A Water Absorption Model Based on Self- Consistent Theory for Solid Buoyancy Materials
Sarunas Paulikas* and Arunas Statkus
Department of Computer Science and Communications Technologies, Vilnius Gediminas Technical University, Lithuania
Submission: October 09, 2020;Published: October 22, 2020
*Corresponding author:Sarunas Paulikas, Department of Computer Science and Communications Technologies, Vilnius Gediminas Technical University, Naugarduko 41, Vilnius, Lithuania
How to cite this article:Sarunas P, Arunas S. Improving of TCP Performance of Embedded Network Devices. Eng Technol Open Acc. 2020; 3(4): 555620 10.19080/ETOAJ.2020.03.555620
Abstract
The water absorption of solid buoyancy material seriously endangers the safety and usability of deep manned/unmanned submersibles. However, the theoretical models which can accurately predict the compressive strength and water absorption of solid buoyancy materials have not been found yet. This seriously hinders the safe operation of deep manned/unmanned submersibles. Therefore, a comprehensive experimental and numerical study is carried out on the solid buoyancy materials used in full ocean depth submersibles. In this paper, a water absorption model according to self-consistent theory for solid buoyancy materials is proposed. Then the water absorption model is verified by comparing with the experimental data. The model shows that the water absorption behavior can be divided into two stages. In the first stage, the water absorption is directly proportional to the volume fraction of fractured hollow glass beads, while in the second stage, the relationship becomes exponential due to the fragmentation propagation of hollow glass beads.
Keywords:Deep sea submersible; Buoyancy material; Compressive strength; Water absorption; Crack propagation
Introduction
Solid buoyancy material is one of the key technologies in submersible development [1-3]. Its importance in deep sea exploration has been widely recognized [4-7]. In the operation of the full ocean depth manned/unmanned submersibles, the issue how to check the status of the buoyancy material becomes one of the main obstacles because that buoyancy materials fully satisfied the classification society rule requirement of 1.5 safety factor do not exist[1]. There are two hazards which come from the water absorption of solid buoyancy material to the safety and reliability of deep manned/unmanned submersibles [8]. The first is reduction of compressive strength of solid buoyancy materials. The water absorption causes the hydrolysis and swelling of epoxy resin, which reduces the bonding strength of hollow glass beads and epoxy resin [9-12]. Then the compressive strength of solid buoyancy material is reduced, which endangers the safety of deep manned/unmanned submersibles. The second is that it reduces the ability of solid buoyancy materials providing buoyancy compensation. Water absorption increases the density of solid buoyancy materials [8], which reduces the abilities of deep manned/unmanned submersibles to float and dive. Therefore, studying the water absorption behavior of solid buoyancy materials is very important in submersible design and operation.
Solid buoyancy materials are usually used in deep sea environment [13,14]. The hollow glass beads fracture under water pressure. At this moment, the solid buoyancy material can be regarded as a porous medium after the hollow glass beads fracturing. It means that the water absorption behavior of solid buoyancy materials is mainly caused by the fractured hollow glass beads. However, previous researchers mainly focused on how the volu Figure me fraction [15-17], compressive strength [18], ratio of wall thickness to radius [19-21] and density [22] of hollow glass beads influence the failure behavior of solid buoyancy materials. The compressive strength model for solid buoyancy materials has not been found. Liang has done some researches on the fracture behavior of hollow glass beads [23-26]. However, these hollow glass beads are not used in solid buoyancy materials. Only two papers focusing on the water absorption of solid buoyancy materials have been found [8,27]. These two papers mainly investigated the influencing factors of water absorption behavior. The effect of fractured hollow glass bead on the water absorption behavior of solid buoyancy materials has not been recognized. The theoretical model which can accurately predict the water absorption has not been found yet. The difficulties in proposing a model come from the randomness of particle size and spatial distributions and the complexities of crack propagation in multiphase and multi-scale.
In this paper, based on the self-consistent theory, a water absorption model for the solid buoyancy materials is proposed. Then the water absorption model is verified by comparing with the experimental data. The theoretical model shows that the water absorption behavior of solid buoyancy materials can be divided into two stages. In the first stage, the water absorption is directly proportional to the volume fraction of fractured hollow glass beads, while in the second stage, the relationship becomes exponential due to the fragmentation propagation of hollow glass beads.
Theoretical Derivation
Basic assumptions
In order to build the mathematical model, the following six assumptions for epoxy resin, glass beads and the buoyancy failure mode are made:
a) The hollow glass beads and the epoxy resin are isotropic and homogeneous.
b) The main failure mode of full ocean depth solid buoyancy materials is the fragmentation of hollow glass beads. The water absorption behavior of solid buoyancy materials is mainly caused by the fractured hollow glass beads.
c) The particle sizes of hollow glass beads follow a normal distribution.
d) The density of all hollow glass beads of identical types is a constant that is independent of particle size.
e) The failure of hollow glass beads under uniform pressure is similar to a mode-II failure.
f) It is difficult to characterize the shapes and sizes of the initial defects in hollow glass beads. Therefore, it is assumed that there is an average value to represent the effect of initial defects on the compressive strength of hollow glass beads.
The compressive strength distribution model
In this section, a unit cell is assumed to be wrapped in a medium large enough in deep sea environment. If the medium is isotropic and homogeneous and its properties are the same as solid buoyancy material. Then the stress at anywhere in the medium is equal to the water pressure it bears. This means that the pressure on the surface of concentric spherical shell is also equal to the water pressure. At this time, the stresses in the unit cell wrapped by the medium is similar to that in the unit cell of solid buoyancy material.
According to the elastic theory, the stresses and strains at the interface between hollow glass bead and epoxy resin should follow the stress balance and displacement coordination conditions. Therefore, there are:
Where, σrg and σrm are the radial stresses in hollow glass bead and epoxy resin, respectively. εTg and εTm are the tangential strains of hollow glass bead and epoxy resin at their interface, respectively.
In the unit cell, the internal pressure of the hollow glass bead is 0. According to the elastic theory, the radial and tangential normal stresses in spherical shell under a uniform load is as follows:
Where, σTg is the tangential stress in the hollow glass bead. σri is the radial stress at the interface. rg is the inner radius of the hollow glass bead. ri is the radius of interface. r is a parameter in Eq. (2) ranges from rg to ri . Unlike the hollow glass bead, the thin shell made of epoxy resin bears internal and external pressures. Therefore, the radial and tangential normal stresses distribute as follows:
Where,σTm is the tangential stress in the epoxy resin. rm is the outer radius of the epoxy resin. r is a parameter in Eq. (3) ranges from ri to rm . p is the uniform pressure on the surface of unit cell. Substituting r = ri into Eq. (2) and Eq. (3) yields:
Eq. (4) shows that the stress balance condition at the interface is automatically satisfied. Aiming to simplify the calculation process, several parameters are defined:
Where, is the hollowness of the glass bead. While is the volume fraction of hollow glass bead in the unit cell. Substituting i r = r and Eq. (5) into Eq. (2) and Eq. (3) yields the tangential normal stresses of hollow glass bead and epoxy resin:
Both hollow glass bead and epoxy resin in the unit cell are isotropic materials. Therefore, the tangential normal strains of hollow glass bead and epoxy resin at their interface are:
Where, Eg and Em are the elastic modulus of hollow glass bead and epoxy resin, respectively. μg and μm are the Poisson’s ratios of hollow glass bead and epoxy resin, respectively. According to Eq. (1), the tangential normal strains of hollow glass bead and epoxy resin are equal at the interface. Substituting Eq. (6) and Eq. (7) into Eq. (1) yields the radial normal stress at the interface:
Substituting Eq. (8) into Eq. (2) yields the radial and tangential normal stresses in the hollow glass bead. Now, we can obtain the radial distribution of the maximum shear stress in the hollow glass bead:
Obviously, when r = a , the maximum shear stress is the maximum. It means that the crack source will occur on the inner surface of hollow glass beads in the unit cell. Substituting Eq. (5) and Eq. (8) into Eq. (9) yields the fracture strength of the unit cell:
Where, th τ is the shear strength of the unit cell. Griffith proposed a theoretical expression of the crystal fracture strength for crystal materials:
where E is the elastic modulus of the material, b is the equilibrium lattice constant, and γ is the surface energy. Eq. (11) shows that the fracture strength of crystal materials is proportional to the square root of surface energy, while the surface energy increases with the specific surface area [28-29]. The force, displacement and modulus in Eq. (11) are generalized quantities. Therefore, the fracture strength of solid materials under shear stress is assumed to be proportional to the square root of surface energy:
Hollow glass beads are spherical shell structures, but compared to the parameters of hollow glass beads, it is more reasonable to regard hollow glass beads as solid spheres when calculating the specific surface area. Therefore, the specific surface areas are:
According to Eq. (12) and Eq. (13), the fracture strength under shear stress is inversely proportional to the root of the radius. We define k as a proportional constant, and we obtain:
Substituting Eq. (14) into Eq. (10) and transposing it yields the failure pressure of the hollow glass bead:
Then Eq. (15) can be simplified as:
Eq. (17) expresses the failure pressure of the unit cell. The compressive strength probability density function and distribution function of the buoyancy material are:
Where, th p is the uniform pressure when the unit cell cracks. μ and σ are the average and standard deviation of the outer radius for the hollow glass bread.
The water absorption model
According to Eq. (17), the radius range of fractured hollow glass beads under a certain water pressure can be known. Then, according to the radius, the volume fraction of the fractured hollow glass beads can be calculated. Based on the assumption about the relationship between the water absorption of solid buoyancy material and the volume fraction of fractured hollow glass beads, the water absorption caused by fractured hollow glass beads can be obtained. According to the relationship between the radius and volume of the spherical shell, there is:
Substituting Eq. (19) into the probability integral formula of normal distribution yields:The probability distribution function and probability density function of hollow glass bead volume is shown in Eq. (20). It should be noted that F(Vth) is the probability of the remaining hollow glass beads. Therefore, the probability of fractured hollow glass beads is 1 − F( Vth).The volume fraction of the fractured hollow glass beads is:
After the fragmentations, only the hollow part of glass beads can absorb water. The volume fraction of the hollow part is:
If the hollow part of the fractured glass beads can be completely filled with water, then the water absorption caused by fractured hollow glass beads is:
Where, ρs and ρw are the densities of solid buoyancy material and water, respectively. The hollowness h V is usually an unknown parameter. Substituting into Eq. (23) yields:
It is assumed that the water absorption behavior of solid buoyancy materials can be divided into two stages. In the first stage, the water pressure is still low. The crack does not propagate after the hollow glass beads fracturing. On the other hand, due to the short time of testing time, the hollow part of the fractured glass beads cannot be completely filled with water. Therefore, it is assumed that the water absorption is directly proportional to the volume fraction of fractured hollow glass beads. In the second stage, the fragmentations of hollow glass beads propagate under a certain water pressure. Then the water absorption is exponentially related to the volume fraction of the fractured hollow glass beads. Based on these assumptions, the water absorption model for the solid buoyancy material is:
Where, k ' is a proportional constant. It represents the ratio of water absorption to the volume fraction of the hollow part of the fractured glass beads. p0 is a certain water pressure which is the boundary between the first and the second stages.
Model Verification
Experimental objects and results
In this section, some water absorption experiments have been done to verify the water absorption model for the solid buoyancy material. The test object is HZ-42 solid buoyancy material produced by “ESS” company (Engineered Syntactic Systems). For this solid buoyancy material, the volume fraction of epoxy resin is 0.3. The density is 0.67±0.3g/cm3. There are 3 kinds of samples which numbered E1, E2 and E3, respectively. The quantities of E1, E2 and E3 are 4, 3 and 3, respectively. The size of E1 is 610mm×305mm×100mm. The sizes of E2 and E3 are 305mm×305mm×100mm. Six groups of water pressure experiments have been done. The test object, water pressure and holding time in every experiment are given in Table 1. The processes of water pressure experiment are as follows:
a) Weigh the masses of solid buoyancy material samples by electronic balance (in gram). Then measure the volumes of the solid buoyancy material samples by drainage method.
b) Place the solid buoyancy material samples in a 180MPa pressure chamber after counterweighing. Then check the relevant equipment and prepare for the pressure test.
c) The pressure is uniformly pressurized to the test water pressure at a rate of 2-3MPa/min. Then after holding for two hours, the water pressure is released to 0MPa at a rate of 2-3MPa/ min.
d) After completion of test, take out the buoyancy samples as soon as possible. Investigate the surface. Then dry the sample surfaces with absorbent paper. Weigh the masses of samples again. The whole process should be completed within 15 minutes.
It is shown that small cracks appear on the solid buoyancy material surface under 160MPa pressure. While obvious cracks appear on the surface of samples under 165MPa pressure. The water absorptions tested in these experiments are shown in Table 2. It shows that the water absorptions remain at a very low level before 145MPa. While they increased rapidly after 160MPa.
Verification of theoretical models
In this section, the water absorption model is verified by comparing with the experimental results. The material properties of hollow glass beads and epoxy resin have been tested. The results show that the properties of hollow glass beads are similar to that of im16K. The material parameters of im16K and epoxy resin are shown in Table 3. The size distributions, densities, compressive strengths and other parameters of im16K in Table 4 were provided by a representative of 3M. The hollow glass beads with radius of [2~14μm] are selected as the research objects. According to the calculation, the proportion of hollow glass beads in this range is P{2 ≤ r ≤14} = 99.988%.
According to the uniform pressure under which 90% of im16k remain intact (see Table 4), by looking up the cumulative probability function table of standard normal distribution, we yields Then, Substituting the value of k and other material parameters shown in Table 3 into Eq. (16) yields
The relationship between the compressive strength of unit cell and the radius of hollow glass bead is shown in Figure 1. It shows that the compressive strengths of the unit cells decrease with the radii of hollow glass bead increasing.
t3 t4f
The radius range of the fractured hollow glass beads under a certain water pressure can be obtained according to Figure 2. Then volume fractions of the fractured hollow glass beads can be calculated. Taking 115.0MPa as an example, the radius of the fractured hollow glass beads is 10.00μm. The corresponding volume is 4188.79μm3. The radii and volumes under other pressures are shown in Table 5.
The volume probability density distribution and cumulative probability distribution of hollow glass beads are shown in Figure 3. It shows that the probability density is asymmetrically distributed. Therefore, it does not follow the normal distribution. The volume fraction of fractured hollow glass beads can be obtained according to the cumulative probability distribution.
The water absorption caused by fractured hollow glass beads can be obtained referring to the calculation process from Eq. (21) to Eq. (24). The probability and the volume fraction of the fractured hollow glass beads, the volume fraction of the hollow part, water absorption caused by the fractured glass beads fragmentation are given in Table 6. Compared with the experimental data in Table 2, the unknown parameters in Eq. (25) can be obtained and listed in Eq. (26).
Substituting Eq. (26) into Eq. (25) yields the water absorption model of solid buoyancy materials. Comparison between theoretical values and experimental results is shown in Figure 4. It can be seen that the theoretical values agree with the experimental data very well. This verified the theoretical model. The model shows that 128.4MPa is a certain water pressure after which the water absorption is exponentially related to the volume fraction of the fractured hollow glass beads. It means that the fractured hollow glass bead will become the source and then cause crack propagation under this water pressure.
Analysis
Deviation analysis
The probability of fractured hollow glass beads in Table 6 is 97.48% under 165MPa. Experimental results show that the solid buoyancy materials do not fail so serious. This deviation may mainly come from the volume fraction change during the fracture process of hollow glass beads. With hollow glass beads fracturing, the volume fraction of epoxy resin increases (see Figure 5). This makes the epoxy resin to share more pressure which is significant to alleviate the fracture of hollow glass beads. However, simply introducing the volume fraction increase of epoxy resin into the derivation process may be inconsistent with the fact. Therefore, to study how the volume fraction change of epoxy resin influences the fracture of hollow glass beads, further study is required.
Stress distributions
An im16K hollow glass bead with a radius of 8μm is selected as the research object in this section. The inner radius of hollow glass bead and the outer radius of unit cell can be obtained according to the hollow degree and the volume fraction of hollow glass beads. The uniform pressure and the sizes of unit cell are shown in Table 7.
The radial and tangential normal stress distributions along radial direction are shown in Figure 6. It can be seen that most of the radial and tangential normal stresses are borne by the hollow glass bead. Figure 6(b) shows that the tangential normal stress distribute along radial direction is discontinuous at interface. The radial normal stress at inner surface of the hollow glass bead is 0. Therefore, the tangential stress may be the main factor leading to the fracture of unit cell.
The maximum shear stress distribution along radial direction is shown in Figure 7. It shows that the maximum shear stress occurs on the inner surface of hollow glass bead. The relationship between the maximum shear stress on the inner surface and the uniform pressure is shown in Figure 8. The curve in this figure is a straight line. In this paper, the slope of the line for the HZ-42 solid buoyancy material is k = 3.40.
Compressive strength distribution
The compressive strength distribution of the unit cell is shown in Figure 9. It can be seen that the compressive strength does not follow the normal distribution through the asymmetry of the curve.
The crack section of solid buoyancy material under 165MPa pressure are shown in Figure 10. The compressive strength of unit cell decreases with the increase of particle size. Therefore, in theory, the hollow glass beads with larger particle sizes will fracture first. However, Figure 10 shows that the radii of hollow glass beads kept intact are generally small. It means that the compressive strength of unit cell is indeed related to the particle size. Figure 10 also shows that some hollow glass beads with smaller particle size also fractured. The reason for this phenomenon may be that the compressive strength of hollow glass beads with small particle size decreases due to defects.
Summary and Conclusion
A water absorption model based on self-consistent theory for solid buoyancy materials has been proposed after investigating the failure behavior of hollow glass beads. Then the water absorption model is verified by comparing with the experimental results. It shows that the failure of solid buoyancy materials is mainly caused by the fractured hollow glass beads. Therefore, the maximum shear stress is the dominant factor in the failure process of solid buoyancy materials. The crack sources in the solid buoyancy materials also occur on the inner surface of hollow glass beads.
The water absorption behavior of solid buoyancy materials is mainly caused by the fractured hollow glass beads. It can be divided into two stages. In the first stage, the water pressure is still low. The crack does not propagate after the hollow glass beads fracturing. On the other hand, due to the short time of testing time, the hollow part of the fractured glass beads cannot be completely filled with water. Therefore, it is assumed that the water absorption is directly proportional to the volume fraction of fractured hollow glass beads. In the second stage, the fragmentations of hollow glass beads propagate under a certain water pressure. Then the water absorption is exponentially related to the volume fraction of the fractured hollow glass beads. The water absorption model shows that, for the HZ-42 solid buoyancy material, the boundary between the first and the second stages is 128.4MPa.
It is shown that the fracture degree predicted by theory is severer than experimental results. Analysis shows that the deviation mainly comes from the volume fraction change during the fracture process of hollow glass beads. Increase of volume fraction makes the epoxy resin to share more pressure. This has not been considered in the model derivation process. Therefore, further study is required.
The distributions of stresses, compressive strength and volume has been investigated in this part. It can be seen that most of the radial and tangential normal stresses are borne by the hollow glass bead. The maximum shear stress occurs on the inner surface of hollow glass bead. The crack section of solid buoyancy material under 165MPa pressure shows that the radii of hollow glass beads kept intact are generally small. This proves that the compressive strengths of unit cells are related to its particle sizes. However, some hollow glass beads with small particle sizes also fractured. This phenomenon may be caused by the defects.
Acknowledgement
This work is supported by the General Program of National Natural Science Foundation of China ‘Study on water absorption characteristics of buoyancy material in full ocean depth manned submersible’ (Grant No.51879157), the National Natural Science Foundation of China for Youth Science Project ‘Mechanism of stressmoisture- creep coupling on compressive strength of composites in deep-sea’ (Grant No.11902288) and the “Construction of a Leading Innovation Team” project by the Hangzhou Municipal government.
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