On the determination of the cohesive zone parameters for the modeling of micro-ductile crack growth in thick specimens

The paper deals with the determination of the cohesive zone parameters (separation energy, , and cohesive strength, T _max) for the 3D finite element modeling of the micro-ductile crack growth in thick, smooth-sided compact tension specimens made of a low-strength steel. Since the cohesive zone parameters depend, in general, on the local constraint conditions around the crack tip, their values will vary along the crack front and with crack extension. The experimental determination of the separation energy via automated fracture surface analysis is not accurate enough. The basic idea is, therefore, to estimate the cohesive zone parameters, and T _max, by fitting the simulated distribution of the local crack extension values along the crack front to the experimental data of a multi-specimen J _IC-test. Furthermore, the influence of the cohesive zone parameters on the crack growth behavior is investigated. The point of crack growth initiation is determined only by the magnitude of . Both and T _max affect the crack growth rate (or the crack growth resistance), but the influence of the cohesive strength is much stronger than that of the separation energy. It turns out that T _max as well as vary along the crack front. In the center of the specimen, where plane strain conditions prevail, the separation energy is lower and the cohesive strength is higher than at the side-surface.     

On the directional stability of wedging

The directional stability of the crack path during wedging of a strip is investigated, using finite element methods. Linearly elastic material and plane conditions are considered. No dynamic effects are included. Stable crack growth in the direction of maximum mode I stress intensity factor at the tip of the crack is assumed. It is found that directional stability seems to prevail if the thickness of the wedge at the foremost point of contact between the wedge and the crack surfaces is less than about 1.69 K _Ic% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca% WG3bGaai4laiaadweaaSqabaaaaa!3A93!\[\sqrt {w/E}\], where K _Ic is the fracture toughness, w half the width of the strip and E the modulus of elasticity.     

Strength characteristics and fatigue crack growth in a composite with long aligned fibers

Strength values and fatigue crack growth on a specially made composite material are reported. The composite specimen consisted of an epoxy matrix and one layer of long aligned glass fibers that were equally spaced. The results on strength showed that for a range of fiber spacing , the composite's strength _c, scaled with a fiber spacing in the form of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa!39C2!\[\sigma \] c % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca% qG7oaaleqaaaaa!395B!\[\sqrt {\text{\lambda }} \] = % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOUdaaa!393F!\[{\text{\kappa }}\]. Based on dimensional arguments the constant was found proportional to the fracture toughness of the matrix. Fatigue crack propagation experiments were performed on specimens with different fiber spacing and applied loads. The crack speed reached a steady mode of propagation in specimens where relation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa!39C2!\[\sigma \] c % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca% qG7oaaleqaaaaa!395B!\[\sqrt {\text{\lambda }} \] = % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeOUdaaa!393F!\[{\text{\kappa }}\] was satisfied. The same mode of propagation was reached for the debonding along fibers in the bridging zone as well as the crack opening displacement. The crack opening displacement at a fiber location and the corresponding debonding were linearly related. Within the resolution of the observations, no fiber fracture was seen in the bridging zone. Using a standard Green's function, stress intensity factor simulations were carried out for different types of tractions on the fibers in the bridging zone. When the fibers in the bridging zone were under a uniform load, the total stress intensity factor K _t, at the crack tip, was found constant at the steady state and proportional to % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdmhaaa!39C2!\[\sigma \] % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOhIu6aaO% aaaeaacqaH7oaBaSqabaaaaa!3B3F!\[\infty \sqrt \lambda \]. Assuming that K _t is constant during steady crack growth, the results of the simulations were used to correlate steady crack speed in three sets of data. Dimensional analysis of the steady crack speed was carried out as an attempt to identify important parameters and the role of the fiber spacing in the fracture of the composite specimens. The steady crack speeds were correlated with the total stress intensity factor for each fiber spacing. The resulting exponents were found to be about 20 percent different. Assuming that at steady state the energy release rate for an interfacial crack is proportional to t t ² r, where t is the stress carried by a fiber [20], a power expression for the rate of debonding with t % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0Jf9crFfpeea0xh9v8qiW7rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaacq% aHepaDaSqabaaaaa!39DF!\[\sqrt \tau \] was found to have an exponent approximately equal to that for the steady crack speed.     

NMR studies on single crystals of H2: I. The crystalline field energy for ortho-H2 impurities in solid para-H2

Using NMR techniques on two single crystals of H₂, we have determined the crystalline field V _c that splits the rotational ground state of isolated ortho-H₂ molecules in para-H₂. From the temperature-dependent splitting of the NMR doublet peak, we obtain an average % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4baFfea0dXde9vqpa0lb9% cq0dXdb9IqFHe9FjuP0-iq0dXdbba9pe0lb9hs0dXda91qaq-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaGG8bacbaGab8% NvayaaraWaaSbaaSqaaiaa-ngaaeqaaOGaaiiFaiaac+cacaWFRbWa% aSbaaSqaaiaa-jeaaeqaaOGaeyypa0JaaGimaiaac6cacaaIWaGaaG% imaiaaiIdacaaIZaGaeyySaeRaaGimaiaac6cacaaIWaGaaGimaiaa% ikdacaqGGaGaa83saaaa!4563!\[|\bar V_c |/k_B = 0.0083 \pm 0.002{\rm{ }}K\] independent of ortho concentration X over the range between x= 0.1% and 1.5%. The experimental spectra, as a function of both T and the applied field orientation, can be fitted by a model where V _c has a Gaussian distribution about its average value with a standard deviation V _c289-03. This distribution might be caused by strains in the sample. The values of V _c are compared with those from previous experiments.Research supported by a grant from the National Science Foundation.     

Three-dimensional finite element calibration of the short-rod specimen

The short rod is a simple, inexpensive configuration for fracture toughness testing. Since no rigorous analytical stress-intensity factor calibration of this geometry has yet appeared, a three-dimensional finite element study was undertaken. Successively finer meshes were employed to investigate convergence in compliance versus crack length, and the dimensionless calibration constant, A, in the expression% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Teada% WgaaWcbaacbaGaa4xmaiaa-ngaaeqaaOGaeyypa0Jaa8NramaaBaaa% leaacaWFJbaabeaakiaa-feacaGFVaGaa43waiaa-jeadaahaaqcba% uabeaalmaalyaajeaqbaGaaG4maaqaaiaaikdaaaaaaOGaaiikaiaa% igdacqGHsislcaWF2bWaaWbaaSqabKqaafaacaWFYaaaaOGaaiykam% aaCaaaleqajeaqbaWaaSGbaeaacaaIXaaabaGaaGOmaaaaaaGccaGG% Dbaaaa!4A74!\[K_{1c} = F_c A/[B^{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-\nulldelimiterspace} 2}} (1 - v^2 )^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ]\]Quarter-point singular elements were used along the crack front, and a range of crack lengths 0.65 <- a/B <- 1.1 was investigated. Distributed and point loading cases were considered. Polynomials were least-squares fit through the compliance data and were differentiated to yield expressions for average stress-intensity factor along the crack front% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Teada% WgaaWcbaacbaGaa4xmaaqabaGccqGH9aqpdaWadaqaamaalaaabaGa% a8NramaaCaaaleqajeaybaGaa8Nmaaaakiaa-veacaWFNaaabaGaaG% Omaiaa-jgacaWFGaaaamaalaaabaGaa4hzaiaa-neaaeaacaqGKbGa% a8xyaaaaaiaawUfacaGLDbaadaahaaWcbeqcbauaamaalyaabaGaaG% ymaaqaaiaaikdaaaaaaaaa!477B!\[K_1 = \left[ {\frac{{F^2 E'}}{{2b }}\frac{{dC}}{{{\text{d}}a}}} \right]^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \]Minima of this expression were obtained corresponding to critical average stress-intensity factors and crack lengths. The above expressions were then equated to solve for calibration constant values of,% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGynaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaGaaeOlaiaabkdacaqG1aaaaa!4227!\[A = 25.9{\text{ }} \pm {\text{ 1}}{\text{.25}}\]at a _c/B=0.86 for the distributed load case and% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGimaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaaaaa!4004!\[A = 20.9{\text{ }} \pm {\text{ 1}}\]at a _c/B=.69 for the point load case.The distributed load case value of A is in very good agreement with previously reported values of 24.4 ± 1.3 and 25.0, and is about 11 percent higher than the currently recommended value obtained through K _Ic correlation.Preliminary results of a study of stress-intensity factor variation along the crack front are also presented. They show that maximum stress-intensity factor occurs at the edges of the crack front. This observation is consistent with the reverse tunneling phenomenon sometimes observed in short rod testing.Recommendations for further numerical study of the short rod configuration are suggested.

---

Résumé Le barreau court est d'une configuration simple et peu coûteuse pour les essais de ténacité à la rupture. Comme il n'est pas encore apparu de calibrage rigoureux du facteur d'intensité de contrainte relatif à cette géométrie, on a entrepris une étude par éléments finis à trois dimensions. Un maillage de plus en plus fin a été utilisé pour étudier la convergence entre la compliance et la longueur de fissuration ainsi que la constante de calibrage sans dimension A, dans l'expression de K _Ic.Le long du front de fissuration, on a eu recours à des éléments singuliers en quart point et on a examiné une gamme de longueurs de fissure rapportées au diamètre et comprise entre 0,65 et 1,1. On a envisagé des cas de mises en charge réparties et ponctuelles.Les valeurs polynomiales ont été ajustées par la méthode des moindres carrés grâce aux donnés de compliance et ont été différenciées en vue d'aboutir à des expressions d'un facteur d'intensité de contrainte moyen le long du front de fissure.Les minima de la valeur K _I ainsi trouvés ont été obtenus en faisant se correspondre les facteurs d intensité critiques moyens de contrainte et les longueurs de fissuration, on a pu ainsi définir les valeurs de constante de calibration en faisant se correspondre K _Ic et K _I, et l'on a tiré% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGynaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaGaaeOlaiaabkdacaqG1aaaaa!4227!\[A = 25.9{\text{ }} \pm {\text{ 1}}{\text{.25}}\]pour a _c/B=0,86 correspondant à une mise en charge distribuée et% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-feacq% GH9aqpcaaIYaGaaGimaiaac6cacaaI5aGaaeiiaiabgglaXkaabcca% caqGXaaaaa!4004!\[A = 20.9{\text{ }} \pm {\text{ 1}}\]pour a _c/B=0,69 dans le cas d'une mise en charge ponctuelle.La valeur de A correspondant à la mise en charge répartie est en excellent accord avec les valeurs précédemment trouvée 24,4 ± 1,3 et 25,0 et est d'environ 11% supérieure à la valeur généralement recommandée obtenue par une corrélation avec K _Ic.On présente également les résultats préliminaires d'une étude de la variation des facteurs d'intensité de contrainte le long du front de fissuration. Ces résultats montrent que le facteur d'intensité des contraintes passe par un maximum aux bords du front de la fissure. Cette observation est conforme à un phénomène inverse à celui du phénomène tunnel, qui est rencontré parfois dans les essais de barreau court.On suggère des recommandations pour une étude numérique complémentaire de la configuration du barreau court.

     

The effect of composition and neutron irradiation on the upper critical field of Nb3Pt

The upper critical field H _c2 (T) of the A15 compound Nb_100–x Pt _x (19 x 29.1) has been measured as a function of temperature T for various compositions and levels of irradiation with fast reactor neutrons. Compositions ranging from 19 to 29 at % Pt and neutron fluences up to 16 × 10¹⁸ n/cm² (E > 1 Me V) were studied. A stoichiometric sample that had been rapidly quenched from the liquid (splat-cooled) was also investigated. For the unirradiated samples, T _cexhibits a maximum of 10.4 K at the stoichiometric composition, while both H _c2 (0) and % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% GaaeizaiaabIeadaWgaaWcbaGaae4yaiaabkdaaeqaaOGaaeiiaiaa% bIcacaqGubGaaeykaiaab+cacaqGKbGaaeivaiaabYhadaWgaaWcba% Gaaeivaiaab2dacaqGubaabeaakmaaBaaaleaacaqGJbaabeaaaaa!48E5!\[{\rm{dH}}_{{\rm{c2}}} {\rm{ (T)/dT|}}_{{\rm{T = T}}} _{\rm{c}} \] attain their maximum values on the Pt-rich side of stoichiometry. The irradiated samples show a decrease of H _c2 (0) with increasing neutron fluence. The results are discussed in terms of antisite disorder and a strong correlation is obtained between decreasing H _c2 (0) and increasing disorder of the Nb atomic sites.Research supported by the U.S. Department of Energy under Contract No. EY-76-S-03-0034-PA227-3 (SEL and MBM), by Research Corporation (ARS), and by the U.S. Department of Energy under Contract No. EY-76-C-02-0016 (SM).     

Critical scattering around the antiferromagnetic transition in holmium

Detailed analyses of the electrical resistance of holmium near the antiferromagnetic transition T _N in magnetic fields up to 71 kOe are reported. The critical exponent in the expression for the reduced resistance % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaaieGacaWFYbacba% Gaa4hkaiaa-rhacaGFPaGaeyypa0Jaa8NCamaaBaaajeaqbaGaa4hm% aaWcbeaakiabgUcaRiaa-jeacaWF0bGaey4kaSIaa8xqamaaBaaale% aacqGHRaWkaeqaaOGaa8hDamaaCaaaleqabaGaa4xmaiabgkHiTGGa% aiab9f7aHbaakiabgUcaRiaa-feadaWgaaWcbaGaaeiiaiabgkHiTa% qabaGccaGFOaGaeyOeI0Iaa8hDaiaa+LcadaahaaWcbeqaaiaa+fda% cqGHsislcqqFXoqyaaaaaa!51B4!\[r(t) = r_0 + Bt + A_ + t^{1 - \alpha } + A_{{\rm{ }} - } ( - t)^{1 - \alpha } \] for % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacqGHsislcaaIWa% GaaiOlaiaaigdacqGHKjYOcaWG0bGaeyizImQaaGymaiaac6cacaaI% 0aGaaeiiaGqaaiaa-HhacaWFGaGaa8xmaiaa-bdadaahaaWcbeqaai% abgkHiTiaaikdaaaaaaa!469D!\[ - 0.1 \le t \le 1.4{\rm{ }}x 10^{ - 2} \], where % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0xe9qqVa0l% b9peea0lb9sq-JfrVkFHe9peea0dXdarVe0Fb9pgea0xa9pue9Fve9% Ffc8meGabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWG0bGaeyypa0% JaaiikaiaadsfacqGHsislcaWGubWaaSbaaKqaGfaaieaacaWFobaa% leqaaOGaaiykaiaac+cacaWGubWaaSbaaKqaGfaacaWFobaaleqaaa% aa!41DC!\[t = (T - T_N )/T_N \], is –0.27 for the c-axis crystal and is approximately field independent. The critical temperature decreases approximately linearly with increasing field. For T < T _N, the magnetoresistance exhibits a maximum and is negative at higher fields. These results have been interpreted in terms of field-induced changes in the spiral-spin structure and the estimated critical field is found to be in good agreement with published data.This research was supported in part by a grant from the Natural Sciences and Engineering Research Council of Canada.     

Quasi-statically steady crack growth in rate dependent solids-one-parameter field encompassing different constraints

A finite element analysis is presented for quasi-statically steady crack growth in an elastic-viscoplastic material under Mode I, plane strain and small scale yielding conditions. The effects of material rate-sensitivity on the fields in the vicinity of the moving crack tip are examined. Our analysis employs a modified boundary layer formulation whereby the remote tractions are given by the first two-terms of elastic asymptotic stress field, characterized by K _Iand T. When the physical coordinates are scaled by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Hcaca% WFlbWaaSbaaSqaaGqaaiaa+fdaaeqaaOGaai4laiabeo8aZnaaBaaa% leaacaaIWaaabeaakiaacMcadaahaaWcbeqaaiaaikdaaaaaaa!3ECA!\[(K_1 /\sigma _0 )^2 \], where % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaiabeo8aZnaaBa% aaleaacaaIWaaabeaaaaa!3A07!\[\sigma _0 \] is the tensile yield stress, the near-tip fields over a wide range of stress triaxialities are members of a family of self-similar solutions parameterized by % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-rfaca% WFVaGaeq4Wdm3aaSbaaSqaaiaa-bdaaeqaaaaa!3B8E!\[T/\sigma _0 \]. Members of this family are found to collapse into a single near-tip distribution when the physical coordinates are normalized by a characteristic length L _g, which is a significant fraction of the plastic zone length directly ahead of the crack tip. This distribution depends only on the relative crack speed given by the dimensionless number % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-zfaca% WFVaGaa8hkaiaa-XeadaWgaaWcbaGaa83zaaqabaacciGccuGFiiIZ% gaGaamaaBaaaleaacaaIWaaabeaakiaacMcaaaa!3EB2!\[V/(L_g \dot \in _0 )\] where V is the crack speed and % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGGaciqb-HGioB% aacaWaaSbaaSqaaiaaicdaaeqaaaaa!39D6!\[\dot \in _0 \] is the material's viscoplastic strain rate at a reference stress. Near-tip field distributions are obtained for several values of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-zfaca% WFVaGaa8hkaiaa-XeadaWgaaWcbaGaa83zaaqabaacciGccuGFiiIZ% gaGaamaaBaaaleaacaaIWaaabeaakiaacMcaaaa!3EB2!\[V/(L_g \dot \in _0 )\] and material strain rate sensitivity, m. Our results show that strong material rate sensitivity and high crack speed elevate the stress level ahead of the moving crack tip.     

Relaxation of the distribution function branch imbalance Q * in Sn and Sn-In alloys

We have used double tunnel junctions M₁-S₂-N₃ to measure the detector junction (S₂-N₃) voltage at zero detector current V _d(0) vs. generator junction (M₁-S₂) current I _g for samples cooled by helium exchange gas. We find terms both symmetric V _d ^S(0; I _g ²) and asymmetric V _d ^A(0; I _g) in I _g. The term V _d ^S is attributed to a thermoelectric voltage generated in the sample; V _d ^A is produced by Q ^*, the distribution function branch imbalance, in S₂. We have analyzed Q ^* relaxation in these samples and have compared the results with those obtained earlier by Paterson. For Sn and Sn : In alloys we find % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS% baaSqaaGqaaiaa-ffacaWFQaaabeaakiabg2da9Gqaciaa+fdacaGF% UaGaa4xmaiaa+flacaGFWaGaa4Nlaiaa+jdacaGFxdGaa4xmaiaa+b% dadaahaaWcbeqaaiaa+1cacaGFXaGaa4hmaaaakiabfs5aejaacIca% caGFWaGaaiykaiaac+cacqqHuoarcaGGOaGaa8hvaiaacMcaaaa!4C83!\[\tau _{Q*} = 1.1 \pm 0.2 \times 10^{ - 10} \Delta (0)/\Delta (T)\] sec for T - T _c; the data at all temperatures have been compared with the calculations by Chang. The attempt to measure % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aaS% baaSqaaGqaaiaa-ffacaWFQaaabeaaaaa!3964!\[\tau _{Q*} \] in Pb near T _c is discussed and data are presented for V _d ^S for Pb.Supported by grants from the Research Corporation and the National Science Foundation (DMR 74-23661).     

Contribution of bending energy losses to the apparent tear energy

When a strip is torn, energy is expended both in tearing it and in propagating a bend along each torn section. Estimates are given of the contribution of bending energy losses to the apparent tear energy. Experiments with highly-dissipative semi-crystalline polymers, torn with controlled amounts of bending, are then described. The bending energy losses ranged from 5 to 70 percent of the total tear energy, depending upon the degree of bending imposed, the thickness of the strip, and the extent to which it had been partly cut through before tearing. These results were in satisfactory agreement with approximate theoretical estimates. When the torn strips were allowed to take up naturally bent configurations under the action of the tearing force, then the contribution of bending energy losses to the apparent tear energy became rather independent of the strip dimensions and depended principally upon the dissipative nature of the material, represented by the fraction H of deformation energy that is not recovered. A general relationship is proposed between the apparent (G _c) and true (G _c) tear energies in this case: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Deada% qhaaWcbaGaa83yaaqaaiaa-DcaaaGccqGH9aqpcaWFhbWaaSbaaSqa% aiaa-ngaaeqaaOGaai4laiaacIcacaaIXaGaeyOeI0Iaa8hsaGqaai% aa+Lcaaaa!415E!\[G_c^' = G_c /(1 - H)\]. Values of H for the materials examined ranged from 30 to 70 percent. Thus, bending energy losses are expected to increase the tear energy by a factor of 1.4 × to 3.3 × for unconstrained tearing of these semi-crystalline polymers. Somewhat smaller increases were actually observed, ranging from 1.1× to 2×.

---

Résumé Lorsque l'on déchire une bande, l'énergie est dépensée à la fois dans le déchirement et dans la propagation d'une flexion sur chacun des bords de la déchirure.On donne des estimations de la contribution des pertes d'énergie associées à ces flexions, à l'énergie apparente de déchirement. On décrit ensuite des expériences de déchirement de bandes en polymères semi-cristallins à haute dissipation, sous des conditions de flexion contrôlées. On a établi que la dissipation d'énergie associées à la flexion vaut de 5 à 70% de l'énergie totale de déchirement, selon le degré de flexion imposé, l'épaisseur de la bande, et la longueur de la coupe réalisée avant déchirure. Ces résultats ont été trouvés en accord satisfaisant avec les estimations théoriques. Lorsque les portions déchirées adoptent la configuration de flexion qui correspond à une situation naturelle sous l'effet des forces de déchirement, les pertes dues à l'énergie de flexion contribuent à l'énergie apparente de déchirement de manière relativement indépendante des dimensions de la bande, mais principalement dépendante de la nature dissipatoire du matériau, représentée par la fraction H de l'énergie de déformation non récupérée.Une relation générale est proposée dans ce cas entre l'énergie apparente (G _c) et réelle (G _c) de déchirement: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Deada% qhaaWcbaGaa83yaaqaaiaa-DcaaaGccqGH9aqpcaWFhbWaaSbaaSqa% aiaa-ngaaeqaaOGaai4laiaacIcacaaIXaGaeyOeI0Iaa8hsaGqaai% aa+Lcaaaa!415E!\[G_c^' = G_c /(1 - H)\]Les valeurs de H varieront de 0,3 à 0,7 selon les matériaux examinés. Dès lors, on s'attend à ce que les pertes par énergie de flexion accroissent l'énergie de déchirement d'un facteur 4,4× à 3,3× dans le cas d'un déchirement sans rétreint des polymères semi-cristallins étudiés.Dans la réalité, on a observé des accroissements un peu plus faibles, variant de 1,1× à 2×.

---

---

Department of Aerospace Engineering and Mechanics, University of Minnesota     

‘When can we apply LEFM principles to elastic softening materials?’

The paper focusses on the determination of R, the size of the fully developed softening zone associated with a semi-infinite crack in a remotely loaded infinite elastic softening solid. R is a characteristic length for a material, and is important in that if R is less than an appropriate characteristic dimension of a structure, then LEFM principles can be used to describe the structure's failure. With p _c and _c being respectively the maximum stress and displacement within the softening zone, then provided the softening is not particularly pronounced, i.e. the area under the stress (p)-displacement (v) curve is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaacaaeaacq% GH+aGpaiaawoWaaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaamiC% aSGaam4yaOGaeqiTdq2ccaWGJbaaaa!3FB5!\[\widetilde > {\text{ 0}}{\text{.25}}pc\delta c\], it is shown that R 0.4E _0c/P _c and R is relatively insensitive to the precise p-v softening behaviour (E ₀ = E/(1 – v ²) where E is Young's modulus and is Poisson's ratio. However, when the area under the curve is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaacaaeaacq% GH8aapaiaawoWaaiaabccacaqGWaGaaeOlaiaabkdacaqG1aGaamiC% aSGaam4yaOGaeqiTdq2ccaWGJbaaaa!3FB1!\[\widetilde < {\text{ 0}}{\text{.25}}pc\delta c\], then R increases above this 0.4E _0c/P _c value. For this case, and provided most of the area under the p-v curve is not associated with the tail in the softening law, a more appropriate expression for R is R 0.1E ₀ ² / ₀ ² /K ² , with K ² /E ₀ being the area under the p-v curve and K being the stress intensity associated with the full development of a softening zone.     

Observation of deformation and damage at the tip of cracks in adhesive bonds loaded in shear and assessment of a criterion for fracture

The evolution of damage at the tip of cracks in adhesive bonds deforming in shear was monitored in real time using a high-magnification video camera. Brittle and a ductile epoxy resins were evaluated, with the bond thickness t being an experimental variable. An extensive zone of plastic deformation developed ahead of the crack tip prior to fracture. In the case of the brittle adhesive, for relatively thick bonds tensile microcracks formed within that zone. Increased loading caused the microcracks to grow from the interlayer to the interface, which led to a complete bond separation after interface cracks emanating from adjacent microcracks linked. In contrast, for the ductile adhesive the crack always grew from the tip. Strain gradients tended to develop there when the bond thickness was large.The adhesive shear strain was determined from fine lines scratched on the specimen edge. For both adhesives, the average crack tip shear strain at crack propagation rapidly decreased with increasing t. This effect was attributed to the changing sensitivity of the bond to the presence of flaws; thicker bonds can accommodate larger microcracks or microvoids which cause greater stress concentration. For a given bond thickness, the critical crack tip shear strain agreed well with the ultimate shear strain of the unflawed adhesive previously determined using the napkin ring shear test [12]. This suggests that the ultimate shear strain is a key material property controlling crack growth. The critical distortional strain energy/unit area of the unflawed adhesive W _s was determined from the area under the stress-strain curve in the napkin ring test. Good agreement between W _s and the adhesive mode II fracture energy was found for all joints tested except for relatively thick bonds. For the particular case of an elastic-perfectly plastic adhesive, the agreement above implies % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaciaacaqabeaadaqaaqaaaOqaaGqaciaa-Deada% WgaaWcbaacbaGaa4xsaiaa+LeacaGFdbaabeaakiabg2da9iaa-Dfa% daWgaaWcbaGaa83CaaqabaGccqGHHjIUcaWF0bGaeqiXdq3aaSbaaS% qaaiaa-LhaaeqaaOGaeq4SdC2aaSbaaSqaaiaa-zgaaeqaaaaa!463A!\[G_{IIC} = W_s \equiv t\tau _y \gamma _f \].     

Diffusive thermal conductivity of superfluid 3He-B

The diffusive thermal conductivity (t) of superfluid ³He-B is calculated in the s-p-d-wave approximation by solving the Boltzmann equation for the Bogoliubov-Valatin quasiparticles variationally. A new set of Landau para- meters calculated from recent heat capacity data as well as old ones given in Wheatley's review are used to estimate the scattering amplitudes of the collision integral. Landau parameters F ₂ ^s, F ₁ ^a, and F ₂ ^a are treated as free parameters under the constraint that _exact(T _c) = _exp(T _c), where _exact and _exp are the exact theoretical value and the experimental value, respectively. We have varied F ₂ ^s, F ₁ ^a, and F ₂ ^a over a wide range % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaai4Eaiabgk% HiTiaaigdacaaIWaGaeyizImQaaeyqamaaDaaaleaacaqGXaaabaGa% ae4CaiaabYcacaqGHbaaaOGaeyyyIORaaeOramaaDaaaleaacaqGXa% aabaGaae4CaiaabYcacaqGHbaaaOGaai4laiaacUfacaaIXaGaey4k% aSIaaeOramaaDaaaleaacaqGXaaabaGaae4CaiaabYcacaqGHbaaaO% Gaai4laiaacIcacaaIYaGaamiBaiabgUcaRiaaigdacaGGPaGaaiyx% aiabgsMiJkaaigdacaaIWaGaaiyFaaaa!570F!\[\{ - 10 \leqslant {\text{A}}_{\text{1}}^{{\text{s,a}}} \equiv {\text{F}}_{\text{1}}^{{\text{s,a}}} /[1 + {\text{F}}_{\text{1}}^{{\text{s,a}}} /(2l + 1)] \leqslant 10\} \] and found the possible range of the reduced diffusive thermal conductivity % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGafqOUdSMbaG% aacaGGOaGaaeivaiaabMcacaqG9aGaeqOUdSMaaeikaiaabsfacaqG% PaGaaeivaiaab+cacqaH6oWAcaGGOaGaaeivamaaBaaaleaacaqGJb% aabeaakiaacMcacaqGubWaaSbaaSqaaiaabogaaeqaaaaa!46ED!\[\tilde \kappa ({\text{T) = }}\kappa {\text{(T)T/}}\kappa ({\text{T}}_{\text{c}} ){\text{T}}_{\text{c}} \]. The behavior of \~(T) in the s-p-d-wave approximation does not much depend on the values of the Landau parameters, and \~(t) decreases monotonically with decreasing tem- perature.     

Crack layer analysis of fatigue crack propagation in rubber compounds

A methodology to characterize the resistance of rubber compounds to crack propagation (fracture toughness) is presented. A constitutive model based on the crack layer theory is utilized to extract the specific energy of damage ^*, a material parameter characteristic of the material's resistance to crack propagation and the dissipative characteristic, . The model expresses the rate of crack propagation as% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabaGaaiaacaqabeaadaqaaqaaaOqaamaalaaabaGaam% izaGqaciaa-fgaaeaacaWGKbGaa8Ntaaaaaaa!3AFA!\[\frac{{da}}{{dN}}\]= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGak0dh9WrFfpC0xh9vqqj-hEeeu0xXdbba9frFj0-OqFf% ea0dXdd9vqaq-JfrVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr% 0-vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamaalaaabaGaeq% OSdiMaamOsamaaDaaaleaacaaIXaaabaGaaGOmaaaaaOqaaiaadMha% caGGQaGaamOuamaaBaaaleaacaaIXaaabeaakiabgkHiTiaadQeada% WgaaWcbaGaaGymaaqabaaaaaaa!41A5!\[\frac{{\beta J_1^2 }}{{y*R_1 - J_1 }}\]where da/dN is the cyclic rate of fatigue crack propagation (FCP), J ₁ is the energy release rate (tearing energy) and R ₁ is the resistance moment which accounts for the amount of damage associated with the crack advance. Microscopic examination revealed that crack tip microcracking is the dominant damage mechanism. Hence, R ₁ was evaluated as the area (m²) of microcracking surfaces per unit crack advance.Fatigue crack propagation data for a particular rubber compound have been analyzed using the present model. The proposed equation describes the entire FCP history in the compound. According to this model, ^* and for the compound investigated, are found to be 9.3 kJ m^-2 and 9.7×10^-9 m⁴/J-cycle, respectively.

---

Résumé On présente une méthodologie pour caractériser la résistance de composés de caoutchouc à la propagation des fissures du point de vue de la ténacité à la rupture. Un modèle constitutif basé sur la théorie de la couche de fissuration est utilisé pour obtenir l'énergie spécifique d'endommagement ^*, un paramètre du matériau représentatif de sa résistance à la propagation d'une fissure, et une caractéristique de dissipation . Le modèle exprime la vitesse de propagation d'une fissure de fatigue par cycle da/dN en fonction de ces deux paramètres, de la vitesse de relaxation de l'énergie de cisaillement J ₁, et du moment résistif R ₁ qui tient compte de état de l'endommagement associé à la progression de la fissure. Un examen microscopique révèle que la microfissuration à l'extrémité de la fissure est le mécanisme déterminant de l'endommagement. Dès lors, on évalue R ₁ en fonction de l'aire de microfissuration (en m²) par unité de progression de la fissure.Des données de propagation de fissure de fatigue sont analysées à l'aide du présent modèle pour un composé de caoutchouc particulier. L'équation proposée décrit l'entièreté de la propagation de la fissure dans le composé. Des valeurs numériques pour ^* et pour de respectivement 9,3 kJ m^-2 et 9,7×10^-9 m^-4/J-cycle sont trouvées.

     

Estimation of high‐temperature fracture parameters for small punch specimen with a surface crack

An outline of a newly proposed methodology for evaluating creep crack growth (CCG) parameters using cracked small‐punch (SP) specimens is explained. Three‐dimensional finite element analyses were performed to calculate the stress intensity factor along the crack front for a surface crack formed at the centre of a SP specimen. Effects of crack ratio, (a/t); crack aspect ratio, (a/c); and thickness of the specimen, (t), on the fracture parameters were studied. It was observed that the minimum variation of K‐value along the crack front can be achieved when a/c was 0.50 except the location very near the intersection of the crack and free surface. This condition is similar to the case of constant K‐values along the crack front of the conventional compact tension specimen. Thus, it can be argued that the SP specimen with a surface crack is a suitable specimen geometry for CCG testing. The proposed CCG test method was found to be practically applicable for the crack geometry of 0.10 to 0.30 of a/t with constant aspect ratio of 0.50. An estimation of the K and C_t‐parameter under the small scale creep condition was derived. Future work for further development of the suggested CCG testing is discussed.     

Effective slip in numerical calculations of moving-contact-line problems

For many coating flows, the profile thickness h, near the front of the coating film, is governed by a third-order ordinary differential equation of the form h=% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby% kaaa!37B5!\[f\](h), for some given % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby% kaaa!37B5!\[f\](h). We consider here the case of dry wall coating which allows for slip in the vicinity of the moving contact-line. For this case, one such model equation, due to Greenspan, is % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaccaGae8NKby% kaaa!37B5!\[f\](h)=–1+(1+)/(h ²+), where is the slip coefficient. The equation is solved using a finite difference scheme, with a contact angle boundary condition prescribed at the moving contact-line. Using the maximum thickness of the profile as the control parameter, we show that there is a direct relationship between the effective Greenspan slip coefficient and the grid-spacing of the numerical scheme used to solve the model equation. In doing so, we show that slip is implicity built into the numerical scheme through the finite grid-spacing. We also show why converged results with finite film thickness cannot be obtained if slip is ignored.     

Numerical modeling of ductile crack growth in 3-D using computational cell elements

This study describes a 3-D computational framework to model stable extension of a macroscopic crack under mode I conditions in ductile metals. The Gurson-Tvergaard dilatant plasticity model for voided materials describes the degradation of material stress capacity. Fixed-size, computational cell elements defined over a thin layer at the crack plane provide an explicit length scale for the continuum damage process. Outside this layer, the material remains undamaged by void growth, consistent with metallurgical observations. An element vanish procedure removes highly voided cells from further consideration in the analysis, thereby creating new tractionfree surfaces which extend the macroscopic crack. The key micro-mechanics parameters are D, the thickness of the computational cell layer, and f ₀ , the initial cell porosity. Calibration of these parameters proceeds through analyses of ductile tearing to match R-curves obtained from testing of deep-notch, through-crack bend specimens. The resulting computational model, coupled with refined 3-D meshes, enables the detailed study of non-uniform growth along the crack front and predictions of specimen size, geometry and loading mode effects on tearing resistance, here described by J-a curves. Computational and experimental studies are described for shallow and deep-notch SE(B) specimens having side grooves and for a conventional C(T) specimen without side grooves. The computational models prove capable of predicting the measured R-curves, post-test measured crack profiles, and measured load-displacement records.     

Critical currents of SNS junctions with Cu:Mn and Cu:Ni barriers

Critical supercurrent densities j _c have been measured for Pb-Cu₉₁Ni₉-Pb and Pb-Cu_99.8Mn_0.2-Pb junctions over the temperature range 1.3<T<4.2K. Diffraction patterns for the junctions indicate highly uniform current densities in the non-self-field-limited regime. The data for the Cu:Ni barriers provide some indication of a spin-flip depairing temperature% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqySdeMaey% ypa0JaeS4dHGMaai4laiaaikdacqaHapaCcaWGRbWaaSbaaSqaaiaa% dkeadaahaaadbeqaaiabes8a0naaBaaabaGaam4CaaqabaaaaaWcbe% aakiabgIKi7kaaiodacaqGGaacbaGaa83saaaa!4613!\[\alpha = \hbar /2\pi k_{B^{\tau _s } } \approx 3{\rm{ }}K\]and the data for Cu:Mn provide some indication of an 15 K; the measured supercurrent densities, however, are only approximately represented by the derived expression % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8NAam% aaBaaaleaacaWFdbaabeaakiaacIcacaWFubGaaiykaiabg2da9iaa% -jeaieGacaGFLbGaa4hEaiaa+bhacaGF7bGaeyOeI0Iaeq4SdCMaai% ikaiaa-rfacqGHRaWkcqaHXoqycaGGPaGaai4laiaa-rfadaWgaaWc% baGaa83qaaqabaGcdaahaaWcbeqaaiaaigdacaGGVaGaaGOmaaaaki% aac2faaaa!4CEE!\[j_C (T) = Bexp\{ - \gamma (T + \alpha )/T_C ^{1/2} ]\]. The data for Cu:Mn fit the empirically determined % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-qqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xHapdbiqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaGaa8NAam% aaBaaaleaacaWFJbaabeaakiaacIcacaWFubGaaiykaiabg2da9iaa% dQgacaGGOaGaaGimaiaacMcaieGacaGFLbGaa4hEaiaa+bhacaGGBb% GaeyOeI0IaeqOSdiMaaiikaiaa-rfacaGGVaGaa8hvamaaBaaaleaa% caWGJbaabeaakiaacMcadaahaaWcbeqaaiaaiodacaGGVaGaaGOmaa% aakiaac2faaaa!4CD0!\[j_c (T) = j(0)exp[ - \beta (T/T_c )^{3/2} ]\]much better; there is, however, no apparent theoretical basis for this expression. The measured junction resistances are 50% higher than the values implied by the measured resistivity of the barrier material for both the Cu:Ni and the Cu:Mn; this excess resistance is presumably contributed by the two S-N interfaces.Work supported by the Atomic Energy Commission and by NSF Grant DMR 74-23661.     

On the contribution of the superconductor to the resistance of a superconductor-normal metal system

The problem of dc current flow through a superconductor-normal metal interface near the critical temperature T _c is considered. The equations for the Green's functions integrated with respect to the variable = v(p–p₀) are used to calcuate the resistance of the pure and dirty superconductor S. It is shown that the electric field E decays exponentially over a length % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% acbaGaa8xmamaaBaaaleaacaWFIbaabeaakiabg2da9iaacIcadaWc% gaqaaiaa-reacqaHepaDdaWgaaWcbaGaa8xzaaqabaacbiGccaGF0a% Gaa8hvaaqaaiabec8aWjabgs5aejaa-LcadaahaaWcbeqaamaaliaa% baGaa8xmaaqaaiaa-jdaaaaaaaaaaaa!49B4!\[1_b = ({{D\tau _e 4T} \mathord{\left/ {\vphantom {{D\tau _e 4T} {\pi \Delta )^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} }}} \right. \kern-\nulldelimiterspace} {\pi \Delta )^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}} }}\] within the superconductor and at the S-N interface there is a jump of E [strictly speaking, E varies over the correlation length (T)]; here D is the diffusion coefficient, and _e is the energy relaxation time. The magnitude of the jump of E is of the order of or less than the value of E at the boundary of the S region. In the case of a pure superconductor this jump is caused by the Andreev reflection of quasiparticles at the S-N interface.     

Mixed mode I and II fracture of carbon black filled natural rubber

The J-integral was used to characterize initiation and rapid fracture under mixed mode loading conditions in carbon black filled natural rubber. The total critical J was geometry dependent, but the J analysis partitions the energy into that needed locally in the crack tip region and in the bulk. The critical conditions for pure mode II loading could not be determined because of buckling of the specimen; however, values obtained by extrapolation show J _IIc to be about twice J _Ic. The relation % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabiqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaadQeadaWg% aaWcbaGaamysaaqabaaakeaacaWGkbWaaSbaaSqaaiaadMeacaWGJb% aabeaaaaGccaqGGaGaae4kaiaabccadaWcaaqaaiaadQeadaWgaaWc% baGaamysaiaadMeaaeqaaaGcbaGaamOsamaaBaaaleaacaWGjbGaam% ysaiaadogaaeqaaaaakiaabccacaqG9aGaaeiiaiaabgdaaaa!5355!\[\frac{{J_I }}{{J_{Ic} }}{\text{ + }}\frac{{J_{II} }}{{J_{IIc} }}{\text{ = 1}}\]

---

Résumé On utilise l'intégrale J pour caractériser l'amorcage et le déploiement brutal d'une rupture dans du caoutchouc noir naturel soumis à des conditions de sollicitation de mode mixte. Pour une sollicitation purement de Mode II, on n'a pu déterminer les conditions critiques en raison d'un flambage de l'échantillon. Néanmoins, les valeurs obtenues par extrapolation montrent que J _IIc vaut à peu prés le double de J _Ic. Les conditions de rupture sous sollicitation de mode mixte sont décrites par la relation: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqef0uAJj3BZ9Mz0bYu% H52CGmvzYLMzaerbd9wDYLwzYbItLDharqqr1ngBPrgifHhDYfgasa% acOqpw0xe9v8qqaqFD0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8Wq% Ffea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qqQ8frFve9Fve9Ff0dme% aabiqaaiGacaGaamqadaabaeaafiaakeaadaWcaaqaaiaadQeadaWg% aaWcbaGaamysaaqabaaakeaacaWGkbWaaSbaaSqaaiaadMeacaWGJb% aabeaaaaGccaqGGaGaae4kaiaabccadaWcaaqaaiaadQeadaWgaaWc% baGaamysaiaadMeaaeqaaaGcbaGaamOsamaaBaaaleaacaWGjbGaam% ysaiaadogaaeqaaaaakiaabccacaqG9aGaaeiiaiaabgdaaaa!5355!\[\frac{{J_I }}{{J_{Ic} }}{\text{ + }}\frac{{J_{II} }}{{J_{IIc} }}{\text{ = 1}}\].