1. Introduction
Due to their cellular microstructure, metal foams have attracted considerable interest in several industrial applications for their unique morphological characteristics, which allow for joining in a single material an effective combination of structural properties (low density, high capability to absorb energy during deformation), and various functional properties [
1]. The field of possible applications has been further expanded by additive manufacturing technologies, which have revealed a marked capability in fabricating structures characterized by almost all types of cell shapes [
2].
Regarding the mechanical behavior, in the last decades great attention has been focused on the characterization and modelling of closed-cell aluminum foams. Uniaxial stress–strain behavior has been investigated extensively [
3,
4,
5], also taking strain rate sensitivity into consideration [
6,
7], and exploiting the advantages of new manufacturing technologies in cellular structures optimization, in order to obtain greater mechanical efficiency without increasing the relative density [
8].
The compressive properties of cellular materials, which can be expressed by some key parameters of the stress–strain curve, such as the elastic modulus, plateau stress, and final deformation values, are particularly important for the mechanical design of components [
9]. It is a matter of fact that the cellular structure determines an important effect on the compression behavior [
10,
11], because the amount of energy absorbed is directly related to the way the cells collapse [
12]. As a consequence, the aforementioned compression properties are strictly connected to the morphological characteristics of foams, as cells’ size, shape, and wall thickness, which undergo an evolution during deformation, and affect the deformation modes [
13]. This evidence suggests investigating the potential of the morphological properties for predicting the mechanical behavior of cellular structures.
With specific reference to closed-cell Al alloy foams, various models have been proposed to describe their behavior by means of constitutive laws which represent the cellular structures as solid materials, and are based on empirical equations, formulated by processing experimental results [
14].
The representative models for closed-cell foams, instead, allow for the representation of regular periodic structures, by using polyhedra to model the cells and fill the 3D space of the structure. The reference study by Gibson and Ashby [
15] proposes a skeleton cubic cell model, which allows to establish a well-known set of equations to calculate the main parameters of the stress–strain curve of the foam, by performing a dimensional analysis and introducing empirical constants to be fitted on the experimental data depending on the specific foam. Although the equations resulting from this model can be considered widely validated and are frequently used, the geometric schematization of the cellular structure is not very representative of the real structure of some types of foams. For this reason, other representative models have been proposed. The truncated cube model by Santosa and Weirzbicki [
16] assumes the foam to consist of a packed lattice of small and large cells, obtained by assembling symmetric cubic cells truncated at the vertices, and allows to formulate an analytical solution for the crushing strength. As an improvement of the truncated cube model, the cruciform-hemisphere model proposed by Meguid et al. [
17] replaces the pyramidal sections obtained at the vertex of the truncated cubes with hemispherical sections, obtaining a more realistic geometric representation.
The representative models described above are the basis of the most known analytical formulations that allow for calculating the key parameters of the compressive behavior of the closed-cell structures (elastic modulus, plateau stress, and densification strain) [
9]. By means of these parameters it is possible just to construct schematic and approximate stress–strain curves. This is due to the limited number of points in the curve that can be derived by these parameters, and to the significant deviations between their values predicted by analytical calculation and the corresponding values detected experimentally [
18]. Furthermore, these basic formulations do not allow to correlate the mechanical behavior with the morphological properties of the structure, as they are direct functions of the relative density of the foam.
To obtain a complete simulation of the mechanical behavior extended to the whole compression process, these formulations have been widely implemented in Finite Element (FE) numerical models. Numerical computation is, therefore, the tool most used for studying the behavior of cellular structures by representative modeling. This is true not only for geometries based on polyhedra [
19], but also in the case of models that use basic cell shapes (circular, elliptic, rectangular, square) [
20], or combinations of them [
21], and models obtained by positioning spherical cells in cubic units, according to the arrangements that characterize the well-known crystal structures [
22].
However, even in investigations based on numerical modeling, the cases in which the mechanical behavior is directly related to the morphological properties of the structure are limited. The size, aspect ratio, orientation, and anisotropy of the cells, as well as the length, straightness, inclination, and thickness of the cell walls were correlated to the deformation mechanism of closed-cell foam, with particular regard to the initiation of collapse [
23]. A FE model with Representative Volume Elements was set to investigate the effects of cells height and wall thickness on the compressive behavior [
24]. A similar modeling approach was used to obtain a morphological control on cell size, wall thickness, and curvature distributions on metal closed-cell structures, and to simulate their behavior under compressive loading up to the densification stage [
25].
The most complex numerical models used to represent random cells arrangements and geometrical properties variability by means of a large number of elements generated by tessellation algorithms, constitute an advanced approach to the modeling of cellular structures, and requires high computational capacity. They have been recently used for investigating the effects of various morphological properties focusing on some aspects of mechanical behavior. Random Laguerre tessellations was used to generate model foams with strongly varying cell sizes and simulate the effect on elastic stiffness of the structure [
26]. The cell size effects on the compressive properties of Al foams were investigated by Voronoi modeling and statistical analysis of the relationship between the micro-structure parameters and the macro-properties of foams [
27]. The compression behavior of the Voronoi model was also simulated to investigate the influence of cell irregularity on elastic and plastic behavior of closed-cell foams [
28]. The effects of cell-size dispersity on the Young’s modulus of foam were investigated through numerical simulations on polydispersed model obtained by the random Laguerre tessellation algorithm [
29]. Different magnitudes of pore size and misalignment was simulated by imposing random displacement in irregular Voronoi models, so to assess the effects on the compressive response [
30].
As a general observation, the more complex computational approaches based on FE modeling are hardly able to accurately simulate the whole spectrum of regimes that arise during the compression of cellular structures (linear elastic deformation, plastic collapse, densification), generally achieving greater accuracy in the second one, although there are examples in which a good fit of the experimental curve in the densification zone [
31] or an acceptable agreement in the linear elastic region [
32] is obtained.
Therefore, the predictive simulation of the whole stress–strain curve based on the geometrical properties of the cellular structure remains a challenging task, and the analytical approach has not been pushed to this direction.
Focusing on closed-cell Al foams, the present work proposes an analytical model aimed for the prediction of the compressive behavior, by calculating the main parameters of the stress–strain curve and simulating it up to the final densification, based on the two-dimensional analysis of the morphological characteristics of the cellular structure, and their evolution during the collapse process.
Considering that the precursor conditions to produce the metal foams (powder composition and the parameters of the foaming process) can be correlated to the morphological properties of the final cellular structures [
33], the applicative purpose of the model can be addressed to predict with good approximation the mechanical behavior of the structure, by means of a basic two-dimensional detection and analysis of the morphology resulting from the foaming conditions set.
With these purposes, the relationship between the distributions of morphological characteristics of the foam sections longitudinal to the load direction and the mechanical response of the cellular structure, during the compressive progression, was searched for. As a result, an analytical model for morphology–behavior correlation, comprising a procedure for constructing the stress–strain curve, was defined, with the objective of simulating the whole compressive behavior, up to the final densification.
Starting from the consideration that foams generally show a typical non-homogeneous morphology, being characterized by cells having geometry and size different from each other [
34,
35], the correlation between the morphological properties of the foam and the characteristic parameters of the stress–strain curve (elastic modulus, plastic plateau stress, strain parameters of densification) needs to transform the real cellular structure, non-homogeneous at local level, into a virtual reference model consisting of cells ordered in space, with global homogeneous geometric properties. The translation of the local morphological parameters into the global parameters of the reference model was obtained by introducing an intermediate model to consider the asymmetry of the cells and their orientation with respect to the direction of compression.
To investigate the potential of the model for the prediction of compressive behavior based on the morphological analysis, it was first fitted on the experimentally detected behavior for a specific foam, following the evolution of morphology during the progression of deformation; subsequently, it was used to simulate the compression behavior of similar foams. For this purpose, closed-cell Al foams were produced by an in-house process based on the compacted powder method, starting from Al and SiC powders with the addition of TiH2 powder as foaming agent at various contents. By this way it was possible to fit the model on the experimentally detected compression behavior for a foam with a specific powder composition and evaluate its potential in predicting the same behavior for similar foams obtained by different powder compositions.
The morphology of the cellular structure, expressed by the distribution of the values of the equivalent diameters and circularities of the cells through the specimen sections, was investigated by X-ray computed tomography (CT), a technique that, together with basic radioscopy, has already proved particularly suitable for the internal investigation of metal foams, focusing on different aspects: foaming process efficiency [
36], morphological analysis, and microstructural characterization of foamed structures [
34,
35], and their mechanical behavior [
37,
38]. Specifically, in [
39], significant changes in the structure of the internal cells during the deformation process were documented by means of CT images, overcoming previous approaches based on combination of metallographic images and static theory application [
13].
For the specimen used for model fitting, the tomographic observations were carried out at different levels of compression in order to study the evolution of cells’ morphology during deformation up to the final densification, according to an experimental procedure first presented in [
40] for Al foam, and in [
41] for a Cu tube filled with an Al alloy foam.
Once fitted on the basis of the CT observations as compression progresses, the analytical model was used to predict the mechanical behavior of the other similar foams, produced by varying the content of the TiH2 forming agent, and was validated by means of comparison between simulated and experimentally detected stress–strain curves.
The paper has been structured according to three main sections. In
Section 2, the materials and methods are detailed. Particularly, the approach to experimental investigation is treated, specifying specimens’ fabrication and characteristics, the statement of compression testing and tomographic observations, and the method of conducting the morphology analysis.
Section 3 is dedicated to the theoretical development of the morphology–behavior correlation model, reported according to its main steps: the modeling of cellular structure, the simulation of compressive behavior, and the fitting of the correlation model.
Section 4 reports the application of the model and the results obtained, focusing on the predictive use of the fitted model. The significance and implications of the main results are discussed.