Development of a finite element model to study the effect of temperature on erosion resistance of polyurethane elastomers

Author’s Accepted Manuscript Development of a Finite Element Model to Study the Effect of Temperature on Erosion Resistance of Polyurethane Elastomers Hossein Ashrafizadeh, Andre McDonald, Pierre Mertiny www.elsevier.com/locate/wear

PII: DOI: Reference:

S0043-1648(17)30492-1 http://dx.doi.org/10.1016/j.wear.2017.08.009 WEA102229

To appear in: Wear Received date: 17 March 2017 Revised date: 21 August 2017 Accepted date: 25 August 2017 Cite this article as: Hossein Ashrafizadeh, Andre McDonald and Pierre Mertiny, Development of a Finite Element Model to Study the Effect of Temperature on Erosion Resistance of Polyurethane Elastomers, Wear, http://dx.doi.org/10.1016/j.wear.2017.08.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Development of a Finite Element Model to Study the Effect of Temperature on Erosion Resistance of Polyurethane Elastomers Hossein Ashrafizadeh*, Andre McDonald, Pierre Mertiny Department of Mechanical Engineering, University of Alberta, Edmonton, Alberta, T6G 1H9, Canada *

Corresponding Author - Email: [email protected] Phone: +1-780-492-6982, Fax: +1-780-492-2200

ABSTRACT In this study, a numerical model based on the finite element technique was developed to study the effect of temperature on the erosion mechanism of material removal during solid particle impact on polyurethane elastomers. The hyperelastic, isotropic hardening, and Mullins damage criteria were chosen as the material model formulations to account for elastic, plastic, and stress softening behavior of the elastomer, respectively. The model inputs were determined experimentally by tensile testing and cyclic loading. In the finite element modeling approach that was developed the impact of ten erodant particles at a single location on the substrate elastomer at controlled temperatures of 22ºC, 60ºC, and 100ºC was simulated. Erosion testing experiments were conducted to provide data for model verification. The results obtained from the finite element model showed that the final elongation at break and associated ultimate stress have the most significant influence on the erosion rate in cases where the stresses produced exceeded the failure stress during impact. This was the case for PU at 100ºC in which the material had the lowest failure stress. The model also successfully simulated the mechanism of material removal by accumulating residual strains such that detachment of larger pieces of material occurred at the surface. The plastic deformation and Mullins stress softening were found as parameters affecting

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this type of wear. The Mullins damage caused by the impact of particles facilitated the localized detachment of small fragments from the surface upon impact of subsequent particles. Overall, although the model allowed for the study of the wear mechanisms and was capable of predicting the morphology of the eroded surfaces of the PU, the model failed to predict accurately the erosion rates, most probably due to the assumptions that were made to simplify the model.

KEYWORDS Erosion; Finite element; Particle impact; Polyurethane elastomer; Residual strain; Stress softening; Ultimate stress

1. INTRODUCTION In some processes in the oil and gas industry, hard erodant particles such as sand may be mixed with the transporting fluid. The flow of these solid-liquid mixtures in piping, pumps, and other equipment can negatively affect the longevity of the equipment due to the loss of material by solid particle erosion [1]. In solid particle erosion, the material detaches from the target surface due to the damage and fracture of the surface caused by the repeated impact of a stream of erodant particles [2]. This type of wear reduces the lifetime of the equipment and it is considered as one of the leading causes of equipment failure in the oil and gas industry [1, 3, 4]. The resulting economic and environmental costs incurred from unpredicted failures can be significant [5]. The fabrication of protective coatings and liners on the target surfaces provides a solution for extending the longevity of equipment exposed to harsh erosive environments. From among different types of protective liners, soft polyurethane (PU) elastomers have received great 2

attention, owing to their excellent resistance to wear. Additionally, ease in processability, and comparatively lower cost of PU elastomers have enabled the use of PU protective liners for large scale applications such as pipeline liners [6-10]. PU elastomers are organic polymers with a urethane group in their chemical structure that can be synthesized by methods typically used for polymers [11]. Thus, while PU elastomers have superior mechanical properties of vulcanized rubber such as high elongation at break, minimal plastic deformation, and resistance to wear, the fabrication processes of PU elastomers are comparatively simpler than those for vulcanized rubber elastomers [12, 13].

Erosion of elastomers caused by solid particle impact is a complex process due to the high number of factors that can affect the mechanism of material removal. The experimental study of wear resistance of elastomers has been the subject of previous studies [12-21]. In these studies, different parameters have been introduced as the key factors affecting the final wear resistance. For example, Beck et al. [17] showed that PU elastomers with similar softness level had different erosion rates. The difference in fractional energy lost in a deformation cycle and hysteresis was found as the parameter that best correlated with the erosion rate of PU. In a study conducted by Hutchings et al. [18], rebound resilience was introduced as the factor with the greatest influence on the wear resistance of rubber elastomers. The rubber with the highest rebound resilience exhibited the highest resistance to erosion. This study showed that there is no simple relation between the erosion rate of rubber elastomers and their Shore hardness, elongation at break, and ultimate tensile strength [18]. On the other hand, Li et al. [14] showed that the hardness and tensile strength had the greatest effect on the wear resistance of castable PU elastomers. The softest material that had the lowest tensile strength was found to be the most wear resistant elastomer. In contrast, Ping et al. [20] showed that two PU liners with similar 3

tensile and tear strength had different erosion rates, most probably due to differences in elongation at break between the two elastomers. In a recent study by Ashrafizadeh et al. [21], elasto-plastic behavior and elongation at break were introduced as the key parameters affecting the overall resistance of PU elastomers to erosion.

Considering the above mentioned studies, there is an apparent discrepancy in the literature about the parameters that have the greatest effect on the wear resistance of elastomers. This may be due to the fact that in most of the previous studies, the authors limited their attention to a qualitative comparison of the data obtained from experiments without an in-depth analysis of the physics of the problem. The erosion caused by solid particle impact occurs within microseconds, and experimental determination of the stresses and plastic strains produced is not a trivial task. Thus, although experimental studies of the erosion mechanism of elastomers provide important insight into the factors that influence it, such studies may fail to provide a comprehensive understanding of how the key material properties can affect deformation and material removal as a result of impact of erodant paricles. To that end, models that can simulate the erosion caused by solid particle impact are valuable tools when used in conjunction with experiments for studying erosion phenomena. Models permit the detailed examination of the erosion caused by solid particle impact, leading to an improved understanding of the fundamental principles of the material removal mechanisms. Moreover, after verification, the models developed may be used as predictive tools in order to minimize the number of costly and time consuming experiments for further investigation of the effect of various parameters on erosion resistance.

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A number of analytical and numerical models have been developed to date to study wear phenomena [22-24]. From among the different models developed, the finite element (FE) analysis technique has received great attention due to its ability in modeling of complex geometries, material models, and contact algorithms. While most of the previous research studies have focused on the development of FE models to simulate the erosion caused by solid particle impact on ductile metals and brittle ceramics [23-28], fewer studies have focused on modeling the solid particle erosion of soft elastomeric materials such as PU. Martinez et al. [12] developed a two-dimensional (2D) FE model to study the wear mechanism as a result of sliding and contact of PU over rough metal surfaces. The Yeoh strain energy potential function was chosen to describe the hyperelastic behavior of PU. The FE simulation allowed for a micro-level evaluation of the contact between the PU and the rough metal surface. The stresses calculated from the FE model supported the hypothesis that there was a possibility of crack formation below the worn surface. In another study by Gong et al. [29], a three-dimensional (3D) combined mesh-free FE model was developed to simulate the solid particle erosion of PU liners. The mesh-free formulation was chosen to eliminate the adverse effects of element distortion in the FE technique. The Johnson-Cook viscoplastic formulation was taken to be representative of the material deformation model. The large deformation impact area was discretized by so-called smoothed hydrodynamics (SPH) particles while FE meshes were used to model the remaining section. Results obtained from the mesh-free method and a conventional FE model were compared in terms of stresses produced and computation time. It was shown that the difference between the stresses at the impact point, as calculated by the two models, were negligible [29]. However, the computation time of the FE model was approximately four times shorter, which emphasized the advantage of a conventional FE method over a mesh-free formulation in

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modeling the erosion process. In a recent study by Zhang et al. [16], a FE model based on an elastic-plastic constitutive law with linear isotropic hardening was developed to simulate the impact of a single particle on PU liners with different thicknesses. An element removal criterion was defined to simulate the material removal and calculate the wear rate as a result of single particle impact. While the FE model predicted monotonical enhancement of erosion resistance with increasing coating thickness, experiments showed that mass loss initially decreased with increasing coating thickness, but then, beyond an optimum thickness, mass loss increased with thickness. The FE model failed to predict this behavior since it did not account for changes in the PU mechanical properties as a result of temperature increases in thicker PU liners.

A review of the technical literature revealed that even though several researchers employed FE modeling techniques to simulate the wear phenomenon for PU elastomers, the number of models developed for simulating the repeated impact of particles is limited. Additionally, few studies have focused on defining material models that can simultaneously account for key mechanical properties of elastomers such as viscoelasticity, hyperelastic response, plastic deformation, and Mullins stress softening. In particular, the effect of viscoelasticity and strain rate dependency has been neglected in most previous studies to simplify the problem. Shipway and Weston [30] employed an elastoplastic material model that was obtained based on quasi-static contact mechanics theories [31] to model the thermal behavior of polymeric substrates impacted by a stream of particles. It was shown that the temperature rise predicted by the model was in good agreement with experimental data for conditions where the polymeric substrate was impacted by robust glass beads. Arnold and Hutchings [22] neglected viscoelastic effects in the modeling of erosive wear of rubber elastomers impacted by particles at oblique angles. Although the results from the 6

model showed good qualitative agreement with experimental measurements, quantitative agreement was less acceptable most likely due to the simplifying assumptions that were made, including neglecting the viscoelastic properties. Neglecting the viscoelastic properties in modeling of erosive wear of elastomers is not limited to older studies with analytical formulations; this simplifying assumption has also been reported in recent numerical studies that are based on FE formulation. Zhang et al. [16] neglected the effect of viscoelasticity of PU in FE modeling of the erosive wear of PU elastomers. Good agreement between the experimental data and that obtained from simulation in terms of erosion rate and temperature rise within PU during erosive wear was reported. Thus, the literature seems to be inconclusive about the necessity of employing complex material models to account for viscoelasticity in wear modeling of elastomers.

In a recent study, the authors developed an experimental setup for erosion testing at controlled temperatures to investigate the effect of temperature on the erosion resistance of PU elastomers [21]. The authors proposed that the elongation at break and plastic deformation upon loading were the parameters with the strongest impact on the wear resistance of PU elastomers. However, only a qualitative discussion about the effect of these parameters on the stresses/strains in the materials and the material removal mechanism was given. In response to these suppositions, the present study was formulated with the following objectives in mind: (1) develop a FE numerical model to simulate solid particle erosion of PU elastomers, (2) employ the model in an in-depth study of material removal mechanisms of PU elastomers at room temperature and at elevated temperatures, and (3) evaluate the possibility of employing the material properties obtained at low strain rate for modeling the erosive wear of PU elastomers. 7

2. EXPERIMENTAL METHOD 2.1. Erosion Testing The erosion resistance of a castable PU liner with Shore A hardness of 85 (RoCoat3000M-85A, Rosen Group, Lingen, Germany) was evaluated using a custom made erosion testing assembly. The testing equipment was designed based on the provisions of the ASTM Standard G76 [32]. A description of the testing assembly and sample dimensions has been provided by Ashrafizadeh et al. [21]. The experimental data obtained was used for comparison with numerical results and validation of the FE model. The erosion tests were conducted at an impact angle of 30º and three set temperatures of 22ºC, 60ºC, 100ºC.

Garnet sand (Super Garnet, V.V. Mineral, Tamil Nadu, India) with rounded corners impacted the surface of the PU with a maximum velocity of 73 m/s, which was estimated by using a mathematical model based on the fundamentals of compressible gas flowing through a convergent-divergent nozzle [21]. The average diameter of the garnet sand particles was calculated by using an image analysis software (ImagePro, Media Cybernetics, Bethesda, MD, USA) to be 266 ± 49 µm (n = 159) [21]. The corner radii were also approximated by drawing a circle around the corner tips and computing the circle radii by use of image analysis. The average radius of the corners of the erodant particles was estimated to be 67 ± 26 µm (n = 132). Additional details about the erosion testing equipment and the parameters selected can be found elsewhere [21].

2.2. Tensile Testing

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Tensile tests and cyclic loadings were conducted at room and elevated temperatures in order to provide data as input parameters of the selected material formulation in the FE model. PU sheets with a thickness of 1 mm were waterjet cut to dogbone shape samples according to dimensions suggested by ASTM Standard D638-Type V [33]. The tensile tests and cyclic loadings were conducted using a dynamic mechanical analyzer (ElectroForce 3200, TA Instruments, Eden Prairie, MN, USA) at 22°C, 60°C, and 100°C. The experiments were performed at a strain rate of 0.25 s-1 and up to a nominal strain of 350% for tensile tests and to 20%, 50%, 100%, and 200% for cyclic loadings.

2.3. Scanning Electron Microscopy The eroded PU samples were cleaned by using compressed air prior to imaging by using a scanning electron microscope (SEM) (EVO LS15 EP, Carl Zeiss Canada Ltd., Toronto, ON, Canada) to produce images of the eroded surfaces. A thin film of carbon was deposited onto the surface by using a carbon evaporation system (EM SCD 005, Leica Baltec Instrument, Balzers, Liechtenstein). The conductive layer protected the PU surface from charging during SEM imaging. The beam voltage was set to 5 kV in SEM imaging, and the secondary electron mode was employed to evaluate the surface morphology of the eroded PU samples.

3. FINITE ELEMENT SIMULATION The erosion process caused by the impact of garnet sand particles on PU elastomers at different testing temperatures was modeled by employing a finite element (FE) numerical 9

technique. This was done in order to study further the effect of erosion testing temperature on the stresses produced in the PU and to understand better the mechanisms of material removal. The FE model was developed by utilizing a general purpose FE solver (Abaqus Version 6.13, Dassault Systèmes Americas Corp., Waltham, MA, USA) [34]. Details about the simulation formulation, material model, contact algorithm, and model boundary conditions are discussed in the following sections.

3.1. Finite Element Explicit Formulation Erosion caused by solid particle impact is a high velocity dynamic phenomenon. Therefore, the FE explicit dynamic formulation was employed in this study to ensure the convergence of the solution. The discretized equilibrium equation for the explicit dynamic formulation is [23]:

Mu  F Ext  F Int ,

(1)

where M is the lumped (diagonal) mass matrix, u is the nodal acceleration at each time step, FExt is the externally applied force vector at each node, and FInt is the internal force vector as determined from the stress domain. The nodal acceleration at each time step was calculated as [34]: 1 Ext Int ut  M t ( Ft  Ft ) .

(2)

The central difference explicit time integration method was used to calculate the displacement and velocity at each time step [34]. The time increment was automatically determined by the Abaqus solver in each time step based on the dilatational wave speed, which is a function of the properties of the material and minimum element size in the model [34].

3.2 Material Model 10

In the FE model, the garnet sand particles were modeled as rigid particles rather than deformable objects since the PU was much softer than the garnet sand. A preliminary analysis was conducted that supported this hypothesis. It was found that the stress field produced upon impact of rigid and deformable impacting particles was identical. In the modeling of rigid particles, no stress field is calculated in each time step of the explicit solution and, consequently, the computational effort is reduced. A hyperelastic material model based on the Marlow formulation [34] was chosen to predict the nonlinear elastic response of the PU elastomer. The Marlow formulation was selected as recommended by the Abaqus solver user manual for the experiemental testing condition that was employed to characterize the material response (see Section 2.2) [34]. A close fit between the mechanical response predicted by this material model and that of experimental data was verified. The formulation of the selected hyperelastic material model is described by the strain energy potential, and details regarding the formulation can be found elsewhere [34].

In order to model the plastic deformation of PU elastomers, the isotropic hardening von Mises plasticity material model was employed. The use of this material model is common for modeling the plasticity of ductile materials mainly due to its simplicity in algebraic equations that allows for a straightforward determination of the explicit equation for the material stiffness matrix [16, 34]. Details on the equations and theory of isotropic hardening plasticity can be found elsewhere [16, 34]. The input data for the isotropic hardening model were obtained from experimental cyclic tensile tests and were defined as yield stress and plastic strain in tabular format.

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In order to incorporate the effect of Mullins stress softening and damage caused by previous loadings, the Mullins damage criteria was defined as a part of the material formulation. Similar to the hyperelastic material model, the Mullins damage formulation is defined based on the strain potential energy [34]. However, in contrast to the hyperelastic material model in which the strain potential energy is only a function of the deformation gradient tensor, the Mullins damage formulation is a function of both deformation and a scalar variable that describes the damage [34]. The damage variable is a function of elastomer properties and controls the fraction of energy that is dissipated due to damage to differentiate the unloading and subsequent reloading from that of the primary (initial) loading path and reflect stress softening [34]. In this study, the Mullins damage parameters were determined from the automatic calibration of the uniaxial tensile cyclic test data parameters by the solver. Experimental loading-unloading stress-strain data were used as the input to the model. The solver computed the material parameters from the graph of experimental stress-strain data by using a nonlinear least-squares curve fitting algorithm [34]. It should be noted that similar to previous studies, the viscoelasticity of the PU was neglected for simplifications [16, 22, 30]. Evaluation of the validity of this assumption was one of the goals of this work and is discussed in the Results and Discussion Section of this manuscript.

The material removal caused by the solid particle erosion was modeled by defining an element deletion criterion that was based on equivalent plastic strain. The definition of this technique has been suggested in previous studies for FE modeling of the erosion process [16, 23, 24]. In this method, a state variable is defined as [34]

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D  

d

pl

(3)

 D pl

where ωD, εpl and εDpl are the damage variable, equivalent plastic strain and equivalent plastic strain onset of damage, respectively. The damage criteria is met when ωD = 1 and the element will be deleted from the model [16, 34]. The plastic failure strain was determined based on data obtained from the tensile test experiments since εDpl is equal to the residual strain at failure in uniaxial deformation [16].

3.3 Contact Algorithm The contact between the eroding particles and the PU surface was defined by the penalty contact formulation [34]. For simplicity, it was assumed that the contact was independent of the sliding rate, pressure, and temperature and followed Coulomb’s rule of friction, defined as [23] Ft  Fn ,

(4)

where Ft is the tangential force, Fn is the normal force, and µ is the coefficient of friction, which was assumed to be 0.2 [23]. This assumption was made based on previous FE impact studies and the suggestion of use of low values of the dynamic friction coefficient in these types of analyses [23, 35]. The contact was defined between the outer surface of the erodant particles and the elements of the area to be eroded. Upon deletion of the failed elements from the model, the contact algorithm was transferred to the newly exposed elements.

3.4 Model Geometry and Boundary Conditions In erosion caused by the impact of solid particles, damage of the target surface and final removal of material occurs due to the repeated impingement of erodant particles. To account for the effect of repeated impact of erodant particles in this study, ten particles were positioned at a 13

distance of 1.6 mm from each other at the beginning of the simulation. Impact of several particles allows for studying the effect of stress softening and plastic deformation on the erosion resistance. The assumption of impact of the erodant particles on a single area on the PU surface permitted rapid damage development at the surface, and hence, modeling of a lower number of impacting particles with reduced computational time. The 1.6 mm distance between the particles was chosen to ensure that a subsequent incoming particle does not impact the surface while another particle was still in contact with the substrate. Figure 1a shows a view of the model assembly. A symmetry boundary condition was employed to permit modeling one half of the particle-target configuration to reduce the computation effort. The full description of the model boundary conditions, involving symmetry with respect to the xz plane (see Fig. 1a), zero displacement of the bottom face of the PU substrate, and eroding particle initial velocity, is as follows: 

Symmetry condition: u y i  0 and rxi  rzi  0 at y  0 ,

(5)



Zero displacement on bottom surface of PU material: u x i  u y i  u z i  0 at z  0 ,

(6)



Initial condition of particle nodes: u xi  vinix , u yi  0 , u zi  viniz at t  0 .

(7)

where u, r and v represent nodal displacement, rotation and velocity, respectively, with coordinates x, y, z as defined in Fig. 1a.

The erodant particles were assumed to be identical in shape and dimensions to the particle shown in Fig. 1b. This assumption was made based on the values of the average dimensions of the erodant particles that were measured from SEM images (see Section 2.1). The target PU material was modeled as two zones, namely Part A and Part B as shown in Fig. 1a. The position of the erodant particles in the model assembly was adjusted so that impact occurred on 14

Part A, which was meshed with refined elements. Part B was added to the model to provide support for Part A and to ensure that the stresses caused by the impact of erodant particles were not influenced by any discontinuities within the target material. Part B was meshed with larger elements and it was attached to Part A with a tie constraint that imposed zero displacement between the nodes of the two zones that were in contact. An 8-node 3D brick element type was used to discretize Parts A and B. The element size for Part A was chosen as 10 µm for areas to be eroded while its outer areas and Part B were meshed with an element length of 30 µm. Overall, Parts A and B were discretized with 40,800 and 9,234 elements, respectively. The erodant particles were meshed with an element length of 10 µm. The garnet sand particles were discretized with both brick and quadrilateral elements due to the complexity of the particle geometry. Each garnet sand particle was meshed with 1,032 elements. A number of preliminary trial simulations were performed to test the sensitivity of the model and to ensure that the results obtained in terms of equivalent plastic strain and equivalent stress were not a function of the PU sample dimensions, constraints, and element sizes.

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(a)

(b)

Figure 1 FE model geometry showing dimensions of a) the area to be eroded on a PU target surface and b) the erodant particle

3.5. Stress-Strain Response Predicted by the FE Material Model To compare the response of the selected material model with that of experiments, a simple FE model with one single element subjected to cyclic displacement up to a strain value of 50% was developed. Figure 2 shows typical stress-strain curves of the 3000M-85A PU at 22ºC that was loaded and unloaded to 50% nominal strain for two cycles. As shown, the chosen material model correctly predicted the hyperelastic behavior and residual strain upon unloading. Beyond the yield point (approximately 5 MPa), the numerical model predicted a softer response compared to that of experiments (see Fig. 2) due to the simultaneous use of isotropic hardening and hyperelastic material models [34]. Moreover, since an ideal Mullins damage formulation in the FE simulation was employed, the FE simulation predicted a softer response compared to that of the experiments. In the ideal Mullins damage formulation, cyclic loadings after the initial loading overlapped with the first unloading curve [36]. Consequently, although the selected

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material models enables an accounting of the plastic deformation and stress softening of the elastomer, an overestimation in stress softening occurs, which may lead to an overestimation of the erosion rate by the FE model when simulating the repeated impact of erodant particles.

Figure 2 Experimental and numerical nominal stress-strain respond of first and second cycles of loading-unloading of 3000M-85A PU at 22ºC for elongations up to 50%

4. RESULTS AND DISCUSSION 4.1. Experimental Evaluation of Erosion Resistance of PU at Selected Temperatures The erosion resistance of the 3000M-85A PU liner was evaluated at PU temperatures of 22ºC, 60ºC, and 100ºC. The non-dimensional erosion rates measured are summarized in Table 1. The erosion rate was defined as the ratio of the PU mass loss to the mass of erodant particles that impacted the target surface. As shown in Table 1, an initial improvement in erosion resistance occurred up to a temperature of 60°C, followed by an increased erosion rate beyond 60°C for the PU elastomer that was studied. The fact that the final elongation at break and associated 17

ultimate stress of the PU continuously decreased at elevated temperatures (see Table 1) suggested that these properties do not simply correlate with PU erosion resistance. It will be shown later in this article that the FE model not only permitted an explanation of the behavior and changes in erosion rate as a result of temperature variation, it also allowed for an in-depth understanding of the role of final elongation at break and ultimate stress on the erosion resistance of PU elastomers. It should be noted that a thermocouple that was inserted 1.5 mm below the surface verified that the temperature rise caused by repeated deformation and friction forces of the erodant particles was negligible [21].

Table 1 Erosion rate, nominal stress and strain failure data of 3000M-85A PU elastomer [21] Temperature

Ultimate stress

Elongation at break

Erosion rate

(°C)

(MPa)

(m/m %)

(mg/g)

22

76.6 ± 9.3 (n = 3)

210 ± 18 (n = 3)

0.053 ± 0.004 (n = 3)

60

26.4 ± 3.0 (n = 3)

200 ± 11 (n = 3)

0.034 ± 0.001 (n = 3)

100

4.9 ± 0.2 (n = 3)

58 ± 7 (n = 3)

0.063 (n = 2)

4.2. Rebound of Impacting Solid Particle from PU Surface The softness and high elongation at break of elastomers are two of the main causes of their superior resistance to wear according to previous studies [14, 20]. The ability of elastomers to sustain large deformation may allow for gradual absorption of the kinetic energy of the impacting particles to protect the surface from extreme loads upon impact. In this study, the impact of a garnet sand particle on the PU elastomer from the beginning of the impact until complete detachment from the surface was studied by the FE model. Figure 3 shows the impact and rebounding of an erodant particle on the PU surface. The stress values shown in the figure represent the von Mises stress. As shown, the PU deformed upon impact of the erodant particle 18

due to the impact force as a result of the variation in momentum of the particle. Within the initial microseconds of the impact (see Fig. 3a to b), the erodant particle penetrated into the PU and the PU absorbed the kinetic energy of the particle to reduce the particle velocity. The maximum stresses were produced within the initial stage of the impact as shown in Fig. 3b. Some elements of the numerical mesh in the PU were forcibly dragged at the beginning of the impact (t = 1 µs to 2 µs) and were later compressed as the particle penetrated into the PU (t = 3 µs). The elements that represented the PU material close to the surface experienced higher stresses and, therefore, were subjected to plastic deformation and possible damage caused by Mullins stress softening. The stresses produced were smaller than the ultimate failure stress of the PU (see Table 1 and Fig. 3). Thus, no removal of material from the surface occurred. This behavior is in agreement with previous experimental studies in which no material removal occurred due to impact of a single erodant particle. The kinetic energy absorbed and stored as potential energy was released later during rebounding of the particle as shown in Fig. 3d.

(a)

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(b)

(c)

(d) Figure 3 Stresses produced during the impact of a single particle with initial velocity of 73 m/s on 3000M-85A PU at 60ºC at a) t = 1 µs, b) t = 3 µs, c) t = 5 µs, and d) t = 7 µs

4.3. Effect of PU Softness on the Impact Stresses

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The impact time affects the forces and stresses produced as well as the overall resistance to erosion. As the impact time becomes longer, the impact force decreases since the force is the time derivative of momentum as given by the impulse formula:

  mv   Fimpact dt ,

(8)

where m is the particle mass, Δv is variation in particle velocity vector, Fimpact is the impact force, and t is time. Thus, the impact force is smaller on softer materials due to the longer impact duration. In order to evaluate the effect of PU softness on the stresses produced therein, the stress produced upon impact of the first erodant particle on the 3000M-85A PU elastomer at 22ºC and at 60ºC, when the material was softer [21], was evaluated. Figures 3 and 4 show the stresses produced within the first few microseconds of the impact. As presented in these figures, the stresses produced were higher for the PU at 22ºC due to the higher stiffness of the material compared to when it was heated to 60ºC. Thus, even though the softer substrate allowed for larger deformation of the PU surface, the impact force and, therefore, the generated stresses were smaller. The results obtained by the model in terms of lower stresses in a softer material corroborates the approach by Li et al. [14] in which elastomer softness was introduced as a factor with significant influence on erosion resistance. The higher stress values leads to greater Mullins damage and stress softening that may activate the failure criterion in subsequent impacts. This shows the simultaneous importance of the softness and elongation at break for the final wear resistance of elastomers. The high elongation at break of a softer elastomer can protect the elastomer from material failure as a result of excessive deformation.

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(a)

(b) Figure 4 Stresses produced during the impact of a single particle with initial velocity of 73 m/s on 3000M-85A PU at 22ºC at a) t = 1 µs, and b) t = 3 µs

4.4. Mechanism of Material Removal The FE model was used to study the material removal mechanism. Ten erodant particles were positioned to impact a single spot on the PU surface. Figure 5 shows the equivalent plastic strain that was produced after the impact of four erodant particles on the surface of the 3000M-85A PU at 60°C. It is observed that after the impact of the first garnet sand particle (see Fig. 5a), the elements on the PU surface did not meet the plastic strain failure criterion, and hence, no material removal occurred. However, some parts of the PU deformed plastically as 22

shown in Fig. 5a. Based on the plastic strain failure criterion, failure occurs when the equivalent plastic strains accumulated and exceeded the failure or fracture strain of the target material. In this study, the failure strain of the PU material was measured by conducting tensile testing (see Section 2.2). Upon impact of the second particle (see Fig. 5b), the failure criterion was met by a few elements on the top surface, and these particles were removed from the model. The Mullins stress softening damage during the impact of the first particle enabled the removal of damaged elements during the impact of the second particle. The present results validate findings reported by Beck et al. [17] regarding a higher erosion rate of elastomers as a consequence of larger hysteresis loop and Mullins damage. This shows the importance of considering the Mullins damage criteria in FE simulation of solid particle erosion of elastomers. Depressions formed as a result of localized removal of small material fragments can also be observed in the SEM images taken from the top surface of the experimentally eroded surfaces. Those areas are indicated by circles in Fig. 6. It should be noted that the spherically shaped features observed in Fig. 6 are cavities that were formed in the PU elastomer during its fabrication as a result of possible air entrapment and formation of gases during curing. Two of these typically spherical defects on the surface of the PU are indicated by arrows in Fig. 6.

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(a)

(b)

24

(c)

(d) Figure 5 Equivalent plastic strain after impact of erodant particles with initial velocity of 73 m/s on 3000M-85A PU at 60ºC for the a) first particle, b) second particle, c) third particle, and d) fourth particle [37]

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Figure 6 SEM images of the eroded 3000M-85A PU surface at 60ºC

The localized removal of material by the impact of the second particle increased the maximum value of the equivalent plastic strain as shown in Fig. 5b and enlarged the area in which plastic deformation occurred. The residual strain increased further upon impact of the third and fourth particles (see Fig. 5c and d), and satisfied the plastic strain failure criterion of the PU material at the specified temperature. As a matter of fact, upon impact of the third particle, a larger area was subject to failure and was removed from the model. The erosion rates calculated based on the removed elements from the FE model supported the former discussion. The erosion rate increased from 0.657 mg/g to 5.186 mg/g after the impact of the third particle. This provides further support for the material removal mechanism that involved the accumulation of residual strains resulting in failure and detachment of material that was discussed by the present authors in a previous study [21]. Figure 7 shows a typical SEM image taken from the top surface of an eroded PU sample. In this figure, some of the areas suspected of removal of material as a result 26

of the accumulation of residual strains and detachment of the asperities that were formed are marked with circles. Interestingly, the shape of the asperity shown in Fig. 5d by an oval, as predicted by the FE model, is similar to that indicated in Fig. 7 by circles and also shown in the side view SEM image of Fig. 8 [21], which was taken from the ridges in an eroded PU elastomer. In Fig. 8, the arrow shows the impact direction. Interestingly, the results obtained by the FE model provided further support for the erosion mechanism of soft polymers proposed by Friedrich [38]. In this work it was proposed that the ridges were formed due to significant plastic deformation, together with material removal around the deformed tips caused by cracking and localized material loss. The plastically deformed ridge, coupled with areas in which localized material loss occurred, is shown in Figs. 5d and 8.

Although the FE model predicted the shape of the asperities as being similar to those observed on the eroded surfaces obtained from experiments, the model overestimated the material removal from the substrate. Thus, the calculated erosion rates was also overestimated by the model. It was presumed that this was due to the assumptions that were made in the formulation of the model with respect to the material model (see Section 4.2), initial impact velocity of the erodant particles, repeated impact of erodant particles on a single spot, and the deletion of elements upon meeting the failure criterion.

At the cold spray nozzle exit, the high velocity air mixes with the stationary ambient air and that causes deceleration of the air jet. Therefore, the erodant particles are decelerated as well and thus, the velocity profile of the erodant particles has a range from the maximum velocity at the center of impact to lower velocities at areas distal from the center. In experiments, the erosion rate of the entire eroded area was measured, whereas the FE model only accounts for

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impacting particles with maximum velocity (73 m/s). Additionally, in order to reduce the computation effort by minimizing the number of particles that needed to be modeled, it was assumed that the erodant particles repeatedly impact the target in a single spot, maximizing the damage that is imparted on the material. However, it is unlikely that particles repeatedly impact in the same exact location in actual experiments. This assumption together with impacting of erodant particles with maximum velocity are considered possible causes for the overestimation of the erosion rate by the FE model. Moreover, in the simulation the material in the form of failed elements was removed entirely from the substrate surface, whereas in reality, detached fragments will largely remain on the sample surface. Those detached fragments hinder further penetration of the erodant particle into the PU surface. This observation is a significant finding of this study as it reveals that assumptions made for the FE erosion modeling of ductile metals and brittle ceramics [16, 23, 24] cannot be adopted indiscriminately for soft elastomers that experience large deformations upon impact. It should also be noted that in the developed model, no viscoelastic behavior was defined. This can also be one of the causes for overestimation of the erosion rate since viscoelastic elastomers may exhibit a stiffer response at high strain rates [39]. This emphasizes the importance of considering viscoelastic and rate dependent properties of elastomers when modeling erosive wear. Consequently, although in this study an advanced material model was employed to account for the hyperelasticity, isotropic hardening, and Mullins stress softening, development of a model that can also account for viscoelasticity is essential to obtain quantitative agreement between simulation results and experiments.

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Figure 7 SEM images of the eroded 3000M-85A PU surface at 60ºC

Figure 8 SEM image of an eroded PU surface at 60ºC [21]

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The plastic strains produced in the 3000M-85A PU material after the impact of 10 particles with initial velocity of 73 m/s at 22ºC was also evaluated and is shown in Fig. 9. Similar to the PU at 60ºC, the repeated impact of subsequent particles led to localized removal of material from the surface and an increase in the value of the equivalent plastic strain and the area with high residual strains. The model predicted the area of the PU at 22ºC with plastic strain to be larger than that of the PU that was modeled at 60ºC (see Fig. 5a). This behavior was due to the greater ability of PU material at 60°C to revert back to its initial state with less residual strains upon unloading compared to that of PU at 22°C as evidenced by the stress-strain graphs of cyclic loadings up to a strain of 50% (see Fig. 10). As seen in these graphs, the material behavior of 3000M-85A PU varied with elevated temperature in such a way as to improve its ability to revert to its initial condition upon unloading as compared to samples at room temperature. Although impact of ten erodant particles was not sufficient to reach the point of detachment of larger fragments from the surface in modeling of PU at 22°C, the results obtained by the FE model show that upon reaching the failure criterion, a larger amount of material will be removed compared to the PU at 60°C, in which there was a smaller area that was plastically deformed. The results obtained from the FE model provide an explanation of the improvement in erosion resistance of the PU that was held at 60°C as observed from experiments (see Table 1) and shows how plastic deformation can affect the erosion resistance of PU elastomers.

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(a)

(b)

31

(c) Figure 9 Equivalent plastic strain after impact of erodant particles with initial velocity of 73 m/s on 3000M-85A PU at 22ºC for the a) first particle, b) third particle, and c) tenth particle

Figure 10 First cycle of loading-unloading at various temperatures for elongation up to 50% for 3000M-85A PU elastomer [21]

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The evaluation of the stresses produced in PU at 100°C by an impacting erodant particle (see Fig. 11) revealed that upon initiation of impact, the stresses were higher than the ultimate stress of the PU material at 100°C. This behavior was due to the significant reduction in ultimate stress of PU held at 100°C (see Table 1), which led to removal of elements that were in contact with the erodant particle. In fact, instead of storing the kinetic energy of the colliding particle, the elements on the PU surface achieved the element removal criterion and were deleted from the model. This behavior emphasizes the importance of material ultimate stress on erosion resistance and provides justification for the increase in erosion rate of PU that was observed at 100°C in this study and also in PU elastomers with low ultimate stress in previous studies [14]. The erodant particle penetrated the surface and remained attached to the PU substrate (see Fig. 11d). Although the model successfully predicted significant loss of material, even by the impact of the first erodant particle, would occur as evidenced qualitatively in Fig. 11, the model overestimated the penetration of the erodant particle into the PU. The images taken by SEM from the experimentally eroded surfaces did not reveal the existence of any deep holes or particles trapped in the PU surface. As discussed earlier, this behavior is likely due to the assumptions that were made about the removal of failed elements from the analysis and also neglecting the viscoelastic behavior of the PU elastomer. In the experiments, damaged PU fragments will remain on the sample surface where they impede impacting particles from excessive material removal and penetration into the PU surface. Also, the fact that the material may have higher yield stress at high strain rates [40] emphasizes the importance of considering the viscoelastic rate dependent behavior of elastomers when modeling the wear of elastomers caused by the impact of particles.

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(a)

(b)

(c)

34

(d) Figure 11 Stresses produced during the impact of a single particle with initial velocity of 73 m/s on 3000M-85A PU at 100ºC at a) t = 1 µs, b) t = 3 µs, c) t = 5 µs, and d) t = 7 µs [37]

5. CONCLUSIONS In this study, a numerical model based on the finite element technique was developed to study the mechanism of material removal during solid particle impact on PU elastomers, and to study the effect of erosion testing temperature on the produced stresses. The FE model simulated the impact of 10 erodant particles at a single location on the substrate elastomer at controlled temperatures of 22ºC, 60ºC, and 100ºC. The results obtained from the FE model showed that the stresses produced upon the initiation of particle impact at 100ºC were higher than the ultimate stress of the PU material, which led to the removal of elements that were in contact with the erodant particle. This behavior provided support for the importance of ultimate failure stress and elongation at break of the material as parameters affecting the erosion rate. The model also successfully simulated the mechanism of material removal as the accumulation of residual strains up to the detachment of larger pieces from the substrate surface. The study of eroded surfaces modeled showed that the model was capable of predicting the surface morphology with formed 35

asperities similar to those observed in SEM images taken from the top surface of experimentally eroded samples.

The FE model developed in this study was more comprehensive than previous models in terms of being able to simulate repeated impact of erodant particles and also advancements in material formulation. Hyperelasticity, isotropic hardening, and Mullins stress softening material behavior were accounted for in the PU elastomer for an in-depth study of the factors that affect the resistance of PU elastomers to erosive wear. The results obtained from this study aid in explaining the discrepancies that are present in the technical literature with respect to the principal factors that affect the erosion resistance of PU elastomers.

Although the results obtained from the model provided a good qualitative comparison with that observed from the experiments, the model failed to predict accurately the erosion rates due to the simplifying assumptions that were made. The assumption of a complete material removal associated with failed elements from the model surface upon reaching the failure criterion led to an overestimation of the erosion rate caused by the impact of erodant particles. This is one of the significant finding of this study showing that assumptions made for the FE erosion modeling of ductile metals cannot be transferred indiscriminately to soft elastomers that experience large deformations upon impact. The assumption of neglecting the viscoelastic response of PU elastomer could have been another source affecting the model accuracy in predicting the erosion rate. This emphasizes the importance of accounting for viscoelastic and rate dependent behavior of elastomers when simulating erosive wear caused by impact of particles. Future work will

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be undertaken to develop models that can also account for the viscoelasticity of PU elastomers. Such models will enable studies of viscoelastic and strain rate dependency of PU on the resistance of PU elastomers to erosion caused by solid particle impact.

6. ACKNOWLEDGMENTS The authors would like to acknowledge the support by the Natural Sciences and Engineering Research Council of Canada (NSERC), Rosen Group, and Syncrude Canada Limited. The authors would also like to thank Mr. Bernie Faulkner (Machine Shop, Department of Mechanical Engineering, University of Alberta) for his technical support.

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Highlights



FE model developed enabled in-depth analysis of the erosion mechanism of PU



Effect of temperature on erosion mechanism of PU and stresses produced was studied



FE model provided support of how elongation at break affect PU erosion resistance



FE model successfully predicted shape of asperities similar to that from experiments

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