is corrected, and the method generalized to apply to any matrix material. Calculations using the corrected theory show that despite the error, a table of shear modulus values published with the original theory is sufficiently correct for most practical purposes. Applicability of the generalized method to the large class of composites having compressible matrices is discussed. Shear moduli calculated by the corrected and extended method are compared with corresponding values calculated by other methods currently used.
The elasticity of a polycrystalline aggregate is expressed in terms of the elasticity of the individual grains. The stress within each grain is estimated with the aid of an analysis of the stress distribution around a spherical cavity in an isotropic medium. The strain within each grain is expressed in terms of the average stress in the polycrystal as a whole by pseudoelastic constants which are related to the actual elastic constants. The calculated elasticities for physical tests and for x-ray diffraction measurements in polycrystals are given for a few cubic metals.
A self-consistent model is used to calculate the elastic moduli of two-phase materials with the assumption that the inclusions are spheroidal. The limiting cases of needle and disk shaped inclusions are explicitly shown. It is found that the latter has the greatest effect in changing the elastic moduli of the matrix.RésuméUn modèle auto-uniforme sert à calculer les coéfficients d'élasticité de matières biphasées en supposant que les inclusions soient sphéroidales. Les cas limites d'inclusions en forme d'aiguille ou de disque sont nettement indiqués. On a trouvé que cette dernière cause le plus grand changement de coéfficients de la matrice.ZusammenfassungEin in sich konsistentes Modell wird zur Errechnung des Elastizitätsmoduls zweiphasiger Werkstoffe benützt unter der Voraussetzung, dass die Einschlüsse sphäroider Form sind. Grenzfälle nadel-und scheibenförmiger Einschlüsse werden ausdrücklich gezeigt. Es wurde festgestellt, dass die letzteren die stärkste Wirkung in Bezug auf die Veränderung des Elastizitätsmoduls der Matrix ausüben.РефератДaeтcя пpимeнeниe caмocoвмecтимoй мoдeли для вычиcлeния мoдuлн uпpuгocти двuчфaзoвыч вeщecтв, пpeдпoлaгaя cфepoиднocть влoжeний. Пoдpoбнo paзpaбoтaниы кpaйниe cлuчaи иглы и диcкa. ucтaнoвлeннo, чтo мaкcимaльнoe измeнeниe мoдuля uпpuгocти мaтpиcы пocтигaeтcя в лимитиpuющeм cлuчae диcкa.
An investigation is made of void growth in a viscous metal. First, the effective viscosity μ and expansion viscosity κ of a viscous body containing spherical voids is estimated. Then the growth of a typical void embedded in a compressible viscous body with μ and κ is studied. The analytical tool used in this study is a combination of Eshelby's equivalent inclusion method and Mori-Tanaka's back stress analysis. By Mori-Tanaka's back stress analysis the interaction among voids which is enhanced at larger volumn fractions of void can be taken into account in our analysis.
This paper examines the effects of fibre orientation on the effective elastic moduli, thermal conductivities and thermoelastic constants of short-fibre composites. The short fibres are assumed to be ellipsoidal in shape and uniformly distributed in the matrix. Both two- and three-dimensional fibre distributions are considered. Numerical results and their comparisons with some experimental data are presented. The special case of random fibre distribution is also examined.
A self-consistent method of estimating effective macroscopic elastic constants for inhomogeneous materials with ellipsoidal inclusions is developed using elastic-wave scattering theory. The method is a generalization of the method for spherical inclusions presented in part I. The results are compared to the Kuster-Toksoz estimates for the elastic moduli and to the rigorous Hashin-Shtrikman bounds and Miller bounds. For general ellipsoidal inclusions, the self-consistent estimates satisfy both the Hashin-Shtrikman bounds and the more stringent Miller bounds. The method is also briefly compared to other self-consistent scattering theory approaches.
A theoretical study has been made to predict overall thermal expansion characteristics of a two-phase material consisting of a matrix and aligned ellipsoidal inclusions. The calculation is based on Eshelby's theory on the transformation problem; an approximate method is Introduced to account for the effect of finite concentrations of the inclusions. The following two cases are treated: elastic matrix-elastic inclusions and elastoplastic matrix- elastic inclusions. The results are explicitly obtained when shapes of the inclusions are spherical, disc-shaped, and fiber-shaped. The calculated over all thermal expansion coefficient for the case of elastic matrix-elastic spheres agrees with Kerner's prediction.
A matrix is considered with elastic moduli K//1 and G//1 (bulk and shear moduli respectively) in which are embedded particles with elastic moduli K//2 and G//2. The fractional volume of the particles is c//2 and the fractional volume of the matrix is c//1 (c//1 plus c//2 equals 1). This two-phase suspension is assumed to be statistically homogeneous and isotropic. The effective bulk and shear moduli are K* and G* respectively. It is shown that, if (G//2 minus G//1)(K//2 minus K//1) greater than equivalent to 0 (most usual cases), and for dilute suspensions, the spherical inclusion provides the minimum reinforcing effect, and the disk-shaped inclusion the maximum one. When the suspension is not dilute, a self-consistent scheme is used. It is proved that disk-shaped inclusions still provide the maximum reinforcing effect.
Compressibility of porous material is greater than that of solid material of the same composition, and the difference is shown to be equal to rate of change of porosity with pressure, for any pore shape or concentration. Expressions for compressibility are given for two special cases for material of low pore concentration: for spherical pores and for narrow cracks. Comparison of the two cases shows that a crack increases compressibility nearly as much as a spherical pore of the same diameter as the length of the crack, although porosity in these two cases differs enormously. For material in which all porosity occurs as narrow cracks, it is shown that porosity can, in certain cases, be determined quite precisely from compressibility measurements.
It is supposed that a region within an isotropic elastic solid undergoes a spontaneous change of form which, if the surrounding material were absent, would be some prescribed homogeneous deformation. Because of the presence of the surrounding material stresses will be present both inside and outside the region. The resulting elastic field may be found very simply with the help of a sequence of imaginary cutting, straining and welding operations. In particular, if the region is an ellipsoid the strain inside it is uniform and may be expressed in terms of tabulated elliptic integrals. In this case a further problem may be solved. An ellipsoidal region in an infinite medium has elastic constants different from those of the rest of the material; how does the presence of this inhomogeneity disturb an applied stress-field uniform at large distances? It is shown that to answer several questions of physical or engineering interest it is necessary to know only the relatively simple elastic field inside the ellipsoid.
A formulation to compute the effective thermal expansion coefficients (αc ) of an anisotropic short fiber-reinforced composite and the thermal stress (σ) induced in and around the fiber is developed. The formulation is based on the Eshelby’s equivalent inclusion method. Main emphasis is placed on short Carbon fiber/Aluminum. The thermal stress due to a uniform temperature rise ΔT is computed at points just outside the fiber. The effects of various parameters on αc and σ are also investigated.
THIS PAPER EXAMINES THE EFFECTIVE LONGITUDINAL YOUNG'S MODULUS OF COMPOSITES CONTAINING MISORIENTED SHORT FIBERS. THE ANALYSIS IS BASED ON THE ESHELBY'S EQUIVALENT INCLUSION METHOD AND THE AVERAGE INDUCED STRAIN APPROACH OF TAYA, MURA, ANDCHOU. THE PRESENT APPROACH IS UNIQUE IN THAT IT TAKES INTO ACCOUNT THE INTERACTIONS AMONG FIBERS AT DIFFERENT ORIENTATIONS. NUMERICAL RESULTS ARE PRESENTED TO DEMONSTRATE THE EFFECTS OF FIBER ELASTIC PROPERTY, ASPECT RATIO, VOLUME FRACTION, AND ORIENTATION DISTRIBUTION FUNCTION ON COMPOSITE YOUNG'SMODULUS. FIBER ORIENTATION DISTRIBUTION HAS A MORE SIGNIFICANT EFFECT ON COMPOSITE LONGITUDINAL YOUNG'S MODULUS THAN FIBER VOLUME FRICTION, WITHIN THE RANGE EXAMINED.
Bounds and expressions for the elastic moduli of two or many phase nonhomogeneous materials are obtained by an approximate method based on the variational theorems of the theory of elasticity and on a concentric-spheres model. Theoretical results are in good agreement with experimental results for a two-phase alloy.
A model study is conducted on the prediction of the stiffness and fracture toughness of an unidirectional short-fiber reinforced composite containing numerous cracks in the matrix. It is assumed in our model that short-fibers are aligned in the uniaxial loading direction and that cracks in the matrix are perpendicular to the loading axis and are penny-shaped. Then the longitudinal Young's modulus of the composite weakened by those cracks and the energy release rate of a representative penny-shaped crack are com puted by a combination of Eshelby's equivalent inclusion method and Mori- Tanaka's back stress analysis. The interaction between fibers and that be tween fibers and cracks are taken into account by Mori-Tanaka's back stress analysis. Hence our results are valid even for a large volume fraction of fiber and crack.
An attempt has been made to predict the thermal expansion coefficients of particle-filled polymers in which particles have various types of orienta tion distributions. The calculation is based on Eshelby's theory and a kind of self-consistent method is introduced to estimate the effect of the orien tation distribution together with the interaction among particles. The ex pression for the space random orientation of disk-shaped particles with infinitesimal aspect ratio, which is derived here, agrees with lower bound of isotropic composites predicted by Schapery. The calculated results are also compared with existing experimental data for glass particle-epoxy resin composites, and these are almost in good agreement.
One of the experimental findings on short-fiber reinforced composite materials is that the fiber-ends act as a crack initiator. The effect of the fiber-end crack on the overall stiffness and the strength of the composite are investigated. Emphasis is placed on the weakening longitudinal Young's modulus by the fiber-end crack which is assumed to be penny-shaped. The energy release rate of the penny-shaped crack at the fiber-end under a uniaxial applied stress is calculated for a fracture criterion. It is assumed that short fibers are all aligned in the loading direction and the penny-shaped crack at the fiber-end extends in the direction perpendicular to the fiber axis. The analytical technique is a combination of J. E. Eshelby's equivalent inclusion method and Mori-Tanaka's back stress analysis so that the results are valid even for a large volume fraction of fibers.
For elastic-plastic composites at finite strains and rotations. Hill's self-consistent method is used to estimate the overall instantaneous elastic-plastic moduli. Results are applied to a porous metal whose matrix constitutive relations are characterized by a generalized J2 plasticity model. As an illustration, uniaxial deformation of a porous elastic-plastic solid is considered, and conditions under which the overall uniform deformation becomes unstable by localized deformations, are examined.
Composite materials are reviewed fromthe applied mechanics and engineering science point of view. Analyses are presented of composite materials properties such as: elasticity, thermal expansion, mositure swelling, viscoelasticity, conductivity (which includes, by mathematical analogy, dielectrics, magnetics, and diffusion) static strength, and fatigue failure.
In a continuation of a previous paper, it is shown here how the gross bulk and shear moduli of a composite material consisting of a suspension of grains or a compact of grains may be deduced. The grains are assumed to be perfectly bonded to the suspending medium or to each other, and are taken to be spheres in the mean. By using an averaging procedure due to Bruggeman, and analysing the effect of a uniform hydrostatic compression and of a uniform tension on an average grain, a pair of de-coupled equations for the gross moduli is found for suspensions. When the suspending medium vanishes and the grains are packed, these equations become coupled and there is exhibited a discontinuity in the gross moduli. The bulk coefficients of linear expansion of the two kinds of composites are found from an analysis of the dilatation and bulk stress for average spherical grains when the composite as a whole is subjected to some small temperature change. All results are free of any limitation on the number of components.
Mit Hilfe des Begriffes der elastischen Polarisierbarkeit knnen die elastischen Konstanten des makroskopisch isotropen Vielkristalls aus den Einkristallkonstanten exakt ausgerechnet werden. Im Falle des Aggregats aus kubischen Kristalliten, in dem die Bestimmung des Kompressionsmoduls trivial ist, folgt der Schubmodul aus einer Gleichung 3. Grades [Gl. (22)], in der Kombinationen der Einkristallhauptkonstanten als Koeffizienten auftreten. Die experimentellen Vergleichswerte weichen von den berechneten etwas ab, was beweist, da die Experimente nicht den von der Theorie geforderten idealen Bedingungen gengt haben. Weitere Anwendungsmglichkeiten der Methode werden besprochen.
Variational principles in the linear theory of elasticity, involving the elastic polarization tensor, have been applied to the derivation of upper and lower bounds for the effective elastic moduli of quasi-isotropic and quasi-homogeneous multiphase materials of arbitrary phase geometry. When the ratios between the different phase moduli are not too large the bounds derived are close enough to provide a good estimate for the effective moduli. Comparison of theoretical and experimental results for a two-phase alloy showed good agreement.
Solutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.
The title problem concerns two isotropic phases firmly bonded together to form a mixture with any concentrations. An elementary account of several theoretical methods of attack is given, among them the derivation of inequalities between various moduli. The approach is completely general and exact. Additionally, the problem is fully solved when the phases have equal rigidities but different compressibilities, the geometry being entirely arbitrary.
Calculations on the basis of the self-consistent method are made for the elastic moduli of bodies containing randomly distributed flat cracks, with or without fluid in their interiors. General concepts are outlined for arbitrary cracks and explicit derivations together with numerical results are given for elliptic cracks. Parameters are identified which adapt the elliptic-crack results to arbitrary convex crack shapes. Finally, some geometrical relations involving randomly distributed cracks and their traces on cross-sections are presented.
Bounds on the overall elastic and instantaneous elastoplastic moduli of composites with periodic microstructures are found using the extremum principles of Hashin and Shtrikman (1962) and the analytic solution of Nemat-Nasser et al. (1982). The bounds contain terms which depend on the geometric properties of the constituent materials and the corresponding interaction effects. Examples are presented for composites whose constituents are elastically isotropic having isotropic and kinematic plastic hardening responses.
Having noted an important role of image stress in work hardening of dispersion hardened materials, (1,3) the present paper discusses a method of calculating the average internal stress in the matrix of a material containing inclusions with transformation strain. It is shown that the average stress in the matrix is uniform throughout the material and independent of the position of the domain where the average treatment is carried out. It is also shown that the actual stress in the matrix is the average stress plus the locally fluctuating stress, the average of which vanishes in the matrix. Average elastic energy is also considered by taking into account the effects of the interaction among the inclusions and of the presence of the free boundary.
Themacroscopic elastic moduli of two-phase composites are estimated by a method that takes account of the inhomogeneity of stress and strain in a way similar to the Hershey-Kröner theory of crystalline aggregates. The phases may be arbitrarily aeolotropic and in any concentrations, but are required to have the character of a matrix and effectively ellipsoidal inclusions. Detailed results arc given for an isotropic dispersion of spheres.
A heuristic analysis is given for the determination of the elastic moduli of a composite material, the several constituents of which are each isotropic and elastic. The results are intended to apply to heterogeneous materials composed of contiguous, more-or-less spherical grains of each of the phases.
The overall elastic moduli of some solid composite materials are evaluated, first by bounding them precisely, and secondly by a ‘self-consistent’ estimate. Transversely isotropic inclusions of ‘needle’ and ‘disc’ shapes are particularly considered, at both random and aligned orientations, and at arbitrary volume concentration.
Bounds of Hashin-Shtrikman type and self-consistent estimates for the overall properties of composites, which may be anisotropic, are developed. Bodies containing aligned ellipsoidal inclusions are considered particularly, generalizing previously known results. The overall thermal conductivity of a body containing aligned spheroidal inclusions is discussed as an example including, as limiting cases, bodies containing highly-conducting aligned needles and bodies containing aligned pennyshaped cracks.
A differential scheme to compute the effective moduli of composites is presented. The method is based on the idea of realizability, i.e. the composite is constructed explicitly from an initial material through a series of incremental additions. The construction process is uniquely specified by parametrizing the volume fractions of the included phases. The properties of the final composite depend upon the construction path taken and not just on the final volume fractions. Assuming the grain shapes are ellipsoidal, a system of ordinary differential equations for the moduli is obtained which is integrated along the path. The present method includes as special cases of paths or endpoints the differential scheme of Roscoe-Boucher and the self-consistent scheme of Kroner-Hill, respectively. The method includes a realization of the Hashin-Shtrikman bounds for a two-phase composite with . For example, the upper bounds are achieved by imbedding disks of the stiffer material in a matrix of the more compliant material.
The problem of two kinds of ellipsoidal inhomogeneities embedded in an elastic body is formulated with an application to a hybrid (three-phase) composite. The analytical tool used in this study is a combination of Eshelby's equivalent inclusion method[1] and Mori-Tanaka's back stress analysis[2], and therefore the results are valid for large volume fraction of inhomogeneities. As a demonstration, two types of hybrid composites are examined: (i) fiber-fiber; and (ii) fiber-particulate systems.
Based on Mori and Tanaka's concept of “average stress” in the matrix and Eshelby's solutions of an ellipsoidal inclusion, an approximate theory is established to derive the stress and strain state of constituent phases, stress concentrations at the interface, and the elastic energy and overall moduli of the composite. Both “stress-free” strain (polarization strain) and “strain-free” stress (polarization stress) are employed in these derivations under the traction- and displacement-prescribed conditions. The theory was developed first for a general multiphase, anisotropic composite with arbitrarily oriented anisotropic inclusions; explicit results are then given for a suspension of uniformly distributed, multiphase isotropic spheres in an isotropic matrix. Numerical results for stress concentrations in the spherical inclusions and at the interface are given for a 2-phase composite. Further, it is shown that the derived moduli are related to the Hashin-Shtrikman bounds and that, when the shear moduli are equal, the overall bulk modulus of a 2-phase composite reduces to Hill's exact solution. As compared with experimental data, the theory also provides reasonably accurate estimates for the Young's modulus of some 2- and 3-phase composites.
The overall moduli of a composite with an isotropic elastic matrix containing periodically distributed (anisotropic) inclusions or voids, can be expressed in terms of several infinite series which only depend on the geometry of the inclusions or voids, and hence can be computed once and for all for given geometries. For solids with periodic structures these infinite series play exactly the same role as does Eshelby's tensor for a single inclusion or void in an unbounded elastic medium.For spherical and circular-cylindrical geometries, the required infinite series are calculated and the results are tabulated. These are then used to estimate the overall elastic moduli when either the overall strains or the overall stresses are prescribed, obtaining the same results. These results are compared with other estimates and with experimental data. It is found that the model of composites with periodic structure yields estimates in excellent agreement with the experimental observations.
The overall moduli of a 2-phase linearly elastic composite are estimated by the differential scheme. The phases may be arbitrarily anisotropic and 1 phase is regarded as similar ellipsoidal inclusions at any concentration embedded homogeneously in a matrix. For an isotropic dispersion of spheres and for a fibre reinforced material it is shown that the estimates of the overall moduli lie between their Hashin-Shtrikman bounds. Also some known exact results are reproduced using this scheme.
A differential scheme for the effective moduli of compositesCorrection and extension of Van der Poel's method for calculating the shear modulus of a par-ticulate composite
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A study of the differential scheme for composite materials
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