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1

The Effect of Zirconium as an Alloying Additive on the Microstructure and Properties of AlSi9Mg Alloy Cast in Sand Moulds

Kamińska Jadwiga, Angrecki Michał, Dudek P.

Archives of Foundry Engineering

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2022

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Vol. 22, iss. 4

5--13

EN

The aim of the study was to select the optimal content of zirconium introduced as an alloying additive to obtain the best strength properties of Al-Si alloy. A technically important disadvantage is the tendency of silumins to form a coarse-grained structure that adversely affects the mechanical properties of castings. To improve the structure, modification processes and alloying additives are used, both of which can effectively refine the structure and thus increase the mechanical properties. According to the Hall-Petch relationship, the finer is the structure, the higher are the mechanical properties of the alloy. The proposed addition of zirconium as an alloying element has a beneficial effect on the structure and properties of silumins, inhibiting the grain growth. The starting material was an aluminium-silicon casting alloy designated as EN AC-AlSi9Mg (AK9). Zirconium (Zr) was added to the alloy in an amount of 0.1%, 0.2%, 0.3%, 0.4% and 0.5% by weight. From the modified alloy, after verification of the chemical composition, samples were cast into sand moulds based on a phenolic resin. The first step in the research was testing the casting properties of alloys with the addition of Zr (castability, density, porosity). In the next step, the effect of zirconium addition on the structure and mechanical properties of castings was determined.

2

Study on Reclamation of CO2-hardened Sodium Silicate Used Sands by CaO Powder

Lu Jijun, Yang Lei, Qian Jiangbing, He Wei, Wang Huafang

Archives of Foundry Engineering

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2022

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Vol. 22, iss. 3

99--106

EN

Aiming at the problems of wet reclamation consuming a lot of water, dry (mechanical) reclamation having wear and power consumption, this paper to find suitable reclamation reagents to reduce the influence of harmful substances in used sodium silicate sands. By comparing the reclamation effect of CaO, Ca(OH)2 and Ba(OH)2 reclamation powder reagents, it was concluded that CaO had the best reclamation effect. Through the single factor experiment, the influence of CaO on the reclamation effect was explored: 1. addition amount of CaO; 2. the additional amount of water; 3. reclamation time. The orthogonal results showed that the CaO reclamation effect was the best when the amount of CaO was 1.5%, the amount of sodium silicate was 4.0%, the amount of water added was 6.0%, and the reclamation time was 12.0h. In this experiment, 82.2% carbonate and 75.0 % silicate in used sands can be removed. The microscopic analysis of the reclamation sands was carried out by scanning electron microscope (SEM); The surface was relatively smooth, without large area cracks and powder accumulation. Compared with the used sands, the instant, 24h ultimate, and residual strengths of the reclaimed sands were increased by 536.5%, 458.1%, and 89.8%, respectively, which was beneficial to the reclamation of the CO2 sodium silicate used sands.

3

Mechanical performance of ultra-high-performance strain-hardening cementitious composites according to binder composition and curing conditions

Kim Min-Jae, Oh Taekgeun, Yoo Doo-Yeol

Archives of Civil and Mechanical Engineering

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2022

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Vol. 22, no. 2

art. no. e63, 1--12.

EN

This study investigated the mechanical properties and microstructures of three ultra-high-performance strain-hardening cementitious composites (UHP-SHCCs) with different mix proportions and curing conditions. The binders comprised ordinary Portland cement (OPC), silica fume, and ground granulated blast furnace slag (GGBS); the specimens were cured under air and wet curing conditions for 28 and 91 days, respectively. Compressive and direct tensile tests were performed, along with subsequent microstructural analyses using the particle packing theory and scanning electron microscopy, on the composite matrix and reinforcing polyethylene (PE) fibers. The test results indicate that the inclusion of GGBS, more than 50% (by weight of OPC), leads to a decrease in compressive and tensile strength by up to 35.7% but an increase in ductility by up to 55.9%. In addition, a higher content of GGBS resulted in larger deviations based on the curing conditions. The wet curing condition was more effective for the development of a higher energy absorption capacity than the air curing condition at a curing age of 28 d. By contrast, 91 d of wet curing resulted in the lowest strain energy in this study, mainly because of the considerably reduced strain capacity.

4

Effect of grinding conditions on the friction and wear performance of Ni based singlecrystal superalloy

Xu Yunchao, Gong Yadong, Zhang Weiqiang, Wen Xuelong, Xin Bo, Zhang Huan

Archives of Civil and Mechanical Engineering

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2022

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Vol. 22, no. 2

art. no. e102, 1--16

EN

In this work, four types of surfaces were prepared as follows: untreated one, dry grinding (DG), wet grinding (WG) and minimum quantity lubrication grinding (MQLG) for Ni-based single crystal superalloy. The effects of grinding conditions on the surface roughness and microstructure evolution were studied. Dry sliding tests of ground surfaces were carried out at room temperature. Through the quantitative characterization of the wear rate, the area, width and depth of the worn profiles, the friction and wear mechanism of superalloy prepared by different grinding conditions were analyzed. The results show that the MQLG surface with low surface roughness and work hardening behavior has the best wear resistance. The element transfer behavior from the GCr15 ball to the worn surface was detected by EDS analysis. The wear type is mainly abrasive wear, accompanied by slight adhesive wear and oxidation wear. It is shown that high-quality surface with nanocrystalline and high density dislocation structure produced by MQLG improves the tribological properties of superalloy, which provide theoretical guidance for the surface machining of single crystal blade to reduce fretting wear.

5

Ductility and formability of Al-Mn ultrafine-grained plates processed by incremental equal channel angular pressing

Ciemiorek Marta, Goliński Jacek, Majchrowicz Kamil, Lewandowska Małgorzata

Archives of Civil and Mechanical Engineering

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2022

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Vol. 22, no. 4

art. no. e169, 2022

EN

The article discusses the ductility and formability of ultrafine-grained 3003 aluminum alloy plates processed using incremental equal channel angular pressing. The influence of temperature and strain rate is evaluated by means of tensile tests and cupping tests under various conditions. It is reported that tensile elongation increases two-fold at elevated temperature, without excessive grain growth. With the right selection of processing conditions, the formability of the plate, expressed as cup height deformed in a cupping test, can be enhanced—as much as 62% compared with room temperature when the processing takes place at 150 °C. The improvement in ductility was attributed to a reduced apparent activation volume due to grain refinement, which translated into improved strain rate sensitivity.

6

Rozwój mechanicznego utwardzania nawierzchni drogowych z tłucznia

Skowron Marek

Drogownictwo

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2021

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nr 3

89--92, III okł.

PL

W artykule przedstawiono rozwój mechanicznego utwardzania nawierzchni drogowych z tłucznia w niektórych krajach europejskich w XIX w. Opisano wybrane rodzaje walców drogowych z napędem konnym i parowym. Podano również główne szczegóły ich konstrukcji oraz ich wady i zalety.

EN

This paper presents the development of mechanical drystone road paving in some European countries in the 19th century. Selected types of horse and steam powered road rollers are described. The main details of their design and their advantages and disadvantages are also given.

7

Początki mechanicznego utwardzania nawierzchni drogowych na ziemiach polskich

Skowron Marek

Drogownictwo

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2021

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nr 7-8

226--230

PL

W artykule przedstawione zostały początki mechanicznego utwardzania nawierzchni drogowych na ziemiach polskich. Podano szczegóły konstrukcji powszechnie używanego walca konnego systemu Majewskiego. Opisano także pierwsze próby zastosowania walców parowych.

EN

This paper presents the development of mechanical hardening of road pavement in Poland. Selected types of horse and steam powered road rollers are described. Details of the construction of the commonly used Majewski system horse roller are given. The first attempts to use steam rollers were also described.

8

Use of the Precipitation Hardening to Improve the Properties of Combustion Engine Pistons Made of Eutectic Silumins

Trepczyńska-Łent Małgorzata, Műller K.

Archives of Foundry Engineering

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2021

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Vol. 21, iss. 4

77--81

EN

The study presents methods to be used for improving the performance parameters of car engine pistons made of EN AC-AlSi12CuNiMg alloy according to the PN-EN 1706: 2011. Pistons of slow sucking and turbocharged engines were researched. A solution heat and ageing treatments were applied according to four variants. Temperatures of the solution heat treatment were: 550 ±5°C; 510°C ±5°C; and alternate: 276 ±5°C/510 ±5°C. The solution time ranged from 6 min to 4 h. Temperatures of the ageing heat treatment were 20°C and 250°C, while the ageing time ranged from 1,5 to 3h. Natural ageing was performed in 5 days. Measurements of hardness HRB and the piston diameters were performed. An improvement in the performance parameters of combustion engines was observed. Three solution heat treatment and ageing variants, allowed to obtain the pistons with hardness equal/higher than pistons of the turbocharged engines. The test results confirmed the possibility of providing a piston with properties exceeding the high load parameters specified by the manufacturer. Further studies will make it possible to improve the effects of the proposed solutions.

9

Inter-relationship between microstructure evolution and mechanical properties in inertia friction welded 8630 low-alloy steel

Banerjee Amborish, Ntovas Michail, Da Silva Laurie, Rahimi Salaheddin, Wynne Bradley

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 4

244--262

EN

The evolution of microstructure and mechanical properties in AISI 8630 low-alloy steel subjected to inertia friction welding (IFW) have been investigated. The effects of three critical process parameters, viz. rotational speed, friction and forge forces, during welding of tubular specimens were explored. The mechanical properties of these weld joints, including tensile and Charpy V-notch impact were studied for determining the optimum welding parameters. The weld joints exhibited higher yield strength, lower hardening capacity and ultimate tensile strength compared to base metal (BM). The maximum strength and ductility combination was achieved for the welds produced under a nominal weld speed of ~ 2900–3100 rpm, the highest friction force of ~ 680–720 kN, and the lowest axial forging load of ~ 560–600 kN. The measured hardness distribution depicted higher values for the weld zone (WZ) compared to the thermo-mechanically affected zone (TMAZ), heat-affected zone (HAZ) and BM, irrespective of the applied welding parameters. The substantial increase in the hardness of the WZ is due to the formation of microstructures that were dominated by martensite. The observed microstructural features, i.e. the fractions of martensite, bainite and ferrite, show that the temperature in the WZ and TMAZ was above Ac3, whereas that of the HAZ was below Ac1 during the IFW. The fracture surface of the tensile and impact-tested specimens exhibited the presence of dimples nucleating from the voids, thus indicating a ductile failure. EBSD maps of the WZ revealed the formation of subgrains inside the prior austenite grains, indicating the occurrence of continuous dynamic recrystallisation during the weld. Analysis of crystallographic texture indicated that the austenite microstructure (i.e. FCC) in both the WZ and TMAZ undergoes simple shear deformation during IFW.

10

Influence of curing conditions on the mechanical performance of ultra-high-performance strain-hardening cementitious composites

Kim Min-Jae, Oh Taekgeun, Yoo Doo-Yeol

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 3

607--617

EN

This study investigated the influence of curing conditions and the inclusion of ground granulated blast furnace slag (GGBS) on the mechanical performance of ultra-high-performance strain-hardening cementitious composites (UHP-SHCC). Air- and wet-curing conditions were applied for 28 and 91 days, respectively. Compressive strength and direct tensile tests were performed, and the microstructure of the tested cementitious matrix and surface of the polyethylene (PE) fibers were inspected using scanning electron microscopy. The results showed that 3 months of wet-curing notably deteriorated the tensile performance of UHP-SHCC with or without GGBS as compared to those at the curing age of 1 month, whereas the 3 months of air-curing further enhanced the tensile performance. Therefore, the 3 months air-cured specimens, using binders consisting only of ordinary portland cement (OPC) or OPC with GGBS, could develop the highest tensile strength and strain capacity of up to 12.1 MPa and 9.1% or 13.6 MPa and 9.1%, respectively. The inclusion of GGBS led to a higher rate of stress development as well as tensile strength at the air-curing age of 3 months, resulting in the highest energy absorption capacity of 985 kJ/m3 measured in this study.

11

Experimental and numerical investigation on plasticity and fracture behaviors of aluminum alloy 6061-T6 extrusions

Zhu Peihua, Zhang Qilin, Xu Hongjun, Ouyang Yuanwen

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 3

13--30

EN

Aluminum alloy 6061-T6 (AA 6061-T6) extrusions have been widely applied in large-span reticulated shells. However, researches on behaviors of the aluminum alloy under large deformation and fracture by means of meso-mechanics are still insufficient. This paper focuses on the plastic and fracture behaviors of AA 6061-T6 extrusions over a wide range of stress states. Experiments on smooth and notched round bars, grooved plates and shear plates are conducted. It is observed that the yield stress decreases as the notch or groove radius decreases. The yield stress under plane strain or pure shear is lower than that under axisymmetric tension. In addition, true stress–true strain data obtained from tensile coupon tests overestimate the hardening behavior at large deformation. To fully describe these experimental observations, a new plastic model considering the pressure dependence, Lode angle effect, and hardening with post-necking correction is proposed. Regarding the ductile fracture, modified Mohr–Coulomb fracture model with damage-induced softening is adopted to predict crack initiation and propagation under different stress states. The material models of plasticity and fracture are numerically implemented into FE code ABAQUS/explicit by the material subroutine VUMAT. It is found that both the overall nominal stress–strain curves and fracture patterns for all specimens are well predicted by the material models. For practical engineering design, recommended reduced values of shear strength and high-triaxiality tensile strength of AA 6061-T6 extrusions are given.

12

Elastic-plastic torsion problem with non-linear hardenings using the method of fundamental solution

Moayyedian Farzad, Grabski Jakub Krzysztof

Archives of Civil and Mechanical Engineering

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2021

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Vol. 21, no. 4

334--354

EN

In the current research, a torsion of isotropic prismatic rods with elastic–plastic behavior under non-linear hardening behavior, such as Swift, Voce, and Ramberg-Osgood relations, is investigated with the method of fundamental solutions. Based on the Saint-Venant displacement assumption and deformation theory of plasticity for the stress-strain relation, the non-linear boundary value problem for the stress function is formulated. The purpose of the current research is study the elastic–plastic torsion problem with non-linear hardenings in a new simple form and solving the presented equations with the method of fundamental solutions and radial basis functions. The non-linear torsion problem is solved by means of the Picard iteration method. The proposed algorithm is based on solution of the linear Poisson equation at each iteration step.

13

Factors Influencing Cavitation Erosion of NiCrSiB Hardfacings Deposited by Oxy-Acetylene Powder Welding on Grey Cast Iron

Szala Mirosław, Walczak Mariusz, Hejwowski Tadeusz

Advances in Science and Technology. Research Journal

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2021

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Vol. 15, no 4

376--386

EN

The work presents the results of a study on cavitation erosion (CE) resistance of two NiCrSiB self-fluxing powders deposited by oxy-acetylene powder welding on cast iron substrate grade EN-GJL-200. The mean hardness of deposits A-NiCrSiB, C-NiCrSiB is equal to 908 HV, 399 HV and exceeds those of EN-GJL-200 and X5CrNi18-10 reference specimens 197 HV and 209 HV, respectively. To study CE, the vibratory apparatus has been used and tests were conducted according to the ASTM G32 standard. Cavitation eroded surfaces were examined using a profilometer, optical and scanning electron microscopy. The research indicated that the CE resistance, expressed by the cumulative mass loss decreased in the following order C-NiCrSiB > A-NiCrSiB > X5CrNi18-10 > EN-GJL-200. Therefore, hardfacings were characterised by lower cumulative mass loss, in turn, higher CE resistance than the reference sample and therefore they may be applied as layers to increase resistance to cavitation of cast iron machine components. Results indicate that in the case of multiphase materials, hardness cannot be the main indicator for CE damage prediction while it strongly depends on the initial material microstructure. To qualitatively estimate the cavitation erosion damage (CEd) of NiCrSiB self-fluxing alloys at a specific test time, the following factors should be considered: material microstructure, physical and mechanical properties as well as surface morphology and material loss both estimated at specific exposure time. A general formula for the CEd prediction of NiCrSiB deposits was proposed.

14

Renowacje infrastruktury wod-kan są eko

Borkowski Marek

Wodociągi - Kanalizacja

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2020

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nr 5

32--34

15

Analysis of Hard Facing Material Designed for Application on the Forestry Tools

Ťavodová Miroslava, Hnilicová Michaela, Miturska Izabela

Advances in Science and Technology. Research Journal

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2019

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Vol. 13, no 4

111--119

EN

The article deals with the assessment specific hard facing material with W2C high content – filled welding rod for manual flame hard facing RD 571. Is was applied in two layers on the samples (16MnCr5 steel) of the tool for crushing undesirable advance growths, frequently used in forestry technologies. The tools work in a heterogeneous environment. They are loaded by abrasive impression of wood mass, which are characterized by various hardness. The material loss on the functional parts is significant. The wear degradation of the tool and its subsequent early decommissioning represents economic losses for forest companies. The investigations conducted under laboratory conditions involved hardness measuring HV10 and HRC, wear resistance testing of materials by friction against loosely fixed abrasive parts, with assessment profile and depth of the mark as well as assessment of quality mixing of the layers, microstructure assessment by SEM with EDX analyses. By evaluating the laboratory tests, the conclusions were drawn to determine the suitability or the non-suitability of the selected hardfacing material in service. Whether the material increases the service life of the undesirable crusher tool, will by verified by performing field trials.

16

Influence of the Reclaim Addition into the Matrix on the Hardening Process Kinetics of the Moulding Sand with Furfuryl Resin

Jamrozowicz Ł., Kolczyk J., Wojtuń P.

Archives of Foundry Engineering

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2018

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Vol. 18, iss. 1

41--46

EN

Measurements of the hardening process course of the selected self-hardening moulding sands with the reclaimed material additions to the matrix, are presented in the hereby paper. Moulding sands were produced on the „Szczakowa” sand (of the Sibelco Company) as the matrix of the main fraction FG 0,40/0,32/0,20, while the reclaim was added to it in amounts of 20, 50 and 70%. Regeneration was performed with a horizontal mechanical regenerator capacity of 10 t/h. In addition, two moulding sands, one on the fresh sand matrix another on the reclaimed matrix, were prepared for comparison. Highly-fluid urea-furfuryl resin was used as a binder, while paratoluensulphonic acid as a hardener. During investigations the hardening process course was determined, it means the wave velocity change in time: cL = f(t). The hardening process kinetics was also assessed (dClx/dt = f(t)). Investigations were carried out on the research stand for ultrasound tests. In addition strength tests were performed.

17

Fenantropterydyna i 12-karboksydibenzo[a,c]fenazyna jako fotoinicjatory polimeryzacji rodnikowej triakrylanów

Pyszka I., Kucybała Z.

Przemysł Chemiczny

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2017

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T. 96, nr 4

908--911

PL

Zsyntezowano fenantropterydynę i 12-karboksydibenzo[ a,c]fenazynę i zastosowano jako fotoinicjatory polimeryzacji rodnikowej triakrylanów działające w zakresie światła widzialnego. Do testowania skuteczności fotoinicjatora charakteryzowanej szybkością początkową polimeryzacji stosowano monomer TMPTA. W badanych układach fotoinicjujących polimeryzację barwnik pełnił funkcję akceptora elektronu lub akceptora atomu wodoru, a koinicjator polimeryzacji rodnikowej funkcję donora elektronu lub atomu wodoru. Dobierając warunki polimeryzacji triakrylanu TMPTA (PETTA), można było obserwować odchylenie od typowego kształtu krzywych fotoinicjowanej polimeryzacji.

EN

Phenantropteridine and 12-carboxydibenzo[a,c]phenazine were synthesized and used as photoinitiators for radical polymerization of 2-ethyl-2-hydroxymethyl-1,3-propandiol and pentaerythritol triacrylates in the visible light. In the studied systems, a photoinitiating polymerization dye acting as an electron acceptor or hydrogen acceptor, and a coinitiator of radical polymerization acting as an electron donor or H donor were also used. Deviations from the typical shape of the photoinitiated polymerization curves (“humps”) were obsd.

18

Epoksydowane oleje roślinne w wytwarzaniu materiałów powłokowych

Sałaciński Ł., Milchert E.

Przemysł Chemiczny

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2017

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T. 96, nr 5

1156--1161

PL

Przedstawiono sposoby wytwarzania materiałów powłokowych, opierając się na epoksydowanych olejach roślinnych, produktach ich przekształceń do estrów glicydolowych epoksydowanych kwasów tłuszczowych, lub wielokarboksylowych kwasów cykloalifatycznych w reakcji z alkoholem allilowym, epoksydacji epichlorohydryną lub glicydolem. Wykazano możliwości epoksydacji produktów transestryfikacji olejów eterem monoglikolu etylenowo-winylowego i epoksydacji norbornanów. Omówiono wyniki utwardzania epoksydowanych olejów i produktów ich funkcjonalizacji za pomocą bezwodników kwasów karboksylowych i poliamin.

EN

A review, with 59 refs., of the methods for prepg. the title paint coatings by using carboxylic acid anhydrides and polyamines.

19

Proces wiązania poprzez polimeryzację metakryloamidu prowadzony in situ na osnowie mineralnej

Grabowska B.

Prace Instytutu Odlewnictwa

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2017

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Vol. 57, no. 2

115--124

PL

Proces wiązania poprzez polimeryzację metakryloamidu prowadzono in situ na osnowie kwarcowej w temperaturze 25°C oraz w polu mikrofal (150°C; 800 W; 2,45 GHz). Skład mieszaniny reakcyjnej, oprócz metakryloamidu – jako monomeru, zawierał: inicjator (nadsiarczan amonu), aktywator (N,N,N’,N’-tetrametyloetylenodimina), upłynniacz (sorbitol) oraz osnowę kwarcową. Zmiany strukturalne powstałe wskutek oddziaływań makromolekularnych w mieszaninie reakcyjnej podczas procesu polimeryzacji metakryloamidu, a dalej konsolidacji ziaren osnowy, określono metodą spektroskopii w podczerwieni przy użyciu spektrometru z transformacją Fouriera (FTIR). Stwierdzono, że podczas sieciowania powstają nowe wiązania w obrębie grup polarnych, typu: -OH, >NH, >C=O oraz silanolowych (Si-O). Ponadto przeprowadzono termoanalizę układu reakcyjnego w zakresie temperatury 25−500°C z zastosowaniem metody spektroskopii rozproszonego odbicia (FTIR-DRS). Badania te miały na celu określenie zakresu temperatury, w którym dochodzi do zmian związanych z sieciowaniem układu, a jeszcze nie jego termodestrukcją. Dla otrzymanego układu poreakcyjnego wykonano badania mikroskopowe metodą SEM. Uzyskana wiedza na temat procesu wiązania, poprzedzonego polimeryzacją metakryloamidu prowadzoną in situ na osnowie kwarcowej, pozwoli na podjęcie dalszych prac w obszarze technologii formowania i konsolidacji układów polimerowo-krzemianowych metodą druku 3D.

EN

The process of bonding by polymerization of methacrylamide was performed in situ on a quartz matrix at 25°C as well as in a microwave field (150°C; 800 W; 2.45 GHz). The composition of the reaction mixture, beside methylacrylamide as the monomer, was: the initiator (ammonium persulfate), the activator (N,N,N’,N’-tetramethylethylenediamine), the fluidizer (sorbit) and the quartz matrix. The structural changes occurring as a result of macromolecular reactions in the reaction mixture during the process of methacrylamide polymerization and then consolidation of the matrix grains, were determined by means of Fourier transform infrared spectroscopy (FTIR). It was found that during the cross-linking, new bonds are formed within the polar groups, of the following types: -OH, >NH, >C=O and silanol (Si-O). Also, a thermoanalysis of the reaction system was performed in the temperature range of 25−500°C with the use of dispersed reflection spectroscopy (FTIR-DRS). These studies aimed at determining the temperature range in which we can observe changes connected with the cross-linking of the system, and not its thermodestruction. For the obtained post-reaction system, SEM microscopic examinations were performed. The obtained knowledge of the bonding process, preceded by methacrylamide polymerization performed in situ on a quartz matrix, will make it possible to conduct further research in the scope of the technology of forming and consolidating polymer-silicate systems with the method of 3D printing.

20

Cam-clay models in mechanics of granular materials

Ilyashenko A. V., Kuznetsov S. V.

Mechanics and Mechanical Engineering

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2017

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Vol. 21, nr 4

813--821

EN

The mathematical models for granular materials utilizing concept of the critical state, is reviewed. Several extensions of the critical state Modified Cam-Clay (MCC) models are reviewed, including kinematic hardening with bounding surface (BS), the general plasticity (GP) model, extension of the MCC model to include finite strain, and different variants of the pressure hardening rule, including bi-modulus extension, hypoplastic, and the hyperelastic potential extension. The associated flow rules coupled with different hardening equations are considered. In the review the main attention is paid to the case of the infinitesimal strains.