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At least 109 records · Page 6

Reinforcing of thermoplastic polycarbonate and polysulfone with carbon fibers: Production and characteristics of UD-compound objects

The production and characteristics of the carbon fiber reinforced thermoplastics polycarbonate and polysulfone are described. The production of prepregs from defined polymer solutions is emphasized along with methods of optimizing the production of compounds. The characteristics of unidirectionally reinforced thermoplastics, such as shear strength, bending strength, and impact resistance are compared with regard to fracture behavior, the influence of intermediate layers, and the behavior under cryogenic conditions and under slightly elevated temperatures. The problem of adhesion between high strength carbon fibers and thermoplastics is examined, taking into account the effect of moisture on the shear strength and the impact resistance.

Fitzer, E.↗

Thermoplastic coated carbon fibers for textile preforms

A continuous process for producing prepreg from carbon fiber and thermoplastic matrix is described. After the tow has been spread using a pneumatic device, the process utilizes a fluidized bed to apply thermoplastic powder to the bundle. Finally, direct electrical heating of the coated fiber tow melts the polymer on the individual fibers, creating a uniform and extremely flexible prepreg. The efficiency of the process was evaluated during initial trials in which a thermoplastic polyimide, LaRC-TPI, was applied to T-300, 3K (3000 filament) carbon fiber tow. The physical properties of unidirectional composite specimens fabricated from this prepreg were measured, and the matrix uniformity and void content of the samples was determined. The results of these evaluations are detailed and discussed.

Allen, L. E.↗

Thermoplastic coating of carbon fibers

A process is being developed which evenly coats individual carbon fibers with thermoplastic polymers. In this novel, continuous coating process, the fiber tow bundle is first spread cover a series of convex rollers and then evenly coated with a fine powder of thermoplastic matrix polymer. Next, the fiber is heated internally by passing direct current through the powder coated fiber. The direct current is controlled to allow the carbon fiber temperature to slightly exceed the flow temperature of the matrix polymer. Analysis of the thermoplastic coated carbon fiber tows produced using this continuous process indicates that 30 to 70 vol pct fiber prepregs can be obtained.

Edie, D. D.↗

Advanced thermoplastic resins, phase 1

Eight thermoplastic polyimide resin systems were evaluated as composite matrix materials. Two resins were selected for more extensive mechanical testing and both were versions of LaRC-TPI (Langley Research Center - Thermoplastic Polyimide). One resin was made with LaRC-TPI and contained 2 weight percent of a di(amic acid) dopant as a melt flow aid. The second system was a 1:1 slurry of semicrystalline LaRC-TPI powder in a polyimidesulfone resin diglyme solution. The LaRC-TPI powder melts during processing and increases the melt flow of the resin. Testing included dynamic mechanical analysis, tension and compression testing, and compression-after-impact testing. The test results demonstrated that the LaRC-TPI resins have very good properties compared to other thermoplastics, and that they are promising matrix materials for advanced composite structures.

Hendricks, C. L.↗

High Pressure Gas Permeation and Liquid Diffusion Studies of Coflon and Tefzel Thermoplastics

The life of fluid-carrying flexible or umbilical pipes during service at elevated temperatures and pressures depends inter alia on their resistance to attack by the fluids present and the rate at which these fluids are absorbed by the pipe lining materials. The consequences of fluid ingress into the thermoplastic lining could mean a) a reduction in its mechanical strength, to increase chances of crack formation and growth and thus a loss of integrity, b) the occurrence of permeation right through the lining material, with pressure build- up in the outer pipe wall construction (of flexible pipes) or chemical attack (from a hostile permeant) on outer layers of reinforcements. Therefore it is important within this project to have relevant permeation data for Coflon and Tefzel thermoplastics: the former is plasticised, the latter is not. A previous report (CAPP/M.2) described experimental equipment and techniques used by MERL when measuring high pressure (up to 5000 psi) gas permeation and liquid diffusion through thermoplastic samples cut from extruded bar or pipe, and provided the basic theory involved. Norsk Hydro are also performing gas permeation tests on pipe sections, at up to 100 bars (1450 psi) pressure or so, and reporting separately. Some comparisons between data from Norsk Hydro and MERL have been made herein. The tests should be considered as complementary, as the Norsk Hydro test has the obvious benefit of using complete pipe sections, whilst MERL can test at much higher pressures, up to 1000 bar if necessary. The sophisticated analytical measuring equipment of Norsk Hydro can distinguish the individual components of mixed gases and hence the various permeation-linked coefficients whereas MERL, in using pressure increase at constant volume to determine permeation rate, is limited to obtaining single gas data, or apparent (or representative) coefficients for a mixed gas as a whole. Except for the initial fluid diffusion data for Tefzel described in CAPP/M.2, the present report covers all aspects of fluid permeation and diffusion for Coflon and Tefzel, including all the pen-neation data accumulated in the project to date. Test gases have mainly been methane (CH4) and carbon dioxide (CO2). More high pressure (HP) gas permeation tests have been performed since the last issue of this report, most being concerned with changes in permeation characteristics brought about by ageing in various relevant fluids. This revision supersedes previous issues.

Morgan, G. J.↗

Process for preparing tapes from thermoplastic polymers and carbon fibers

The instant invention involves a process for use in preparing tapes or rovings, which are formed from a thermoplastic material used to impregnate longitudinally extended bundles of carbon fibers. The process involves the steps of (a) gas spreading a tow of carbon fibers; (b) feeding the spread tow into a crosshead die; (c) impregnating the tow in the die with a thermoplastic polymer; (d) withdrawing the impregnated tow from the die; and (e) gas cooling the impregnated tow with a jet of air. The crosshead die useful in the instant invention includes a horizontally extended, carbon fiber bundle inlet channel, means for providing melted polymer under pressure to the die, means for dividing the polymeric material flowing into the die into an upper flow channel and a lower flow channel disposed above and below the moving carbon fiber bundle, means for applying the thermoplastic material from both the upper and lower channels to the fiber bundle, and means for withdrawing the resulting tape from the die.

Chung, Tai-Shung↗

Fiber reinforced thermoplastic resin matrix composites

Polyimide polymer composites having a combination of enhanced thermal and mechanical properties even when subjected to service temperatures as high as 700.degree. F. are described. They comprise (a) from 10 to 50 parts by weight of a thermoplastic polyimide resin prepared from 2,2-bis[4-(4-aminophenoxy)phenyl]hexafluoropropane and (b) from 90 to 50 parts by weight of continuous reinforcing fibers, the total of (a) and (b) being 100 parts by weight. Composites based on polyimide resin formed from 2,2-bis[4-(4-aminophenoxy)phenyl]hexafluoropropane and pyromellitic dianhydride and continuous carbon fibers retained at least about 50% of their room temperature shear strength after exposure to 700.degree. F. for a period of 16 hours in flowing air. Preferably, the thermoplastic polyimide resin is formed in situ in the composite material by thermal imidization of a corresponding amide-acid polymer prepared from 2,2-bis[4-(4-aminophenoxy)phenyl]hexafluoropropane. It is also preferred to initially size the continuous reinforcing fibers with up to about one percent by weight of an amide-acid polymer prepared from 2,2-bis[4-(4-aminophenoxy)phenyl]hexafluoropropane. In this way imidization at a suitable elevated temperature results in the in-situ formation of a substantially homogeneous thermoplastic matrix of the polyimide resin tightly and intimately bonded to the continuous fibers. The resultant composites tend to have optimum thermo-mechanical properties.

Jones, Robert J.↗

Progressive Damage and Failure Analysis of Thermoplastic Composites in Low Velocity Impact Using MAT299

As part of the NASA Hi-Rate Composite Aircraft Manufacturing (HiCAM) Project, state-of-the-art progressive damage and failure analysis (PDFA) tools developed for use with thermosets are being evaluated for use in modeling alternative material systems, like thermoplastics. Experimental low-velocity impact data of a thermoplastic material system, AS4D/PEKK-FC, is presented and includes characterization of the impact damage mechanisms as well as associated load and displacement data. Following the presentation of experimental results, two simulation approaches using the PDFA tool MAT299 in the commercial off-the-shelf finite element software LS-DYNA are employed to predict damage area and force-displacement responses of thermoplastic panels subjected to various impact energies. The first modeling method uses solid elements with a high-density mesh and a ply-by-ply modeling approach similar to previously published work for thermosets. This method has the capability of capturing individual crack development, progression, and delamination on a per ply basis. The second modeling method uses TSHELL elements that have in-plane dimensions that are an order of magnitude larger than the elements used in the solid element approach and reduces the number of elements through the thickness of the laminate. The second method produces a lower-fidelity model with reduced run times that is incapable of monitoring every delamination plane and damage within each ply. Results from both simulation methods are compared to experiment, and limitations of the methods are discussed.

Composite material↗

Impact of Thermoplastic Composites: Testing and Modeling

Thermoplastic Composites (TPCs) are being increasingly considered for aerospace applications given their faster manufacturing cycles and lower cost. The NASA High-Rate Composite Aircraft Manufacturing (HiCAM) project aims to evaluate and mature thermoplastic manufacturing technologies to achieve a 4X – 6X increase in the production rate of Next Generation Single Aisle (NGSA) commercial aircraft. Within the purview of the NASA HiCAM project, this work investigates the response of thermoplastic composites under High Energy Dynamic Impact (HEDI) conditions. HEDI tests were conducted on panels fabricated with a carbon fiber reinforced low melt semi-crystalline resin TC1225 LMPAEK T700G (T700/LMPAEK) material system. A blunt metal projectile was chosen as the impactor which impacted the T700/LMPAEK panels over a range of impact velocities imparted by a gas gun test setup. As expected, lower velocity impacts caused the projectile to rebound, whereas higher velocity impacts resulted in the projectile penetrating the test panels. For the test cases in which the projectile rebounded, the predominant damage modes recorded with Ultrasonic (UT) scans included interlaminar delaminations which exhibited a “rotating fan” like structure when viewed in the through-thickness direction. Additionally, the dynamic deflection of the center point of the back face of the test panels and the velocity of the projectile was also measured during the test. Finite element models were developed to model a test case in which the projectile rebounded. These models were developed using LS-DYNA® wherein the primary objective was to evaluate the ability of the material model MAT299 to capture the dynamic response and damage modes in the T700/LMPAEK test panels. MAT299 is a Deformation Gradient Decomposition (DGD) based Continuum Damage Mechanics (CDM) material model. The interlaminar delaminations were modeled using the cohesive contact formulation available within LS-DYNA, wherein the mixed-mode fracture is captured via the Benzegaggh-Kenane (B-K) mode-dependent fracture energy interpolation law. The above-mentioned modeling methods have mostly been applied to thermoset composites and adapting them to TPCs involves addressing challenges associated with appropriately representing material behavior. Therefore, the proposed paper would discuss the systematic approach taken to adapt thermoset composites modeling practices for applications to TPCs, while appropriately addressing material behavior. The proposed paper would show that the predicted back face center point deflection correlated reasonably well with experimental data and the peak deflection was predicted to be within 5% of the experimental measurements. The projectile rebound velocity, while predicted to be higher than the experimental measurement, was within reasonable bounds. Additionally, it will be shown that the predicted delamination shapes were in reasonable agreement with experimental data.

Composite Materials↗

Progressive Damage and Failure Analysis of Thermoplastic Composites in Low Velocity Impact Using MAT299

As part of the NASA Hi-Rate Composite Aircraft Manufacturing (HiCAM) Project, state-of-the-art progressive damage and failure analysis (PDFA) tools developed for use with thermosets are being evaluated for use in modeling alternative material systems, like thermoplastics. Experimental low-velocity impact data of a thermoplastic material system, AS4D/PEKK-FC, is presented and includes characterization of the impact damage mechanisms as well as associated load and displacement data. Following the presentation of experimental results, two simulation approaches using the PDFA tool MAT299 in the commercial off-the-shelf finite element software LS-DYNA are employed to predict damage area and force-displacement responses of thermoplastic panels subjected to various impact energies. The first modeling method uses solid elements with a high-density mesh and a ply-by-ply modeling approach similar to previously published work for thermosets. This method has the capability of capturing individual crack development, progression, and delamination on a per ply basis. The second modeling method uses TSHELL elements that have in-plane dimensions that are an order of magnitude larger than the elements used in the solid element approach and reduces the number of elements through the thickness of the laminate. The second method produces a lower-fidelity model with reduced run times that is incapable of monitoring every delamination plane and damage within each ply. Results from both simulation methods are compared to experiment, and limitations of the methods are discussed.

Composite material↗

Thermal Modelling of the In-Situ Consolidation of Automated Fiber Placement Of Thermoplastic Composites

NASA is developing carbon fiber reinforced thermoplastic composites processing methods under the Hi-rate Composites Aircraft Manufacturing (HiCAM) project. The in-situ consolidation automated fiber placement (AFP) of thermoplastics (ICAT) process has high potential to increase manufacturing throughput and lower costs as it combines the repeatability and fast lay down rates of AFP with the thermoformability of thermoplastic materials to achieve part consolidation out of the autoclave. Physics-based process models are employed in the ICAT process development to understand, fundamentally, the thermal response of the carbon fiber (CF)/polyaryletherketone (PAEK) material during the rapid heating and cooling associated with laser-assisted AFP. A one-dimensional (1-D) closed-form model was developed capable of predicting the temperature profile through-the-thickness of the tape material. In addition, a two-dimensional (2-D) model was developed to predict the heat transfer through-the-thickness and in the laydown x-direction of the AFP head motion. The solution of the Fourier heat-transfer equations in the 2-D model is approximated using the explicit finite difference numerical method. The temperature profile during placement of PAEK slit-tape materials at various laser target temperatures and placement speeds were measured during ICAT process development trials at Electroimpact®, Inc. †† and the resulting experimental data is compared with the predictions of the 1-D and 2-D models.

Christopher J. Stelter↗

Nanocellular thermoplastic foam and process for making the same

Prepare a thermoplastic polymer foam having a porosity of 70% or more and at least one of: (i) an average cell size of 200 nanometers or less; and (ii) a nucleation density of at least 1.times.1015 effective nucleation sites per cubic centimeter of foamable polymer composition not including blowing agent using a foamable polymer composition containing a thermoplastic polymer selected from styrenic polymer and (meth)acrylic polymers, a blowing agent comprising at least 20 mole-percent carbon dioxide based on moles of blowing agent and an additive having a Total Hansen Solubility Parameter that differs from that of carbon dioxide by less than 2 and that is present at a concentration of 0.01 to 1.5 weight parts per hundred weight parts thermoplastic polymer.

36 MATERIALS SCIENCE↗

Synthesis of High-Performance Lignin-Based Inverse Thermoplastic Vulcanizates with Tailored Morphology and Properties

In this work, we report synthesis of a high-strength renewable phenolic composition with linear large deformation strain without a thermoplastic-like yielding while retaining thermal processability. Small molecule carboxylic acid derivatives with varying molecular architectures act as esterifying crosslinkers in an equal mass mixture of lignin and acrylonitrile–butadiene copolymers in a highly scalable, solvent-free process. These “inverse thermoplastic vulcanizates” (iTPVs)—unique in their approach of crosslinking the rigid lignin phase rather than the soft phase—exhibit ordered self-assembly, tunable nanoscale morphology, and processability. The first of its kind iTPV compositions exhibit engineering stress–strain curves with two- to sixfold linear extensibility, a twofold rise in strength, and an order of magnitude enhanced modulus compared to a simple lignin-rubber blend. Viscoelastic properties correlate well with crosslinker architecture and the resulting morphology, allowing competing properties of toughness and stiffness to be tuned. This research finds a path for identifying the potential of lignin as a sustainable feedstock.

36 MATERIALS SCIENCE↗

Rheological survey of carbon fiber-reinforced high-temperature thermoplastics for big area additive manufacturing tooling applications

Carbon fiber (CF)-reinforced thermoplastic composites have been widely used in different structural applications due to their superior thermal and mechanical properties. The big area additive manufacturing (BAAM) system, developed at Oak Ridge National Laboratory’s Manufacturing Demonstration Facility, has been used to manufacture several composite components, demonstration vehicles, molds, and dies. These components have been designed and fabricated using various CF-reinforced thermoplastics. In this study, the dynamic rheological and mechanical properties of a material commonly used in additive manufacturing, 20 wt% CF-acrylonitrile butadiene styrene (ABS), as well as three CF-reinforced high-temperature polymers, 25 wt% CF-polyphenylsulfone (PPSU), 35 wt% CF-polyethersulfone (PES), and 40 wt% CF-polyphenylene sulfide (PPS), used to print molds were investigated. The viscoelastic properties, namely storage modulus, loss modulus, tan delta, and complex viscosity, of these composites were studied, and the rheological behavior was related to the BAAM extrusion and bead formation process. The results showed 20 wt% CF-ABS and 40 wt% CF-PPS to display a more dominant elastic component at all frequencies tested while 25 wt% CF-PPSU and 35 wt% CF-PES have a more dominant viscous component. This viscoelastic behavior is then used to inform the deposition and bead formation process during extrusion on the BAAM system.

36 MATERIALS SCIENCE↗

Multi-Source Machine Learning and Thermoplastics Enhanced Aerostructure Manufacturing (mTEAM)

RTX Technology Research Center (RTRC), together with Collins Aerospace (Collins) and Oak Ridge National Laboratory (ORNL) has developed an Artificial Intelligence (AI) / Machine Learning (ML) guided solution to advance the manufacturing and assembly of high performance and lightweight thermoplastic composite (TPC) aerospace products. The solution aims to lower risk, cost and lead time for induction heating based welding and consolidation processes for TPC structure. The cost and lead time of part and material specific process development for induction welding (IW) and induction consolidation will be reduced by replacing traditional empirical methods with optimization methods that merge AI/ML and physics-based process simulations and process experiments with sensing and controls. TPC-IW process development is empirical in nature, and uncertainties in material & process behavior exist near & far from the induction coil. Physics-based simulations can be leveraged directly for process optimization but can be too computationally expensive to run in high fidelity and real time to do robust process optimization. The key impact of successful TPC induction consolidation and welding is cost & lead time reduction for part & material specific consolidation and welding recipes. This is an enabler for more rapid deployment of TPC structures via joining assembly, which can reduce energy & cost intensive usage of autoclaves & ovens. The solution aimed to advance the U.S. Department of Energy’s interests in using thermoplastics and automation in composite manufacturing for improvement of products for existing markets via increased production speeds, reduced costs, and lowered use of energy. Welded TPC structures can offer significant weight & energy savings for high-value commercial aerospace & industrial applications compared to metal & thermoset composite structures assembled by mechanical fastening and/or adhesive bonding. The project was organized into two Budget Periods. Budget Period 1 (BP1) was 15 months and its goal was to perform ML process optimization framework development & deployment on lab-coupon aerostructure components. A Go/No-Go Review was performed at the end of BP1 to verify fulfilment of key tasks & milestones to justify a Go Decision to move into the next Budget Period. Budget Period 2 (BP2) was 12 months and its goal was the deployment of the ML framework for ML process optimization of pilot industrial scale aerostructure components. The overall project aim was to develop & demonstrate ML-enhanced modeling framework that learns process-property mapping from multiple data sources at different fidelities. During BP1, the team accomplished key tasks & milestones to demonstrate the concept of multi-source ML for TPC aerostructure consolidation and assembly. First, the team completed documentation of induction based TPC heating requirements including baseline metrics to compare measured results against. Next the team completed demonstration of data generation from physics-based simulations for ML surrogate model generation and demonstrated the integration of physics-based simulation data into multi-source AI/ML algorithms. In parallel, the team established the lab-coupon scale induction welding system and completed a process to label and reduce generated data from physics-based simulation and experiments for ML surrogate models to enable multi-source ML model training & testing. To complete BP1, the team integrated physics-based simulation data and experimental data into multi-source ML algorithms. This was based on the team completing ML deployment of the induction welding on a lab system at RTRC and AI/ML deployment on existing induction welding line at Collins. ORNL visited both Collins and RTRC sites to witness the TPC induction welding process. Then, ORNL designed and constructed a new version of their vision-based sensing system better adapted to acquire process signals of the TPC induction welding process for process anomaly and defect detection. In BP2, the team accomplished key tasks & milestones to scale up multi-source ML for TPC aerostructure consolidation and assembly from the lab-coupon scale to the pilot-industrial scale. In BP2, the team demonstrated real time anomaly & defect detection via experiments performed by ORNL & RTRC. The team completed ML-optimization heating trials for TPC induction consolidation at Collins, and the team confirmed pilot industrial scale experimental data from Collins was compatible with the developed ML pipeline from RTRC. The team completed sub-element scale ML process optimization demonstration at RTRC, where the team leveraged RTRC’s robotic TPC welding setup to de-risk the ML process optimization by performing ML analysis of recorded temperatures to account for complex part features. Then, the team applied its ML-derived control strategies and ML process optimization framework at Collins to the pilot-industrial scale on a demo skin-stiffener part representative of a nacelle aerostructure fan cowl section. The key innovation is the AI/ML framework enabling effective process development of high performance, lightweight, energy efficient TPCs for composite aircraft structures.

36 MATERIALS SCIENCE↗

Field testing of thermoplastic encapsulants in high‐temperature installations

Abstract Recently there has been increased interest in using thermoplastic encapsulant materials in photovoltaic modules, but concerns have been raised about whether these would be mechanically stable at high temperatures in the field. Recently, this has become a significant topic of discussion in the development of IEC 61730 and IEC 61215. We constructed eight pairs of crystalline‐silicon modules and eight pairs of glass/encapsulation/glass thin‐film mock modules using different encapsulant materials, of which only two were formulated to chemically crosslink. One module set was exposed outdoors with thermal insulation on the back side in Mesa, Arizona, in the summer (hot‐dry), and an identical module set was exposed in environmental chambers. High‐precision creep measurements (±20 μ m) and electrical performance measurements indicate that despite many of these polymeric materials operating in the melt or rubbery state during outdoor deployment, no significant creep was seen because of their high viscosity, lower operating temperature at the edges, and/or the formation of chemical crosslinks in many of the encapsulants with age despite the absence of a crosslinking agent. Only an ethylene‐vinyl acetate ( EVA ) encapsulant formulated without a peroxide crosslinking agent crept significantly. In the case of the crystalline‐silicon modules, the physical restraint of the backsheet reduced creep further and was not detectable even for the EVA without peroxide. Because of the propensity of some polymeric materials to crosslink as they age, typical thermoplastic encapsulants would be unlikely to result in creep in the vast majority of installations.

14 SOLAR ENERGY↗

Investigation of Droplet Deposition for Suspensions Usable for Thermoplastic 3D Printing (T3DP)

Thermoplastic 3D printing (T3DP) is an additive manufacturing (AM) technology, which can be used for the production of dense single- and especially multi-material components. This becomes possible because of the combination of the precise deposition of small droplets of molten thermoplastic suspensions containing ceramic or metal particles, and a curing mechanism caused on cool down increasing the viscosity. In this paper, the droplet formation behavior of zirconia suspensions for T3DP (82 and 84 wt.%) was investigated. The droplet fusion factor (dff) is introduced to calculate the necessary distance between two droplets to form filament-like structures by fusion of adjacent droplets. Filament-like structures with a smooth surface and a nearly homogeneous cross section were manufactured for both suspensions with a dff of 44% or higher.

36 MATERIALS SCIENCE↗

A validation of the fibre orientation and fibre length attrition prediction for long fibre-reinforced thermoplastics

To improve the mechanical performance of polymeric parts, fibre reinforcement has established in industrial applications during the last decades. Next to the widely used Short Fibre-reinforced Thermoplastics (SFT) the use of Long Fibre-reinforced Thermoplastics (LFT) is increasingly growing. Especially for non-polar polymeric matrices like polypropylene (PP), longer fibres can significantly improve the mechanical performance. As with every kind of discontinuous fibre reinforcement the fibre orientations (FO) show a high impact on the mechanical properties. On the contrary to SFT where the local fibre length distribution (FLD) can be often neglected, for LFT the FLD show a high impact on the material’s properties and has to be taken into account in equal measure to the FOD. Recently numerical models are available in commercial filling simulation software and allow predicting both the local FOD and FLD in LFT parts. The aim of this paper is to compare i.) the FOD results and ii) the FLD results from available orientation- and fibre length attrition-models to those obtained from experimental data. The investigations are conducted by the use of different injection moulded specimens made from long glass fibre reinforced PP. In order to determine the FOD, selected part sections are examined by means of Computed Tomographic (CT) analyses. The fully three dimensional measurement of the FOD is then performed by digital image processing using grey scale correlation. The FLD results are also obtained by using digital image processing after a thermal pyrolytic separation of the polymeric matrix from the fibres. Further the FOD and the FLD are predicted by using a reduced strain closure (RSC) as well as an anisotropic rotary diffusion - reduced strain closure model (ARD-RSC) and Phelps-Tucker fibre length attrition model implemented in the commercial filling software Moldflow, Autodesk Inc., San Rafael, CA, USA.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗