1. Introduction
Material science has undergone great evolution in recent years, representing an extremely important field for the development of many technological areas for several reasons, such as those related to a sustainable economic and environmental nature. The introduction of the functional gradient concept in the context of composite particulate materials has contributed to the design of advanced materials able to meet specific objectives, through spatial variation in composition and/or microstructure [
1,
2].
Functional gradient materials (FGMs) were developed in Japan in the late 1980s for thermal insulation coatings [
2]. With more than three decades of history, and being a part of a wide variety of composite materials, materials with functional gradients continue to be the object of attention. This is due to their tailorability, arising from a gradual and continuous microstructure evolution and, consequently, of locally varying material properties in one or more spatial directions. Therefore, FGMs can be appropriately idealized to meet certain specifications [
3,
4].
Composite materials can generally be described as systems composed of a matrix and a reinforcement, the material properties of which surpass those presented by each constituent individually [
5]. The gradual and continuous evolution of their material properties provide FGMs with better mechanical behavior than traditional composite materials [
6]. Among other characteristics possessed by some FGMs, such as low thermal conductivity and low residual stresses, these materials also make it possible to minimize the stress discontinuities of conventional laminated composites [
5,
7,
8]. FGMs’ high resistance to temperature and the absence of interface problems gives them great importance in several engineering applications, which is why these composite materials have been extensively studied and used in the past two decades in a wide range of science fields requiring improved mechanical and thermal performances [
5,
6,
7]. This broad range of applications justifies the need to study and predict the responses of FGMs’ components [
9]. In the past year, Li et al. developed several studies involving, among other topics, a stability and buckling analysis of functionally graded structures such as cylinders and arches [
10,
11,
12,
13,
14].
Considering the continuous mixture concept behind their composition, FGMs can be constituted of two or more material phases. Hence, the resulting local material properties depend on this mixture evolution, allowing for a design according to the desired functions and specifications [
15]. From the literature, the most common FGMs are constituted of two material phases, often one ceramic and one metal [
6,
11].
The manufacturing process of sintering, which is common in the production of FGMs, is responsible for the formation of micro voids or porosities within the materials, making important the introduction of porosity effects at the structures’ design stage [
9]. The literature contains various studies which include dynamic and static analyses of functionally graded porous plates and beams [
16,
17,
18,
19,
20,
21]. In addition, there are several studies considering ring and arch structures, especially those developed by Li et al. in the past year [
13,
22,
23,
24,
25,
26,
27].
Functionally graded porous materials (FGPs) combine both porosity and functional gradient characteristics, where the porosity may have a graded evolution across the volume, providing desirable properties for some applications (as in biomedical implants), and undesirable in others where voids may cause serious problems (as in the aeronautical sector). The change in porosity in one or more directions can be caused by local density effects or pore size alteration. Functionally graded porous materials possess a cell-based structure, which can be classified as open or closed (i.e., containing interconnected or isolated pores, respectively) [
15].
Porous gradient materials also present a multifunctional character, where, among other aspects, one may highlight a high performance-to-weight ratio and resistance to shock; nevertheless, it is important to note that pores imply a local loss of stiffness. The latest advances in manufacturing processes allow the consideration of the development of porous materials with a functional gradient, using methods such as additive 3D printing. Thus, it becomes possible to design porous structures with designed variable stiffness, which can be customized for specific engineering applications, optimizing performance and minimizing weight [
28]. Due to the relevant role that such materials have in a range of applications, it is important to have a wider perspective of the contexts where one can find them.
Mechanical or more generically structural components made of porous materials, bioinspired materials, can be designed for sensitive and very precise operating conditions—for example, robotic, prosthetic, and aeronautical components, among others [
28]. Most of the materials used in engineering are dense; however, porous materials are also of great interest and applicability in fields such as in membranes [
3]. Bioinspired materials thus have great potential for current technologies, as their unique characteristics allow them to meet various design requirements [
28]. Natural or man-made, porous compacts or foams—the types of porous materials are many and diverse. Bones, wood, ceramics, and aluminum foams are just a few examples [
29]. In the field of biomaterials, the inclusion of porosities allows diversification of their applicability, with ceramic and polymeric scaffolds being examples of this [
3].
In the biomedical field, a bone implant must guarantee a functional gradient that mimics real bone stiffness variations. With functionally graded porous scaffolds it is possible to obtain the variations in mechanical and biological properties required for bone implants, as the presence of a porosity gradient is imperative for bone regeneration. Thus, functionally graded porous scaffolds for bone implants are designed to achieve the ideal balance between porosity and mechanical properties [
30]. Since bone implants have various requirements, it is important to have ceramic porous materials with a wide and diversified range of pore sizes and morphologies in order to accomplish these requirements. Therefore, different porous hydroxyapatite structures have been developed to mimic natural bone’s bimodal structure [
31]. Since implants with no porosity show weak tissue regeneration and implant fixation capability, the introduction of a pore distribution in these alloys’ structures leads to bone-like mechanical properties, allowing cellular activity [
32].
In another field [
33], membranes produced according to the Fuji process present a structure with relatively wide pore surface, followed by a gradually tightening pore size and a clearing of pores, finishing with an isotropic structure. Thus, these membranes can be mentioned as an example of an asymmetric structure from the porosity perspective, being implemented in several applications, with filtration and medical diagnosis being some examples. Porous membranes are the object of research related to materials and structure optimization [
34].
Membranes can also be used for gas separation applications, typically possessing a microporous substrate, a mesoporous intermediate layer (or more), and a microporous top layer (or more). Regarding materials, α-Al
2O
3 is the most frequent, but TiO
2, ZrO
2, SiC, and their combinations are also very common [
35]. For industrial wastewater treatment applications, ceramic microfiltration membranes made of Al
2O
3, SiO
2, ZrO
2, TiO
2, and their composites present excellent behavior. These specific membranes possess multi-layered asymmetric structures, with a macroporous support followed by intermediate layers of graded porosity [
36]. The use of ceramic membranes extends to catalytic membrane reactors due to the huge resistance to high temperatures and aggressive chemical environments. In this case, the membranes usually present disk-like, planar, tubular, and hollow fiber designs [
37]. According to Sopyan et al., the material properties of porous ceramics (e.g., Young’s modulus and flexural strength) depend exponentially on the ceramics’ total porosity [
31].
Secondary batteries technology uses porous membranes to isolate cathode and anode from each other, preventing a short circuit, and to allow the charge transport. Therefore, these membranes should be simultaneously excellent electric insulators and good ion conductors, presenting a great safety level. In this field of action there are microporous, ultrafiltration. and nanofiltration membranes, with organic polymers being the most used materials [
34].
Metallic foams are another example of materials whose mechanical properties depend on the porosity characteristics. Recently, they have been gaining use among applications of aluminum and other alloys since the combination of properties intrinsic to metal alloys with the effects of porosity are of great interest, highlighting the low density and high energy absorption. The change in the porosity characteristics of these materials (e.g., pore size) makes it possible to obtain properties suitable for specific applications. Aluminum foams find use in structural applications, as well as automotive and aeronautics industries, as examples [
38].
A well-defined spatial porosity gradient is a requirement of solid foams for some specific applications like filtration, energy adsorption, and tissue engineering. Therefore, control over porosity in terms of morphology, pore size, and pores’ connectivity is a challenge in the development of fabrication processes, since these parameters have a great influence on the porous materials’ performance [
39]. In their work, Costantini et al. mentioned that a pore size gradient confers an increased strength and energy absorption to a material, and that this kind of material needs a more precise characterization of the porosity gradient concerning the mechanical properties [
39].
Since pores can have different dimensions and distributions, porous materials can appear with different porosity gradients. In a typical rectangular plate, there are several possible porosity gradient configurations. Regardless of the specific distribution, the relevance should be placed on the correspondence to the design requirements [
3], knowing that the heterogeneity and spatial gradient characteristic of porous materials will play an extremely important role in the resulting mechanical properties [
40].
The Young modulus and shear modulus are strongly influenced by several factors, from the manufacturing process, to the size, shape, and distribution of the pores. Consequently, the analytical prediction of porous materials’ properties is not simple because of the randomness present in their structures, and the need of a knowledge of the microstructure that is as accurate as possible in order to obtain a significant numerical prediction [
29].
Concerning porosity distributions, Nguyen et al. [
41] studied the mechanical behavior of porous FGPs. To this purpose they took into account two different porosity distributions, varying both through the thickness direction (namely, the so-called even and uneven distributions). Zhang and Wang [
15] produced eight different porous material structures with different pore distributions, including gradient distributions, and subjected them to some mechanical tests in order to evaluate important materials properties like Young’s modulus. With this work they developed techniques to estimate the effective Young’s modulus of functionally gradient porous materials. Having verified that there is an obvious relation between this material property and porosity, the relationship between both is not necessarily linear. However, the experimental results constitute a good basis for validating material properties obtained through theoretical models.
With this introductory section, the importance of porous materials becomes clear, particularly regarding the development of porosity distribution models that best represent the effects on the respective effective material properties.
Hence, the present work presents three porosity distribution models, two of which are based on the reference literature, and respective estimates of material properties. For these cases, we performed a set of parametric studies focused on the static behavior of porous plates with a functional gradient in order to characterize the influence of the shear correction factor, associated with the use of the first-order shear deformation theory. These studies were performed via the finite element method considering an equivalent single layer approach. To the best of our knowledge, there are no previously published works focusing on the assessment of the influence of the shear correction factor in the static bending behavior of porous plates. Hence, this study addresses this, considering the characterization of the neutral surface deviation from the mid-plane surface, which also provides an illustrative measure of the medium’s heterogeneity, typical of graded mixture and porous materials.