A unified theory for brittle and ductile shear mode fracture

A unified theory captures both brittle and ductile fracture. The fracture toughness is proportional to the applied stress squared and the length of the crack. For purely brittle solids, this criterion is equivalent to Griffith's theory. In other cases, it provides a theoretical basis for the Irwin-Orowan formula. For purely ductile solids, the theory makes direct contact with the Bilby-Cottrell-Swinden model. The toughness is highest in ductile materials because the shielding dislocations in the plastic zone provide additional resistance to crack growth. This resistance is the force opposing dislocation motion, and the Peach-Koehler force overcomes it. A dislocation-free zone separates the plastic zone from and the tip of the crack. The dislocation-free zone is finite because molecular forces responsible for the cohesion of the surfaces near the crack tip are not negligible. At the point of crack growth, the length of the dislocation-free zone is constant and the shielding dislocations advance in concert. As in Griffith's theory, the crack is in unstable equilibrium. The theory shows that a dimensionless variable controls the elastoplastic behaviour. A relationship for the size of the dislocation-free zone is derived in terms of the macroscopic and microscopic parameters that govern the fracture.     

Toughness mechanism in polypropylene composites: Polypropylene toughened with elastomer and calcium carbonate

Polypropylene (PP) was modified with elastomer or CaCO₃ particles of two different sizes (1 μm and 50 nm) in various volume fractions. The dispersion morphology and mechanical properties of the two systems were investigated as functions of the particle size and volume fraction of the modifier. The brittle‐to‐tough transition occurred when the matrix ligament thickness was less than the critical ligament thickness, which was about 0.1 μm for the PP used here, being independent of the type of modifier. At the same matrix ligament thickness, the improvement of the toughness was obviously higher with the elastomer rather than with CaCO₃, but adding CaCO₃ increased the modulus of PP. © 2004 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 42: 1656–1662, 2004     

一些特殊图类乘积的离散度

连通图的离散度是用s(G)来表示的,s(G)=max{ω(G-S)-|S|:ω(G-S)>1,SV(G)}.给出了两个完全图乘积的和一个完全图与路的乘积的离散度.还给出了两个完全图乘积的坚韧度.     

图的边韧性度

文[1]中,定义图G(V,E)的边韧性度定义为min{(|S|+T(G-S))/(ω(G-S)):S?E(G)},这里,T-(G-S)和ω(G-S)分别表示G-S中最大分支的顶点数和连通分支数.这是一个能衡量网络图稳定性较好的参数,因为它不仅考虑到了图G-S的分支数也考虑到了它的阶数.在以前的工作中,作者得到了边韧性度图的一个充要条件.利用这些结果证明了K-树是严格边韧性度图,并找到了边韧性度与较高阶的边坚韧度和边坚韧度之间的关系.     

Divinylsiloxane‐bisbenzocyclobutene‐based polymer modified with polystyrene–polybutadiene–polystyrene triblock copolymers

Divinylsiloxane‐bisbenzocyclobutene (DVS‐bisBCB) polymer has very low dielectric constant and dissipation factor, good thermal stability, and high chemical resistance. The fracture toughness of the thermoset polymer is moderate due to its high crosslink density. A thermoplastic elastomer, polystyrene–polybutadiene–polystyrene triblock copolymer, was incorporated into the matrix to enhance its toughness. The cured thermoset matrix showed different morphology when the elastomer was added to the B‐staged prepolymer or when the elastomer was B‐staged with the DVS‐bisBCB monomer. Small and uniformly distributed elastomer domains were detected by transmission electron micrographs (TEM) in the former case, but TEM did not detect a separate domain in the latter case. A high percentage of the polystyrene–polybutadiene–polystyrene triblock copolymer could be incorporated into the DVS‐bisBCB thermoset matrix by B‐staging the triblock copolymer with the BCB monomer. The elastomer increased the fracture toughness of DVS‐bisBCB polymer as indicated by enhanced elongation at break and increased K1c values obtained by the modified edge‐lift‐off test. Elastomer modified DVS‐bisBCB maintained excellent electrical properties, high T_g and good thermal stability, but showed higher coefficient of linear thermal expansion values. © 2006 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem 44: 1591–1599, 2006     

Independence of the brittle–ductile transition from the rubber particle size for impact‐modified poly(vinyl chloride)

A series of acrylic impact modifiers (AIMs) with different particle sizes ranging from 55.2 to 927.0 nm were synthesized by seeded emulsion polymerization, and the effect of the particle size on the brittle–ductile transition of impact‐modified poly(vinyl chloride) (PVC) was investigated. For each AIM, a series of PVC/AIM blends with compositions of 6, 8, 10, 12, and 15 phr AIM in 100 phr PVC were prepared, and the Izod impact strengths of these blends were tested at 23 °C. For AIMs with particle sizes of 55.2, 59.8, 125.2, 243.2, and 341.1 nm, the blends fractured in the brittle mode when the concentration of AIM was lower than 10 phr, whereas the blends showed ductile fracture when the AIM concentration reached 10 phr. It was concluded that the brittle–ductile transition of the PVC/AIM blends was independent of the particle size in the range of 55.2–341.1 nm. When the particle size was greater than 341.1 nm, however, the brittle–ductile transition shifted to a higher AIM concentration with an increase in the particle size. Furthermore, the critical interparticle distance was found not to be the criterion of the brittle–ductile transition for the PVC/AIM blends. © 2006 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys 44: 696–702, 2006     

聚氯乙烯接枝丙烯酸稀土的研究

将无机稀土有机化后与聚氯乙烯(PVC)进行接枝共聚。探讨了氧化稀土与丙烯酸反应、PVC接枝丙烯酸稀土的原理和方法，并通过红外光谱、扫描电镜对接枝共聚物进行定性表征，用微量热天平对接枝共聚物的热性能进行测定。实验结果表明，经丙烯酸稀土接枝共聚后的PVC，其耐热分解性能和韧性有了明显的提高。     

Diglycidyl ether of bisphenol-A epoxy resin–polyether sulfone/polyether sulfone ether ketone blends: phase morphology, fracture toughness and thermo-mechanical properties

The properties of diglycidyl ether of bisphenol-A epoxy resin toughened with poly(ether sulfone ether ketone) (PESEK) and poly(ether sulfone) (PES) polymers were investigated. PESEK was synthesised by the nucleophilic substitution reaction of 4,4’-difluorobenzophenone with dihydroxydiphenylsulfone using sulfolane as solvent and potassium carbonate as catalyst at 230 °C. The T _g–composition behaviour of the homogeneous epoxy resin/PESEK blend was modelled using Fox, Gordon–Taylor and Kelley–Bueche equations. A single relaxation near the glass transition of epoxy resin was observed in all the blend systems. From dynamic mechanical analysis, the crosslink density of the blends was found to decrease with increase in the thermoplastic concentration. The storage modulus of the epoxy/PESEK blends was lower than that of neat resin, whilst it is higher for epoxy/PES blends up to glass transition temperature, thereafter it decreases. Scanning electron microscopic studies of the blends revealed a homogeneous morphology. The homogeneity of the blends was attributed to the similarity in chemical structure of the modifier and the cured epoxy network and due to the H-bonding interactions between the blend components. The fracture toughness of epoxy resin increased on blending with PESEK and PES. The increase in fracture toughness was due to the increase in ductility of the matrix. The thermal stability of the blends was comparable to that of neat epoxy resin.     

Effect of Hyperbranched Polyester on Modification of Epoxy Resins Cured with Anhydride

Epoxy resins are widely used in coatings, adhesives and polymer composites, but the applications of cured epoxy resins are often restricted by their poor toughness. HBP can be used as toughener to improve the toughness of epoxy resins due to its high-dens…     

Curing behavior and toughening performance of epoxy resins containing hyperbranched polyester

The effects of the hyperbranched polyester with hydroxyl end groups (HBPE‐OH) on the curing behavior and toughening performance of a commercial epoxy resin (diglycidyl ether of bisphenol A, DGEBA) were presented. The addition of HBPE‐OH into DGEBA strongly increased its curing rate and conversion of epoxide group due to the catalytic effect of hydroxyl groups in HBPE‐OH and the low viscosity of the blend at curing temperature. The improvements on impact strength and critical stress intensity factor (or fracture toughness, K_1c) were observed with adding HBPE‐OH. The impact strength was 8.04 kJ m^?1 when HBPE‐OH reached 15 wt% and the K_1c value was approximately two times the value of pure epoxy resin when HBPE‐OH content was 20 wt%. The morphology of the blends was also investigated, which indicated that HBPE‐OH particles, as a second phase in the epoxy matrix, combined with each other as the concentration of HBPE‐OH increased. Copyright © 2004 John Wiley & Sons, Ltd.