Consistency in Young’s Modulus of Powders: A Review with Experiments

Abstract

This review, complemented by empirical investigations, delves into the intricate world of industrial powders, examining their elastic properties through diverse methodologies. The study critically assesses Young’s modulus (E) across eight different powder samples from various industries, including joint filler, wheat flour, wheat starch, gluten, glass beads, and sericite. Employing a multidisciplinary approach, integrating uniaxial compression methodologies—both single and cyclic—with vibration techniques, has revealed surprising insights. Particularly notable is the relationship between porosity and Young’s modulus, linking loose powders to the compacts generated under compression methods. Depending on the porosity of the powder bed, Young’s modulus can vary from a few MPa (loose powder) to several GPa (tablet), following an exponential trend. The discussion emphasizes the necessity of integrating various techniques, with a specific focus on the consolidation state of the powder bed, to achieve a comprehensive understanding of bulk elasticity. This underscores the need for low-consolidation methodologies that align more closely with powder technologies and unit operations such as conveying, transport, storage, and feeding. In conclusion, the study suggests avenues for further research, highlighting the importance of exploring bulk elastic properties in loose packing conditions, their relation with flowability, alongside the significance of powder conditioning.

1. Introduction

Powders have played essential roles in human and societal evolution, whether as food (wheat, flour, sugar, coffee, etc.), in medicines (active principles, pills, inhalators, etc.), or in construction materials for our homes (sand, cement, wood, etc.). They are omnipresent in our daily life, industry, and nature. Paradoxically, despite their prevalence, understanding the intricate behavior of powders remains a formidable challenge.

Powders exhibit versatile states, transitioning between solid (in a sandpile), liquid (like sand flowing in an hourglass), and gas forms (as a sand/dust devil whirlwind), resulting in a rich phenomenology ([1,2,3]) marked by phenomena like negligible thermal fluctuations [4] and highly dissipative interactions [2,4,5]. The sheer number of particles in a small volume poses challenges for discrete approximations, blurring the lines between microscopic and macroscopic scales.

The densification of granular media is crucial industrially, influencing material transportation and storage [6,7,8,9,10,11,12]. When one thinks of the vast cargoes of sand—over 50 billion metric tons—and wheat—over 765 million metric tons—to be stored and transported annually worldwide, the importance of the consolidation of powders becomes self-evident. When energy is transferred into a granular material, the grains begin to reorganize themselves in a denser structure. Densification, compressibility, and compactness are grain properties arising from the collective forces acting on the media. It is important to outline that powder densification and consolidation are two physically different processes that can be accomplished through vibration and loading of powders. Densification is characterized by the increased bulk density due to the reduction of porosity as a result of the reorganization—repacking—of particles or due to plastic flow. If dynamic loading is accompanied not only by porosity reduction but also by the creation of physical bonding between contacting particles, then it is considered a consolidation process. In different scientific communities, the terms compression, consolidation, and compaction may carry varying meanings. Here, we use compression or consolidation to refer to instances where powder bed densification occurs through a uniaxial force to form a tablet or compact. The term compaction is reserved for procedures where densification occurs at a free surface or does not involve the creation of a tablet.

In the realm of consolidation, a fundamental mechanical characteristic is elasticity, typically quantified by the elastic modulus, also known as Young’s modulus. Despite its fundamental importance, determining the elastic properties of granular materials remains a non-trivial task. Various methodologies have been employed to determine the elastic modulus of powders. The three- or four-point beam bending technique, a well-known method, involves pre-compacting powder into tablets that act as beams. These tablets are then subjected to bending, similar to solid materials, using a three-point [13,14,15,16] or four-point bend rig [17,18], depending on the chosen methodology and experimental conditions. Typical values obtained through this technique range up to a few tens of GPa, contingent on factors such as the bending geometry and applied load. Despite its reproducibility and seemingly reliable results, the measured elastic modulus may not effectively describe the flow behavior of particulate materials. This limitation is likely attributed to the tablet’s consolidated nature, which may not offer a direct representation of individual powder particles.

To better address the needs of powder technologies, particularly in industries like pharmaceuticals, new methodologies based on compression have been developed for determining the elastic modulus of powders. These compression methodologies exhibit various variations, particularly in the compaction phase and its linearity. Options include compressing individual particles [19,20] or, more commonly, bulk compression [19,21,22,23,24,25,26,27,28,29]. The diversity in compression approaches is detailed in Table 1.

Consolidating a powder involves both plastic (${\epsilon }^{pl}$) and elastic deformation (${\epsilon }^{el}$). Initially, particles undergo rearrangements through translation and rotation movements, potentially resulting in deformations at contact points. As stress intensifies, the particles experience a combination of elastic and plastic stresses. When unloading the sample to zero stress, it expands, representing a reversible elastic deformation. Most compression studies characterize the elastic behavior of powders during unloading, known as elastic recovery [29,30,31]. Approaches to analyzing elastic recovery can be divided into two groups: those leveraging the linear behavior of a portion of the stress-deformation curve [21,28], and those treating it as entirely non-linear [24,29]. The latter is argued to be more precise as it eliminates the need to determine the point where the unloading ceases to be linear.

More recently, a shift towards utilizing the loading phase has emerged (Figure 1). This involves employing pre-compaction stages to eliminate plastic deformations resulting from rearrangement in prior partial compression. It allows for the observation of a linear behavior on the loading curve during the main compaction [22,23,26,27].

Alternative approaches have emerged, with a particular emphasis on individual particles. One notable method is the Indentation method. Within its various adaptations, the high-temperature Microindentation test on pelletized powders of Al-Si alloys has demonstrated typical values in the range of several tens of GPa for loads ranging from 0.2 to 1.5 N [32]. Another variant is Nanoindentation where E values typically range from a few GPa under loads of a few mN [33,34,35,36,37]. Recent advancements in nanomechanical testing instruments have enabled analysis from single crystals/particles to tablets, offering insights into structure–property relationships. However, the technology still faces significant challenges in sample preparation, necessitating flat sample surfaces and equipment capable of applying loads down to a few piconewtons for pharmaceutical ingredients [36].

Table 1. Summary of elastic moduli (E) of powders determined by compression methods.

Material	E [GPa]	Device, Methodology and Comments
Lactose	4.6	Device: Instrumented Frogerais OA machine; cylindrical die 1 cm2 × 1 cm. Methodology: maximal stress of 250 MPa; strain rate = 0.005 ${\mathrm{s}}^{-1}$; linear part of unloading curve [28].
Avicel PH 101	3.0–12.9	Device: Lloyd testing machine (JJ Lloyd T30 K). Methodology: compression of compacts; tablets with diameter from 5 to 13 mm; crosshead rate of 1 mm/min; Eo by Spriggs [25].
Starch 1500	1.5–4.1
Calcium phosphate	6.3–928
Avicel PH-101	0.25–25.0	Device: high-speed compaction simulator (Phoenix Calibration & Services Ltd.); 8 mm instrumented die. Methodology: maximal stress of 250 MPa; compaction speed of 0.1 mm/s; linear part of unloading curve [24].
Eudragit L100–55	1.6 ± 0.2	Device: diametrical compression of a single particle; probe with a diameter of 50 $\mathsf{\mu }$m. Methodology: maximal stress = load required to cause fracture; displacement speed of 2 $\mathsf{\mu }$m/ms; compression till nominal strain of $4.3\%$ or $6\%$ [19].
Eudragit L100	1.2 ± 0.5
Eudragit S100	0.6 ± 0.1
Advantose S100	0.9 ± 0.4
Calcium carbonate	0.3 ± 0.1
Starlac	1.8 ± 0.9
Granulated sugar	0.0013–0.12	Device: triaxial apparatus; Methodology: Young’s modulus in unloading cycles: combination of Poisson’s ratio, determined by triaxial test (pressures of 100, 200 and 300 kPa, compression rate of 0.51 mm/min), and oedometric modulus, determined by an oedometer (11 to 360 kPa); hygroscopic moisture between 0.1 and 14% [38].
Confectioners’ sugar	0.00053–0.23
Barley flour	0.00020–0.023
Maize flour	0.00032–0.094
Soja flour	0.00054–0.043
Wheat flour	0.00050–0.055
CRM 116 limestone	3.0–3.7	Device: uniaxial tester. Methodology: Low pressure (10–130 kPa); cyclic compression; ignores Poisson effect [29].
Silica	0.034–0.196
Alumina	5.8–5.2
MCC Vivapur 12	0.1–2.6	Device: Stylcam 200R (Medelpharm). Methodology: double compaction method; Precompression up to 90 % of main compression; main compression with elastic behavior [26].
Anhydrous calcium phosphate	0.9–6.5
Avicel PH200	0.5–2.2, $\left(E_0\right)$ 3.2	Device: Stylcam 200R (Medelpharm). Methodology: double compaction method; precompression up to 90% of main compression; Young’s modulus from main compression, with linear (elastic) behavior; apparent Young’s modulus from precompression as a function of porosity and applied pressure [27].
Anhydrous calcium phosphate	1.8–5.8, $\left(E_0\right)$ 11.1
Ibuprofen DTP	1.2–2.6, $\left(E_0\right)$ 4.3
Mannitol	1.6–5.0, $\left(E_0\right)$ 190
Lactose monohydrate	4.38–5.37	Device: compaction simulator Styl’One Evolution (Medelpharm); diameter of 11.28 mm. Methodology: fourfold compaction; precompression of 100 MPa or 200 MPa, two more compressions at pressure lower than the first one; main compression with displacement between 50 and 100 $\mathsf{\mu }$m; strain rates between 0.001 and 1 s−1 [22].
MCC Vivapur 200	1.43–2.52
Starch	0.63–1.11
Anhydrous calcium phosphate	7.34–10.00
MCC Vivapur 102	0.56–1.36	Device: parallelepiped die compaction (1.56 × 5.4 × 4.0 cm); Methodology: elastic behavior considered linear, isotropic and density-dependent; pressure from 5 to 35 MPa [21]
API A	1.8–2.7	Device: servo-hydraulic compaction simulator (HB100). Methodology: elastic properties were determined using a cyclic compression approach ranging from 50 MPa to 300 MPa at 50 MPa increments. The compression data was acquired from the reloading region of a reloaded compact [39].
API B	2.5–4.5
Mannitol	1.4–7.4
Avicel PH102	0.9–2.7

Table 1. Summary of elastic moduli (E) of powders determined by compression methods.

Material	E [GPa]	Device, Methodology and Comments
Lactose	4.6	Device: Instrumented Frogerais OA machine; cylindrical die 1 cm2 × 1 cm. Methodology: maximal stress of 250 MPa; strain rate = 0.005 ${\mathrm{s}}^{-1}$; linear part of unloading curve [28].
Avicel PH 101	3.0–12.9	Device: Lloyd testing machine (JJ Lloyd T30 K). Methodology: compression of compacts; tablets with diameter from 5 to 13 mm; crosshead rate of 1 mm/min; Eo by Spriggs [25].
Starch 1500	1.5–4.1
Calcium phosphate	6.3–928
Avicel PH-101	0.25–25.0	Device: high-speed compaction simulator (Phoenix Calibration & Services Ltd.); 8 mm instrumented die. Methodology: maximal stress of 250 MPa; compaction speed of 0.1 mm/s; linear part of unloading curve [24].
Eudragit L100–55	1.6 ± 0.2	Device: diametrical compression of a single particle; probe with a diameter of 50 $\mathsf{\mu }$m. Methodology: maximal stress = load required to cause fracture; displacement speed of 2 $\mathsf{\mu }$m/ms; compression till nominal strain of $4.3\%$ or $6\%$ [19].
Eudragit L100	1.2 ± 0.5
Eudragit S100	0.6 ± 0.1
Advantose S100	0.9 ± 0.4
Calcium carbonate	0.3 ± 0.1
Starlac	1.8 ± 0.9
Granulated sugar	0.0013–0.12	Device: triaxial apparatus; Methodology: Young’s modulus in unloading cycles: combination of Poisson’s ratio, determined by triaxial test (pressures of 100, 200 and 300 kPa, compression rate of 0.51 mm/min), and oedometric modulus, determined by an oedometer (11 to 360 kPa); hygroscopic moisture between 0.1 and 14% [38].
Confectioners’ sugar	0.00053–0.23
Barley flour	0.00020–0.023
Maize flour	0.00032–0.094
Soja flour	0.00054–0.043
Wheat flour	0.00050–0.055
CRM 116 limestone	3.0–3.7	Device: uniaxial tester. Methodology: Low pressure (10–130 kPa); cyclic compression; ignores Poisson effect [29].
Silica	0.034–0.196
Alumina	5.8–5.2
MCC Vivapur 12	0.1–2.6	Device: Stylcam 200R (Medelpharm). Methodology: double compaction method; Precompression up to 90 % of main compression; main compression with elastic behavior [26].
Anhydrous calcium phosphate	0.9–6.5
Avicel PH200	0.5–2.2, $\left(E_0\right)$ 3.2	Device: Stylcam 200R (Medelpharm). Methodology: double compaction method; precompression up to 90% of main compression; Young’s modulus from main compression, with linear (elastic) behavior; apparent Young’s modulus from precompression as a function of porosity and applied pressure [27].
Anhydrous calcium phosphate	1.8–5.8, $\left(E_0\right)$ 11.1
Ibuprofen DTP	1.2–2.6, $\left(E_0\right)$ 4.3
Mannitol	1.6–5.0, $\left(E_0\right)$ 190
Lactose monohydrate	4.38–5.37	Device: compaction simulator Styl’One Evolution (Medelpharm); diameter of 11.28 mm. Methodology: fourfold compaction; precompression of 100 MPa or 200 MPa, two more compressions at pressure lower than the first one; main compression with displacement between 50 and 100 $\mathsf{\mu }$m; strain rates between 0.001 and 1 s−1 [22].
MCC Vivapur 200	1.43–2.52
Starch	0.63–1.11
Anhydrous calcium phosphate	7.34–10.00
MCC Vivapur 102	0.56–1.36	Device: parallelepiped die compaction (1.56 × 5.4 × 4.0 cm); Methodology: elastic behavior considered linear, isotropic and density-dependent; pressure from 5 to 35 MPa [21]
API A	1.8–2.7	Device: servo-hydraulic compaction simulator (HB100). Methodology: elastic properties were determined using a cyclic compression approach ranging from 50 MPa to 300 MPa at 50 MPa increments. The compression data was acquired from the reloading region of a reloaded compact [39].
API B	2.5–4.5
Mannitol	1.4–7.4
Avicel PH102	0.9–2.7

Recent studies have emphasized the value of employing single-particle compression and nanoidentation methods to gain insights into powder compression processes. However, it remains clear that the elastic behavior of a single particle does not necessarily describe the compression behavior of the bulk powder [40]. The unknown intricate relationship between the microscale properties of the grains and the macroscale properties of bulk materials remains a significant challenge in understanding granular materials behavior.

Another set of methodologies involves the analysis of bulk elastic properties through dynamic measurement methods based on vibration [41,42,43,44,45,46,47]. In the earliest 90, Okudaira group [41] pioneered the investigation of the dynamic properties of loosely packed powder beds using the sound of white noise. These methods rely on assessing bed properties either through the dissipation of energy in powder beds or by evaluating sound transmission during vibration, both conducted at low acceleration levels. As illustrated in

Table 2. Summary of elastic moduli (E) of powders determined using vibration methods.

Material	d [$\mathsf{\mu }$m]	E [MPa]	Methodology and Comments
White carbon	83.2	0.15/0.16	Elastic coefficient through vibration Top Cap Method (TCM) (first row) and through sound absorption (second row); acrylic test cell (84 mm inner diameter, 20 mm height, bottom 20 mm thick); top cap: 76 mm diameter iron disc (71 g); pre-packing of powders by tapping and compacting under a 0.618 kg iron disc; acceleration from 0.01 to 3.16 m/s2; RH between 50 and 60% [41].
Nylon powder	85.5	0.80/0.99
Fused spherical silica	19.3	0.25/0.28
Lycopodium	32.3	0.61/0.74
Mica	37.5	0.55/0.68
Vermiculite	93.1	0.16/0.25
Glass beads	232	20–35	Elastic coefficient through vibration (TCM); Acrylic test cell (75 mm inner diameter, 50 mm height, bottom 10 mm thick); Top cap: 50 mm diameter iron disc (generating axial stresses from 150–1400 Pa); peak acceleration of 0.1 g [45].
Glass beads	463	13–26
Glass beads	1060	33–64
Polyethylene powder	395	0.8–1.4
Glass beads	4417	15–28	Elastic coefficient through vibration (TCM); acrylic test cell (78 mm inner diameter, 20 mm height, bottom 10 mm thick); top cap: 70 mm diameter disc (generating axial stresses from 150–1400 Pa); peak acceleration of 0.02 g [43].
Glass beads	1049	15–28
Glass beads	459	8–15
Polyethylene powder	462	0.5–1.0
Polyethylene powder	320	0.3–0.9
Black resin powder	219	0.5–1.0
Grey crumb powder	382	0.15–0.31
Sand	169	8–11
Polyethylene powder	459	0.15	Elastic coefficient through sound absorption; acrylic test cell (inner diameter from 34.5 to 149 mm, height from 10 to 80 mm); acceleration peak from 0.02 to 0.75 g [44].
Glass beads	462	5.76
Sand	169	3.66
Grey crumb powder	382	0.01
China clay-kaolin	10.5	0.25
Glass beads	335	0.12, TCM 0.21	Elastic coefficient through vibration (TCM as [41]) and through sound absorption as [43]; acrylic test cell (74 mm inner diameter, 38 mm height); top cap of 13.9 g; RH = 68 ± 5% [45].
Polyethylene powder	400	0.18, TCM 0.29
	125	2.2–37.8	Elastic coefficient through transmission of acoustic waves; freely poured samples; elastic modulus increases with the height of the powder bed (from 15 to 100 mm); frequency range 200–6400 Hz; RH = 40% [47].
Cocoa powder	238	1.1–12.8
	303	1.8–19.3

, typical E values reach a few MPa with no discernible relationship to particle size distribution. In contrast to compression methods, there is limited scientific literature available on vibration methods. Indeed, the acoustic response of granular materials has been extensively explored by mechanics groups to determine damping and energy dissipation, with major applications in aeronautics [48]. To our knowledge, powder technology groups have not extensively utilized this type of technology, with recent studies primarily focused on pharmaceutical applications, particularly tablets rather than the powder itself [46,49,50].

In essence, this study delves into the complex nature of powders, illuminating their multifaceted behaviors and the inherent challenges in quantifying their elastic properties. The objective of this investigation, through a combination of review and experimental efforts, is to ascertain the Young’s modulus of industrial powders using various methodologies and powder types. The primary focus lies on evaluating the consolidation state of the powder bed at which its elastic behavior is quantified. Vibration-based techniques enable the exploration of powders in loosely packed conditions, while compression methods elucidate the elastic behavior of the tablet or compact formed under consolidation. By employing these methodologies alongside thorough analysis, we aim to deepen the understanding of powders and provide valuable insights into their myriad applications.

2. Theoretical Background: Linear Elasticity

Many objects we handle daily are elastic bodies, for instance, mattresses, nylon, gum, rubber bands, and wool. When an elastic solid is subjected to stress, the sample will deform as a function of the stress/strain response of the material. For Hookean solids, the relationship between strain $\epsilon$ and stress $\sigma$ is linear. In an isotropic and homogeneous material, there are no internal directions defined and stresses are rather complex mathematical objects—like a second-order tensor—that do not arbitrarily depend on the unit normal of any particular plane. The most general linear tensor relation between stress and strain comes from the works of Cauchy 1822 and Lamé 1852:

$${\sigma }_{ij}=2\mu {\epsilon }_{ij}+\lambda {\delta }_{ij}\sum _k{\epsilon }_{kk}$$

(1)

where $\mu$ and $\lambda$ are material constants called Lamé coefficients measured in units of pressure. ${\delta }_{ij}$ is the Kronecker-delta multiplied by the trace ${\sum }_k{\epsilon }_{kk}$ the only scalar quantity that can be formed from a linear combination of strain tensor components. Explicitly, the diagonal elements of the stress tensor are

$${\sigma }_{xx}=(2\mu +\lambda ){\epsilon }_{xx}+\lambda ({\epsilon }_{yy}+{\epsilon }_{zz})$$

(2)

$${\sigma }_{yy}=(2\mu +\lambda ){\epsilon }_{yy}+\lambda ({\epsilon }_{xx}+{\epsilon }_{zz})$$

(3)

$${\sigma }_{zz}=(2\mu +\lambda ){\epsilon }_{zz}+\lambda ({\epsilon }_{xx}+{\epsilon }_{yy})$$

(4)

and the off-diagonal elements of the stress tensor

$${\sigma }_{xy}={\sigma }_{yx}=2\mu {\epsilon }_{xy}=G{\gamma }_{xy}$$

(5)

$${\sigma }_{yz}={\sigma }_{zy}=2\mu {\epsilon }_{yz}=G{\gamma }_{yz}$$

(6)

$${\sigma }_{zx}={\sigma }_{xz}=2\mu {\epsilon }_{zx}=G{\gamma }_{zx}$$

(7)

In engineering, $\mu$ is also known as the shear modulus or rigidity modulus G, as it controls the magnitude of shearing strain and is usually denoted as ${\gamma }_{ij}=2{\epsilon }_{ij}$.

The elastic properties of continuous materials emerge from the molecular forces that bind solids together. Describing the relationship between mechanical properties and molecular structure is a complex task. Fortunately, several materials can be characterized by constants determined through macroscopic experiments that relate stress and strain. In the case of isotropic, elastic materials, these constants include Young’s modulus and Poisson’s ratio.

Young’s modulus (E), named after Thomas Young (1807), measures stiffness or resistance to stretching. A higher Young’s modulus indicates increased difficulty in stretching or bending the material, with typical values on the order of 100 GPa. Poisson’s ratio ($\nu$), introduced by Siméon Poisson in 1829, quantifies the Poisson effect: when a stretching force is applied lengthwise to a solid, the solid stretches and reduces its cross-section. In isotropic materials, atomic bonds exist in all directions; when these bonds are stretched, a transverse tension is generated, necessitating a contraction of the material. Poisson’s ratio typically ranges between 0.2 and 0.4 for ceramics, while a widely adopted global value is 0.3 for pharmaceutical powders, metals, and cannot exceed 0.5 in isotropic materials [13].

The engineering constants $G,E$, and $\nu$ are determined from experiments. These coefficients are related to the Lamé constants by

$$\mu =G=\frac{E}{2(1+\nu )}$$

(8)

$$\lambda =\frac{\nu E}{(1+\nu )(1-2\nu )}$$

(9)

For an elastic solid, E is the slope of the stress–strain diagram in the linearly elastic region. If the material is subject to hydrostatic stress or pressure $p=1/3{\sigma }_{kk}$, the bulk modulus K establishes a relation between p and the volumetric strain $\theta ={ϵ}_{kk}$:

$$K=\frac{\theta }{p}=\frac{E}{3(1-2\nu )}$$

(10)

The bulk modulus (K) indicates the material’s incompressibility. The limit of incompressibility manifests as $K\to \infty$ when $\nu \to 0.5$, indicative of the volumetric strain diminishing under applied pressure.

Equations (8) and (10) let us to the well-known relation

$$E=2G(1+\nu )=3K(1-2\nu )$$

(11)

In the case of powders, a more complex situation arises. When an external stress is applied to the powder, it is transmitted through the mass of powder from particle to particle via contact points. Consequently, particles deform at these contact points under the transmitted force, and this deformation can follow Hooke’s law. However, as the contact area between particles increases with deformation and the local deformation varies over the contact area, Hooke’s law may no longer be applicable [29].

2.1. Elastic Modulus of Powders Determined under Compression Methodologies

If we examine grains as a continuum medium instead of focusing on forces between individual particles, the system can be described by forces or stresses acting on boundary areas of volume elements. Assuming that the strain is uniform in the packing and that every particle mean displaces in accordance with the applied strain of the assembly field, in the elastic range, the behavior of isotropic materials follows the incremental Hooke’s law and Equation (1) can be written as

$$d{\epsilon }_i^{el}=\frac{d{I}_1}{9K}{\delta }_{ij}+\frac{d({\sigma }_i-(I_1/3){\delta }_{ij})}{2G}$$

(12)

with $I_1={\sigma }_{kk}=3p$ the first stress invariant of Cauchy stress tensors. As the focus in this work is on the elastic component ${\epsilon }^{el}$, the elastic strain increment will be simply noted $d\epsilon$.

2.1.1. Uniaxial Compression

In a uniaxial strain test within a rigid cylindrical die, the limitations imposed by the wall can be expressed as:

$$d{\epsilon }_r=d{\epsilon }_{\theta }=0$$

(13)

Due to cylindrical symmetry, ${\sigma }_r={\sigma }_{\theta }$. The incremental Hooke’s law is then rewritten as follows:

$$d{\epsilon }_r=0=\frac{d{I}_1}{9K}+\frac{d({\sigma }_r-I_1/3)}{2G}$$

(14)

$$d{\epsilon }_{\theta }=0=\frac{d{I}_1}{9K}+\frac{d({\sigma }_{\theta }-I_1/3)}{2G}$$

(15)

$$d{\epsilon }_z=\frac{d{I}_1}{9K}+\frac{d({\sigma }_z-I_1/3)}{2G}$$

(16)

Rearranging the incremental radial strain equation (Equation (14)) leads to

$$\frac{d{I}_1}{9K}=\frac{d{I}_1}{6G}-\frac{d{\sigma }_r}{2G}$$

(17)

Injecting this into the equation for $d{\epsilon }_z$ results in

$$d{\epsilon }_z=\frac{d{\sigma }_z-d{\sigma }_r}{2G}$$

(18)

Rearranging the incremental radial strain equation once again yields

$$\frac{1}{2G}=\frac{d{I}_1}{3K(d{\sigma }_z-d{\sigma }_r)}$$

(19)

Injecting this expression into the equation for $d{\epsilon }_z$ gives

$$d{\epsilon }_z=\frac{d{\sigma }_z+2d{\sigma }_r}{3K}$$

(20)

Equating the two equations, (18) and (20), we can relate the increment in axial stress to the increment in radial stress:

$$d{\sigma }_z=\frac{3K+4G}{3K-2G}d{\sigma }_r$$

(21)

This equation explicitly shows that, for K and G constants, the relation between axial and radial stresses is linear. The relationship between the increment in axial stress and the increment in axial strain can be found by injecting this equation into Equation (18):

$$d{\sigma }_z=\left(K+\frac{4}{3}G\right)d{\epsilon }_z$$

(22)

Young’s modulus $\left(E\right)$ and Poisson’s ratio $\left(\nu \right)$ can be extracted from K (Equation (8)) and G (Equation (10)) and Poisson’s ratio can be defined by the equation

$$\nu =\frac{3K-2G}{2(G+3K)}$$

(23)

and, the Young’s modulus E can be described as

$$E=\frac{9GK}{3K+G}$$

(24)

The determination of Young’s modulus involves resolving a system of two equations Equations (21) and (22) with two variables (K and G). Experimentally, Poisson’s ratio is calculated as the negative ratio of lateral strain to axial strain, expressed as $\nu =-{\epsilon }_r/{\epsilon }_z$. These variables can be quantified by placing a radial strain gauge in the die [51]. True strain ($\epsilon$) is defined as the absolute value of the logarithm of the ratio between the final and initial length in either direction, $\epsilon =|ln(L/L_o\left)\right|$. Since the die wall pressure, radial stress, has not been measured in this study, the calculation of Young’s modulus (E) necessitates an estimation of Poisson’s ratio ($\nu$).

2.1.2. Cyclic Compression at Low Pressure (50 N)

This approach is based on a slightly modified version of Hooke’s law for ordinary solid bodies, where the deformation is squared. The elastic modulus reflects the total force $\left(F\right)$ transmitted through the powder sample, which arises from numerous contact forces $F_p$ ($F=n{F}_p$) with n representing the number of force-transmitting contact points between particles, and thus increases with increasing compaction and decreasing porosity [29].

In the experimental procedure, cyclic compression is applied at incrementally low pressures with relaxation states at maximum stresses. Assuming the conservation of the number of contact points during deformation, no breakage, and neglecting the Poisson effect, the Young’s modulus of each cycle can be determined through the relationship between axial stress and elastic deformation, expressed as

$$\sigma =E_p{\epsilon }^2$$

(25)

where $E_p$ represents the modulus of elasticity of the powder, reflecting a large number of contact forces ($E_p=n/m{F}_p$) with m the ratio between total cross sectional area of a loaded powder plug and the projected area of a force transmitting particle [29].

This methodology explores the unloading phase of compression at low pressures, seeking to unveil the elastic characteristics of powders subjected to slight deformations. This is particularly relevant for low-pressure operations such as gravity flow.

2.1.3. Cyclic Compression at High Pressure

A novel experimental approach was developed in this work that involves integrating two approaches: one consisting of short-duration cycles of increasing compression stress [26], and the other incorporating four cycles of low strain rates with decreasing compaction stress [22].

The fourfold compaction method [22] introduces two intermediary compression cycles to eliminate potential plastic or visco-plastic deformations. However, due to the decreasing pressure cycles, it provides the elastic properties of a tablet rather than the powder itself. On the other hand, the double compression method [26] features short compression cycles, leading to high strain rates and a non-linear evolution of axial stress with radial stress, conflicting with the work of [52].

Therefore, our hybrid method consists of employing fourfold compression cycles at low strain rates during crescent stress compaction cycles. Similar to the compression method in [39], only the first compaction cycle is treated as a pre-compaction cycle involving significant plastic deformation. Consequently, Young’s modulus ($E_{ch}$) is estimated for each of the three subsequent cycles using the linear part of loading in the main compression cycle. The elastic zone of each cycle is limited to the maximum pressure of its predecessor. The theoretical development and equations for the elastic modulus follow those of the single compaction at high pressure method, as the same hypotheses and conditions are applicable.

2.2. Elastic Modulus Determined under Vibration Methodologies

Methodologies for determining the elastic modulus of particles through vibration are primarily based on two principles: Transmissibility and Sound Absorption (Table 2). Transmissibility involves studying the acceleration wave through the powder bed, treating the powder bed as a damped harmonic oscillator with mechanical properties approximated to a linear spring and equivalent viscous damper. This system can be characterized by its dominant resonant frequency, $f_r$. The Top Cap Method (TCM), introduced by Okudaira [41] and further investigated by Yanagida [53], involves placing a top cap on the powder bed surface, to which an accelerometer is attached. The transfer function of acceleration from the damper at the die bottom to the powder bed top is determined experimentally using a Fast Fourier Transform (FFT) analyzer. By analyzing the peaks in the transfer function at resonant frequencies, the stiffness constant (k) can be calculated using the formula

$$k={\left(2\pi f_r\right)}^2{m}_e$$

(26)

where $m_e$ is the effective mass.

Notably, $m_e$ encompasses the mass of the top cap ($m_s$), the accelerometer mass ($m_a$), and the powder mass ($m_p$) as $m_e=m_s+m_a+(1/3)m_p$ following Rayleigh’s method. The factor 1/3 converts the distributed mass of the powder bed to the concentrated mass at the top cap. The Young’s modulus ($E_{TCM}$) is then estimated from the stiffness using the formula

$$E_{TCM}=k\frac{h_p}{A}$$

(27)

where $h_p$ is the bed height, and A is the cross-sectional area of the powder bed.

The TCM necessitates a preconsolidated powder bed to support the added mass of the top cap, leading to beds that exhibit relatively higher density compared to naturally packed beds. Conversely, the sound absorption method enables the measurement of the stiffness of loosely packed powder beds using a sweep vibration at low accelerations, quantified in terms of vibrating apparent mass.

The apparent mass is defined as the ratio between the cell bottom force and its acceleration [44], peaking when the frequency is resonant. Once the resonant frequency is identified, the longitudinal elastic modulus $E_a$ is directly calculated using the equation

$$E_a={\rho }_b{\left(4h_p{f}_{r,a}\right)}^2$$

(28)

where ${\rho }_b$ is the bulk density of the bed, and $f_{r,a}$ is the frequency from the apparent mass function peak.

It is worth noting that $E_a$, also known as a constrained modulus, is slightly greater than the actual elasticity of the materials due to the system’s limitation in allowing the lateral expansion of the samples, attributed to the presence of the vessel wall.

When comparing both vibration methods, there are no significant differences in the elastic modulus values, except for the increment caused by the load from the top cap’s weight (confer to Table 2). In the subsequent sections, only the “apparent mass” methodology will be discussed.

3. Materials and Methods

Industrial Powders

Eight powders widely used in various industrial sectors were selected according to their size distribution, flow behavior, and particle shape: microcrystalline cellulose (MCC), sericite (S), wheat flour (WF), joint filler (JF) and glass beads (GB). The two microcrystalline cellulose powders (MCC), Avicel® PH-102 and Avicel® PH-105 (FMC Biopolymer®), were chosen as key excipients in pharmaceutical products and cosmetics. This choice is attributed to their ability to compact, differing in size and, consequently, in cohesion. The wheat flour (WF) we used is an artisan-made T45 WF (Poinsignon flour mill, Fouligny, France). WF is commonly and widely used in the human diet and is thus an irreplaceable part of the food industry. WF is known as the nightmare of granular physics scientists due to its complexity. In order to gain a deeper understanding of its mechanical behavior, a focused investigation was conducted on its primary components, namely wheat starch and gluten.

The physical properties of the samples are presented in

Table 3. Physical properties of samples.

Powder	Particle Size Distribution [$\mu$m]					True Density *
	${\mathit{d}}_{\mathbf{10}}$	${\mathit{d}}_{\mathbf{50}}$	${\mathit{d}}_{\mathbf{90}}$	${\mathit{d}}_{\mathbf{43}}$	${\mathit{d}}_{\mathbf{32}}$	[g/mL]
Avicel® PH-102	35	111	225	122	69	1.55
Avicel® PH-105	7	21	45	24	13	1.58
Glass beads	4	24	60	29	8	2.66
Wheat flour	17	83	177	91	25	1.46
Joint filler	2	18	72	29	6	2.81
Sericite	3	9	27	17	6	2.88
Wheat starch	8	18	30	19	11	1.50
Gluten	18	74	124	74	13	1.30

* Envelope density of the particles (equal to the pycnometric density for non-porous particles).

and the SEM micrographs are displayed in the Appendix A. Samples were conditioned in a sealed desiccator under strict relative humidity (RH) conditions at 30% RH for a minimum of 3 days before the experiments. The conditioning time was validated by monitoring water activity measurements to ensure that equilibrium was achieved.

4. The Protocols

4.1. Single and Cyclic Compression at High Pressure, 250 MPa

The compression tests were conducted using an Instron 5569 electromechanical press, with the evolution of the powder bed height monitored by a video extensometer from National Instruments (Figure 2). A cylindrical steel die with a height of 16.2 mm and a diameter of 10 mm was employed for the press. The powder was freely poured into the die by gravity to set the initial powder bed height as the height of the die. The system underwent calibration at the beginning of each testing day. Calibration involved establishing the “zero force” in the upper punch and determining the “zero height” as the point at which the upper punch reached the base plate without the die.

In the single compression tests, the methodology involved a single compression cycle. The process included loading the powder to a maximum stress level of 250 MPa, holding a 30 s relaxation period, and then unloading. It is worth noting that the chosen high-stress level resulted in the formation of a compact or tablet for most powders. Previous works, such as those by Saheb et al. [52], indicate that the relationship between axial and radial stresses is linear at low speeds but becomes non-linear with increasing compaction speed. Similar observations were made by Imole et al. [54] when studying stress relaxation in cohesive powders such as cocoa and limestone. To adhere to this restriction, all experiments in this work, both loading and unloading, were conducted at low compression rates between 0.01 mm/s and 0.1 mm/s.

For the cyclic compression tests, four cycles of compression with increasing loads of 80, 100, 200, and 250 MPa were executed. Each cycle followed a similar procedure to the single compression, using the same strain rate of 0.01 mm/s and a 30 s relaxation time at the maximum stress. A relaxation time of 10 s was set between each cycle.

4.2. Porosity

The methodology involved a single load cycle with a strain rate of 0.1 mm/s for maximum loads of 80, 100, 200, and 250 MPa. The strain rate was set at 0.1 mm/s, which was faster than the test speed but remained within the limit of linearity between radial and axial stresses. Although the linearity limit was respected, it was not essential for measuring porosity, as the height of the tablet after ejection, and thus the final porosity, depended solely on the load, not the speed of the upper punch.

Tablet heights were measured through photos taken in front of a graduated ruler with a precision of 0.25 mm, and the image processing was performed using ImageJ®, https://imagej.net/ij/ (accessed on 1 March 2024). Each experiment was replicated three times, and the mass of the tablets was measured using a high-precision balance with a sensitivity of 0.0001 g.

The porosity, $\phi$, is calculated based on true density ${\rho }_p$, or envelope density of the powder, determined using a helium pycnometer (Table 3), and the density of the tablets $\rho$, obtained from height measurements, as follows:

$$\phi =\frac{V_{pores}}{V_{total}}=1-\frac{V_{solid}}{V_{total}}=1-\frac{\rho }{{\rho }_p}$$

(29)

Here, $V_{pores}$ represents the volume of the pores, $V_{solid}$ denotes the solid volume, and $V_{total}$ the total volume of the tablet.

It should be noted that, before conducting true density analysis, the samples were conditioned at low relative humidity (RH) conditions (about 10% RH) to prevent water desorption during measurements or swelling of the samples, particularly observed in the case of MCC powders.

Given that the elastic modulus reflects immediate elastic recovery, photos were taken less than three minutes after ejection to capture only the prompt relaxation of tablets. It is worth noting that, for some samples, volume expansion after compression can continue for hours or even days due to viscous phenomena [27].

4.3. Resonance Frequency of the Powder Bed

Figure 3 illustrates the experimental setup, comprising a powder cell, a shaker, and an electronic system for excitation and measurement. The excitation system consists of a homemade signal generator connected to a power amplifier (B&K, type 2728) and the shaker (B&K, type 4810). The cell, inspired by previous works [41,44], is a cylindrical acrylic with a 74 mm inner diameter, 17 mm height. The measurement system consists of a force transducer (B&K, type 8320) positioned between the cell and the shaker, and an accelerometer (DeltaTron B&K, type 4508B) attached to the side of the cell. The signals from the accelerometer and force transducer are transmitted to a homemade signal amplifier for analysis.

The Fast Fourier Transform (FFT) of the ratio between the force and acceleration signals is calculated simultaneously using MatLab® to determine the apparent mass function ($F/a=m_{am}$). Post-treatment data analysis requires the use of a max-peak-search program developed with MatLab^®, allowing the determination of resonance frequencies. A typical analysis is presented in Figure 4, where the lower peaks at high frequencies correspond to the powder cell (wall and base).

To ensure accuracy and repeatability, each experiment involved filling the powder cell completely. A sweep signal from 1 Hz to 1500 Hz, with a rate of 3 Hz/s (500 s duration) and a sampling frequency of 10 kHz, was employed to identify the resonant frequency of the powder. For consistency, sweeps were repeated 3 to 5 times for each material.

It is crucial to highlight that the experiments were performed with an acceleration of 10 m/s² peak-to-peak to mitigate noise. This acceleration is an order of magnitude higher than the values indicated in prior studies, typically described as 0.1 g. The rationale behind this heightened limit is to prevent particle movement. Considering that the majority of our samples exhibit cohesion and show no signs of displacement even at 1 g acceleration levels, it was anticipated that this adjustment would not negatively impact the results.

5. Results and Discussions

5.1. Porosity

Porosity values for the samples range from 0.15 to 0.46, depending on the specific sample and stress load, as presented in

Table 4. Porosity determined at the consolidation stress of 80, 100, 200, and 250 MPa.

Powder	Porosity, $\mathit{\phi }$
	80 MPa	100 MPa	200 MPa	250 MPa
Avicel® PH-102	0.33 ± 0.01	0.31 ± 0.01	0.22 ± 0.02	0.15 ± 0.02
Avicel® PH-102 (60% RH)	0.26 ± 0.02	0.20 ± 0.01	0.19 ± 0.01	0.18 ± 0.01
Avicel® PH-105	0.33 ± 0.01	0.29 ± 0.01	0.21 ± 0.01	0.19 ± 0.01
Wheat flour	0.25 ± 0.01	0.22 ± 0.01	0.18 ± 0.01	0.17 ± 0.00
Wheat flour (60% RH)	0.23 ± 0.01	0.22 ± 0.01	0.20 ± 0.01	0.21 ± 0.01
Sericite	0.46 ± 0.01	0.45 ± 0.01	0.40 ± 0.01	0.38 ± 0.02
Joint filler	0.37 ± 0.00	0.35 ± 0.01	0.32 ± 0.00	0.31 ± 0.00
Wheat starch	0.37 ± 0.02	0.34 ± 0.01	0.23 ± 0.01	0.18 ± 0.01
Gluten	0.26 ± 0.00	0.23 ± 0.00	0.17 ± 0.01	0.16 ± 0.00
Glass beads	0.35 ± 0.00	0.33 ± 0.00	0.30 ± 0.00	0.28 ± 0.00

. Notably, tablets formed with glass bead powders tend to undergo deagglomeration upon removal from the die. The porosity calculation for glass beads is determined based on the tablet height without ejection, specifically at the first point of no contact between the upper punch and the material. Additionally, the compression of wheat starch powders results in tablets with two distinct zones separated by a fracture, one of which exhibits a yellowish color. The thickness of the yellowish zone, attributed to plastic deformation, appears to depend on the maximum pressure applied, increasing with stress. Photos of all the tablets used for porosity quantification are provided in the Appendix B.

The expected trend was a reduction in porosity with an escalation in stress levels. Notably, a clear linear relationship between porosity and compression load is evident for sericite, wheat starch, and Avicel^® PH-102. In contrast, the remaining samples—wheat flour, gluten, Avicel^® PH-105, joint filler, and glass beads—exhibit an asymptotic behavior between stress load and porosity. While an asymptotic behavior could suggest a maximum compaction state with no further rearrangement or plasticity possible under compression, the linear relationship implies the opposite. The progressive decrease in porosity observed for sericite can be attributed to its lamellar structure. In contrast, Avicel^® PH-102, although exhibiting elastic behavior, does not reach maximum compaction within the evaluated range, as expected for an excipient intended for direct compression. The behavior of wheat starch is unique (Appendix B), and as previously discussed, the emergence of fractures complicates the characterization of its porosity using this methodology.

The Table 4 also illustrates the impact of water activity on the bulk porosity for Avicel^® PH-102 and wheat flour. While the influence of air humidity on powder behavior is well-established and powder-dependent, these results underscore the significance of air-humidity conditioning and help explain the low statistical error [55,56]. Furthermore, the porosity of the powder bed appears to be unaffected by particle size distribution, as no discernible trends or correlations were identified between porosity and either powder size distribution or powder true density (Table 3). Similar findings have been reported by others in previous studies [16].

5.2. Poisson’s Ratio

Poisson’s ratio cannot be directly determined with our experimental setup. For some of the samples, its value is selected based on the available literature. For instance, for wheat flour, $\nu$ is set at 0.20 [38], and for glass beads, it is set at 0.21 [57]. Recent studies have investigated Avicel PH102 using a cyclic compression method, uncovering a notable fluctuation in Poisson’s ratio with varying axial pressure: 0.12 at 80 MPa, 0.22 at 100 MPa, 0.32 at 200 MPa, and 0.35 at 250 MPa [39]. Given that Avicel PH105 is a size-reduced variant of PH102 and that size distribution does not seem to impact Young’s modulus [16], similar values of $\nu$ were attributed to this powder. Due to the lack of individual studies for the remaining powder samples, we conducted a sensitivity analysis between typical $\nu$ values between 0.20 and 0.40 and porosity. Mathematically, the Young’s modulus decreases with increasing Poisson’s ratio. In our cases, there is a higher variation for joint filler, with $E_{sh}=7.5$ GPa at 80 MPa and $\nu =0.2$, while at $\nu =0.4$, $E_{sh}$ is 3.9 GPa. Taking into account the consolidation state, and thus porosity, at 200 MPa and $\nu =0.4$, $E_{sh}$ is 7.1 GPa. In general, Young’s modulus calculated using $\nu =0.30$ is approximately 20% smaller than when using $\nu =0.20$ and about 40% larger compared to $\nu =0.40$. For the following sections, the Poisson’s ratio of joint filler, gluten, and sericite is set at 0.30, a value typically adopted for pharmaceutical powders.

5.3. Single Compression at High Pressure (250 MPa), $E_{sh}$

Four materials were investigated using this method: Avicel^® PH-102, Avicel^® PH-105, Wheat Flour, and Sericite. Generally, it is observed that the loading phase of the compression cycle is non-linear, indicating inelastic or plastic behavior in the powders due to grain rearrangement and/or plastic deformations. The unloading phase of the cycle consists of a linear zone at higher stress levels and a non-linear zone under 50 MPa as depicted in Figure 5). The Young’s modulus values obtained from unloading are presented in

Table 5. Young’s modulus ($E_{sh}$) determined during the unloading phase for single compression at 250 MPa.

Powder	$\mathit{\phi }$	$\mathit{\nu }$	${\mathit{E}}_{\mathit{sh}}$ [GPa]
Avicel® PH-105	0.19 ± 0.01	0.35	4.54 ± 0.14
Avicel® PH-102	0.15 ± 0.02	0.35 [39]	5.07 ± 0.14
Sericite	0.38 ± 0.02	0.30	5.53 ± 0.10
Wheat flour	0.17 ± 0.00	0.2 [38]	7.12 ± 0.57

, indicating that wheat flour is the stiffest powder. Additionally, no statistical difference was found between the “most elastic” samples, Avicel^® PH-105 and sericite.

It is worth noting that the only zone where unloading exhibits slopes inferior to or similar to the loading phase is below 30 MPa (Figure 5). Given the low stresses applied to the powder bed in this zone, non-linear behavior is anticipated, indicating that deformation is not elastic. Consequently, this zone is not utilized in the analyses.

Interestingly, comparable values can be identified in the literature, employing the 4-point beam bending technique for Avicel PH-102 (8.67 GPa) and PH-105 (9.43 GPa) [17]. This alignment can be attributed to the methodology focusing on the unloading phase of compression, where, at a pressure of 250 MPa, the powder bed has already formed a compact.

5.4. Cyclic Compression at High Pressure (250 MPa), $E_{ch}$

The Young’s modulus values obtained through the cyclic high-pressure methodology for all samples are presented in

Table 6. Variation of Young’s Modulus determined through the cyclic high-pressure method ($E_{ch}$) with respect to pre-consolidation stress levels: 80, 100, and 200 MPa. Analysis of the loading zone.

Powder	$\mathit{\nu }$	${\mathit{E}}_{\mathit{ch}}$ [GPa]
		80 MPa	100 MPa	200 MPa
Sericite	0.30	1.55 ± 0.02	2.03 ± 0.09	3.62 ± 0.33
Wheat flour	0.20 [38]	2.05 ± 0.05	2.90 ± 0.10	4.77 ± 0.17
Wheat flour (60%RH)	0.20 [38]	2.35 ± 0.05	3.45 ± 0.10	6.24 ± 0.17
Avicel® PH-105	0.22/0.32/0.35 *	2.44 ± 0.11	2.64 ± 0.42	4.25 ± 0.40
Avicel® PH-102	0.22/0.32/0.35 *	2.69 ± 0.01	2.52 ± 0.15	4.10 ± 0.25
Avicel® PH-102 (60%RH)	0.22/0.32/0.35 *	2.84 ± 0.26	2.90 ± 0.12	3.47± 0.26
Gluten	0.30	2.53 ± 0.37	3.28 ± 0.33	4.85 ± 0.11
Joint filler	0.30	6.19 ± 0.12	7.61 ± 0.63	11.23 ± 0.37
Glass beads	0.21	2.25 ± 0.09	3.12 ± 0.12	4.62 ± 0.37

* data from [39] taking into account he maximum stress level for each pre-consolidation stress.

. The table reveals that the joint filler stands out as notably stiffer, while, conversely, sericite exhibits the highest elasticity among the samples. The elastic behavior of the other powder samples is practically identical within the statistical error range. Surprisingly, glass beads exhibit similar behavior, contrary to expectations. Glass is typically considered a perfectly elastic material (70 GPa) that undergoes no permanent deformation until breakage. The unexpected elasticity of glass beads may be attributed to their smaller particle size and the high polydispersity of the powder (Table 3 and Appendix A). Evidence of particle reorganization is observed through changes in porosity, decreasing from 0.36 to 0.27 with pressure, determined without ejection—impossible otherwise, as no compact is formed. While the hypothesis of sudden breakage during the test is not considered, given their resistance to breakage under higher consolidations based on our group’s experience. This observation not only adds an interesting dimension to our understanding of their behavior but also provides insights into their industrial applications as road markers, shedding light on their resistance to breakage under higher consolidations.

From Table 6, it is observed that higher bed porosity is associated with smaller $E_{ch}$ values. However, the evolution of $E_{ch}$ with porosity is highly dependent on the nature of the powder and interestingly does not appear to be influenced by grain size. For instance, when comparing MCC powders, such as Avicel PH-105 ($d_{32}=13$ $\mathsf{\mu }$m) and PH102 ($d_{32}=69$ $\mathsf{\mu }$m), no statistical differences are found despite their differences in flowability and cohesiveness, with Avicel PH-102 being more flowable and less cohesive [12]. It is worth noting that despite using the $\nu$ values obtained in [39] for Avicel PH-102 and employing similar consolidation cycles, our values are approximately twice as large. $E_{ch}$ is 4.10 GPa at 200 MPa, compared to about 2.6 GPa in [39].

It is also observed from Table 6 that the conditioning of samples can affect their elastic behavior. At 60% relative humidity, wheat flour consistently exhibits a higher Young’s modulus across all experimental conditions compared to its performance at lower humidity levels. In contrast, Avicel^® PH-102 shows statistically similar elastic moduli regardless of the stress levels.

Noteworthy is the relationship observed among wheat flour, wheat starch, and gluten. Flour comprises two distinct structures—gluten, a larger shattered matrix, and starch, smaller globule-like particles (Appendix A). Our results indicate that under the specified conditions, the Young’s modulus of gluten is statistically equivalent to that of flour. Conversely, the findings related to starch are unexpected, as there was no linear zone to exploit during the compression cycles (Figure 6), most likely due to the tablets exhibiting lamination during compression.

Generally, in both single and cyclic compression methods, the slope of the unloading phase is visibly higher than that of the loading phase throughout the compression cycle. Mathematically, this results in E values obtained from the unloading phase being up to twice as high as those obtained during the loading phase of compression (not presented herein). When comparing both compression methods (Table 5 and Table 6), it is noted that single compression methods give rise to significantly higher E values than those obtained using the loading part of cycles with pre-compaction stages. Other than the differences in the slope of the unloading and loading phases, a clear difficulty of the single compression method is finding the linear zone of these curves. The phenomena occurring during the unloading phase appear to generate non-linearity, and the elastic modulus calculated from it could be distorted by the presence of plastic deformations. Additionally, it could be expected that attractive forces between particles may have different effects during compression and expansion phases.

5.5. Vibration, $E_a$

The measurements obtained through the apparent mass method are presented in

Table 7. The Young’s modulus of powders determined using the apparent mass vibration methodology is denoted as $E_a$.

Powder	$\mathit{\phi }$	${\mathit{f}}_{\mathit{r},\mathit{a}}$ [Hz]	${\mathit{m}}_{\mathit{a}}$ [g]	${\mathit{E}}_{\mathit{a}}$ [MPa]
Avicel® PH-102	0.79	587	24.4	0.53
Joint filler	0.71	378	59.9	0.54
Wheat flour	0.67	581	35.5	0.76
Glass beads	0.50	792	96.7	3.84
Sericite	0.88	318	25.7	0.16
Avicel® PH-105	0.81	396	22.5	0.22
Gluten	0.62	500	35.8	0.57
Wheat starch	0.67	410	36.4	0.39

. The porosity values reflect the loose packing state of the powder bed, with no evolution observed during analysis. From the table, it is evident that glass beads exhibit the highest stiffness, whereas sericite powder displays the highest elasticity. With the exception of wheat flour, and potentially gluten by extension, the elastic characteristics of the powders align with our expectations. Furthermore, the methodology facilitated the study of the elastic behavior of wheat starch and enabled differentiation between MCC powders and between wheat flour and gluten. Notably, Avicel^® PH-105 (0.22 MPa) exhibited greater elasticity than Avicel^® PH-10 (0.53 MPa), both with similar bed porosity, illustrating the effect of size reduction on elasticity for powders of similar nature.

When comparing the elastic modulus ($E_a$) of glass beads to those reported in the literature (Table 2), no clear trends can be depicted, highlighting the significance of the method itself, including factors such as the powder container and excitation energy, both relating to energy dissipation. A more appropriate approach would be to characterize the system based on the bed porosity. Indeed, the void ratio is an essential parameter in the transmission of mechanical waves.

The validation of $E_a$ values relies on their applicability. When comparing the $E_a$ values with powder flowability using the Hausner ratio (HR) [55,56], a clear pattern emerges: powders with higher E values, indicating lower elasticity, demonstrate better flowability following a power-law trend (Figure 7). Interestingly, the HR describes wheat flour as a fair flow powder while Avicel Ph-102 as an acceptable flow powder, which is contrary to naked-eye observations from simply pouring the powders (observable in Appendix A). Indeed, wheat flour and Avicel^® Ph-105 are cohesive powders, while Avicel^® Ph-102 flows freely. Elasticity might clarify the discrepancies in HR analysis, elucidating why Avicel^® PH-102 appears less flowable than wheat flour, because of its higher elasticity and greater densification tendency. Indeed, HR describes flowability through densification studies under vibration at free surface conditions and was originally validated for studying metallic powders, which are typically stiffer and less influenced by their mechanical properties during densification. Furthermore, no discernible correlation between $E_a$ and true density values is apparent, challenging the assumption that “heavier particles, driven by gravity, flow more easily”. This straightforward analysis underscores the sensitivity of flow to the stiffness and inelasticity of the powder.

While it might not be immediately evident to assume a connection between densification, cohesion, and elasticity, there is a connection through the powder contact network. Recent theoretical approaches highlight the influence of particle elastic properties on flow dynamics. For instance, studies utilizing the discrete element method (DEM) to describe the rheological behavior of cohesive grains flowing down an inclined plane [58], as well as in fluidized beds [59,60], have shed light on this relationship. The theoretical description of contact mechanics between non-spherical particles is complex owing to the heterogeneous shapes of grains and the intricacies of the contact region, and remains one of the bottlenecks in granular flow understanding.

Indeed, another notable advantage of this type of measurement lies in its capacity to provide valuable insights into a multitude of mechanical parameters. For example, the shape of resonant peaks offers crucial information about the viscoelastic nature of the sample, while the phase difference between displacement and force signals provides insight into energy dissipation within the granular media [48]. Particularly noteworthy is the observation that viscoelastic powders, such as gluten, exhibit significantly broader peaks in the spectrum during our experiments. Although our current understanding remains largely qualitative, as evidenced by previous studies on tablets [46], it suggests considerable potential for future advancements and deeper exploration of viscoelastic properties.

From an experimental standpoint, the setup should involve a consolidated recipient to avoid multiple resonances, as the use of low-dissipativity materials is highly recommended. It should be noted that both frequency sweep methods, such as the one used in this work, and impulse excitation methods enable the study of the frequency response of the system.

5.6. Exploring the Complexity: Consistency and Challenges in Young’s Modulus Determination for Powders

When a solid material undergoes fragmentation into numerous individual particles, the elastic modulus of the bulk diminishes in correlation with its consolidation state. This phenomenon appears to be effectively described through the bulk porosity, as depicted in Figure 8. From the Figure, it is evident that the evolution of E with porosity is powder-dependent, with sericite displaying the highest elasticity–porosity relationship.

Depending on the porosity of the powder bed, the Young’s modulus can vary from a few MPa, as observed in the vibration method, to several GPa, as seen in the cyclic compression method, representing a thousandfold increase. This variation follows an exponential trend described by $E=A{e}^{-b/\phi }$, reminiscent of the initial empirical descriptions originally proposed in the 1950s for porous brittle solids, where A represents the elastic modulus of a nonporous specimen and b stands as an empirical constant [61,62]. However, it should be noted that such an assertion is not validated by our data. For instance, A-values for glass beads are about five orders of magnitude larger than the Young’s modulus of soda–lime glass (72 GPa).

While numerous empirical or semi-empirical relationships between the Young’s modulus of porous solids, tablets or compacts and porosity have been [16,39,63,64], to the best of our knowledge, such an assertion has not been extended to powders from loose packing conditions to tablet formation. Therefore, the $E=A{e}^{-b/\phi }$ empirical relationship from Figure 8 requires further experimental work to fully explore and exploit its potential. Both approaches, $E_{ch}$ and $E_a$, not only offer complementary information but also open new avenues for exploration and understanding. It should be noted that the single compression at high pressure method does not follow the observed trend due to differences from the cyclic compression method, as explained earlier.

To encompass a broad spectrum of consolidation states, diverse methodologies, and interdisciplinary approaches are indispensable. Uniaxial compression methodologies have been extensively investigated (Table 1), particularly by research groups focused on the pharmaceutical industry, given that tablets are the most common dosage form. From a mechanical standpoint, high consolidation states encounter limitations in determining elastic properties when applied to powders that are unable to form a compact/tablet without fracturing, as observed in the case of wheat starch. However, such limitations could also be encountered using low consolidation techniques to determine elastic properties [21]. While vibration methods, inspired by mechanical and acoustic engineers, have been explored for determining elastic properties (Table 2), their application in this context remains relatively understudied. More recent studies have applied vibration methods to study the elastic and viscoelastic behavior of tablets [46,49,50]. To the authors, the development of low-consolidation methodologies appears highly relevant, considering many situations where powders are in loose packing conditions, such as during transport, storage, and feeding. This is where the vibration methodology, through resonance peaks analysis, proves to be an interesting choice for in-depth exploration. Additionally, it offers new insights into Discrete Element Modeling of dense flows, where single-particle elastic properties and tablet elasticity may not properly reflect the elastic behavior of the bulk (work in progress).

Consequently, the consistency of Young’s modulus appears closely linked to the porosity of the powder bed. Indeed, indicative of its consolidation state, the porosity seems to function as an average parameter at the mesoscale level of the grains, reflecting the combined influence of multiple factors, including particle elasticity, particle size distribution, density, morphology, moisture content, and interparticle interactions as well as geometrical factors like confinement and particle distribution, thereby describing the contact network itself. Therefore, the use of diverse methods is not only relevant but also imperative for achieving a thorough comprehension of powder elastic properties. The call for further research underscores the significance of establishing consistent methodologies capable of accurately describing powder elastic properties in less-compacted states.

It is important to emphasize that methods allowing for the determination of the elastic modulus of powders within medium bulk porosity levels (0.30–0.55), such as the low-pressure cyclic compression methodology, could enhance our understanding and validate previous statements. Unfortunately, technical constraints prevented the authors from pursuing the low-pressure cyclic compression methodology in this study. Specifically, the FT4 powder rheometer, a well-established instrument in powder technology, could have facilitated such experiments. Nonetheless, despite its high force sensitivity, the FT4 imposes a force plateau at 5 kPa, as illustrated in Figure 9, irrespective of the user-defined force protocol. This limitation impacts the resulting curves during both loading and unloading phases, making it challenging to establish a clear relationship between the applied stress and its corresponding strain.

6. Conclusions

This work sheds light on the complexities of determining the Young’s modulus of powders (YMP), which, unlike solid materials, does not emerge as an intrinsic property of powders. Guided by the insights gained from the review, and to provide meaningful insights, three techniques were used: single compression at high pressure, cyclic compression at high pressure, and vibration, to estimate the YMP of eight powder samples as varied as wheat flour, wheat starch, gluten, sericite, Avicel PH102, Avicel PH105, joint filler, and glass beads. This diverse selection aimed to capture a broad spectrum for analysis. Our qualitative analysis revealed varying elastic behaviors, underscoring the need for a nuanced understanding of powder mechanics. Notably, sericite consistently emerged as the most elastic powder across methodologies, while microcrystalline cellulose exhibited method-dependent variations. The investigation exposed qualitative relationships among the samples, indicating differing values and classifications across different methods. Glass beads and wheat flour exhibited significant fluctuations in elastic behavior. In contrast, wheat flour’s elastic modulus displayed unclear variations across different methodologies, posing challenges in definitively positioning it in comparison to other powders, especially considering both vibration and compression methods. Organic samples such as gluten presented difficulties due to mechanical properties that do not allow YMP quantification under compression methods.

The synthesis of the existing literature with our experimental findings has unveiled the intricate dependencies of YMP on various factors. One crucial insight gleaned from our experimental investigations is the importance of correlating Young’s modulus with the porosity of the bed, regardless of the methodology employed. In uniaxial compression methods, variations in results are significantly influenced by both Poisson’s ratio and the porosity of the bed. The sensitivity to porosity and stretchability varies depending on the sample. For instance, among our samples, the joint filler exhibited the most substantial variation in E values with $\nu$, while sericite and glass beads were comparatively less affected. To ensure the validity of the values attributed to the elastic modulus, it is imperative to measure the die-wall signal to determine Poisson’s ratio and to present Young’s modulus measurements as a function of the bed porosity. Additionally, and irrespective of the method used to determine the elastic properties, it is crucial to condition the powder under controlled environmental conditions, particularly with regards to air humidity but also temperature, depending on the sensitivity of powder properties to these parameters. Furthermore, given that porosity values depend on the accuracy of the true density measurements, it is highly recommended to condition the samples at low relative humidity conditions (10% RH) before measuring true density values. This precaution helps prevent water release during measurement or the analysis of swelled particles.

The exploration of vibrational methods has proven insightful, particularly in studying the YMP under loose packing conditions, providing valuable insights into flowability. Looking ahead, the extrapolation of the method to determine viscoelasticity, damping, and energy dissipation, other important mechanical properties of granular materials, along with the suggestion to explore cyclic compression at low pressures, opens new avenues for further research, offering potential applications in understanding phenomena such as gravity flow.

In summary, our work contributes to both the review of existing knowledge and the expansion of insights through experimental investigations. The challenges and nuances identified pave the way for future research directions in understanding the elastic properties of powders.