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Quantifying the uncertainties in modeling soft composites via a multiscale approach
Abstract
The goal of this paper is to evaluate the uncertainties in soft composites composed of a soft solid as the matrix filled by incompressible liquid inclusions. The surface stresses at the liquid–solid interfaces can significantly impact the elastic deformation of the bulk composite at the macroscale. Depending on the characteristic dimension, a stiffening or softening can happen. We model the elastic behavior employing a stochastic multiscale approach and counting on well established rules for composites. Insights on the mechanics of elasto-capillary coupling are provided considering the heterogeneity at different length scales. Global sensitivity analysis is performed to quantify the effect of variation in the material properties, surface tension, the size, and the random dispersion and agglomerations of the inclusions. The influence of the model choice at the mesoscale on the uncertainty of the response is also taken into account as an additional source of uncertainty. The results show that the relative stiffness is mainly influenced by the uncertainties in the size of liquid inclusions and the surface tension, while the remaining parameters show a significant interaction effects.
... where ϕ mn,x 1 x 1 represents the second-order partial derivative of the mn-th mode with respect to x 1 , and ϕ mn,x 1 x 2 represents the second-order mixed partial derivative of the mn-th mode with respect to x 1 and x 2 . Upon obtaining the self-power spectral densities as presented in Equations (40) and (42) to (44), the mean square value of the arbitrary response u i (x 1 , x 2 , t) can be obtained by frequency domain integration, i.e., ...
... This section presents a comparative analysis between the outcomes of a 3D finite element analysis (3D-FEA) and the VAM-based 2D equivalent Reissner-Mindlin model (2D-ERM) to assess its accuracy in analyzing the dynamic characteristics of CSP-TCH. The effects of selecting the model class are quantified through dynamic numerical examples [44], wherein the relative error signifies the variance between the 2D-ERM and 3D-FEA outcomes. The dynamic analyses of both models are implemented using the linear perturbation procedure (frequency and random response) in the Abaqus finite element software. ...
A full triangular chiral (Tri-Chi) honeycomb, combining a honeycomb structure with triangular chiral configuration, notably impacts the Poisson's ratio (PR) and stiffness. To assess the random vibration properties of a composite sandwich panel with a Tri-Chi honeycomb core (CSP-TCH), a two-dimensional equivalent Reissner-Mindlin model (2D-ERM) was created using the variational asymptotic method. The precision of the 2D-ERM in free and random vibration analysis was confirmed through numerical simulations employing 3D finite element analysis, encompassing PSD curves and RMS responses. Furthermore, the effects of selecting the model class were quantified through dynamic numerical examples. Modal analysis revealed that the relative error of the first eight natural frequencies predicted by the 2D-ERM consistently remained below 7%, with the modal cloud demonstrating high reliability. The PSD curves and their RMS values closely aligned with 3D finite element results under various boundary conditions, with a maximum error below 5%. Key factors influencing the vibration characteristics included the ligament-rib angle of the core layer and layup modes of the composite facesheets, while the rib-to-ligament thickness ratio and the aspect ratio exert minimal influence. The impact of the ligament-rib angle on the vibration properties primarily stems from the significant shift in the core layer's Poisson's ratio, transitioning from negative to positive. These findings offer a rapid and precise approach for optimizing the vibration design of CSP-TCH.
... where ϕ mn,x 1 x 1 represents the second-order partial derivative of the mn-th mode with respect to x 1 , and ϕ mn,x 1 x 2 represents the second-order mixed partial derivative of the mn-th mode with respect to x 1 and x 2 . Upon obtaining the self-power spectral densities as presented in Equations (40) and (42) to (44), the mean square value of the arbitrary response u i (x 1 , x 2 , t) can be obtained by frequency domain integration, i.e., ...
... This section presents a comparative analysis between the outcomes of a 3D finite element analysis (3D-FEA) and the VAM-based 2D equivalent Reissner-Mindlin model (2D-ERM) to assess its accuracy in analyzing the dynamic characteristics of CSP-TCH. The effects of selecting the model class are quantified through dynamic numerical examples [44], wherein the relative error signifies the variance between the 2D-ERM and 3D-FEA outcomes. The dynamic analyses of both models are implemented using the linear perturbation procedure (frequency and random response) in the Abaqus finite element software. ...
A full triangular chiral (Tri-Chi) honeycomb, combining a honeycomb structure with triangular chiral configuration, notably impacts the Poisson’s ratio (PR) and stiffness. To assess the random vibration properties of a composite sandwich panel with a Tri-Chi honeycomb core (CSP-TCH), a two-dimensional equivalent Reissner–Mindlin model (2D-ERM) was created using the variational asymptotic method. The precision of the 2D-ERM in free and random vibration analysis was confirmed through numerical simulations employing 3D finite element analysis, encompassing PSD curves and RMS responses. Furthermore, the effects of selecting the model class were quantified through dynamic numerical examples. Modal analysis revealed that the relative error of the first eight natural frequencies predicted by the 2D-ERM consistently remained below 7%, with the modal cloud demonstrating high reliability. The PSD curves and their RMS values closely aligned with 3D finite element results under various boundary conditions, with a maximum error below 5%. Key factors influencing the vibration characteristics included the ligament–rib angle of the core layer and layup modes of the composite facesheets, while the rib-to-ligament thickness ratio and the aspect ratio exert minimal influence. The impact of the ligament–rib angle on the vibration properties primarily stems from the significant shift in the core layer’s Poisson’s ratio, transitioning from negative to positive. These findings offer a rapid and precise approach for optimizing the vibration design of CSP-TCH.
... Vu-Baca et al. employed a multiscale model to study the effect of these uncertainties on the Young's modulus [23]. Consequently, due to its statistical properties aligning well with practical observations, the elastic modulus is commonly modeled as a Gaussian random process [24][25][26][27]. The Central Limit Theorem states that when the number of contributing factors is sufficiently large, the aggregate effect tends to follow a Gaussian distribution, making it a natural choice for modeling the variability in these properties. ...
Accurate prediction of material properties is essential in structural engineering design to ensure reliability and safety. Traditional approaches often rely on Gaussian distributions to model material variability. However, our research reveals limitations with Gaussian assumptions, particularly when covariance parameters exceed certain thresholds, leading to physically unrealistic negative values for material properties. To overcome these limitations, we investigate an alternative approach using lognormal distributions for material property modeling. Through Monte Carlo simulations employing Cholesky decomposition, we compare the performance of lognormal distributions with Gaussian counterparts. Our findings demonstrate that lognormal distributions offer a viable alternative, providing more accurate representations of material variability while maintaining computational efficiency. Furthermore, we utilize finite element method (FEM) data from Monte Carlo simulations to predict beam deflection using deep neural networks (DNNs). Leveraging inverse modeling techniques, we showcase the ability to predict elastic modulus from beam deflection data under both normal and lognormal distribution assumptions. By integrating lognormal modeling and inverse modeling techniques into structural analysis, we enhance the realism and accuracy of predictions, thereby improving reliability in engineering design. This paper discusses the implications of our findings and emphasizes the importance of considering alternative probability distributions for material property modeling in structural engineering applications.
... Agglomeration, the clustering of particles, plays a significant role in influencing the overall material properties and the effectiveness of the HIP process. When particles agglomerate, they form clusters that can disrupt the uniform distribution of material, leading to inconsistencies during the pressing process [57,58]. These clusters can create localized areas of higher or lower density, which in turn can result in non-uniform density distributions throughout the final product. ...
This study focuses on developing a machine learning (ML) model capable of predicting relative density and equivalent strain in samples produced through hot isostatic pressing (HIP) of Ti-6Al-4 V powders. The model is trained using data from numerical simulations, incorporating processing parameters and powder size and distribution as input features. Results demonstrate strong predictive performance, with R² values of 0.951 and 0.911 for relative density and equivalent strain, respectively. The findings also reveal that the effectiveness of ML predictions is greatly influenced by the weight functions assigned to processing parameters as input features, while the impact of powder size and distribution weighting on optimal prediction is comparatively minimal. This suggests that particle behavior may demonstrate a higher level of consistency in response to the HIP process compared to the variability created by processing parameters. The outcomes of the ML predictions are further utilized to provide a detailed discussion on how variations in temperature, pressure, and powder size and distribution impact changes in relative density and equivalent strain in a specimen.
... Global sensitivity index considers the average impact of random input over the entire distribution region on output statistical characteristics. Using global sensitivity analysis, random inputs are ranked in order of importance, and the important inputs of affecting the output statistical characteristics can be determined to guide the effective modulation of the interested output characteristics (Zhou et al. 2021;Liu and Li 2020;Hamid and Ghasemi 2022). For the more important input screened by the global sensitivity analysis, more resources should be allocated to adjust its uncertainty to effectively achieve the desired output statistical characteristics, while for the unimportant input, they can be fixed at their mean value to simplify the analytical model of the output statistical characteristics. ...
Variance global sensitivity (VGS) is defined by the mean square difference between output expectation and conditional one on input realization, and it can calculate the mean contribution of the input within its distribution region and guide the effective modulation of output variance. The Monte Carlo simulation (MCS) and quasi MCS are commonly used to estimate VGS, but they are time-consuming respectively due to double-loop framework and computation related to input dimension. Thus, a novel method is proposed to estimate VGS by elaborately using Bayesian updating. In the proposed algorithm, the input realizations are firstly treated as observations to construct a likelihood function. Then by Bayesian updating, all conditional output expectations on different input realizations, which are required in estimating VGS and most time-consuming, can be obtained as the posterior and estimated by the sample of simulating the output expectation. The proposed algorithm shares the sample of solving output expectation to obtain all conditional ones required for solving VGS, which makes the computational effort of estimating VGS equivalent to that of estimating output expectation, thus improving the efficiency of estimating VGS. Numerical and engineering examples fully substantiate the novelty and effectiveness of this algorithm.
... The solid-liquid interfacial free energy (σ SL ) has key role in controlling the microstructure of alloys and compounds during solidification and semi-solid processing, where liquid melt serves as the parent phase [8][9][10][11][12][13]. In the other hand, the elastic behaviour of a soft composite infiltrated with liquid pockets/droplets is dependent on the characteristics of the solid-liquid interface present within the composite, wherein the effect of interfacial property (surface energy) on the material stiffness is dependent on the size of the liquid inclusion inserted into the solid matrix [14]. Coming back to the present problem of melt solidification, the solid-liquid interfacial free energy is defined as the energy per unit area stored in a solid-liquid interface and therefore, it influences the evolution of solid nucleus from the melt. ...
A molecular dynamics-based model of semi-solid formation from liquid melt of the novel Al–15Mg2Si–4.5Si composite is reported in the present study. Growth kinetics of Mg2Si phase is studied here at the pre-nucleation stage i.e., prior to the emergence of a stable nucleus. Moreover, insight has been grabbed on semi-solid processing of the said composite at atomic level, wherein the dynamic evolution of the Mg–Si clusters has been studied during cooling/semi solid slurry generation as well as during isothermal holding stage. The effect of variation of cooling rate on the number of clusters, average size of clusters and density of clusters has been investigated. The average size of clusters decays exponentially with cooling rate, whereas the average atomic density of clusters decays following a power-law relation with cooling rate. The change of the thermodynamic parameters (enthalpy and potential energy) during the phase transformation from liquid to semi-solid state are determined here, which are then employed to determine the value of interfacial free energy of Mg2Si-melt interface in the Al–15Mg2Si–4.5Si composite, along with the parameters obtained from the dynamic evolution of Mg-Si clusters. The paper also reports the temporal evolution of solid content within the melt, corresponding to the Mg2Si phase, during slurry generation as well as during isothermal holding stage.
... In this context, the elastic modulus of materials plays a pivotal role in determining the response characteristics of structures. While traditional approaches often assume deterministic material properties, overlooking the intrinsic variability observed in real-world scenarios, Young's modulus has been reported to display randomness rather than consistency in various investigations [1]- [5]. ...
This paper investigates the influence of random elastic modulus on beam Eigen frequencies using multiple simulation techniques: Monte Carlo simulations (employing Cholesky decomposition and Kosambi Karhunen-Loeve expansion), Polynomial Chaos expansion, and a Proposed Random Sampling method. Anomalies in Monte Carlo simulations, where normally distributed elastic modulus led to negative values and imaginary Eigen frequencies, were effectively addressed by adopting a log-normal distribution. Comparative analyses focused on covariance variation of first three Eigen frequencies with correlation length and standard deviation of the random field, highlighting nuanced differences between normal and log-normal distributions. Polynomial Chaos expansion exhibited distinct responses, showcasing variations in covariance with different distributions. The study culminates in Eigen frequency estimation the proposed Random Sampling Method, (RSM) wherein the beam is discretized into n elements with randomly assigned elastic moduli. The mean and variance of Eigen frequencies are compared with existing methods, which represents an alternative method for achieving similar outcomes.
... Elastic modulus are often modeled as Gaussian random processes due to their statistical properties aligning with Gaussian distributions in many practical scenarios [22][23][24][25]. The Central Limit Theorem states that when the number of contributing factors is sufficiently large, the aggregate effect tends to follow a Gaussian distribution, making it a natural choice for modeling the variability in these properties. ...
Accurate prediction of material properties is essential in structural engineering design to ensure reliability and safety. Traditional approaches often rely on Gaussian distributions to model material variability. However, our research reveals limitations with Gaussian assumptions, particularly when covariance parameters exceed certain thresholds, leading to physically unrealistic negative values for material properties. To overcome these limitations, we investigate an alternative approach using lognormal distributions for material property modeling. Through Monte Carlo simulations employing Cholesky decomposition , we compare the performance of lognormal distributions with Gaussian counterparts. Our findings demonstrate that lognormal distributions offer a viable alternative, providing more accurate representations of material variability while maintaining computational efficiency. Furthermore, we utilize finite element method (FEM) data from Monte Carlo simulations to predict beam deflection using deep neural networks (DNNs). Leveraging inverse modeling techniques, we showcase the ability to predict elastic modulus from beam deflection data under both normal and lognormal distribution assumptions. By integrating lognormal modeling and inverse modeling techniques into structural analysis, we enhance the realism and accuracy of predictions, thereby improving reliability in engineering design. This paper discusses the implications of our findings and emphasizes the importance of considering alternative probability distributions for material property modeling in structural engineering applications .
... It was assumed that the material properties of belts could be considered constant in our initial approach. • According to [36], a sensitivity analysis can be used for quantification. However, in this article, it was not performed. ...
This article serves as a continuation of our previously published work and focuses on loose material transport via sandwich belt conveyors. Experimental, analytical, stochastic, and numerical approaches are used to obtain and utilize the moduli of a bilateral Winkler elastic foundation that represent a loose material, which is wheat (Triticum aestivum) that is free of bran in this case. The solutions were obtained for a uniformly and nonuniformly distributed loose material. The task of the conveyor with loose material is simplified into a symmetric task, i.e., a beam on an elastic bilateral Winkler foundation, for an analytical solution and stochastic solution (Anthill and Matlab sw). In a numerical approach, this is considered a plane strain problem within the finite element method (Ansys and MSC.Marc sw). The experimental data are evaluated and used to obtain the functions of Winkler elastic foundation moduli, which are further considered in the numerical solution. The finite element method mainly serves as a verification tool. The acquired histograms of the elastic foundation moduli can be further applied in various scientific and research fields.
... Quantifying uncertainty: In future work, quantifying uncertainty in material properties and other input parameters, such as geometry, should be considered to provide a more realistic assessment of the influence of materials [46]. ...
The objective of this study was to analyse the influence of materials and their position within the upper assembly on the behaviour of casual footwear using finite element simulation tools. The study was carried out on three models of casual footwear, which are identical in terms of design lines, varying only in the materials of the upper assembly, namely calfskin leather (M1), knitted fabric (M2), and combination of knitted fabric and calfskin leather (M3). The footwear models were designed according to the design constraints specific to casual footwear. The foot was reconstructed based on the shoe last obtained based on anthropometric data. Material definition, 3D models editing, setting up analysis conditions, and constraints were performed using the Ansys 17.2 software. Gait biomechanics were taken into account to define the loading model, force distribution, force values, and constraints. The study evaluates footwear behaviour in terms of directional deformation (Z axis), equivalent von Mises stress, and equivalent elastic strain distribution. This paper explores a methodology that has the potential to enhance the footwear design and manufacturing process, providing designers with information about the deformations and stress distribution on upper parts of the footwear product.
... In this work, the variability and uncertainty of the geometry and material properties of the uvulopalatal system is not investigated. A further study considering these uncertainty effects [37] is planned. ...
Obstructive Sleep Apnea Syndrome (OSAS) is a common sleep-related disorder. It is characterized by recurrent partial or total collapse of pharyngeal upper airway accompanied by induced vibrations of the soft tissues (e.g., soft palate). The knowledge of the tissue behavior subject to a particular airflow is relevant for realistic clinic applications. However, in-vivo measurements are usually impractical. The goal of the present study is to develop a 3D fluid-structure interaction model for the human uvulopalatal system relevant to OSA based on simplified geometries under physiological conditions. Numerical simulations are performed to assess the influence of the different breathing conditions on the vibrational dynamics of the flexible structure. Meanwhile, the fluid patterns are investigated for the coupled fluid-structure system as well. Increasing the respiratory flow rate is shown to induce larger structural deformation. Vortex shedding induced resonance is not observed due to the large discrepancy between the flow oscillatory frequency and the natural frequency of the structure. The large deformation for symmetric breathing case under intensive respiration is mainly because of the positive feedback from the pressure differences on the top and the bottom surfaces of the structure.
The wetting phenomenon of composite substrates in hypergravitational environment has a huge application in electronic devices and astronaut healthcare in aerospace missions. In the present contribution, the governing equation of high-G droplets on the composite substrate is firstly established in the hypergravitational environment. Meanwhile, the apparent contact angles at the contact line between droplets and substrates with different stiffness gradients are achieved. Then, we analyze the effects of hypergravity factor and the substrate stiffness on the wetting profile of high-G droplets. By introducing the droplet volume and contact angle into the Bond number, the scaling law of the high-G droplet profile is established, and we find that the contact radius of the droplet R / S0.5 has a linear relationship with ρω2rl2S / (γLVθ), while the droplet height H / S0.5 has a power-law relationship with ρω2rl2S / (γLVθ). Finally, we explain the profiles of high-G droplets during the wetting process by illustrating energy components of the entire system and find that the substrate with positive triangular stiffness and inverted triangular stiffness show opposite evolution laws. On a substrate with inverted triangular stiffness, the gravitational potential energy is more dominant.
In the oxygen-enriched side-blown smelting furnace for the treatment of electroplating sludge, fluent was used to simulate the gas–liquid two-phase flow process. The relationship between the lance diameter, lance inclination, bath depth, and the bath evaluation indicators were studied, and the oxygen lance spacing was optimized. The results show that the high velocity and high gas rate areas were near the oxygen lance, while the stirring dead zones with low velocity appeared in the central and bottom areas of the molten pool. The key parameters were optimized using single-factor analysis and multifactor comprehensive optimization. The results showed that the bath evaluation indicators were all at good levels under the optimal parameter conditions. These were comprehensively obtained as the following: the lance diameter was 25 mm, the lance inclination was 15°, the lance spacing was 1050 mm, and the bath depth was 1500 mm. The industrial test carried out in an environmental protection enterprise in Guangdong achieved satisfactory results. The test shows that the electroplating sludge containing 7.24% Cu can be melted at 1300~1400 °C to obtain matte. Compared with industrial copper slag, the smelting slag has a higher CaO and a lower Fe content.
The current research assesses the consequences of various damages (crack and delamination) and high-strain loading conditions on the fibre-reinforced composite structure's elasto-plastic stress-strain (EPSS) characteristics. The constitutive responses are obtained numerically (finite element discretization) using higher-order polynomials (to count the deformation) with the help of the MATLAB platform. The EPSS responses are evaluated via the modified Ludwik equation and Cowper Symonds model under high strain rate loadings. The model accuracy has been tested through the comparison between the present numerical and the published experimental data available in open domain. Furthermore, various numerical examples present the effect of damages (debond and/or crack), forces and stress-strain characteristics for a wide range of strain rates (ranging from 0.001 s−1 to 50 s−1). The stochastic constitutive model is obtained to show the modelling importance and the corresponding analysis parameters including the uncertainty quantification. Finally, a detailed understanding of the damages and geometries on the polymeric composite structure are enumerated via the delve model for the futuristic design.
This paper presents a reliability analysis for fracture toughness using a multilevel refinement on a hierarchy
of computational models. A 2D finite element model discretized by quadrilateral elements is developed to
analyze the stress intensity with the presence of an initial edge crack. The multilevel simulations are obtained
considering a non-uniform sequence of mesh refinement in the vicinity of the crack tip. We set the probabilistic
problem accounting for applied stress and crack size uncertainties. We analyze several error tolerances using
the standard and multilevel Monte Carlo methods combined with the selective refinement procedure. The
probability of failure is estimated by expanding it in a telescoping sum of an initial approximation at the
coarsest mesh and a series of incremental corrections between the subsequent levels. In our analysis, we
take on two common fracture problems; a single-edge notched tension to investigate the pure mode-I and
an asymmetric four-points bending to consider the mixed mode-I/II. The results show significant savings in
the computation cost.
In this paper, a multiscale optimization approach for composite material design is presented. The objective is to find different material designs for a short fiber reinforced polymer (SFRP) with a desired effective (in general anisotropic) viscoelastic behavior. The paper extends the work of Staub et al. (2012) and proposes a combination of material homogenization, surrogate modeling, parameter optimization and robustness analysis. A variety of microstructure design parameters including the fiber volume fraction, the fiber orientation distribution, the linear elastic fiber properties, and the temperature dependent material behavior are considered. For the solution of the structural optimization problem, a surrogate-based optimization framework is developed. The individual steps of that framework consist of using design of experiments (DoE) for the sampling of the constraint material design space, numerical homogenization for the creation of a material property database, a surrogate modeling approach for the interpolation of the single effective viscoelastic parameters and the use of differential evolution (DE) for optimization. In the numerical homogenization step, creep simulations on virtually created representative volume elements (RVEs) are performed and a fast Fourier transform (FFT)-based homogenization is used to obtain the effective viscoelastic material parameters. For every identified optimal design, the robustness is evaluated. The considered Kriging surrogate models of Kriging type have a high prediction accuracy. Numerical examples demonstrate the efficiency of the proposed approach in determining SFRPs with target viscoelastic behavior. An experimental validation shows a good agreement of the homogenization method with corresponding measurements. During the manufacturing of composite parts, the results of such optimizations allow a consideration of the local microstructure in order to achieve the desired macroscopic viscoelastic behavior.
Experimental and theoretical results of late have pointed to elastomers filled with various types of liquid inclusions as a promising new class of materials with unprecedented properties. Motivated by these findings, the first of two objectives of this paper is to formulate the homogenization problem that describes the macroscopic mechanical response of elastomers filled with liquid inclusions under finite quasistatic deformations. The focus is on the non-dissipative case when the elastomer is a hyperelastic solid, the liquid making up the inclusions is a hyperelastic fluid, the interfaces separating the solid elastomer from the liquid inclusions feature their own hyperelastic behavior (which includes surface tension as a special case), and the inclusions are initially spherical in shape. The macroscopic behavior of such filled elastomers turns out to be that of a hyperelastic solid, albeit one that depends directly on the size of the inclusions and the constitutive behavior of the interfaces. It is hence characterized by an effective stored-energy function W¯(F¯) of the macroscopic deformation gradient F¯. Strikingly, in spite of the fact that there are local residual stresses within the inclusions (due to the presence of initial interfacial forces), the resulting macroscopic behavior is free of residual stresses, that is, ∂W¯(I)/∂F¯=0. What is more, in spite of the fact that the local moduli of elasticity in the bulk and the interfaces in the small-deformation limit do not possess minor symmetries (due to the presence of residual stresses and initial interfacial forces), the resulting effective modulus of elasticity does possess the standard minor symmetries, that is, L¯ijkl=L¯jikl=L¯ijlk, where L¯ijkl≔∂2W¯(I)/∂F¯ij∂F¯kl. The second objective is to implement and deploy a finite-element scheme to numerically generate solutions for the macroscopic response of a basic class of elastomers filled with liquid inclusions, that of isotropic suspensions of incompressible liquid inclusions of monodisperse size embedded in incompressible Neo-Hookean elastomers wherein the interfaces feature a constant surface tension. With guidance from the numerical solutions, the last part of this paper is devoted to proposing a simple explicit approximation for the effective stored-energy function W¯(F¯).
With many frameworks based on message passing neural networks proposed to predict molecular and bulk properties, machine learning methods have tremendously shifted the paradigms of computational sciences underpinning physics, material science, chemistry, and biology. While existing machine learning models have yielded superior performances in many occasions, most of them model and process molecular systems in terms of homogeneous graph, which severely limits the expressive power for representing diverse interactions. In practice, graph data with multiple node and edge types is ubiquitous and more appropriate for molecular systems. Thus, we propose the heterogeneous relational message passing network (HermNet), an end-to-end heterogeneous graph neural networks, to efficiently express multiple interactions in a single model with ab initio accuracy. HermNet performs impressively against many top-performing models on both molecular and extended systems. Specifically, HermNet outperforms other tested models in nearly 75%, 83% and 69% of tasks on revised Molecular Dynamics 17 (rMD17), Quantum Machines 9 (QM9) and extended systems datasets, respectively. In addition, molecular dynamics simulations and material property calculations are performed with HermNet to demonstrate its performance. Finally, we elucidate how the design of HermNet is compatible with quantum mechanics from the perspective of the density functional theory. Besides, HermNet is a universal framework, whose sub-networks could be replaced by other advanced models.
Graphene-reinforced polymer composites may fail the expected response when the load transmission at the graphene–polymer interface is not carried out properly. Therefore, a careful examination should be allocated to the behavior of this region. As the load transmission can be either perpendicular or tangential to the graphene sheet, normal or shear forces can cause a failure. We analyze the effect of functionalization on the mechanical behavior of the interface. Throughout the simulations, six types of functional groups (COOH–, CH3–, OH–, NH2–, C2H5–, and H–) with different percentages are attached to graphene in the interface. Our results indicated that the shear and normal forces increased with an increment of the coverage percentage (up to 21%) and then, a saturation occurred whereupon no further change is observed in the interfacial properties. Among these functional groups, results stated that COOH has the greatest effect while OH, CH3, and NH2 have relatively equal effects and H has the lowest influence on the enhancement of interfacial mechanics. During polymer detachment, our simulations exhibited that increasing the coverage percentage of the functional group can improve the attachment of graphene to the polymer at the interfacial layer up to point of saturation and further functionalization is redundant.
Graphical abstract
While the forward and backward modeling of the process-structure-property chain has received a lot of attention from the materials’ community, fewer efforts have taken into consideration uncertainties. Those arise from a multitude of sources and their quantification and integration in the inversion process are essential in meeting the materials design objectives. The first contribution of this paper is a flexible, fully probabilistic formulation of materials’ optimization problems that accounts for the uncertainty in the process-structure and structure-property linkages and enables the identification of optimal, high-dimensional, process parameters. We employ a probabilistic, data-driven surrogate for the structure-property link which expedites computations and enables handling of non-differential objectives. We couple this with a problem-tailored active learning strategy, i.e., a self-supervised selection of training data, which significantly improves accuracy while reducing the number of expensive model simulations. We demonstrate its efficacy in optimizing the mechanical and thermal properties of two-phase, random media but envision that its applicability encompasses a wide variety of microstructure-sensitive design problems.
Liquid droplets in solid soft composites have been attracting increasing attention in biological
applications. In contrary with conventional composites, which are made of solid elastic inclusions, available material
models for composites including liquid droplets are for highly idealized configurations and do not include all material
real parameters. They are also all deterministic and do not address the uncertainties arising from droplet radius, volume
fraction, dispersion and agglomeration. This research revisits the available models for liquid droplets in solid soft
composites and presents a multiscale computational material model to determine their elastic moduli, considering nearly
all relevant uncertainties and heterogeneities at different length scales. The effects of surface tension at droplets interface,
their volume fraction, size, size polydispersity and agglomeration on elastic modulus, are considered. Different
micromechanical material models are incorporated into the presented computational framework. The results clearly
indicate both softening and stiffening effects of liquid droplets and show that the model can precisely predict the
effective properties of liquid droplets in solid soft composites.
Double-network (DN) hydrogels, consisting of two contrasting and interpenetrating polymer networks, are considered as perhaps the toughest soft-wet materials. Current knowledge of DN gels from synthesis methods to toughening mechanisms almost exclusively comes from chemically-linked DN hydrogels by experiments. Molecular modeling and simulations of inhomogeneous DN structure in hydrogels have proved to be extremely challenging. Herein, we developed a new multiscale simulation platform to computationally investigate the early fracture of physically-chemically linked agar/polyacrylamide (agar/PAM) DN hydrogels at a long timescale. A “random walk reactive polymerization” (RWRP) was developed to mimic a radical polymerization process, which enables to construct a physically-chemically linked agar/PAM DN hydrogel from monomers, while conventional and steered MD simulations were conducted to examine the structural-dependent energy dissipation and fracture behaviors at the relax and deformation states. Collective simulation results revealed that energy dissipation of agar/PAM hydrogels was attributed to a combination of the pulling out of agar chains from the DNs, the disruption of massive hydrogen bonds between and within DN structures, and the strong association of water molecules with both networks, thus explaining a different mechanical enhancement of agar/PAM hydrogels. This computational work provided atomic details of network structure, dynamics, solvation, and interactions of a hybrid DN hydrogel, and a different structural-dependent energy dissipation mode and fracture behavior of a hybrid DN hydrogel, which help to design tough hydrogels with new network structures and efficient energy dissipation modes. Additionally, the RWRP algorithm can be generally applied to construct the radical polymerization-produced hydrogels, elastomers, and polymers.
Magnetically responsive soft materials are soft composites where magnetic fillers are embedded into soft polymeric matrices. These active materials have attracted extensive research and industrial interest due to their ability to realize fast and programmable shape changes through remote and untethered control under the application of magnetic fields. They would have many high-impact potential applications in soft robotics/devices, metamaterials, and biomedical devices. With a broad range of functional magnetic fillers, polymeric matrices, and advanced fabrication techniques, the material properties can be programmed for integrated functions, including programmable shape morphing, dynamic shape deformation-based locomotion, object manipulation and assembly, remote heat generation, as well as reconfigurable electronics. In this review, an overview of state-of-the-art developments and future perspectives in the multifunctional magnetically responsive soft materials is presented.
Soft materials with a liquid component are an emerging paradigm in materials design. The incorporation of a liquid phase, such as water, liquid metals, or complex fluids, into solid materials imparts unique properties and characteristics that emerge as a result of the dramatically different properties of the liquid and solid. Especially in recent years, this has led to the development and study of a range of novel materials with new functional responses, with applications in topics including soft electronics, soft robotics, 3D printing, wet granular systems and even in cell biology. Here a review of solid–liquid composites, broadly defined as a material system with at least one, phase‐separated liquid component, is provided and discussed their morphology and fabrication approaches, their emergent mechanical properties and functional response, and the broad range of their applications. Soft solid–liquid composites (SLCs) are an emerging paradigm in materials design. Their unique characteristics, which cannot be achieved with traditional rigid composites, enable a broad range of applications across soft matter engineering, including soft electronics, robotics and various chemical and biological systems. Here, a progress‐report of SLCs is provided, and discuss their morphology and fabrication, emergent properties, and diverse applications.
The successful discovery and isolation of graphene in 2004, and the subsequent synthesis of layered semiconductors and heterostructures beyond graphene have led to the exploding field of two-dimensional (2D) materials that explore their growth, new atomic-scale physics, and potential device applications. This review aims to provide an overview of theoretical, computational, and machine learning methods and tools at multiple length and time scales, and discuss how they can be utilized to assist/guide the design and synthesis of 2D materials beyond graphene. We focus on three methods at different length and time scales as follows: (i) nanoscale atomistic simulations including density functional theory (DFT) calculations and molecular dynamics simulations employing empirical and reactive interatomic potentials; (ii) mesoscale methods such as phase-field method; and (iii) macroscale continuum approaches by coupling thermal and chemical transport equations. We discuss how machine learning can be combined with computation and experiments to understand the correlations between structures and properties of 2D materials, and to guide the discovery of new 2D materials. We will also provide an outlook for the applications of computational approaches to 2D materials synthesis and growth in general.
Beyond recent related literature, which focused on spherical incompressible liquid inclusions, the present work studies an elliptical compressible liquid inclusion in an infinite elastic plane under static remote mechanical loading. Here, it is assumed that the change of pressure inside the liquid inclusion is linearly related to the change of inclusion volume with the bulk modulus of the liquid as the proportionality coefficient. Also, the role of the liquid surface tension on the solid-liquid interface is examined especially when the size of the liquid inclusion is comparable to or smaller than the elastocapillary length. Our results show that both the surface tension and the change of liquid pressure have a significant effect on reducing the stress concentration factor at the endpoints of an elliptical liquid inclusion. In addition, the pressure change inside the liquid inclusion is studied when a uniaxial remote stress is applied perpendicular or parallel to the major axis of the elliptical liquid inclusion. In particular, the effective plane-strain Young's modulus of a solid-liquid composite containing circular liquid inclusions predicted by the present model is linearly related to the volume fraction of the liquid inclusions, in reasonable agreement with existing experimental data.
This article provides a brief introduction to micromechanics using linear elastic materials as an example. The fundamental micromechanics concepts including homogenization and dehomogenization, representative volume element (RVE), unit cell, average stress and strain theories, effective stiffness and compliance, Hill-Mandel macrohomogeneity condition. This chapter also describes the detailed derivations of the rules of mixtures, and three full field micromechanics theories including finite element analysis of a representative volume element (RVE analysis), mathematical homogenization theory (MHT), and mechanics of structure genome (MSG). Theoretical connections among the three full field micromechanics theories are clearly shown. Particularly, it is shown that RVE analysis, MHT and MSG are governed by the same set of equations for 3D RVEs with periodic boundary conditions. RVE analysis and MSG can also handle aperiodic or partially periodic materials for which MHT is not applicable. MSG has the unique capability to obtain the complete set of 3D properties and local fields for heterogeneous materials featuring 1D or 2D heterogeneities.
Eshelby's theory is the foundation of composite mechanics, allowing
calculation of the effective elastic moduli of composites from a knowledge of
their microstructure. However it ignores interfacial stress and only applies to
very dilute composites -- i.e. where any inclusions are widely spaced apart.
Here, within the framework of the Mori-Tanaka multiphase approximation scheme,
we extend Eshelby's theory to treat a composite with interfacial stress in the
non-dilute limit. In particular we calculate the elastic moduli of composites
comprised of a compliant, elastic solid hosting a non-dilute distribution of
identical liquid droplets. The composite stiffness depends strongly on the
ratio of the droplet size, R, to an elastocapillary length scale, L.
Interfacial tension substantially impacts the effective elastic moduli of the
composite when . When (R=3L/2) liquid inclusions
stiffen (cloak the far-field signature of) the solid.
In the dilute limit Eshelby's inclusion theory captures the behavior of a
wide range of systems and properties. However, because Eshelby's approach
neglects interfacial stress, it breaks down in soft materials as the inclusion
size approaches the elastocapillarity length L. Here, we use a three-phase
generalized self-consistent method to calculate the elastic moduli of
composites comprised of an isotropic, linear-elastic compliant solid hosting a
spatially random monodisperse distribution of spherical liquid droplets. As
opposed to similar approaches, we explicitly capture the liquid-solid
interfacial stress when it is treated as an isotropic, strain-independent
surface tension. Within this framework, the composite stiffness depends solely
on the ratio of the elastocapillarity length L to the inclusion radius R.
Independent of inclusion volume fraction, we find that the composite is
stiffened by the inclusions whenever . Over the same range of
parameters, we compare our results with alternative approaches (dilute and
Mori-Tanaka theories that include surface tension). Our framework can be easily
extended to calculate the composite properties of more general soft materials
where surface tension plays a role.
The standard models of solid mechanics were developed to describe the behavior of stiff materials. Our recent experiments have shown that the Johnson–Kendall–Roberts theory of adhesion, Young–Dupre relation of wetting, and Eshelby theory of composites each fail when elastic moduli drop below a size-dependent scale. In all cases, surface tension drives new phenomena. In this Soft Matter paper, we reformulate Eshelby's theory of composites to account for surface tension.
Variance-based approaches are widely used for Global Sensitivity Analysis (GSA) of environmental models. However, methods that consider the entire Probability Density Function (PDF) of the model output, rather than its variance only, are preferable in cases where variance is not an adequate proxy of uncertainty, e.g. when the output distribution is highly-skewed or when it is multi-modal. Still, the adoption of density-based methods has been limited so far, possibly because they are relatively more difficult to implement. Here we present a novel GSA method, called PAWN, to efficiently compute density-based sensitivity indices. The key idea is to characterise output distributions by their Cumulative Distribution Functions (CDF), which are easier to derive than PDFs. We discuss and demonstrate the advantages of PAWN through applications to numerical and environmental modelling examples. We expect PAWN to increase the application of density-based approaches and to be a complementary approach to variance-based GSA.
Open access: www.sciencedirect.com/science/article/pii/S1364815215000237
Hydrogels have become very popular due to their unique properties such as high water content, softness, flexibility and biocompatibility. Natural and synthetic hydrophilic polymers can be physically or chemically cross-linked in order to produce hydrogels. Their resemblance to living tissue opens up many opportunities for applications in biomedical areas. Currently, hydrogels are used for manufacturing contact lenses, hygiene products, tissue engineering scaffolds, drug delivery systems and wound dressings. This review provides an analysis of their main characteristics and biomedical applications. From Wichterle’s pioneering work to the most recent hydrogel-based inventions and products on the market, it provides the reader with a detailed introduction to the topic and perspective on further potential developments.
Eshelby's theory of inclusions has wide-reaching implications across the
mechanics of materials and structures including the theories of composites,
fracture, and plasticity. However, it does not include the effects of surface
stress, which has recently been shown to control many processes in soft
materials such as gels, elastomers and biological tissue. To extend Eshelby's
theory of inclusions to soft materials, we consider liquid inclusions within an
isotropic, compressible, linear-elastic solid. We solve for the displacement
and stress fields around individual stretched inclusions, accounting for the
bulk elasticity of the solid and the surface tension (\textit{i.e.} isotropic
strain-independent surface stress) of the solid-liquid interface. Surface
tension significantly alters the inclusion's shape and stiffness as well as its
near- and far-field stress fields. These phenomenon depend strongly on the
ratio of inclusion radius, R, to an elastocapillary length, L. Surface
tension is significant whenever inclusions are smaller than 100L. While
Eshelby theory predicts that liquid inclusions generically reduce the stiffness
of an elastic solid, our results show that liquid inclusions can actually
stiffen a solid when . Intriguingly, surface tension cloaks the
far-field signature of liquid inclusions when R=3L/2. These results are have
far-reaching applications from measuring local stresses in biological tissue,
to determining the failure strength of soft composites.
Surface tension tends to minimize the area of interfaces between pieces of
matter in different thermodynamic phases, be they in the solid or the liquid
state. This can be relevant for the macroscopic shape of very soft solids, and
lead to a roughening of initially sharp edges. We calculate this effect for a
neo-Hookean elastic solid, with assumptions corresponding to actual
experiments, namely the case where an initially sharp edge is rounded by the
effect of surface tension felt when the fluid surrounding the soft solid (and
so surface tension) is changed at the solid/liquid boundary. We consider two
opposite limits where the analysis can be carried to the end, the one of a
shallow angle and the one of a very sharp angle. Both cases yield a
discontinuity of curvature in the state with surface tension although the
initial state had a discontinuous slope.
From bone and wood to concrete and carbon fibre, composites are ubiquitous
natural and engineering materials. Eshelby's inclusion theory describes how
macroscopic stress fields couple to isolated microscopic inclusions, allowing
prediction of a composite's bulk mechanical properties from a knowledge of its
microstructure. It has been extended to describe a wide variety of phenomena
from solid fracture to cell adhesion. Here, we show experimentally and
theoretically that Eshelby's theory breaks down for small liquid inclusions in
a soft solid. In this limit, an isolated droplet's deformation is strongly
size-dependent with the smallest droplets mimicking the behaviour of solid
inclusions. Furthermore, in opposition to the predictions of conventional
composite theory, we find that finite concentrations of small liquid inclusions
enhance the stiffness of soft solids. A straight-forward extension of Eshelby's
theory, accounting for the surface tension of the solid-liquid interface,
explains our experimental observations. The counterintuitive effect of
liquid-stiffening of solids is expected whenever droplet radii are smaller than
an elastocapillary length, given by the ratio of the surface tension to Young's
modulus of the solid matrix.
We study the elastic properties of soft solids containing air bubbles.
Contrary to standard porous materials, the softness of the matrix allows for a
coupling of the matrix elasticity to surface tension forces brought in by the
bubbles. Thanks to appropriate experiments on model systems, we show how the
elastic response of the dispersions is governed by two dimensionless
parameters: the gas volume fraction and a capillary number comparing the
elasticity of the matrix to the stiffness of the bubbles. We also show that our
experimental results are in good agreement with computations of the shear
modulus through a micro-mechanical approach.
The Johnson-Kendall-Roberts theory is the basis of modern contact mechanics. It describes how two deformable objects adhere together, driven by adhesion energy and opposed by elasticity. Here we characterize the indentation of glass particles into soft, silicone substrates using confocal microscopy. We show that, whereas the Johnson-Kendall-Roberts theory holds for particles larger than a critical, elastocapillary lengthscale, it fails for smaller particles. Instead, adhesion of small particles mimics the adsorption of particles at a fluid interface, with a size-independent contact angle between the undeformed surface and the particle given by a generalized version of the Young's law. A simple theory quantitatively captures this behaviour and explains how solid surface tension dominates elasticity for small-scale indentation of soft materials.
In this paper several methods for model assessment considering uncertainties are discussed. Sensitivity analysis is performed to quantify the influence of the individual model input parameters. In addition to the well-known analysis of a single model, a new procedure for quantifying the influence of the model choice on the uncertainty of the model prediction is proposed. Furthermore, a procedure is presented which can be used to estimate the model framework uncertainty and which enables the selection of the optimal model with the best compromise between model input and framework uncertainty. Finally Bayesian methods for model selection are extended for model assessment without measurements using model averaging as reference.
In this paper, we consider the probabilistic modeling of media exhibiting uncertainties on material symmetries. More specifically, we address both the construction of a stochastic model and the definition of a methodology allowing the numerical simulation (and consequently, the inverse experimental identification) of random elasticity tensors whose mean distance (in a sense to be defined) to a given class of material symmetry is specified. Following the eigensystem characterization of the material symmetries, the proposed approach relies on the probabilistic model derived in Mignolet and Soize (2008), allowing the variance of selected eigenvalues of the elasticity tensor to be partially prescribed. In this context, a new methodology (regarding in particular the parametrization of the model) is defined and illustrated in the case of transversely isotropic materials. The efficiency of the approach is demonstrated by computing the mean distance of the random elasticity tensor to a given material symmetry class, the distance and projection onto the space of transversely isotropic tensors being defined by considering the Riemmanian metric and the Euclidean projection, respectively. It is shown that the methodology allows the above distance to be (partially) reduced as the overall level of statistical fluctuations increases, no matter the initial distance of the mean model used in the simulations. A comparison between this approach and the initial nonparametric approach introduced in Soize (2008) is finally provided.
Sobol' sensitivity indices,used in variance based global sensitivity analysis of model output,are compared with the Analysis of Variance in classical factorial design. Monte Carlo computation of Sobol' indices is described briefly, and a bootstrap approach is presented,which can be used to produce a confidence interval for the true,unknown indices.
Biological tissues have the remarkable ability to remodel and repair in response to disease, injury and mechanical stresses. Synthetic materials lack the complexity of biological tissues, and man-made materials that respond to external stresses through a permanent increase in stiffness are uncommon. Here we report that polydomain nematic liquid crystal elastomers increase in stiffness by up to 90% when subjected to a low-amplitude (5%), repetitive (dynamic) compression. Elastomer stiffening is influenced by liquid crystal content, the presence of a nematic liquid crystal phase and the use of a dynamic as opposed to static deformation. Through rheological and X-ray diffraction measurements, stiffening can be attributed to a mobile nematic director, which rotates in response to dynamic compression. Stiffening under dynamic compression has not been previously observed in liquid crystal elastomers and may be useful for the development of self-healing materials or for the development of biocompatible, adaptive materials for tissue replacement.
We propose an approach to measure surface elastic constants of soft solids. Generally, this requires one to probe interfacial mechanics at around the elastocapillary length scale, which is typically microscopic. Deformations of microscopic droplets embedded in soft solids are particularly attractive, because they avoid intrinsic nonlinearities associated with previous experiments such as the equilibrium of contact lines and the relaxation of patterned surfaces. We derive analytical solutions for the shape of droplets under uniaxial deformation and for the radius of droplets upon hydrostatic inflation. We couple mechanical deformations to the dissolution of droplets to assess experimental sensitivities. Combined with experimental data from both modes of deformation, one should be able to reliably extract the complete set of isotropic surface material parameters following a specific minimization procedure.
Soft composites with a soft solid as the matrix and incompressible liquid inclusions as reinforcements have been synthesised in recent times. These soft composites can be designed to respond to external stimuli and offer the prospect of providing a targeted combination of stiffness and toughness. In fact, counter-intuitively, soft solids, when reinforced with liquid inclusions, can become stiffer than the matrix material. This effect is attributed to liquid like surface stresses at the liquid–solid boundaries and is therefore, accentuated when the inclusions are very small. We perform computational homogenisation on liquid inclusion reinforced soft solids with a view to understand the effect of surface stresses on their overall stiffness and initiation toughness. Especially, we look at the hitherto unexplored effect of the surface strain dependent part of surface stresses on the fracture behaviour of soft solids reinforced with fluid inclusions. Finite deformation based hyperelasticity is the computational framework used. Results indicate that, when surface stresses are sensitive to surface strains, stress concentrations at the microscopic level are alleviated. Also, in pre-cracked composites, though initiation of microcracks from the main crack occurs quite early in the deformation history, the microcracks are deflected off the symmetry plane by the liquid inclusions, indicating a possible significant increase in the propagation toughness of the material.
The mechanical effects of dilute liquid inclusions on the solid-liquid composite are explored, based on an analytical circular inclusion model incorporating the internal pressure change of the liquid and the surface tension of the interface. Several simple explicit dependences of the stress field and effective stiffness on the bulk modulus and the size of the liquid, the surface tension, and Poisson’s ratio of the matrix are derived. The results show that the stresses in the matrix are reduced, and the stiffness of the solid-liquid composite is enhanced with the consideration of either the surface tension or the internal pressure change. Particularly, the effective Young’s modulus predicted by the present model for either soft or stiff matrices agrees well with the known experimental data. In addition, according to the theoretical results, it is possible to stiffen a soft solid by pressured gas with the presence of the surface tension of the solid-gas interface.
Even in the simple linear elastic range, the material behavior is not deterministic, but fluctuates randomly around some expectation values. The knowledge about this characteristic is obviously trivial from an experimentalist's point of view. However, it is not considered in the vast majority of material models in which “only” deterministic behavior is taken into account.
One very promising approach to the inclusion of stochastic effects in modeling of materials is provided by the so-called Chaos Polynomial Expansion. It has been used, for example, to derive the so-called stochastic finite element method. This method yields results that are exactly of the desired kind, but unfortunately at increased numerical costs.
This contribution aims to propose a new ansatz that is also based on a stochastic series expansion along with an appropriate relaxation procedure at the Gauβ point level. Energy relaxation provides a synthesized (deterministic) stress measure, while simultaneously offering stochastic properties such as the variance. The total procedure only needs negligibly more computation effort than a simple elastic calculation.
Elastic composites containing liquid inclusions exist widely in rocks, food, tissues and hydrogels. We investigate a single ellipsoidal compressible liquid inclusion embedded in an infinite elastic matrix, such as an isolated cell embedded in an extracellular matrix or an oil or gas pocket embedded within shale. We first derive the displacement and stress fields in the matrix under far field loading. For the special case of a spherical inclusion, we arrive at simple, explicit expressions for these fields. We next focus on the shape evolution of the liquid inclusion and the stress concentration in the matrix, from which we identify when the effect of liquid compressibility is most significant. Finally, we classify common examples of liquid inclusions in nature and engineering. According to our theoretical results, we estimate the importance of liquid compressibility in these examples and provide guidelines for further application of the theory of liquid inclusions in practical situations.
Solid-Solid interface mechanism understanding of composite inclusions, when extended to solid-liquid interface design of composite using Eshelby theory, indicates a possibility of decreasing effective stiffness with increasing liquid inclusion in a solid matrix. In contrast, experimental evidence in the current paper suggests high stiffness and enhanced dynamic energy absorption in a soft polymer (polydimethylsiloxane (PDMS)) with high bulk modulus liquid inclusions (gallium). The basic deformation mechanism is governed by hydrostatic stress causing shape change of the liquid inclusion in large deformation regime and strain hardening of a soft polymer matrix. In addition, dynamic viscoelasticity and fluid motion also play a significant role. We develop these understandings here based on analytical modeling and a detailed finite element with smooth particle hydrodynamic simulations. The large deformation with viscoelasticity of gallium composite shows higher energy absorption and dissipation. Similar strategies of liquid reinforcement to compliant solid matrices are abundant in nature, for example, the intervertebral discs in the spinal cord and deep sea animal skin and lungs.
The liquid inside a solid material is one of the most common composite materials in nature. The interface between solid–liquid plays an important role in unique deformation. Here, model systems of two polymers (polydimethylsiloxane–polyvinylidenefluoride) are used to make sphere of solid with liquid inside it.
Surface tension is an important factor in the behavior of fluids but typically has a minimal or negligible effect in solids. However, when a solid is soft and its characteristic dimension is small, forces due to surface tension can become important and significantly affect elastic deformation, leading to interesting elasto-capillary phenomena. We have developed a finite-element formulation accounting for surface tension and large deformations in three-dimensional settings and demonstrate the simulation capability by examining a class of problems involving fluid-filled droplet inclusions in a soft solid matrix. Specifically, we (1) consider the response of isolated droplets under far-field loading and (2) micromechanically model composite materials made up of a finite volume fraction of fluid-filled inclusions in a soft solid matrix. In the latter case, recent experimental work in the literature has shown that when the matrix material is sufficiently compliant, the presence of droplets leads to stiffening–counter to the intuitive notion of the presence of fluid-filled inclusions leading to a more compliant composite material. We show that our numerical simulation capability predicts all experimentally observed phenomena related to fluid-filled inclusions in soft solids. Furthermore, we consider the large-deformation response of composite materials with fluid-filled inclusions–a situation difficult to address using analytical methods.
a Investigations about the physico-chemical behaviour of fluid inclusions by means of thermal treatments and slow crystal dissolution lead to the evidence of two different kinds of gas bubbles. The results are reproducible for several organic and inorganic crystals obtained in aqueous solutions or organic solvents. Experiments under various pressures highlighted that the solubility of dissolved gases can play a significant role in the number of formed vacuoles. In addition, the nature of the dissolved gas in the mother liquor had an influence on the type of trapped bubbles. Raman spectroscopy confirmed that gaseous bubbles in fluid inclusions were rich in the gas that was dissolved in the mother liquor. Moreover, the pressure inside frozen aqueous fluid inclusions was estimated by measuring the temperature of the non-congruent fusion of the CO 2 clathrate that formed inside the vacuoles at −60 °C. Interferometry experiments were performed in order to visualise the consequences of the presence of vacuoles at the surface of the crystals. This study about fluid inclusion behaviours sheds new light on the formation of 3D defects in single crystals , whose mechanismIJs) of formation is (are) still difficult to understand.
A solid-liquid self-adaptive composite (SAC) is synthesized using a simple mixing-evaporation protocol, with Poly(dimethylsiloxane) (PDMS) and Poly(vinylidene fluoride) (PVDF) as active constituents. SAC exists as a porous solid containing a near equivalent distribution of the solid (PVDF)-liquid (PDMS) phases, with the liquid encapsulated and stabilized within a continuous solid network percolating throughout the structure. The pores, liquid and solid phases form a complex hierarchical structure, which offers both mechanical robustness and a significant structural adaptability under external forces. SAC exhibits attractive self-healing properties during tension, and demonstrates reversible self-stiffening properties under compression with a maximum of seven-fold increase seen in the storage modulus. Compared to existing self-healing and self-stiffening materials, SAC offers distinct advantages in the ease of fabrication, high achievable storage modulus, and reversibility. Such materials could provide a new class of adaptive materials system with multi-functionality, tunability, and scale-up potentials.
A sensitivity analysis (SA) has been conducted to examine the influence of uncertain input parameters on the fracture toughness of polymeric clay nanocomposites (PNCs). In order to predict the macroscopic properties of the composite, a phase-field approach has been employed considering six input parameters. For computationally efficiency, the SA is performed based on a surrogate model.
Screening methods of the Standardized Regression Coefficients and the Regionalized Sensitivity Analysis are applied first. Then, quantitative methods, i.e. Sobol', EFAST, and PAWN are employed. Moreover, we have presented an improvement to the PAWN method that reduces the computational cost. The efficiency, robustness, and repeatability are compared and evaluated comprehensively of the five SA methods. The convergence of the sensitivity indices is achieved through the bootstrapping technique.
The matrix Young's modulus is the most important input parameter affecting the macroscopic fracture toughness, whereas the volume fraction of the clay and the fracture energy of the matrix have a moderate importance. On the other hand, the aspect ratio, the radius of curvature, and the Young's modulus of the clay have negligible effects. Finally, fixing the uncertainties in the important input parameters reduces the coefficient of variation (COV) from 16.82% to 1.97%.
The Halpin-Tsai equations are based upon the ″self-consistent micromechanics method″ developed by Hill. Hermans employed this model to obtain a solution in terms of Hill's ″reduced moduli″ . Halpin and Tsai have reduced Hermans' solution to a simpler analytical form and extended its use for a variety of filament geometries. The development of these micromechanics relationships, which form the operational bases for the composite analogy of Halpin and Kardos for semi-crystalline polymers are reviewed.
We measured the shape change of periodic ridge surface profiles in gelatin organogels resulting from deformation driven by their solid-vapor surface stress. A gelatin organogel was molded onto poly-dimethylsiloxane (PDMS) masters having ridge heights of 1.7 and 2.7 μm and several periodicities. Gel replicas were found to have a shape deformed significantly compared to their PDMS master. Systematically larger deformations in gels were measured for lower elastic moduli. Measuring the elastic modulus independently, we estimate a surface stress of 107 ± 7 mN m(-1) for the organogels in solvent composed of 70 wt% glycerol and 30 wt% water. Shape changes are in agreement with a small strain linear elastic theory. We also measured the deformation of deeper ridges (with height 13 μm), and analysed the resulting large surface strains using finite element analysis.
Models and Sensitivity AnalysisMethods and Settings for Sensitivity Analysis – an IntroductionNonindependent Input FactorsPossible Pitfalls for a Sensitivity AnalysisConcluding RemarksExercisesAnswersAdditional ExercisesSolutions to Additional Exercises
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.
The theorem that an integrable function can be decomposed into summands of different dimensions is proved. The Monte Carlo algorithm is proposed for estimating the sensitivity of a function with respect to arbitrary groups of variables.
We show that a drop of liquid a few hundred microns in diameter placed under a solid, elastic, thin film (∼10 μm thick) causes it to bulge by tens of microns. The deformed shape is governed by equilibrium of tensions exerted by the various interfaces and the solid film, a form of Neumann's triangle. Unlike Young's equation, which specifies the contact angles at the junction of two fluids and a (rigid) solid, and is fundamentally underdetermined, both tensions in the solid film can be determined here if the liquid-vapor surface tension is known independently. Tensions in the solid film have a contribution from elastic stretch and a constant residual component. The residual component, extracted by extrapolation to films of vanishing thickness and supported by analysis of the elastic deformation, is interpreted as the solid-fluid surface tension, demonstrating that compliant thin-film structures can be used to measure solid surface tensions.
Owing to their superior mechanical and physical properties, carbon nanotubes seem to hold a great promise as an ideal reinforcing material for composites of high-strength and low-density. In most of the experimental results up to date, however, only modest improvements in the strength and stiffness have been achieved by incorporating carbon nanotubes in polymers. In the present paper, the stiffening effect of carbon nanotubes is quantitatively investigated by micromechanics methods. Especially, the effects of the extensively observed waviness and agglomeration of carbon nanotubes are examined theoretically. The Mori-Tanaka effective-field method is first employed to calculate the effective elastic moduli of composites with aligned or randomly oriented straight nanotubes. Then, a novel micromechanics model is developed to consider the waviness or curviness effect of nanotubes, which are assumed to have a helical shape. Finally, the influence of nanotube agglomeration on the effective stiffness is analyzed. Analytical expressions are derived for the effective elastic stiffness of carbon nanotube-reinforced composites with the effects of waviness and agglomeration. It is found that these two mechanisms may reduce the stiffening effect of nanotubes significantly. The present study not only provides the relationship between the effective properties and the morphology of carbon nanotube-reinforced composites, but also may be useful for improving and tailoring the mechanical properties of nanotube composites.
The classical formulation of Eshelby (Proc. Royal Society, A241, p. 376, 1957) for em-bedded inclusions is revisited and modified by incorporating the previously excluded surface/interface stresses, tension and energies. The latter effects come into prominence at inclusion sizes in the nanometer range. Unlike the classical result, our modified formu-lation renders the elastic state of an embedded inclusion size-dependent making possible the extension of Eshelby's original formalism to nano-inclusions. We present closed-form expressions of the modified Eshelby's tensor for spherical and cylindrical inclusions. Eshelby's original conjecture that only inclusions of the ellipsoid family admit uniform elastic state under uniform stress-free transformation strains must be modified in the context of coupled surface/interface-bulk elasticity. We reach an interesting conclusion in that only inclusions with a constant curvature admit a uniform elastic state, thus restrict-ing this remarkable property only to spherical and cylindrical inclusions. As an immediate consequence of the derivation of modified size-dependent Eshelby tensor for nano-inclusions, we also formulate the overall size-dependent bulk modulus of a composite containing such inclusions. Further applications are illustrated for size-dependent stress concentrations on voids and opto-electronic properties of embedded quantum dots.
The fundamental framework of micromechanical procedure is generalized to take into account the surface/interface stress effect at the nano-scale. This framework is applied to the derivation of the effective moduli of solids containing nano-inhomogeneities in conjunction with the composite spheres assemblage model, the Mori–Tanaka method and the generalized self-consistent method. Closed-form expressions are given for the bulk and shear moduli, which are shown to be functions of the interface properties and the size of the inhomogeneities. The dependence of the elastic moduli on the size of the inhomogeneities highlights the importance of the surface/interface in analysing the deformation of nano-scale structures. The present results are applicable to analysis of the properties of nano-composites and foam structures.
The following techniques for uncertainty and sensitivity analysis are briefly summarized: Monte Carlo analysis, differential analysis, response surface methodology, Fourier amplitude sensitivity test, Sobol' variance decomposition, and fast probability integration. Desirable features of Monte Carlo analysis in conjunction with Latin hypercube sampling are described in discussions of the following topics: (i) properties of random, stratified and Latin hypercube sampling, (ii) comparisons of random and Latin hypercube sampling, (iii) operations involving Latin hypercube sampling (i.e. correlation control, reweighting of samples to incorporate changed distributions, replicated sampling to test reproducibility of results), (iv) uncertainty analysis (i.e. cumulative distribution functions, complementary cumulative distribution functions, box plots), (v) sensitivity analysis (i.e. scatterplots, regression analysis, correlation analysis, rank transformations, searches for nonrandom patterns), and (vi) analyses involving stochastic (i.e. aleatory) and subjective (i.e. epistemic) uncertainty.
A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.
This paper deals with computations of sensitivity indices in sensitivity analysis. Given a mathematical or computational model y=f(x1,x2,…,xk), where the input factors xi's are uncorrelated with one another, one can see y as the realization of a stochastic process obtained by sampling each of the xi from its marginal distribution. The sensitivity indices are related to the decomposition of the variance of y into terms either due to each xi taken singularly (first order indices), as well as into terms due to the cooperative effects of more than one xi. In this paper we assume that one has computed the full set of first order sensitivity indices as well as the full set of total-order sensitivity indices (a fairly common strategy in sensitivity analysis), and show that in this case the same set of model evaluations can be used to compute double estimates of: •the total effect of two factors taken together, for all such couples, where k is the dimensionality of the model;•the total effect of k−2 factors taken together, for all such (k−2) ples.We further introduce a new strategy for the computation of the full sets of first plus total order sensitivity indices that is about 50% cheaper in terms of model evaluations with respect to previously published works.We discuss separately the case where the input factors xi's are not independent from each other.
Structural reinforcement through liquid encapsulation
- A C Chipara
- P S Owuor
- S Bhowmick
- G Brunetto
- S S Asif
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