Model of the cavitation wear resistance of metal powder filled epoxy composites

. A powder-filled epoxy compound was tested for cavitation wear in the form of bronze particles BraZHNMc9-4-4-1. The particles were obtained by sawing a bronze bar. The composition of the compound was as follows: 100 parts of resin K-153 and 12 parts of hardener (polyethylene polyamine). The specimens were shaped as cylinders, 15.5 and 15 mm in diameter and height, respectively. The tests were carried out with an ultrasonic magnetostrictive vibrator in fresh water at water temperature 20 ± 3 ° C. The frequency and amplitude of oscillations of the vibrator hub end were approximately 22 kHz and 28 µm, respectively. The distance between the end of the concentrator and the end of the specimen wear was set to 0.5 mm. A second-order model describing the cavitation wear as a function of the bronze particle size, its content in the composite and the specific surface area of the particles was constructed by the least-squares method. The cavitation wear of the composite increases with increasing values of these parameters. The greatest influence on composite cavitation wear has the total surface area of bronze microparticles in volume unit.


Introduction
The use of polymeric compositions in machinery and ship repair is becoming more widespread [1], in particular, in restoration of cavitation wear spots on propeller blades (PB).On displacement vessels, cavitation wear occurs mostly in the blade tips [2], quality of the blade tips surface decreases considerably which, in its turn, negatively influences the propeller efficiency [3,4].
As a rule, PB repairs are carried out using epoxy-based repair compounds.Their use is conditioned by good adhesion to metal surfaces and high processability [5,6].Nevertheless, the cavitation resistance of epoxy compounds is considerably lower than cavitation resistance of many thermoplastic polymers [7][8][9][10].For this reason, the addition of powder fillers in the form of metal alloy particles or other inorganic substances to epoxy compounds does not always seem justified.Unlike thermoplastic polymers which break viscously when exposed to cavitation, epoxy compounds, which have a mesh structure, are prone to brittle fracture when exposed to cavitation.The presence of powder filler particles in the epoxy matrix can play both positive [11] and negative roles [12,13].In the first case, the metal filler particles may serve as a barrier for brittle cracks and reduce their propagation speed.In the second case the interface of metal particle -polymer matrix may be a weak point triggering crack formation.Unfortunately, in scientific literature this question is not discussed in detail, there are contradictory data.Therefore the aim of this work is to analyse the effect of powder metallic filler on the cavitation resistance of an epoxy-based composite.For achievement of this purpose it is necessary to solve the following tasks: 1) determine characteristics of metal filler, that influence on cavitation wear resistance of the composite; 2) construct a model describing cavitation wear resistance of the composite, depending on the characteristics of the metal filler.

Methods and materials
Cavitation wear tests were carried out on an ultrasonic magnetostriction vibrator (MSV) in fresh water at water temperature 20±3 оС.The frequency and amplitude of oscillations of the concentrator face 2 were approximately 22 kHz and 28 μm, respectively (Fig. 1a).The distance Z between the concentrator end face and the wear surface of cylindrical specimen 1 was set at 0.5 mm.

a b
Fig. 1.Test scheme (a) and appearance of epoxy composite specimen (b) with irregular distribution of bronze particles in the specimen, with the lowest concentration at the top end and the highest at the bottom end.
Test samples had a cylindrical shape (Fig. 1b).They were made of a composite based on epoxy compound with aluminium bronze particles BraZHNMc9-4-4-1 (GOST 18175-78) as a filler.The particles were made by sawing of a bar from the mentioned bronze.To obtain particles of different size files with different number of cuts per unit length were used.The composition of the compound was as follows: 100 parts of resin K-153 and 12 parts of hardener (polyethylene polyamine).Samples were made by pouring the compound with addition of bronze powder into tubular shapes of about 15.5 mm diameter and about 15 mm height, cut from a metal polymer pipe.After polymerization of the epoxy binder, specimen end faces were grinded on abrasive sandpaper of different grits without removal from the forms and then polished on wet felt.After that the samples were pressed out of the moulds and their end surfaces were analyzed using a metallographic microscope equipped with an object micrometer.The object micrometer was calibrated beforehand.
Preliminary experiments showed that under the cavitation action on the composite filled with bronze particles (Fig. 2a), wear, as shown in Photo Fig. 2 (b), is realized in the form of separation of the bronze particles (arrows 1), breaking of the boundaries of pre-existing pores on the surface (arrows 2), separation of the polymer binder adjacent to the boundaries of the particles (arrows 3).In line with this, the following factors are identified as determinants of cavitation wear: -bronze filler particle size; -bronze particle content in the composite; -the surface area of the filler particles.
Considering the possibilities of the analytical apparatus of stereometric metallography it was decided to estimate -particle size by the value of the average chord length of the particle cross section; -bronze particle content according to the value of the relative volume fraction of the metal filler in the composition; -surface area of filler particles -the value of total surface area of bronze microparticles per unit volume of the composition.
These parameters -average chord length of bronze particles section; relative volume fraction of metal filler in the composition; total surface area of bronze particles per unit volume of the composition -were determined using basic stereometric relations derived in [14].The average chord length was calculated as follows: here: hi is chord length of the i-th particle; z is the number of chords.By a chord we mean a segment of a random secant line obtained by intersection of a secant line with a bronze particle The relative volume fraction of the metal filler in the composition was calculated according to the first general stereometric relation [11]: Vrel = H/L = z·hav/(n·l), mm3/ mm3.
The total surface area of the bronze microparticles per unit volume of the composite was calculated according to the second basic steoreometric relation [11]: n is total number of random secant lines; l -length of secant line.
Since during the polymerisation of the samples there was a gradual deposition of metal particles in the liquid polymer binder, the parameters hav, Vrel and Ssp had different values on the two ends of the same sample.As it can be seen in fig.1(b) and in the table, in the layers of composite adjacent to the upper end of the sample polymerization the concentration of bronze particles is lower and their size is smaller.This circumstance made it impossible to use planning methods for mathematical modelling as it was impossible to design a model with known compositions.For the same reason it was impossible to carry out simultaneous experiments for increased accuracy, because due to randomness of metal particles settling in a liquid polymeric binder it was not possible to obtain several samples with equal values hav, Vrel and Ssp.Therefore, it was decided to carry out a sufficient number of experiments with a random variation of the factors within a given range, and to use the least squares method for data processing.The structure analysis was performed on two faces of each sample and each sample was tested for cavitation wear, first on one end face and then on the other.The duration of each end test was 45 minutes.After testing, the sample was wiped dry with a paper towel and weighed on an analytical weight with a reading range of 0.1 mg after being exposed to air for 30 minutes.

Model development
Analysis of preliminary experimental data shows that the dependencies of cavitation wear on each of the above factors (hav, Vrel and Ssp) are non-linear and can be approximated by ( As can be seen from equation ( 1), there are a total of 11 unknown parameters in the model ao, a1,…, a10, so, the minimum number of compounds to be tested is 11.As mentioned above, during the polymerization of the samples, until the viscosity of the compound increases significantly, sedimentation of the bronze particles takes place.Therefore, in order to study the cavitation resistance of filled composites it was not possible to run more than one experiment in parallel and plan with known compositions.It was decided to test 24 compositions with different values hav, Vrel and Ssp, which allowed for averaging using the least-squares method.
For a space-saving formulation of equations ( 1), we use the following notation: hav = zi1; Vrel = zi2; Ssp = zi3; havVrel = zi4; havSsp = zi5; VrelSsp = zi6; havVrelSsp = zi7; ℎ + = zi8; , -./ + = zi9; 1 23 + = zi10, here i is the number of the experiment (equation).A system of 24 equations was constructed from the results of the tests, which, using the new notations, look as follows: The solution to system (2) according to the least squares procedure [15] is as follows: where A is a vector-column of required coefficients ao, a1,…, a10 representing approximating function (1); Z is a matrix of all factor values z1, z2, z3, z4, z5, … z10 used in the experiments, including the first single column; Zт is a matrix transposed in relation to the matrix Z; Y is a vector-column of experienced values of the value y under study.By solving the system (2), the model can be written in the following form: The graphical representation of the response function is shown in Fig. 3 for three values of the average chord length of the particle cross section (10, 25 and 40 μm).
The adequacy of the model was evaluated using Fisher's criterion [15]: where 1 D + is total variance (variance of the mean) and 1 -.2 + -residual variance.
The obtained value of F was compared with the table value of Fisher's criterion [16].The comparison showed that the obtained model (3) adequately describes the results of the experiment at a significance level of 20 %.This seemingly relatively high level of significance can be explained for two reasons: 1) each value of depreciation yi in the i-th experiment for a particular combination of values hav, Vrel and Ssp was determined only once, i.e., no parallel experiments were carried out; 2) at wearing of composite surface both bronze and epoxy compound particles separated.It is more reliable to estimate it not by mass but by volume losses, because bronze and epoxy compound density differs in almost 7 times, whereas bronze and compound relative share in wear of different samples is approximately 7 times.
To assess the influence of each of the factors on wear and the tightness of the relationship between the factors zj A multiple correlation analysis was carried out, namely two types of pairwise correlation coefficients were calculated: As can be seen from the values obtained, the greatest influence on the cavitation wear has the total surface area of bronze microparticles in the volume unit of the composition (strong correlation); the relative content of bronze particles in the epoxy compound has a moderate influence (medium correlation), and the size of bronze particles has a weak influence on the wear (weak correlation).These findings are also supported by the values of the correlation coefficients between the factors: E H IJ K LMN = 0.383; E H IJ O PQ = 0.0357; E K LMN O PQ = 0.917.
Thus, between the factors Vrel Ssp there is a strong correlation; between the factors hav and Vrel correlation is weak, and between hav and Ssp there is virtually no correlation.

Conclusion
Cavitation wear of composite based on epoxy compound with bronze additives is determined by three factors: size of bronze filler particles; content of bronze particles in composite; total surface area of bronze particles in composite volume unit.Composite wear can be described by a 2nd order model.The greatest influence on cavitation wear has total surface area of bronze particles in unit volume of composite, and the smallest -size of bronze particles.

Fig. 2 .
Fig. 2. Structure of the composite before test (a) and after test (b).

Fig. 3 .
Fig. 3. Dependence of cavitation wear of epoxy resin composites on relative volume fraction of metal filler in the composition and Total surface area of bronze micro-particles per unit volume of the composite with values of average chord length as a parameter of the particle size equal to 10 (a), 25 (b) and 40 μm (c) E D F -the coefficients determining the closeness of the relationship between the response function y and one of the factors zj; E F G -coefficients determining the closeness of the relationship between the factors zj and zm.The calculations showed the following data: E DH IJ = 0.371.E DK LMN = 0.470.E DO PQ = 0.727.

Table 1 .
Filler characteristics on the top and bottom faces of the composite specimens using several specimens as examples.