Prediction of Fatigue Life of Polyetherimide/Carbon Fiber Particulate Composites at Various Maximum Stresses and Filler Contents.

The objective of this research was to predict the fatigue behavior of polyetherimide-based composites loaded with short carbon fibers 200 µm long under cyclic loads. The weight fraction of the filler was 10, 20, and 30 wt.%, while the maximum stress in a cycle was 55, 65, and 75 MPa. A modified fatigue model based on the obtained experimental results and Basquin equation was developed. The novelty of the results is related to developing a model on the structure-property relationship, which accounts for both the maximum stress in a cycle and the carbon fiber content in the composites. In addition, an "algorithm" for designing such composites according to the fatigue life criterion was proposed. The approach to determine relationships between the composition, structure, and properties of PCMs described in this study can be applied to further expand the model and to improve its versatility in the use of other thermoplastic matrices and fillers. The results of this study can be applied for the design of composites for structural applications with designated fatigue properties.

Application of Neural Network Models with Ultra-Small Samples to Optimize the Ultrasonic Consolidation Parameters for 'PEI Adherend/Prepreg (CF-PEI Fabric)/PEI Adherend' Lap Joints.

The aim of this study was to optimize the ultrasonic consolidation (USC) parameters for 'PEI adherend/Prepreg (CF-PEI fabric)/PEI adherend' lap joints. For this purpose, artificial neural network (ANN) simulation was carried out. Two ANNs were trained using an ultra-small data sample, which did not provide acceptable predictive accuracy for the applied simulation methods. To solve this issue, it was proposed to artificially increase the learning sample by including additional data synthesized according to the knowledge and experience of experts. As a result, a relationship between the USC parameters and the functional characteristics of the lap joints was determined. The results of ANN simulation were successfully verified; the developed USC procedures were able to form a laminate with an even regular structure characterized by a minimum number of discontinuities and minimal damage to the consolidated components.

Ranking Technologies of Additive Manufacturing of Removable Complete Dentures by the Results of Their Mechanical Testing.

In this study, a methodology was developed for ranking manufacturing technologies of removable complete dentures (RCDs) according to the results of their full-scale mechanical tests. The actuality of the study is motivated by establishing the advantages and drawbacks of 3D-printed RCDs in contrast with ones manufactured via an analog protocol. The RCDs were fabricated via four technological routes that included various combinations of subtractive technologies (hot polymerization/HP and CAD/CAM milling) and additive manufacturing (digital light processing/DLP) ones and the installation of commercially available cosmetic denture teeth (DT). In the mechanical tests, different blocks of teeth (incisors, canines, premolars and molars) were loaded. To solve the ranking problem, it was proposed to interpret the results of the mechanical tests in terms of the reliability, durability and compliance/stiffness criteria. For this purpose, the combined AHP-VIKOR method was applied. In addition, a computer simulation of the mechanical loading conditions and the response of the RCDs was performed based on the finite element method (FEM). As the key conclusion, it was stated that additive manufacturing (AM) methods are competitive and cost-effective techniques for the fabrication of RCDs.

Influence of the Matrix Material and Tribological Contact Type on the Antifriction Properties of Hybrid Reinforced Polyimide-Based Nano- and Microcomposites.

This paper addresses peculiarities in the formation and adherence of a tribofilm on the wear track surface of antifriction PI- and PEI-based composites, as well as a transfer film (TF) on a steel counterface. It is shown that during hot pressing, PTFE nanoparticles melted and coalesced into micron-sized porous inclusions. In the PEI matrix, their dimensions were much larger (up to 30 µm) compared to those in the PI matrix (up to 6 µm). The phenomenon eliminated their role as effective uniformly distributed nanofillers, and the content of 5 wt.% was not always sufficient for the formation of a tribofilm or a significant decrease in the WR values. At the loaded content, the role of MoS2 and graphite (Gr) microparticles was similar, although filling with MoS2 microparticles more successfully solved the problem of adhering to a PTFE-containing tribofilm in the point tribological contact. This differed under the linear tribological contact. The higher roughness of the steel counterpart, as well as the larger area of its sliding surface with the same PTFE content in the three-component PI- and PEI-based composites, did not allow for a strong adherence of either the stable PTFE-containing tribofilm on the wear track surface or the TF on the steel counterpart. For the PEI-based composites, the inability to shield the steel counterpart from the more reactive polymer matrix, especially under the conditions of PTFE deficiency, was accompanied by multiple increases in the WR values, which were several times greater than that of neat PEI.

Dental Material Selection for the Additive Manufacturing of Removable Complete Dentures (RCD).

This research addresses the development of a formalized approach to dental material selection (DMS) in manufacturing removable complete dentures (RDC). Three types of commercially available polymethyl methacrylate (PMMA) grades, processed by an identical Digital Light Processing (DLP) 3D printer, were compared. In this way, a combination of mechanical, tribological, technological, microbiological, and economic factors was assessed. The material indices were calculated to compare dental materials for a set of functional parameters related to feedstock cost. However, this did not solve the problem of simultaneous consideration of all the material indices, including their significance. The developed DMS procedure employs the extended VIKOR method, based on the analysis of interval quantitative estimations, which allowed the carrying out of a fully fledged analysis of alternatives. The proposed approach has the potential to enhance the efficiency of prosthetic treatment by optimizing the DMS procedure, taking into consideration the prosthesis design and its production route.

Ultrasonic Welding of PEEK Plates with CF Fabric Reinforcement-The Optimization of the Process by Neural Network Simulation.

The optimal mode for ultrasonic welding (USW) of the "PEEK-ED (PEEK)-prepreg (PEI impregnated CF fabric)-ED (PEEK)-PEEK" lap joint was determined by artificial neural network (ANN) simulation, based on the sample of the experimental data expanded with the expert data set. The experimental verification of the simulation results showed that mode 10 (t = 900 ms, P = 1.7 atm, τ = 2000 ms) ensured the high strength properties and preservation of the structural integrity of the carbon fiber fabric (CFF). Additionally, it showed that the "PEEK-CFF prepreg-PEEK" USW lap joint could be fabricated by the "multi-spot" USW method with the optimal mode 10, which can resist the load per cycle of 50 MPa (the bottom HCF level). The USW mode, determined by ANN simulation for the neat PEEK adherends, did not provide joining both particulate and laminated composite adherends with the CFF prepreg reinforcement. The USW lap joints could be formed when the USW durations (t) were significantly increased up to 1200 and 1600 ms, respectively. In this case, the elastic energy is transferred more efficiently to the welding zone through the upper adherend.

Effect of Polymer Matrix on Inelastic Strain Development in PI- and PEI-Based Composites Reinforced with Short Carbon Fibers under Low-Cyclic Fatigue.

Since the inelastic strain development plays an important role in the low-cycle fatigue (LCF) of High-Performance Polymers (HPPs), the goal of the research was to study the effect of an amorphous polymer matrix type on the resistance to cyclic loading for both polyimide (PI)- and polyetherimide (PEI)-based composites, identically loaded with short carbon fibers (SCFs) of various lengths, in the LCF mode. The fracture of the PI and PEI, as well as their particulate composites loaded with SCFs at an aspect ratio (AR) of 10, occurred with a significant role played by cyclic creep processes. Unlike PEI, PI was less prone to the development of creep processes, probably because of the greater rigidity of the polymer molecules. This increased the stage duration of the accumulation of scattered damage in the PI-based composites loaded with SCFs at AR = 20 and AR = 200, causing their greater cyclic durability. In the case of SCFs 2000 µm long, the length of the SCFs was comparable to the specimen thickness, causing the formation of a spatial framework of unattached SCFs at AR = 200. The higher rigidity of the PI polymer matrix provided more effective resistance to the accumulation of scattered damage with the simultaneously higher fatigue creep resistance. Under such conditions, the adhesion factor exerted a lesser effect. As shown, the fatigue life of the composites was determined both by the chemical structure of the polymer matrix and the offset yield stresses. The essential role of the cyclic damage accumulation in both neat PI and PEI, as well as their composites reinforced with SCFs, was confirmed by the results of XRD spectra analysis. The research holds the potential to solve problems related to the fatigue life monitoring of particulate polymer composites.

Structure and Deformation Behavior of Polyphenylene Sulfide-Based Laminates Reinforced with Carbon Fiber Tapes Activated by Cold Atmospheric Plasma.

Low-temperature plasma treatment with atmospheric discharge with runaway electrons (DRE) was shown to be an efficient way to activate carbon fiber's (CF) surface and subsequently increase its interlayer shear strength (ILSS) values. It was demonstrated that an acceptable ILSS level was achieved after a DRE plasma treatment duration of 15 min. The treatment of CFs resulted in their surface roughness being increased and their functional groups grafting. The XPS data showed a change in the chemical composition and the formation of reactive oxygen-containing groups. SEM examinations of the PPS/CF laminates clearly demonstrated a difference in adhesive interaction at the PPS/CF interface. After the DRE plasma treatment, CFs were better wetted with the polymer, and the samples cohesively fractured predominantly through the matrix, but not along the PPS/CF interface, as was observed for the sample reinforced with the untreated CFs. The computer simulation results showed that raising the adhesive strength enhanced the ILSS values, but reduced resistance to transverse cracking under the loading pin. In general, higher flexural strength of the PPS/CF laminates was achieved with a greater interlayer adhesion level, which was consistent with the obtained experimental data.

Surface Modification of Carbon Fibers by Low-Temperature Plasma with Runaway Electrons for Manufacturing PEEK-Based Laminates.

(1) Background: The paper addresses the effect of carbon fibers (CFs) treatment by low-temperature plasma with runaway electrons on the deformation behavior of the polyetheretherketone (PEEK)-layered composites. (2) Methods: The effect of the interlayer adhesion on the mechanical response of the composites was assessed through the tensile and three-point bending tests. In addition, computer simulations of the three-point bending were carried out with the use of the finite element analysis (FEM) with varying conditions at the "PEEK-CF layers" interface. (3) Results: DRE-plasma treatment during the optimal time of t = 15 min led to formation of a rougher surface and partial desizing of a finishing agent. The shear strength of the layered composites increased by 54%, while the tensile strength and the flexural modulus (at three-point bending) increased by 16% (up to 893 MPa) and by 10% (up to 93 GPa), respectively. (4) Conclusions: The results of the numerical experiments showed that the increase in the stiffness, on the one hand, gave rise to enlarging the flexural modulus; on the other hand, a nonlinear decrease in the strength may occur. For this reason, the intention to maximize the level of the interlayer stiffness can result in lowering the fracture toughness, for example, at manufacturing high-strength composites.

Optimizing Ultrasonic Welding Parameters for Multilayer Lap Joints of PEEK and Carbon Fibers by Neural Network Simulation.

The aim of this study is to substantiate the use machine learning methods to optimize a combination of ultrasonic welding (USW) parameters for manufacturing of multilayer lap joints consisting of two outer PEEK layers, a middle prepreg of unidirectional carbon fibers (CFs), and two energy directors (EDs) between them. As a result, a mathematical problem associated with determining the optimal combination of technological parameters was formulated for the formation of USW joints possessing improved functional properties. In addition, a methodology was proposed to analyze the mechanical properties of USW joints based on neural network simulation (NNS). Experiments were performed, and threshold values of the optimality conditions for the USW parameters were chosen. Accordingly, NNS was carried out to determine the parameter ranges, showing that the developed optimality condition was insufficient and required correction, taking into account other significant structural characteristics of the formed USW joints. The NNS study enabled specification of an extra area of USW parameters that were not previously considered optimal when designing the experiment. The NNS-predicted USW mode (P = 1.5 atm, t = 800 ms, and τ = 1500 ms) ensured formation of a lap joint with the required mechanical and structural properties (σUTS = 80.5 MPa, ε = 4.2 mm, A = 273 N·m, and Δh = 0.30 mm).

Role of Testing Conditions in Formation of Tribological Layers at Line Contacts of Antifriction CF-Reinforced PI- and PEI-Based Composites.

High-strength PI and PEI polymers differ by chemical structure and flexibility of the polymer chains that ensure lower cost and higher manufacturability of the latter. The choice of a particular polymer matrix is of actuality at design of antifriction composites on their basis. In this study, a comparative analysis of tribological behavior of PI and PEI- based composites was carried out with linear contact rubbing. The neat materials, as well as the two- and three-component composites reinforced with chopped carbon fibers, were investigated. The third components were typically used, but were different in nature (polymeric and crystalline) being solid lubricant fillers (PTFE, graphite and MoS2) with characteristic dimensions of several microns. The variable parameters were both load and sliding speed, as well as the counterface material. It was shown that an improvement of the tribological properties could be achieved by the tribological layer formation, which protected their wear track surfaces from the cutting and plowing effects of asperities on the surfaces of the metal and ceramic counterparts. The tribological layers were not formed in both neat polymers, while disperse hardening by fractured CF was responsible for the tribological layer formation in both two- and three component PI- and PEI-based composites. The effect of polymer matrix in tribological behavior was mostly evident in two-component composites (PI/CF, PEI/CF) over the entire P⋅V product range, while extra loading with Gr and MoS2 leveled the regularities of tribological layer formation, as well as the time variation in friction coefficients.

Algorithm for Designing a Removable Complete Denture (RCD) Based on the FEM Analysis of Its Service Life.

(1) Background: The paper addresses the computer simulation and prediction of the service life of the base of removable complete dentures (RCDs) under typical loads caused by biting and chewing food. For this purpose, the finite element method (FEM) was used. It is assumed that various blocks of teeth, such as incisors, canines, premolars and molars, are subjected to cyclic impacts during a human life. (2) Methods: Both symmetric and asymmetric mastication (two- and one-sided loads, respectively) cases were considered. The load level was assumed to be 100 N, which corresponds to the average muscular compression force of typical human jaws. (3) Results: The FEM analysis of the stress-strain state evolution for RCDs under cyclic loads was carried out. Maps of equivalent lines were drawn for the denture base in terms of its durability. A multi-axial criterion was implemented to determine the number of cycles prior to failure by the mechanism of a normal opening mode crack. The FEM-based assessment of the service life of RCDs enabled us to establish the critical stress concentration areas, thereby allowing for further planning for the correction of an occlusal scheme or teeth inclinations. As a result, the service life of RCDs under cyclic loading can be improved. (4) Conclusions: An algorithm for designing RCDs in the case of edentulism based on the FEM simulation using commercial software as part of the procedure is proposed.

An Automated Optical Strain Measurement System for Estimating Polymer Degradation under Fatigue Testing.

(1) Background: this study deals with design of an automated laboratory facility based on a servo-hydraulic testing machine for estimating parameters of mechanical hysteresis loops by means of the digital image correlation (DIC) method. (2) Methods: the paper presents a description of the testing facility, describes the grounds for calculating the elastic modulus, the offset yield strength (OYS) and the parameters of the mechanical hysteresis loops by the DIC method. (3) Results: the developed hardware-software facility was tested by studying the fatigue process in neat polyimide (PI) under various amplitude tension-tension loadings. It was found that the damage accumulation was accompanied by the decrease in the loop areas, while failure occurred when it reduced by at least ~5 kJ/m3. (4) Conclusions: it was shown that lowering the loop area along with changing the secant modulus value makes it possible to estimate the level of the scattered damage accumulation (mainly at the stresses above the OYS level). It was revealed that fractography data, namely the pattern and sizes of the fatigue crack initiation and propagation zones, did not correlate well with the dependences of the parameters of the hysteresis loops.

Estimating Low- and High-Cyclic Fatigue of Polyimide-CF-PTFE Composite through Variation of Mechanical Hysteresis Loops.

The fatigue properties of neat polyimide and the "polyimide + 10 wt.% milled carbon fibers + 10 wt.% polytetrafluoroethylene" composite were investigated under various cyclic loading conditions. In contrast to most of the reported studies, constructing of hysteresis loops was performed through the strain assessment using the non-contact 2D Digital Image Correlation method. The accumulation of cyclic damage was analyzed by calculating parameters of mechanical hysteresis loops. They were: (i) the energy losses (hysteresis loop area), (ii) the dynamic modulus (proportional to the compliance/stiffness of the material) and (iii) the damping capacity (calculated through the dissipated and total mechanical energies). On average, the reduction in energy losses reached 10-18% at the onset of fracture, whereas the modulus variation did not exceed 2.5% of the nominal value. The energy losses decreased from 20 down to 18 J/m3 (10%) for the composite, whereas they reduced from 30 down to 25 J/m3 (17%) for neat PI in the low-cycle fatigue mode. For high-cycle fatigue, energy losses decreased from 10 to 9 J/m3 (10%) and from 17 to 14 J/m3 (18%) for neat PI and composite, respectively. For this reason, the changes of the energy losses due to hysteresis are of prospects for the characterization of both neat PI and the reinforced PI-based composites.

High Performance Polymer Composites: A Role of Transfer Films in Ensuring Tribological Properties-A Review.

The purpose of this review is to summarize data on the structure, mechanical and tribological properties, and wear patterns of composites based on high-performance polymers (HPPs) intended for use in friction units. The review includes three key sections, divided according to the tribological contact schemes regardless of the polymer matrix. In the second part, the analysis of composites is carried out in point contacts. The third section is devoted to the results of studies of HPP-based composites in linear ones. The fourth section summarizes information on flat contacts. Particular attention is paid to the formation of transfer films (TFs) in the contacts and their influence on the tribological patterns of the studied rubbing materials. As a conclusion, it is noted that the challenge of experimental methods for analyzing TFs, stated by K. Friedrich, is effectively solved in recent studies by the XPS method, which enables us to accurately determine their composition. Although this determination is completed after the tribological tests, it allows not only a more accurate interpretation of their results considering specific conditions and loading schemes, but also the ability to design HPP-based composites that form required TFs performing their preset functions.

Effect of Transfer Film on Tribological Properties of Anti-Friction PEI- and PI-Based Composites at Elevated Temperatures.

The structure, mechanical and tribological properties of the PEI- and PI-based composites reinforced with Chopped Carbon Fibers (CCF) and loaded with commercially available micron-sized solid lubricant fillers of various nature (polymeric-PTFE, and crystalline-Gr and MoS2) were studied in the temperature range of 23-180 (240) °C. It was shown that tribological properties of these ternary composites were determined by the regularities of the transfer film (TF) adherence on their wear track surfaces. The patterns of TFs formation depended on the chemical structure of the polymer matrix (stiffness/flexibility) as well as the tribological test temperatures. Loading with PTFE solid lubricant particles, along with the strengthening effect of CCF, facilitated the formation and fixation of the TF on the sliding surfaces of the more compliant PEI-based composite at room temperature. In this case, a very low coefficient of friction (CoF) value of about 0.05 was observed. For the more rigid identically filled PI-based composite, the CoF value was twice as high under the same conditions. At elevated temperatures, rising both CoF levels and oscillation of their values made it difficult to retain the non-polar PTFE transfer film on the sliding surfaces of the PI-based composite. As a result, friction of the ceramic counterpart proceeded over the composite surface without any protecting TF at T ≥ 180 °C. For the sample with the more flexible PEI matrix, the PTFE-containing TF was retained on its sliding surface, providing a low WR level even under CoF rising and oscillating conditions. A similar analysis was carried out for the less efficient crystalline solid lubricant filler MoS2.

Experimental-FEM Study on Effect of Tribological Load Conditions on Wear Resistance of Three-Component High-Strength Solid-Lubricant PI-Based Composites.

The structure, mechanical and tribological properties of the polyimide-based composites reinforced with chopped carbon fibers (CCF) and loaded with solid-lubricant commercially available fillers of various natures were investigated. The metal- and ceramic counterparts were employed for tribological testing. Micron sized powders of PTFE, colloidal graphite and molybdenum disulfide were used for solid lubrication. It was shown that elastic modulus was enhanced by up to 2.5 times, while ultimate tensile strength was increased by up 1.5 times. The scheme and tribological loading conditions exerted the great effect on wear resistance of the composites. In the tribological tests by the 'pin-on-disk' scheme, wear rate decreased down to ~290 times for the metal-polymer tribological contact and to ~285 times for the ceramic-polymer one (compared to those for neat PI). In the tribological tests against the rougher counterpart (Ra~0.2 µm, the 'block-on-ring' scheme) three-component composites with both graphite and MoS2 exhibited high wear resistance. Under the "block-on-ring" scheme, the possibility of the transfer film formation was minimized, since the large-area counterpart slid against the 'non-renewable' surface of the polymer composite (at a 'shortage' of solid lubricant particles). On the other hand, graphite and MoS2 particles served as reinforcing inclusions. Finally, numerical simulation of the tribological test according to the 'block-on-ring' scheme was carried out. Within the framework of the implemented model, the counterpart roughness level exerted the significantly greater effect on wear rate in contrast to the porosity.

Deformation Behavior and Fracture Patterns of Laminated PEEK- and PI-Based Composites with Various Carbon-Fiber Reinforcement.

Laminated composites based on polyetheretherketone (PEEK) and polyimide (PI) matrices were fabricated by hot compression. Reinforcing materials (unidirectional carbon-fiber (CF) tapes or carbon fabric) and their layout patterns were varied. Stress-strain diagrams after three-point flexural tests were analyzed, and both lateral faces of the fractured specimens and fractured surfaces (obtained by optical and scanning electron microscopy, respectively) were studied. It was shown that the laminated composites possessed the maximum mechanical properties (flexural elastic modulus and strength) in the case of the unidirectional CF (0°/0°) layout. These composites were also not subjected to catastrophic failure during the tests. The PEEK-based composites showed twice the flexural strength of the PI-based ones (0.4 and 0.2 GPa, respectively), while the flexural modulus was four times higher (60 and 15 GPa, correspondently). The reason was associated with different melt flowability of the used polymer matrices and varied inter- (intra)layer adhesion levels. The effect of adhesion was additionally studied by computer simulation using a developed two-dimensional FE-model. It considered initial defects between the binder and CF, as well as subsequent delamination and failure under loads. Based on the developed FE-model, the influence of defects and delamination on the strength properties of the composites was shown at different stress states, and the corresponding quantitative estimates were reported. Moreover, another model was developed to determine the three-point flexural properties of the composites reinforced with CF and carbon fabric, taking into account different fiber layouts. It was shown within this model framework that the flexural strength of the studied composites could be increased by an order of magnitude by enhancing the adhesion level (considered through the contact area between CF and the binder).

UHMWPE-Based Glass-Fiber Composites Fabricated by FDM. Multiscaling Aspects of Design, Manufacturing and Performance.

The aim of the paper was to improve the functional properties of composites based on ultra-high molecular weight polyethylene (UHMWPE) by loading with reinforcing fibers. It was achieved by designing the optimal composition for its subsequent use as a feedstock for 3D-printing of guides for roller and plate chains, conveyors, etc. As a result, it was experimentally determined that loading UHMWPE with 17% high density polyethylene grafted with VinylTriMethoxySilane (HDPE-g-VTMS) was able to bind 5% glass fillers of different aspect ratios, thereby determining the optimal mechanical and tribological properties of the composites. Further increasing the content of the glass fillers caused a deterioration in their tribological properties due to insufficient adhesion of the extrudable matrix due to the excessive filler loading. A multi-level approach was implemented to design the high-strength anti-friction 'UHMWPE+17%HDPE-g-VTMS+12%PP'-based composites using computer-aided algorithms. This resulted in the determination of the main parameters that provided their predefined mechanical and tribological properties and enabled the assessment of the possible load-speed conditions for their operation in friction units. The uniform distribution of the fillers in the matrix, the pattern of the formed supermolecular structure and, as a consequence, the mechanical and tribological properties of the composites were achieved by optimizing the values of the main control parameters (the number of processing passes in the extruder and the aspect ratio of the glass fillers).

Effect of Various Type of Nanoparticles on Mechanical and Tribological Properties of Wear-Resistant PEEK + PTFE-Based Composites.

The mechanical and tribological properties of polyetheretherketone (PEEK)- and PEEK + PTFE (polytetrafluoroethylene)-based composites loaded with and four types of nanoparticles (carbonaceous, metallic, bimetal oxide, and ceramic) under metal- and ceramic-polymer tribological contact conditions were investigated. It was found that loading with the nanofillers in a small content (0.3 wt.%) enabled improvement of the elastic modulus of the PEEK-based composites by 10-15%. In the metal-polymer tribological contact, wear resistance of all nanocomposites was increased by 1.5-2.3 times. In the ceramic-polymer tribological contact, loading PEEK with metal nanoparticles caused the intensification of oxidation processes, the microabrasive counterpart wear, and a multiple increase in the wear rate of the composites. The three component "PEEK/10PTFE/0.3 nanofillers" composites provided an increase in wear resistance, up to 22 times, for the metal-polymer tribological contact and up to 12 times for the ceramic-polymer one (with a slight decrease in the mechanical properties) compared to that of neat PEEK. In all cases, this was achieved by the polymer transfer film formation and adherence on the counterparts. The various effects of the four types of nanoparticles on wear resistance were determined by their ability to fix the PTFE-containing transfer film on the counterpart surfaces.