ANALYZING DUCTILE FRACTURE USING DUAL DILATIONAL CONSTITUTIVE EQUATIONS

Abstract— By adopting a suggestion made by Thomason, a new failure criterion for the Gurson-Tvergaard model has been recently introduced by the authors. In this study, a method based on the Gurson-Tvergaard constitutive model and the new failure criterion is applied to the analysis of ductile fracture. The main features of the method are that the material failure is a natural process of the development of Thomason's dual dilational constitutive responses, and the void volume fraction corresponding to the failure by void coalescence is not necessarily a material constant and is not needed to be fitted beforehand. Furthermore, void nucleation parameter(s) can be numerically fitted from experimental tension results. This method has been implemented into the ABAQUS finite element program via a user material subroutine and is applied to the prediction of tension problems conducted by the authors. In the analyses, two strain-controlled void nucleation models have been studied and compared. The void nucleation parameters corresponding to the two models have been calibrated. The crack initiation of both smooth and notched axisymmetric tensile specimens are well predicted by the method. Finally, several critical issues in the analysis of ductile fracture are discussed.     

基于微孔贯通细观损伤模型的金属韧性断裂分析

韧性材料断裂过程通常可看作是材料内部微孔洞的形核、扩展及相互贯通的积累。经典的Gurson- Tvergaard (GT)模型能够很好地模拟具有变形均匀、各向同性的孔洞的萌生及扩展过程;但无法模拟由孔洞贯通而引起的局部变形过程,因此需要对其修正,引入相应的孔洞贯通准则。该文采用两种贯通准则对经典GT模型进行修正,即Thomason的塑性极限载荷准则和临界等效塑性应变准则。借助用户自定义程序UMAT将采用这两种贯通准则修正的GT本构关系嵌入至商用有限元软件ABAQUS中,从而可通过对金属材料应力状态和断裂机理的分析控制孔洞的贯通。以一组含有不同缺口根半径的圆棒拉伸试验件为例,分析了该类金属构件自孔洞萌生至最终断裂的整个损伤演化过程,并与试验数据进行了对比,验证了该模型的有效性。该文还讨论了金属断裂过程中应力三轴度对微裂纹萌生与扩展的影响。     

The role of ε-martensite in the impact toughness of an Fe-17Mn alloy

The influence of -martensite on the cryogenic toughness of an Fe-17 wt% Mn alloy was studied in this work. Alloys were tempered at various temperatures in order to systematically increase the volume fraction of -martensite. This was followed by Charpy impact testing conducted at room temperature and at–196°C. The experimental results indicated that although room-temperature toughness was not influenced by the -martensite content, the cryogenic toughness was strongly dependent on the volume fraction of -martensite. In particular, with the exception of the alloys tempered at 400 and 450°C, the impact toughness consistently increased with -martensite content. Microstructural and fractographic evaluations using SEM and TEM suggested that the toughness improvements were attributed to the stress-induced martensite transformation. No microstructural evidence was found which could be ascribed to an effect of -martensite on the low-temperature embrittlement exhibited by Fe-Mn alloys tempered at 400–450°C.     

Effect of fibre misalignment on fracture behaviour of fibre-reinforced composites

When a matrix crack encounters a fibre that is inclined relative to the direction of crack opening, geometry requires that the fibre flex is bridging between the crack faces. Conversely, the degree of flexing is a function of the crack face separation, as well as of (1) the compliance of the supporting matrix, (2) the crossing angle, (3) the bundle size, and (4) the shear coupling of the fibre to the matrix. At some crack face separation the stress level in the fibre bundle will cause it to fail. Other bundles, differing in size and orientation, will fail at other values of the crack separation. Such bridging contributes significantly to the resistance of the composite to crack propagation and to ultimate failure. The stress on the composite needed to produce a given crack face separation is inferred by analysing the forces and displacements involved. The resulting model computes stress versus crack-opening behaviour, ultimate strengths, and works of failure. Although the crack is assumed to be planar and to extend indefinitely, the model should also be applicable to finite cracks.Glossary of Symbols a radius of fibre bundle - C 2 _f /aE _f - ^* critical failure strain of fibre bundle - _b bending strain in outer fibre of a bundle - _c background strain in composite - _f axial strain in fibre - _s strain in fibre bundle due to fibre stretching = _f - () strain in composite far from crack - E Young's modulus of fibre bundle - E _c Young's modulus of composite - E _f Young's modulus of fibre - E _m Young's modulus of matrix - f() number density per unit area of fibres crossing crack plane in interval to + d - F total force exerted by fibre bundle normal to crack plane - F _s component of fibre stretching force normal to crack plane - F _b component of bending force normal to crack plane - G _m shear modulus of matrix - h crack face opening relative to crack mid-point - h _m matrix contraction contribution to h - h _f fibre deformation contribution to h - h _max crack opening at which bridging stress is a maximum - I moment of inertia of fibre bundle - k fibre stress decay constant in non-slip region - k ₀ force constant characterizing an elastic foundation (see Equation 7) - L exposed length of bridging fibre bundle (see Equation 1a) - L _f half-length of a discontinuous fibre - m, n parameters characterizing degree of misalignment - N number of bundles intersecting a unit area of crack plane - P _b bending force normal to bundle axis at crack midpoint - P _s stretching force parallel to bundle axis in crack opening - Q() distribution function describing the degree of misalignment - s _f fibre axial tensile stress - s _f ^* fibre tensile failure stress - S stress supported by totality of bridging fibre bundles - S _max maximum value of bridging stress - v fibre displacement relative to matrix - v elongation of fibre in crack bridging region - u _coh non-slip contribution to fibre elongation - U fibre elongation due to crack bridging - v overall volume fraction of fibres - v _f volume fraction of bundles - v _m volume fraction matrix between bundles - w transverse deflection of bundle at the crack mid-point - x distance along fibre axis, origin defined by context - X distance between the end of discontinuous fibre and the crack face - X ^* threshold (minimum) value of X that results in fibre failure instead of complete fibre pullout - y displacement of fibre normal to its undeflected axis - Z() area fraction angular weighting function - tensile strain in fibre relative to applied background strain - ^* critical value of to cause fibre/matrix debonding - angle at which a fibre bundle crosses the crack plane - (k ₀/4EI)^1/4, a parameter in cantilever beam analysis - v_m Poisson's ratio of matrix - L (see Equation 9) - shear stress - _* interlaminar shear strength of bundle - _d fibre/matrix interfacial shear strength - _f frictional shear slippage stress at bundle/matrix interface - angular deviation of fibre bundle from mean orientation of all bundles - angle between symmetry axis and crack plane     

Vanadium sesquioxide-polymer composites: the study of electrical conductivity

The electrical conductivity of V₂O₃-polymer composites were studied by examining the dependence of resistivity (conductivity) on the volume fraction of V₂O₃. The experimental data (_m versus ) were fitted (using a computer program) to the GEM equation with satisfactory accuracy. The critical volume fraction and other parameters evaluated by fitting (t, _h and _c) and calculated after fitting L and m , L _f and m _f were analysed by taking into account the geometry, orientation and arrangement of the two components. The physical meanings of L, m and t are further illustrated on the basis of their definitions.     

Mechanical properties of injection moulded styrene maleic anhydride (SMA) Part II Influence of short glass fibres and weldlines

The dependence of the various mechanical and fracture properties on the volume fraction ofshort glass fibres in the styrene maleic anhydride (SMA) polymer was investigated. Special attention has been given to describing the dependence of various mechanical properties on the volume fraction of the glass fibres, f by way of the rule of mixtures. It was found that, strength, elastic modulus and fracture toughness, all follow a simple rule-of-mixtures of the form Qc=Qff+Qm(1–f), where Qc is the measured quantity for the composite, Qm and Qf are the corresponding values for the matrix and the fibre, respectively, and is the overall efficiency of the fibres, taking into account the orientation and the length of the fibres in the composite. It was also found that, while the presence of the weldline had no significant effect upon elastic modulus, its presence significantly reduced tensile strength and the fracture toughness of SMA and its composites. © 1998 Kluwer Academic Publishers     

Interfacial debonding and fibre pull-out stresses

Based on a theoretical model developed previously by the authors in Part II of this series for a single fibre pull-out test, a methodology for the evaluation of interfacial properties of fibre-matrix composites is presented to determine the interfacial fracture toughness G _c, the friction coefficient , the radial residual clamping stress q _o and the critical bonded fibre length z _max. An important parameter, the stress drop , which is defined as the difference between the maximum debond stress _d ^* and the initial frictional pull-out stress _fr, is introduced to characterize the interfacial debonding and fibre pull-out behaviour. The maximum logarithmic stress drop, In(), is obtained when the embedded fibre length L is equal to the critical bonded fibre length z _max. The slope of the In()-L curve for L bigger than z _max is found to be a constant that is related to the interfacial friction coefficient . The effect of fibre anisotropy on fibre debonding and fibre pull-out is also included in this analysis. Published experimental data for several fibre-matrix composites are chosen to evaluate their interfacial properties by using the present methodology.On leave at the Department of Mechanical Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong.     

On internal stress and activation volume in polymers

The two-site model is developed for the analysis of stress relaxation data. It is shown that the product of d In (– )/d and (- _i) is constant where is the applied stress, _i is the (deformation-induced) internal stress and = d/dt. The quantity d In ( )/d is often presented in the literature as the (experimental) activation volume, and there are many examples in which the above relationship with (- _i) holds true. This is in apparent contradiction to the arguments that lead to the association of the quantity d In (– )/d with the activation volume, since these normally start with the premise that the activation volume is independent of stress. In the modified theory presented here the source of this anomaly is apparent. Similar anomalies arise in the estimation of activation volume from creep or constant strain rate tests and these are also examined from the standpoint of the site model theory. In the derivation presented here full account is taken of the site population distribution and this is the major difference compared to most other analyses. The predicted behaviour is identical to that obtained with the standard linear solid. Consideration is also given to the orientation-dependence of stress-aided activation.     

Critical assessment of the application of the J-integral and CTOD concepts to circumferentially cracked copper bars

The aim of this work is to explore experimentally the validity of the concepts of J-integral and crack tip opening displacement for characterizing the stress and strain state at the tip of an axisymmetrical crack in a bar undergoing large plastic strain before crack extension. The tests are made on extruded copper round bars presenting a very high ductility. Three different analytical formulations of the J-integral proposed in the literature for circumferentially cracked bars are compared at initiation of cracking. The limit between shallow crack and deep crack geometries is experimentally demonstrated. It is found that, in neither of these geometries, J and _CTOD are dominant. However, the ratio J_c/_c is constant for deep cracks, which suggests an alternative fracture criterion consisting in postulating the dissipation of an average critical energy per unit volume until crack extension.     

Interaction between a dislocation and an impurity in KCl: Mg2+ single crystals

The interaction between a dislocation and the impurity in KCl: Mg²⁺ (0.035 mol% in the melt) was investigated at 77–178 K with respect to the two models: one is the Fleischer's model and the other the Fleischer's model taking account of the Friedel relation. The latter is termed the F-F. The dependence of strain-rate sensitivity due to the impurities on temperature for the specimen was appropriate to the Fleischer's model than the F-F. Furthermore, the activation enthalpy, H, for the Fleischer's model appeared to be nearly proportional to the temperature in comparison with the F-F. The Friedel relation between effective stress and average length of the dislocation segments is exact for most weak obstacles to dislocation motion. However, above-mentioned results mean that the Friedel relation is not suitable for the interaction between a dislocation and the impurity in the specimen. Then, the value of H(T _c) at the Fleischer's model was found to be 0.61 eV. H(T _c) corresponds to the activation enthalpy for overcoming of the strain field around the impurity by a dislocation at 0 K. In addition, the Gibbs free energy, G ₀, concerning the dislocation motion was determined to be between 0.42 and 0.48 eV on the basis of the following equation ln / = G ₀/(kT_p0)1 – (T/T _c)^{1/2 –1}(T/T _c)^1/2 + ln ₀/where k is the Boltzmann's constant, T the temperature, T _c the critical temperature at which the effective stress due to the impurities is zero, _p0 the effective shear stress without thermal activation, and ₀ the frequency factor.     

The θ projection method and small creep strain interpolations in a commercial Titanium alloy

The paper describes the early and final uniaxial creep behaviour of a Titanium alloy used for manufacturing intermediate power compressor disks and blades. Tests were conducted at the operating temperature (773 K) for such components and for rupture lives up to 3600 hours. Creep curves were fitted using either the conventional 4 model or the recently developed 6 equation. Parameters allowing the interpolation of times to small strains were produced and their accuracy checked against experimental values using distributions found to be most supported by the data. At strains above 0.75% both methods yielded zero mean interpolation errors. At strains above 0.27% and below 0.75% the 4 equation produced systematic errors in interpolation but the 6 function gave errors which were not statistically different from zero. For strains below 0.27% both techniques produced systematic interpolation errors but the 6 interpolations were always significantly better than their 4 counterparts. Both the 6 and 4 techniques produced systematic errors when predicting the failure time using interpolated rupture strains. Unlike the 4 function, the 6 equation produced unbiased predictions of the minimum creep rate and so produced failure time interpolations from the Monkman–Grant relation that were indistinguishable from zero.     

Heat transfer in plane film formation

A mathematical model is obtained for the process of cooling with formation of a planar film. The solution obtained is verified experimentally.Notation mean axial velocity gradient - v_x current axial velocity - v_o initial polymer velocity - v₁ sampling velocity - K draw ratio - deformation rate tensor - x, y, z spatial coordinates - X, Y dimensionless coordinates - L() differential operator - T temperature - T_o initial temperature - T_c temperature of surrounding medium - dimensionless temperature - dimensionless temperature averaged over film thickness - thermal-diffusivity coefficient - 2_o initial film thickness - thermal conductivity - heat-transfer coefficient - f(X) distance function - Bi Biot criterion, Bi_o, Biot criterion calculated for initial film thickness - Gz* modified Graetz criterion - V dimensionless velocity - ₁, ₂, ₃ heat-transfer coefficients produced by radiation, free convection, and forced convection - v_c, _c mean velocity and film half-thickness in formation zone - T₁ calculated temperature value - T₂ experimental temperature value - l formation zone length Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 37, No. 5, pp. 854–858, November, 1979.     

Simulation of the Process of Water Freezing in a Round Pipe

The problem of freezing of pure water in a round pipe is treated with due regard for convection under asymmetric thermal boundary conditions in the absence of motion along the pipe. The problem is solved numerically using the control volume approach, SIMPLER algorithm, and the enthalpy method. Results are obtained for three Grashof (Gr) and six Biot (Bi) numbers: Gr = 1.55 × 10⁶, Bi = 0.305 (0 < ), Bi = 0.044 ( < 2); Gr = 1.24 × 10⁷, Bi = 0.610 (0 < ), Bi = 0.087 ( < 2); Gr = 9.89 × 10⁷, Bi = 1.220 (0 < ), Bi = 0.174 ( < 2). The correctness of calculation of the problem disregarding free-convection flows is analyzed.     

Subgap peak and Tomash-McMillan-Anderson oscillations in the density of states of SNS Bridges

The quasiparticle spectra and the densities of states of superconducting-normal-superconducting junctions are computed from the WKBJ transformed Bogoliubov-de Gennes Equations (BdGE), which are solved by Picard iteration and numerical integration. It is shown that the influence of the proximity effect on the bound states can be modeled by a rectangular pair potential well of effective normal layer thickness 2a*= _–L ^L [1–(z)/]dz, where (z) is the pair potential of the junction, is its asymptotic constant value, and 2L is the total length of the sample. The density of states exhibits a subgap peak at energies less than besides the BCS peak atE=; forE> there are geometrical resonances which are due to electron-hole interferences in finiteS layers of thicknessL-a*.     

Optical thermophysical characteristics of the condensed phase of combustion products and heatproof materials

An unconventional unit for experimentally studying the optical thermophysical properties of materials over a wide temperature range is described. Results are presented of studying the temperature function and dispersion of the absorption index for the condensed phase of the combustion products of a metalbearing fuel and of the emittance of fiber fireproof materials.Notation K spectral absorption coefficient - D spectral transmission coefficient - spectral absorption index - wavelength - d sample thickness - d_s surface layer thickness - I spectral intensity of radiation - , spectral and total emittances Kirov Polytechnical Institute. Kazan Chemical Engineering Institute, Russia. Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 64, No. 3, pp. 330–336. March, 1993     

Cutting, by ‘pressing and slicing,’ of thin floppy slices of materials illustrated by experiments on cheddar cheese and salami

Why it is easier to cut with even the sharpest knife when pressing down and sliding than when merely pressing down alone is explained. A variety of cases of cutting where the blade and workpiece have different relative motions is analysed and it is shown that the greater the slice/push ratio given by (blade speed parallel to the cutting edge/blade speed perpendicular to the cutting edge), the lower the cutting forces. However, friction limits the reductions attainable at the highest . The analysis is applied to the geometry of a wheel cutting device (delicatessan slicer) and experiments with a cheddar cheese and a salami using such an instrumented device confirm the general predictions.     

Convection in a shallow laterally heated cavity with conducting boundaries

Shallow cavity flows driven by horizontal temperature gradients are analysed over a range of Rayleigh numbersR and Prandtl numbers , whereR is comparable in size to the aspect ratioL(1). Eigenvalue calculations show the existence of a critical Prandtl number R > R _c (), below which the parallel core-flow structure is destroyed for Rayleigh numbersR>R _c(). For other Rayleigh numbers and Prandtl numbers the horizontal scale of influence of the end walls of the cavity is determined.     

A note on the similarity solutions for free convection on a vertical plate

Summary The similarity solutions for free convection on a vertical plate when the (non-dimensional) plate temperature is x and when the (non-dimensional) surface heat flux is –x are considered. Solutions valid for 1 and 1 are obtained. Further, for the first problem it is shown that there is a value ₀, dependent on the Prandtl number, such that solutions of the similarity equations are possible only for >₀, and for the second problem that solutions are possible only for >–1 (for all Prandtl numbers). In both cases the solutions becomes singular as ₀ and as –1, and the natures of these singularities are discussed.     

Dynamic reduction of the critical current in a Josephson junction

We have investigated the switching behavior of a Josephson tunnel junction driven by a signal with speed comparable to its plasma period _p . The Josephson junction is assumed initially to be in a static state so that the average voltage across it is zero. Application of a steplike current pulse of rise time _p is shown to induce switching to the finite voltage state even for values of currents much smaller than its critical current. A reduction of the critical current occurs due to the dynamic process. The approximate expressions derived show good agreement with the results obtained from simulations. The extent of dynamical reduction is limited to 27%. Shunting of the junction can eliminate this effect.     

1000 to 1300 K slow plastic compression properties of Al-deficient NiAl

Nickel aluminides containing 37, 38.5 and 40 at % Al have been fabricated by XD^tm synthesis and hot pressing. Such materials were compression tested in air under constant velocity conditions between 1000 and 1300 K. Examination of the microstructures of hot pressed and compression tested aluminides indicated that the structure consisted of two phases, and NiAl, for essentially all conditions, where was usually found on the NiAl grain boundaries. The stress-strain behaviour of all three intermetallics was similar where flow at a nominally constant stress occurred after about two plastic per cent deformation. Furthermore the 1000 to 1300 K flow stress-strain rate properties are nearly identical for these materials, and they are much lower than those for XD^tm processed Ni-50Al [1]. The overall deformation of the two phase nickel aluminides appears to be controlled by dislocation climb in NiAl rather than processes in .