Authors:Ruijie Liu; Ahmed Mostafa; Zhijun Liu Abstract: Abstract Zirconium alloys have been serving as primary structural materials for nuclear fuel claddings. Structural failure analysis under extreme conditions is critical to the assessment of the performance and safety of nuclear fuel claddings. This work focuses on simulating structural failure of Zircaloy tubes with multiple hydride defects through modeling explicit crack propagation in ductile media. First, we developed an integrated cladding failure model by taking into account both crack initiation induced by hydride/matrix interface separation and ligament tearing-off between activated hydride cracks. Second, to accommodate the initiation, propagation, and coalescence of multiple cracks in finite plastic media we incorporated this structural failure model into a coupled continuous/discontinuous Galerkin (DG) based finite element code, a traditionally preferred implicit numerical framework. Third, to improve the adaptive placement of DG interface elements for crack propagation and to identify potential coalescence of cracks due to the interaction between adjacent hydride cracks, we defined a special failure index for the assessment of potential failure zones using both true plastic strain developed and predicted failure strain based on the Johnson–Cook material failure criterion. Finally, by calibrating the proposed material failure model using a cluster of Zircaloy material experimental tests, we successfully simulated a complete failure process of a fuel cladding tube with multiple hydride cracks. PubDate: 2018-09-06 DOI: 10.1007/s10704-018-0312-9

Authors:R. J. Kashinga; L. G. Zhao; V. V. Silberschmidt; R. Jiang; P. A. S. Reed Abstract: Abstract Modelling of crack tip behaviour was carried out for a nickel-based superalloy subjected to high temperature fatigue in a vacuum and air. In a vacuum, crack growth was entirely due to mechanical deformation and thus it was sufficient to use accumulated plastic strain as a criterion. To study the strong effect of oxidation in air, a diffusion-based approach was applied to investigate the full interaction between fatigue and oxygen penetration at a crack tip. Penetration of oxygen into the crack tip induced a local compressive stress due to dilatation effect. An increase in stress intensity factor range or dwell times imposed at peak loads resulted in enhanced accumulation of oxygen at the crack tip. A crack growth criterion based on accumulated levels of oxygen and plastic strain at the crack tip was subsequently developed to predict the crack growth rate under fatigue-oxidation conditions. The predicted crack-growth behaviour compared well with experimental results. PubDate: 2018-09-06 DOI: 10.1007/s10704-018-0311-x

Authors:Shushant Singh; Debashis Khan Abstract: Abstract In the present study, mode I crack subjected to cyclic loading has been investigated for plastically compressible hardening and hardening–softening–hardening solids using the crack tip blunting model where we assume that the crack tip blunts during the maximum load and re-sharpening of the crack tip takes place under minimum load. Plane strain and small scale yielding conditions have been assumed for analysis. The influence of cyclic stress intensity factor range ( \(\Delta \hbox {K})\) , load ratio (R), number of cycles (N), plastic compressibility ( \({\upalpha })\) and material softening on near tip deformation, stress–strain fields were studied. The present numerical calculations show that the crack tip opening displacement (CTOD), convergence of the cyclic trajectories of CTOD to stable self-similar loops, plastic crack growth, plastic zone shape and size, contours of accumulated plastic strain and hydrostatic stress distribution near the crack tip depend significantly on \(\Delta \hbox {K}\) , R, N, \({\upalpha }\) and material softening. For both hardening and hardening–softening–hardening materials, yielding occurs during both loading and unloading phases, and resharpening of the crack tip during the unloading phase of the loading cycle is very significant. The similarities are revealed between computed near tip stress–strain variables and the experimental trends of the fatigue crack growth rate. There was no crack closure during unloading for any of the load cycles considered in the present study. PubDate: 2018-09-01 DOI: 10.1007/s10704-018-0310-y

Authors:Ali Taghichian; Hamid Hashemalhoseini; Musharraf Zaman; Zon-Yee Yang Abstract: Abstract Unconventional drilling and completion architecture includes drilling multilateral horizontal wells in the direction of minimum horizontal stress and simultaneous multistage fracturing treatments perpendicular to the wellbore. This drilling and stimulation strategy is utilized in order to raise the connectivity of the reservoir to the wellbore, thereby remedying the low permeability problem, increasing reserve per well, enhancing well productivity, and improving project economics in this type of reservoir. However, in order to have the highest production with the least cost, an optimization technique should be used for the fracturing treatment. According to the fact that aperture, propagation direction, and propagation potential of hydraulic fractures are of paramount importance in optimization of the fracking treatment, in this research paper, these three major factors are studied in detail, the control variables on these three factors are examined, and the effect of each factor is quantified by proposing a complete set of equations. Using the proposed set of equations, one can make a good estimate about the fracture aperture (directly controlling the fracture conductivity), the stress shadow size (directly controlling the fracture path), and the change of stress intensity factor (directly controlling the fracture propagation potential). A geomechanical optimization procedure is then presented for toughness-dominated and viscosity-dominated regimes based on the proposed equations that can be used for estimation of different optimal fracturing patterns. The most efficient fracturing pattern can be determined afterward via considering the cumulative production using a reservoir simulator e.g. ECLIPSE, Schlumberger. This procedure is likely to offer an optimal simultaneous multistage hydraulic fracture treatment without deviation or collapse, with no fracture trapping, with the highest possible propagation potential in the hydrocarbon producing shale layer, and a predicted aperture for proppant type/size decision and conductivity of the fractures. PubDate: 2018-08-24 DOI: 10.1007/s10704-018-0309-4

Authors:Motomichi Koyama; Takahiro Kaneko; Takahiro Sawaguchi; Kaneaki Tsuzaki Abstract: Abstract We investigated the damage evolution behaviors of binary Fe–28–40Mn alloys (mass%) from 93 to 393 K by tensile testing. The underlying mechanisms of the microstructure-dependent damage evolution behavior were uncovered by damage quantification coupled with in situ strain mapping and post-mortem microstructure characterization. The damage growth behaviors could be classified into three types. In type I, the Fe–28Mn alloy at 93 K showed premature fracture associated with ductile damage initiation and subsequent quasi-cleavage damage growth associated with the \(\upvarepsilon \) -martensitic transformation. In type II, the Fe–28Mn alloy at 293 K and the Fe–32Mn alloy at 93 K showed delayed damage growth but did not stop growing. In type III, when the stacking fault energy was \(>\,\) 19 \(\hbox {mJ/m}^{2}\) , the damage was strongly arrested until final ductile failure. PubDate: 2018-08-24 DOI: 10.1007/s10704-018-0307-6

Authors:Najam Sheikh; Sivasambu Mahesh Abstract: Abstract A shear-lag model of hybrid materials is developed. The model represents an alternating arrangement of two types of aligned linear elastic fibres, embedded in a linear elastic matrix. Fibre and matrix elements are taken to fail deterministically when the axial and shear stresses in them reach their respective strengths. An efficient solution procedure for determining the stress state for arbitrary configurations of broken fibre and matrix elements is developed. Starting with a single fibre break, this procedure is used to simulate progressive fibre and matrix failure, up to composite fracture. The effect of (1) the ratio of fibre stiffnesses, and (2) the ratio of the fibre tensile strength to matrix shear strength, on the composite failure mechanism, fracture energy, and failure strain is characterised. Experimental observations, reported in the literature, of the fracture behaviour of two hybrid materials, viz., hybrid unidirectional composites, and double network hydrogels, are discussed in the framework of the present model. PubDate: 2018-08-16 DOI: 10.1007/s10704-018-0306-7

Authors:Shigeru Hamada; Jiwang Zhang; Kejin Zhang; Motomichi Koyama; Toshihiro Tsuchiyama; Tatsuo Yokoi; Hiroshi Noguchi Abstract: Abstract The effect of shear-affected zone (SAZ), with a stress-concentration source induced by the punching process, on tensile properties was investigated. Tests using honed specimens (which have the same shapes and stress-concentration without any SAZ) and smooth specimens were conducted to compare the effect with that of the punched specimens. Dual-phase steel, which has a high work-hardening ability and low yield strength, and precipitation-hardened steel, which has a low work-hardening ability and high yield strength, were used in the tests. Materials with two tensile strength grades were prepared from both types of steel. Only the precipitation-hardened steel with higher strength grade punched specimen showed a brittle fracture with extremely short fracture-elongation, whereas the other specimens showed a ductile fracture. The fracture surface analysis revealed that cracks initiated in the maximum shear stress plane of the SAZ under tensile loading at first. We call the crack “shear crack.” The steel which showed brittle fracture used in this study easily exhibited plastic-strain localization compared with the other steels. If the shear crack is sharp, then the transition from ductile to brittle failure tends to occur. Furthermore, the strength characteristics of the punched specimen depend on the crack length dependency of the strength resistance and the failure phenomenon of the original material. PubDate: 2018-08-07 DOI: 10.1007/s10704-018-0304-9

Authors:Aurélien Doitrand; Dominique Leguillon Abstract: Abstract A numerical study of crack front segmentation under mode I + III loading is proposed. Facets initiation ahead of a parent crack is predicted through a tridimensional application of the coupled criterion. Crack initiation shape, orientation and spacing are determined for any mode mixity ratio by coupling a stress and an energy criterion using matched asymptotic expansions. The stress and the energy conditions are computed through a 3D finite element modeling of a periodic network of facets ahead of the parent crack. The initiation shape, loading and spacing of facets depend on the blunted parent crack tip radius. A good estimate of facet orientations is obtained based on the direction of maximum tensile stress. The facet shapes, determined using the stress isocontours, are qualitatively similar to those observed experimentally. The order of magnitude of numerical predictions of facets spacing is very close to experimental measurements. PubDate: 2018-08-03 DOI: 10.1007/s10704-018-0305-8

Authors:Aditya Kumar; K. Ravi-Chandar; Oscar Lopez-Pamies Abstract: Abstract In a recent contribution, Kumar et al. (J Mech Phys Solids 112:523–551, 2018) have introduced a macroscopic theory aimed at describing, explaining, and predicting the nucleation and propagation of fracture and healing in elastomers undergoing arbitrarily large quasistatic deformations. The purpose of this paper is to present an alternative derivation of this theory—originally constructed as a generalization of the variational theory of brittle fracture of Francfort and Marigo (J Mech Phys Solids 46:1319–1342, 1998) to account for physical attributes innate to elastomers that have been recently unveiled by experiments at high spatio-temporal resolution—cast as a phase transition within the framework of configurational forces. A second objective of this paper is to deploy the theory to probe new experimental results on healing in silicone elastomers. PubDate: 2018-07-31 DOI: 10.1007/s10704-018-0302-y

Authors:Marcelo Paredes; Junhe Lian; Tomasz Wierzbicki; Mihaela E. Cristea; Sebastian Münstermann; Philippe Darcis Abstract: Abstract A non-associated/associated flow rule coupled with an anisotropic/isotropic quadratic yield function is presented to describe the mechanical responses of two distinct X65 pipeline steels. The first as a product of the cold-rolling forming (UOE) process also known as seam weld pipes and the second as a result of high temperature piercing process called seamless tube manufacturing. The experimental settings consist of a wide range of sample types, whose geometric characteristics represent different state of stresses and loading modes. For low to intermediate stress triaxiality levels, flat specimens are extracted at different material orientations along with notched round bar samples for high stress triaxialities. The results indicate that despite the existing differences in plasticity between materials due to anisotropy induced processes, material failure can be characterized by an isotropic weighting function based on the modified Mohr–Coulomb (MMC) criterion. The non-associated flow rule allows for inclusion of strain directional dependence in the definition of equivalent plastic strain by means of scalar anisotropy (Lankford) coefficients and thus keeping the original capabilities of the MMC model. PubDate: 2018-07-27 DOI: 10.1007/s10704-018-0303-x

Authors:Marianela Ripani; Sonia Vrech; Guillermo Etse Abstract: Abstract This work focuses on the evaluation of temperature effects on concrete failure behavior and modes by means of a realistic thermodynamically consistent non-local poroplastic constitutive model, previously developed by the authors, which is modified in this work. In this regard, two original contributions are presented and discussed. Firstly, and based on significant published experimental results related to this very complex aspect such as the effects of temperature in concrete failure, a temperature dependent non-associated flow rule is introduced to the poroplastic constitutive model to more accurately account for the temperature dependent inelastic volumetric behavior of concrete in post-peak regime. This is crucial for improving overall model accuracy, particularly regarding the temperature effects on concrete released energy during degradation processes. Secondly, and more importantly, the explicit solution of the localization condition in terms of the critical hardening modulus is developed regarding the non-local poroplastic constitutive model reformulated in this work, which allows the analysis of localized failure modes in the form of discontinuous bifurcation of quasi-brittle porous materials like concrete under different temperature, hydraulic and stress state scenarios. Also numerical procedures are followed, which also allow the evaluation of temperature effects on the critical directions for localized failure or cracking which is performed in this work for a wide spectrum of stress states and temperatures. Both, undrained and drained hydraulic conditions are evaluated. The results in this work demonstrate the soundness of the proposed constitutive model modifications and of the derived explicit solution for the critical hardening modulus to accurately predict the temperature effects on both, concrete volumetric behavior, and on the failure modes and related critical cracking direction. They also demonstrate that concrete failure mode and critical localization directions are highly sensitive to temperature, particularly in the compressive regime. PubDate: 2018-07-26 DOI: 10.1007/s10704-018-0301-z

Authors:Gülcan Özerim; Günay Anlaş; Ziad Moumni Abstract: Abstract In a domain of reasonable accuracy around the crack tip, asymptotic equations can provide closed form expressions that can be used in formulation of crack problem. In some studies on shape memory alloys (SMAs), although the pseudoelastic behavior results in a nonlinear stress–strain relation, stress distribution in the vicinity of the crack tip is evaluated using asymptotic equations of linear elastic fracture mechanics (LEFM). In pseudoelastic (SMAs), upon loading, stress increases around the crack tip and martensitic phase transformation occurs in early stages. In this paper, using the similarity in the loading paths of a pseudoelastic SMA and a strain hardening material, the stress–strain relation is represented by nonlinear Ramberg–Osgood equation which is originally proposed for strain hardening materials, and the stress distribution around the crack tip of a pseudoelastic SMA plate is reworked using the Hutchinson, Rice and Rosengren (HRR) solution, first studied by Hutchinson. The size of the transformation region around the crack tip is calculated in closed form using a thermodynamic force that governs the martensitic transformation together with the asymptotic equations of HRR. A UMAT is written to separately describe the thermo-mechanical behavior of pseudoelastic SMAs. The results of the present study are compared to the results of LEFM, computational results from ABAQUS, and experimental results for the case of an edge cracked NiTi plate. Both set of asymptotic equations are shown to have different dominant zones around the crack tip with HRR equations representing the martensitic transformation zone more accurately. PubDate: 2018-07-20 DOI: 10.1007/s10704-018-0300-0

Authors:Harika C. Tankasala; Vikram S. Deshpande; Norman A. Fleck Abstract: Abstract The macroscopic tensile strength of a panel containing a centre-crack or a centre-hole is predicted, assuming the simultaneous activation of multiple cohesive zones. The panel is made from an orthotropic elastic solid, and the stress raiser has both a tensile cohesive zone ahead of its tip, and shear cohesive zones in an orthogonal direction in order to represent two simultaneous damage mechanisms. These cohesive zones allow for two modes of fracture: (i) crack extension by penetration, and (ii) splitting in an orthogonal direction. The sensitivity of macroscopic tensile strength and failure mode to the degree of orthotropy is explored. The role of notch acuity and notch size are assessed by comparing the response of the pre-crack to that of the circular hole. This study reveals the role of the relative strength and relative toughness of competing damage modes in dictating the macroscopic strength of a notched panel made from an orthotropic elastic solid. Universal failure mechanism maps are constructed for the pre-crack and hole for a wide range of material orthotropies. The maps are useful for predicting whether failure is by penetration or kinking. Case studies are developed to compare the predictions with observations taken from the literature for selected orthotropic solids. It is found that synergistic strengthening occurs: when failure is by crack penetration ahead of the stress raiser, the presence of shear plastic zones leads to an enhancement of macroscopic strength. In contrast, when failure is by crack kinking, the presence of a tensile plastic zone ahead of the stress raiser has only a mild effect upon the macroscopic strength. PubDate: 2018-07-05 DOI: 10.1007/s10704-018-0296-5

Authors:I. F. Dias; J. Oliver; O. Lloberas-Valls Abstract: Abstract This paper presents a finite element approach for modelling three-dimensional crack propagation in quasi-brittle materials, based on the strain injection and the crack-path field techniques. These numerical techniques were already tested and validated by static and dynamic simulations in 2D classical benchmarks [Dias et al., in: Monograph CIMNE No-134. International Center for Numerical Methods in Engineering, Barcelona, (2012); Oliver et al. in Comput Methods Appl Mech Eng 274:289–348, (2014); Lloberas-Valls et al. in Comput Methods Appl Mech Eng 308:499–534, (2016)] and, also, for modelling tensile crack propagation in real concrete structures, like concrete gravity dams [Dias et al. in Eng Fract Mech 154:288–310, (2016)]. The main advantages of the methodology are the low computational cost and the independence of the results on the size and orientation of the finite element mesh. These advantages were highlighted in previous works by the authors and motivate the present extension to 3D cases. The proposed methodology is implemented in the finite element framework using continuum constitutive models equipped with strain softening and consists, essentially, in injecting the elements candidate to capture the cracks with some goal oriented strain modes for improving the performance of the injected elements for simulating propagating displacement discontinuities. The goal-oriented strain modes are introduced by resorting to mixed formulations and to the Continuum Strong Discontinuity Approach (CSDA), while the crack position inside the finite elements is retrieved by resorting to the crack-path field technique. Representative numerical simulations in 3D benchmarks show that the advantages of the methodology already pointed out in 2D are kept in 3D scenarios. PubDate: 2018-07-02 DOI: 10.1007/s10704-018-0293-8

Authors:Deepak Sharma; B. Subba Reddy; Praveen Kumar Abstract: Abstract We investigated propagation of a sharp crack in a thin metallic conductor with an edge crack due to electric current induced electromagnetic forces. Finite element method (FEM) simulations showed mode I crack opening in the edge-cracked conductor due to the aforementioned (i.e., self-induced) electromagnetic forces. Mode I stress intensity factor due to the self-induced electromagnetic forces, \(K_{\mathrm{IE},}\) was evaluated numerically as \(K_{\mathrm{IE}}=\upmu l^{2}j^{2}(\uppi a)^{0.5}f(a/w)\) , where \(\upmu \) is the magnetic permeability, l is the length of the conductor, a is the crack length, j is the current density, w is the width of the sample and f(a / w) is a geometric factor. Effect of dynamic electric current loading on edge-cracked conductor, incorporating the effects of induced currents, was also studied numerically, and dynamic stress intensity factor, \(K_{\mathrm{IE,d}}\) , was observed to vary as \(K_{\mathrm{IE,d}} \sim f_{d}(a/w)j^{2}(\uppi a)^{1.5}\) . Consistent with the FEM simulation, experiments conducted using \(12\,\upmu \hbox {m}\) thick Al foil with an edge crack showed propagation of sharp crack due to the self-induced electromagnetic forces at pulsed current densities of \(\ge \) \(1.85\times 10^{9}\,\hbox {A/m}^{2}\) for \(a/w = 0.5\) . Further, effects of current density, pulse-width and ambient temperature on the fracture behavior of the Al foil were observed experimentally and corroborated with FEM simulations. PubDate: 2018-07-02 DOI: 10.1007/s10704-018-0299-2

Authors:Deepak Sharma; B. Subba Reddy; Praveen Kumar Abstract: Abstract Recently, we reported fracture of edge cracked thin metallic conductors upon passing only electric current, i.e., due to the self-induced electromagnetic forces. In this study, effect of simultaneous application of an electric current and a mechanical load on fracture of a thin edge-cracked conductor is investigated. Firstly, finite element method (FEM) simulation was performed to analyze the interaction between stress fields due to the electric current and the far-field mechanical loading. FEM simulations showed that stress fields as well as stress intensity factors due to each stimulus can be linearly superimposed to calculate their respective values under the combined loading. To corroborate the FEM results, experiments were conducted where a mechanical load was applied along with the electric pulse current. The critical current density required to propagate the sharp crack under combined loading decreased and the rate of crack propagation per electric current pulse drastically increased. Effect of mode-mixity on the fracture behavior of thin metallic foil under the combined loading was also studied by applying the mechanical load at different angles relative to the crack, and it was observed that the crack could be deflected at a desired, pre-determined angle by changing the mode mixity. PubDate: 2018-06-30 DOI: 10.1007/s10704-018-0298-3

Authors:F. Tankoua; J. Crépin; P. Thibaux; S. Cooreman; A.-F. Gourgues-Lorenzon Abstract: Abstract This work proposes a quantitative relationship between the resistance of hot-rolled steels to brittle cleavage fracture and typical microstructural features, such as microtexture. More specifically, two hot-rolled ferritic pipeline steels were studied using impact toughness and specific quasistatic tensile tests. In drop weight tear tests, both steels exhibited brittle out-of-plane fracture by delamination and by so-called “abnormal” slant fracture, here denoted as “brittle tilted fracture” (BTF). Their sensitivity to cleavage cracking was thoroughly determined in the fully brittle temperature range using round notched bars, according to the local approach to fracture, taking anisotropic plastic flow into account. Despite limited anisotropy in global texture and grain morphology, a strong anisotropy in critical cleavage fracture stress was evidenced for the two steels, and related through a Griffith-inspired approach to the size distribution of clusters of unfavorably oriented ferrite grains (so-called “potential cleavage facets”). It was quantitatively demonstrated that the occurrence of BTF, as well as the sensitivity to delamination by cleavage fracture, is primarily related to an intrinsically high sensitivity of the corresponding planes to cleavage crack propagation across potential cleavage facets. PubDate: 2018-06-29 DOI: 10.1007/s10704-018-0297-4

Authors:X. L. Ji; L. X. Li Abstract: Abstract Biomaterials use a hierarchical structure to optimize their self-healing behavior, for instance. However, the behavior may be constrained under different mechanical environments. In this paper, a system is suggested that the mechanical environment is modeled as a spring connected in series with the fiber bundle material. For the spring, the elastic behavior with stiffness is obeyed while, for the fiber bundle material, the nonlinear elastic constitutive relation is obeyed according to the Weibull distribution and the Daniels’ theory. Relying on the principle of total potential, the applicability condition is proposed for the system and the critical stiffness is thus derived for the spring. The applicability of hierarchical fiber bundle materials is finally investigated. The results show that the hierarchy can significantly change the critical stiffness, and hence demonstrates quite different applicability to a given mechanical environment. PubDate: 2018-06-12 DOI: 10.1007/s10704-018-0290-y

Authors:A. Markov; S. Kanaun Abstract: Abstract The paper is devoted to the problem of slow crack growth in heterogeneous media. The crack is subjected to arbitrary pressure distribution on the crack surface. The problem relates to construction of the so-called equilibrium crack. For such a crack, stress intensity factors are equal to the material fracture toughness at each point of the crack contour. The crack shape and size depend on spatial distributions of the elastic properties and fracture toughness of the medium, and the type of loading. In the paper, attention is focused on the case of layered elastic media when a planar crack propagates orthogonally to the layers. The problem is reduced to a system of surface integral equations for the crack opening vector and volume integral equations for stresses in the medium. For discretization of these equations, a regular node grid and Gaussian approximating functions are used. For iterative solution of the discretized equations, fast Fourier transform technique is employed. An iteration process is proposed for the construction of the crack shape in the process of crack growth. Examples of crack evolution for various properties of medium and types of loading are presented. PubDate: 2018-05-11 DOI: 10.1007/s10704-018-0284-9

Authors:Sergey Cherny; Vasily Lapin; Dmitriy Kuranakov; Olga Alekseenko Abstract: Abstract The paper presents an extension of authors’ previous model for a 3D hydraulic fracture with Newtonian fluid, which aims to account for the Herschel–Bulkley fluid rheology and to study the associated effects. This fluid rheology model is the most suitable for description of modern complex fracturing fluids, in particular, for description of foamed fluids that have been successfully utilized recently as fracturing fluids in tight and ultra-tight unconventional formations with high clay contents. Another advantage of using Herschel–Bulkley rheological law in the hydraulic fracture model consists in its generality as its particular cases allow describing the behavior of the majority of non-Newtonian fluids employed in hydraulic fracturing. Except the Herschel–Bulkley fluid flow model the considered model of hydraulic fracturing includes the model of the rock stress state. It is based on the elastic equilibrium equations that are solved by the dual boundary element method. Also the hydraulic fracturing model contains the new mixed mode propagation criterion, which states that the fracture should propagate in the direction in which mode \({{\mathrm{\mathrm {II}}}}\) and mode \({{\mathrm{\mathrm {III}}}}\) stress intensity factors both vanish. Since it is not possible to make both modes zero simultaneously the criterion proposes a functional that depends on both modes and is minimized along the fracture front in order to obtain the direction of propagation. Solution for Herschel–Bulkley fluid flow in a channel is presented in detail, and the numerical algorithm is described. The developed model has been verified against some reference solutions and sensitivity of fracture geometry to rheological fluid parameters has been studied to some extent. PubDate: 2018-05-10 DOI: 10.1007/s10704-018-0289-4