تجزیه و تحلیل حساسیت جهانی برای خواص الاستیک از فوم های ساختاری حوزه های توخالی پر شده با استفاده از روش بازنمایی مدل ابعادی بالا

An accurate prediction of the bulk properties of syntactic foams, even for the elastic properties, is difficult due to the microstructure being composed of constituents with strong distinctions in mechanical properties. Moreover, it is very costly and time-consuming to characterize the influence of various parameters on the bulk properties of syntactic foams by experiments. In this study, a microstructure-based finite element simulation approach was developed to predict the elastic mechanical behaviors of hollow spheres filled syntactic foams. Three-dimensional cubic unit cell model with interface simulated by cohesive elements was constructed to capture the microstructure and stress/strain fields in mesoscale. The effective elastic properties of syntactic foams in terms of Young’s modulus and Poisson’s ratio were calculated by means of homogenization method. To get an enhanced understanding of property–structure relations, a global sensitivity analysis was performed based on the high dimensional model representation (HDMR) method. Ten parameters, including geometry and mechanical properties of constituent phases, were selected as input parameters. Independent and cooperative effects of the input parameters on the elastic properties of syntactic foams were investigated by first- and second-order sensitivity indices, respectively. An importance ranking of the input parameters for Young’s modulus and Poisson’s ratio could then be obtained. The procedure proposed in this work provides a powerful tool for design and optimization of syntactic foams.

Syntactic foams, manufactured by filling a polymeric matrix with hollow spheres called microspheres or microballoons, are generally classified as particle reinforced composites. In general, epoxy resins, polyesters, silicones, polyurethanes, and several other polymers are used as binders, while the fillers have been made of glass, carbon, ceramics, polymers, and even metals [1]. A general microstructure of syntactic foams is illustrated in Fig. 1[2]. Compared with standard foams (containing blown gas bubbles only), syntactic foams are preferred when high specific mechanical properties are required, rather than just low density [3]. Due to many advantages over other materials, e.g., low density, high specific strength, excellent compressive properties, low conductivity, low thermal expansion, no corrosion effects, fire resistance, low moisture absorption [1] and [3], syntactic foams have been employed in many engineering applications, ranging from marine equipments for deep sea operations to impact energy absorption components in automotive industry, core materials of sandwiches. And more recently, syntactic foams are also used as manufacturing material of missile canister cover in aerospace industry. Full-size image (43 K) Fig. 1. General microstructure of syntactic foams. Figure options Generally, the levels of the description of material behavior can be divided into nano-, micro-, meso- and macrolevel [4]. Following the pioneering work of Eshelby [5] on the stress field in an isolated ellipsoidal inclusion within an infinite elastic matrix, the mesomechanics has been developing rapidly. Many analytical models were developed to calculate the elastic constants of two-phase composites, e.g., the Mori–Tanaka estimate [6], the Hashin–Shtrikman bounds [7], the self-consistent scheme [8]. Most of the analytical models are simple and explicitly formulated; however, they are limited in scope because of the special assumption for the geometry of reinforcing phase and failure to express the details of the stress and strain fields in mesoscale. Besides, almost all of the analytical models are based on the assumption that the inclusions are perfectly bonded to the matrix material. Actually, it has been well known that an interfacial transition zone (ITZ) may form between inclusions and matrix due to complex chemical and physical actions during the manufacturing process of composites. The interfacial transition zone regarded as an individual interphase whose properties differing from those of constituent phases plays a vital role in the mechanical behaviors of composites. Marur [9] and Bhuiyan et al. [10] demonstrated that analytical models based on perfect interface assumption overestimated the experimentally determined elastic modulus. Rjafiallah and Guessasma [2] and Marur [11] analyzed the interface effect on the elastic properties of syntactic foams. “Computational mesomechanics of materials” that is a generalizing term suggested by Mishnaevsky and Schmauder [4] aims at achieving a much better resolution of mesoscale stress and strain fields than is possible with analytical models, typically by investigating suitable model geometries by means of numerical engineering techniques, especially the finite element method (FEM) [12], [13] and [14]. Syntactic foams can be regarded as macroscopically homogeneous and isotropic medium due to the random distribution of microspheres, while in mesoscale, syntactic foams are heterogeneous materials. Multi-scale unit cell analysis based on homogenization method is applicable to such materials to compute the effective mechanical properties given the microstructures and material properties of constituent phases. The unit cell homogenization methods treat the bulk material as a periodic array of particles suspended in the matrix and solve only a representative volume. Recently, several types of unit cell have been developed and used successfully to model particle reinforced composites, including periodic two-dimensional cylindrical and spherical unit cells [15] and [16], three-dimensional cubic unit cell [15], [16], [17], [18] and [19], three-dimensional random unit cells [18], [19] and [20]. To better understand the bulk properties of syntactic foams, and characterize the influence of various parameters (e.g., volume fraction, wall thickness, inclusion diameter), sensitivity analysis is required. However, the literatures are limited to the independent effects of a few mesoscale parameters on the mechanical properties of syntactic foams, usually by material experiments [21], [22], [23], [24] and [25]. To our best knowledge, a global sensitivity analysis considering the interacting effects of mesoscale parameters for the elastic properties of syntactic foams based on numerical simulation has not been reported. The aim of the present work is to develop a three-dimensional FE unit cell model considering the interface effect to predict the elastic properties of syntactic foams in terms of Young’s modulus and Poisson’s ratio. And a global sensitivity analysis based on the high dimensional model representation (HDMR) method is performed to quantify the importance of various mesoscale parameters and characterize the independent and cooperative effects of these parameters on the elastic properties of syntactic foams.

In this study, a reliable microstructure-based simulation approach has been presented by finite element method to predict the elastic properties of hollow spheres filled syntactic foams in terms of Young’s modulus and Poisson’s ratio. Cohesive elements governed by a bilinear traction-separation constitutive law were introduced to the unit cell model to simulate the mechanical behaviors of the interface between hollow spheres and matrix material. By comparison with the experimental data and two classical analytical methods, the validity of the proposed modeling method was verified. The results of numerical simulations indicate that the FE model without interface and the analytical methods based on perfect interface assumption all overestimate the experimentally determined elastic properties of syntactic foams seriously. Only the results obtained by the FE model with interface are much closer to the experimental data. It suggests that cohesive element is applicable to modeling the mechanical behaviors of interface in syntactic foams. To obtain an accurate prediction, more reliable mechanical parameters of the interface are needed in the simulation. For syntactic foams, interface effect is difficult to measure by existing experimental techniques, e.g., single fiber pull-out, nano-indentation. We believe that an inverse method combining the microstructure-based numerical simulation and the macroscopic mechanical test will be helpful for the parameter identification of the interface. Besides, some other microstructure features, e.g., non-periodic arrangement, clustering of the microspheres, and voids should be taken into account for an accurate prediction by a finer FE model. At least a multi-particle model is required. Although these two topics are beyond the scope of this paper, they are interesting open issues deserving further research. To quantify the importance of various parameters and characterize the independent and cooperative effects of these parameters on the elastic properties of syntactic foams, a global sensitivity analysis was performed using the high dimensional model representation (HDMR) method. RS-HDMR expansions up to second order were verified to be sufficient to represent Young’s modulus and Poisson’s ratio of syntactic foams. According to the sensitivity indices, the most important parameters for Young’s modulus and Poisson’s ratio of syntactic foams were identified and an importance ranking of these parameters was given. The results of the global sensitivity analysis indicate that elastic properties of syntactic foams heavily depend on the mechanical properties of matrix material and the dependence is almost linear. Second only to the matrix, the mechanical properties of interface especially the normal stiffness have a significant impact on the elastic properties of syntactic foams. As a key manufacturing parameter, the volume fraction of filler phase has no independent effect, but mainly influences the Young’s modulus of syntactic foams together with other parameters; for the Poisson’s ratio, the volume fraction of filler phase has both independent and interactive effects. In general, interactions of the parameters have a significant impact on the Young’s modulus of syntactic foams, but in contrast, the Poisson’s ratio of syntactic foams is mostly affected by the parameters acting independently, the interactive effects are quite small and almost negligible. Being capable of providing a straightforward approach to explore the property–structure relations and a very efficient way to calculate the sensitivity indices without the need of a large sample size, the procedure proposed in this work has proved to be a powerful tool for design and optimization of syntactic foams.