Abstract: In unconventional shale reservoirs, the presence of natural fractures is common and widespread. Multi-stage hydraulic fracture treatments are used to generate complex fracture geometry and stimulate reservoirs to unlock hydrocarbon in unconventional reservoirs. Natural fractures further complicate the complexity of the fracture geometry. In this paper, the objective is to investigate the impact of natural fractures on multiple hydraulic fracture propagation using an in-house fracture propagation model. We have developed a complex hydraulic fracture model to simulate multiple fracture propagation in naturally fractured reservoirs. The model can calculate partitioning of fluid between multiple fractures, fracture interaction within a stage and between stages, as well as the interaction between hydraulic and natural fractures. For simultaneous multiple fracture propagation, the exterior fractures generally take the majority of fluid and propagate at the expense of the interior fractures. Natural fractures can retard the growth of hydraulic fractures and change the fluid distribution between fractures and propagation trajectories of hydraulic fractures. In a reservoir with a heterogeneous natural fracture distribution, the interruption of natural fractures can complicate fracture geometry. The higher density or smaller spacing of natural fractures can mitigate stress shadow effects and favor more fluid flowing into the inner fractures. Due to the uncertainty of natural fracture distribution, the reliability of the complex fracture geometry provided by numerical models has been challenged. This paper demonstrates the potential of incorporating diagnostic results to constrain numerical results. Through combining diagnostic results with the fracture model, the accuracy of predicted fracture geometry can be improved. PubDate: 2019-01-16

Authors:David Roucou; Julie Diani; Mathias Brieu; Armel Mbiakop-Ngassa Abstract: In order to estimate mode I fracture strain energy release rate of a rubber upon monotonic loadings, the material is submitted to pure shear and single edge notch tension tests. Catastrophic failure happens suddenly for both tests, revealing mirror-like crack surfaces, assessing the fragile fracture. Nonetheless, Griffith failure analysis could be carried out on pure shear tests only. This analysis leads to an energy release rate value that allows challenging approximate expressions existing in the literature for pure shear and single edge notch tension tests. The pure shear approximate expression provides quantities that match the Griffith analysis. Meanwhile, the strain energy release rate values calculated directly from the single edge notch tension tests differ significantly from the values obtained in pure shear. This discrepancy is explored and possible explanations are discussed showing that pure shear tests should be favored. PubDate: 2019-01-12 DOI: 10.1007/s10704-018-00336-8

Authors:Shinya Matsuda Abstract: This paper presents a theoretical approach to determine the dynamic fatigue strength characteristics of ceramics under variable loading rates on the basis of the slow crack growth (SCG) concept. First, a probabilistic effective inert strength model was derived on the basis of the SCG concept in conjunction with the Weibull distribution for ceramics subjected to multi-stage loading. Second, a four-point bending test was conducted on \(\hbox {Al}_{{2}}\hbox {O}_{{3}}\) under constant and two-stage variable loading rates, and the fracture surface was then observed. The experimental data that depend on loading rates can be unifiedly evaluated after converting the data to the effective inert strength, obeying the three-parameter Weibull distribution. In addition, the Weibull plots of the inert strength, which were calculated from the inclusion size on the fracture surface using the grain fracture model, showed good agreement with the three-parameter Weibull distribution for the converted effective inert strength. These analytical results theoretically indicate that dynamic fatigue under variable loading rates occurs by obeying SCG at the inclusion. Further, the inert strength and its scatter depend on the size and distribution of inclusions. PubDate: 2019-01-08 DOI: 10.1007/s10704-018-00337-7

Authors:Chung Yuen Hui; Jingyi Guo; Mincong Liu; Alan Zehnder Abstract: In previous works we developed a three-dimensional large deformation constitutive theory for a dual cross-link hydrogel gel with permanent and transient bonds. This theory connects the breaking and reforming kinetics of the transient bonds to the nonlinear elasticity of the gel network. We have shown that this theory agrees well with experimental data from both tension and torsion tests. Here we study the mechanics of a Mode III or anti-plane shear crack for this particular class of rate dependent solids. We first show that our constitutive model admits a non-trivial state of anti-plane shear deformation. We then establish a correspondence principle for the special case where the chains in the network obey Gaussian statistics. For this special case there is a one to one analogy between our model and standard linear viscoelasticity. We also study the asymptotic behavior of the time dependent crack tip field, and show that for a wide class of network energy density functions, the spatial singularities of these fields are identical to a hyper-elastic, cracked body with the same but undamaged networks. PubDate: 2019-01-03 DOI: 10.1007/s10704-018-00335-9

Authors:Yong Zhang; Pizhong Qiao Abstract: Peridynamics is a nonlocal theory, and it has been applied to a series of fracture problems based on its two main bond failure criteria: the critical stretch criterion and the critical energy density criterion. In this paper, a new criterion, the critical skew criterion, corresponding to the shear deformation, is for the first time proposed specifically for ordinary state-based peridynamic mode II fracture analysis. The necessity of the critical skew criterion is demonstrated by limitations and inaccuracy of the existing critical stretch and energy density criteria in theoretical and numerical mode II fracture analysis. The validity of the proposed critical skew criterion is illustrated by quantitative analysis of the captured behaviors of the typical mode II fracture tests. The results obtained by the critical skew criterion agree well with the benchmark data from the linear elastic theory, the virtual crack closure technique, and the Griffith’s theory in different aspects of analysis. A simplified formula of the bond energy density is also derived and verified, and it can serves as a fundamental tool in peridynamic fracture analysis. PubDate: 2019-01-01 DOI: 10.1007/s10704-018-00341-x

Authors:A. S. Savinykh; G. V. Garkushin; G. I. Kanel; S. V. Razorenov Abstract: In order to determine the compressive and tensile strength of concrete under conditions of explosive loading, and to develop a methodological framework in this regard, three types of concrete have been investigated: concrete with fine-grained granite in the form of crushed stone having a static compressive strength of 47 MPa, the same concrete with the addition of steel fibers and also the same concrete reinforced by steel bars. The samples were rods of 50 and 100 mm diameter and five to ten diameters in length. The compressive fracture occurs at a relatively small distance of propagation of the load pulse along the rod and is accompanied by fast decay. The measurements of parameters of the compression pulse at the end of the fracture zone allowed us to determine the values of the dynamic compressive strength of the concrete while measurements of the free-surface velocity history at long distances were used for determining the dynamic tensile strength or spall strength values. The values of the dynamic compressive strength were found to be 2.5 times higher than the static strength. The steel fibers increased the dynamic compressive strength by about 10%. The obtained values of the dynamic tensile strength are 3–8 times higher than the values of the static tensile strength. The steel fibers increase the tensile strength by 20–50%. The reinforced sample has shown an increase of dynamic tensile strength by a factor of about 30. PubDate: 2019-01-01 DOI: 10.1007/s10704-018-00342-w

Authors:Chyanbin Hwu; Wei-Ren Chen; Ting-Hsiang Lo Abstract: Green’s function for a two-dimensional anisotropic elastic solid containing a rigid or elastic inclusion has been previously explored. According to the special feature of Stroh formalism for two-dimensional anisotropic elasticity, the same mathematical form of Green’s function can be extended to cases with piezoelectric and magneto-electro-elastic materials by expanding the related matrix dimension. In this paper, we show that some important constant terms are missing in the existing Green’s functions and the solutions should be corrected to ensure the displacement and traction continuity across the inclusion interface. Besides the necessary analytical check, a further verification is provided by applying the corrected Green’s functions to the problems of crack-inclusion interaction. We consider that the cracks exist in smart materials made by composites embedded with piezoelectric and/or magneto-electro-elastic sensors and actuators. Since the anisotropic elastic, piezoelectric and magneto-electro-elastic materials exist simultaneously, an adaptable adjustment technique is proposed. With this technique, the dislocation superposition method and boundary-based finite element methods developed previously for the problems with a single material type can now be extended to study the coupled-field interaction problems. PubDate: 2018-12-17 DOI: 10.1007/s10704-018-00338-6

Authors:Shai Feldfogel; Oded Rabinovitch Abstract: Adhesively bonded FRP patches can be used to strengthen damaged reinforced concrete slabs, which typically exhibit a two-dimensional cracking pattern. The combination of a layered configuration and a cracked substrate designates interfacial debonding as a principal failure mechanism. The present work aims to shed light on the fundamental questions regarding the behavior of this cracked and strengthened layered form, using a high-order multi-layered plate theory and a corresponding finite element formulation, and with emphasis on the modeling of two dimensional plate-type cracks and their impact on the structural response. The analytical–computational platform addresses the manifold challenges involved in modeling the crack induced discontinuities and the two-dimensional evolution of interfacial debonding. The study looks into the behavior of a shear cracked beam and a diagonally cracked and FRP patched square slab. It reveals the unstable, two-dimensional, and not-self-similar nature of crack-induced debonding in two-say slabs. The debonding mechanism is characterized through irregular equilibrium paths and through the interfacial traction profiles that govern its evolution and stability features. The essential differences between the known debonding mechanisms observed in cracked beams and the case at hand are noted and discussed, setting the latter as a new and separate category. PubDate: 2018-12-10 DOI: 10.1007/s10704-018-0333-4

Authors:Valerio Carollo; Marco Paggi; José Reinoso Abstract: The problem of adhesive wear is herein investigated in relation to periodic asperity junction models in the framework of the Archard interpretation suggesting that wear debris formation is the result of asperity fracture. To this aim, the phase field model for fracture is exploited to simulate the crack pattern leading to debris formation in the asperity junction model. Based on dimensional analysis considerations, the effect of the size of the junction length, the lateral size of the asperity, and the amplitude of the re-entrant corner angles \(\gamma \) and \(\beta \) defined by the junction geometry is examined in the parametric analysis. Results show that two failure modes are expected to occur, one with a crack nucleated at the re-entrant corner \(\gamma \) , and another with a crack nucleated at the re-entrant corner \(\beta \) , depending on the dominant power of the stress-singularity at the two re-entrant corner tips. Steady-state adhesive wear, where the initial asperity junction geometry is reproduced after debris formation, is observed for asperity junctions with \(\gamma = 45^\circ \) , almost independently of the lateral size of the asperity and of the horizontal projection of the junction length. PubDate: 2018-12-06 DOI: 10.1007/s10704-018-0329-0

Authors:R. Eckner; L. Krüger; C. Ullrich; M. Wendler; O. Volkova Abstract: The recently developed austenitic-martensitic TRIP cast steel Fe–14Cr–3Ni–3Mn–0.4Si–0.11N–0.15C was subjected to different Quenching & Partitioning (Q&P) treatments in order to achieve a variation of the microstructural and mechanical properties. Subsequently, the fracture properties of three material variants were studied by means of tensile tests and fracture mechanical 3-point-bending tests to determine J– \(\Delta a\) fracture resistance curves. Due to Q&P treatment, the steel achieved considerable strength and ductility values (UTS of about 1500 MPa with a total elongation of almost 30%) which qualify it for the 3rd generation of AHSS. The fracture toughness behavior was significantly influenced by the initial \(\upalpha ^\prime \) -martensite content as well as by the austenite stability, which could be adjusted by varying the Q&P parameters. If the austenite stability was low, the formation of deformation-induced \(\upalpha ^\prime \) -martensite became possible. This TRIP effect is known to be beneficial for fracture toughness of austenitic steels. However, the experimental results suggest that there was a contrary effect of embrittlement due to metastable austenite which undergoes martensitic transformation already in the early stages of deformation. Therefore, the Q&P parameters have to be carefully chosen in order to achieve a remarkable combination of strength, ductility and fracture toughness of the investigated high-alloy austenitic-martensitic TRIP steel. PubDate: 2018-12-01 DOI: 10.1007/s10704-018-0332-5

Authors:Q. Zeng; M. H. Motamedi; A. F. T. Leong; N. P. Daphalapurkar; T. C. Hufnagel; K. T. Ramesh Abstract: Brittle and quasibrittle materials such as ceramics and geomaterials fail through dynamic crack propagation during impact events. Simulations of such events are important in a number of applications. This paper compares the effectiveness of the embedded finite element method (EFEM) and the extended finite element method (XFEM) in modeling dynamic crack propagation by validating each approach against an impact experiment performed on single crystal quartz together with in-situ imaging of the dynamic fracture using X-ray phase contrast imaging (XPCI). The experiment is conducted in a Kolsky bar (generating a strain rate on the order of \(10^3\,\text {s}^{-1}\) ) that is operated at the synchrotron facilities at the advanced photon source (APS). The in situ XPCI technique can record the dynamic crack propagation with micron-scale spatial resolution and sub-microsecond temporal resolution, and the corresponding images are used to extract the time-resolved crack propagation path and velocity. A unified framework is first presented for the dynamic discretization formulations of EFEM and XFEM. This framework clarifies the differences between the two methods in enrichment techniques and numerical solution schemes. In both cases, a cohesive law is used to describe the fracture process after crack initiation. The simulations of the dynamic fracture experiment using the two simulation approaches are compared with the in situ experimental observations and measurements. The performance of each method is discussed with respect to capturing the early crack propagation process. PubDate: 2018-11-17 DOI: 10.1007/s10704-018-0330-7

Authors:Xinpeng Tian; Qun Li; Qian Deng Abstract: The flexoelectric effect is a significant electromechanical coupling phenomenon between strain gradients and electric polarization. Since the design of materials with high flexoelectricity should be accompanied with stress concentration/intensity, the strength and fracture analysis of flexoelectric materials with large strain gradients is desired. The famous J-integral can be used to characterize the singularity at crack tips and predict the fracture behavior of flexoelectric solids. However, the definition of J-integral in flexoelectric solids is lacked or incomplete in the open literature. In this study, an explicit expression of J-integral associated with material configurational forces is derived from the gradient operation of electric enthalpy density function for centrosymmetric flexoelectric solids, where the electric enthalpy density depends not only on the strain and strain gradient, but also on the polarization and polarization gradient. The path-independence of J-integral in flexoelectric solids is also examined through the Gauss–Green’s theorem. Then the derived J-integral is applied to study a cylindrical cavity and a mode III crack problem in flexoelectric solids. The results indicate that, in flexoelectric solids, there is a conservation law of the J-integral. That is, the J-integral defined in a global coordinate system vanishes when the integration contour chosen to calculate the J-integral encloses whole cavity. The present complete expression of J-integral in flexoelectric solids is addressed from the self-consistent theory of flexoelectricity. It corrects the inaccurate definition of J-integral in the previous literature. The J-integral obtained in this paper will provide a useful way to study fracture problems in flexoelectric solids. PubDate: 2018-11-15 DOI: 10.1007/s10704-018-0331-6

Authors:Leo Škec; Giulio Alfano; Gordan Jelenić Abstract: In this work we develop a complete analytical solution for a double cantilever beam (DCB) where the arms are modelled as Timoshenko beams, and a bi-linear cohesive-zone model (CZM) is embedded at the interface. The solution is given for two types of DCB; one with prescribed rotations (with steady-state crack propagation) and one with prescribed displacement (where the crack propagation is not steady state). Because the CZM is bi-linear, the analytical solutions are given separately in three phases, namely (i) linear-elastic behaviour before crack propagation, (ii) damage growth before crack propagation and (iii) crack propagation. These solutions are then used to derive the solutions for the case when the interface is linear-elastic with brittle failure (i.e. no damage growth before crack propagation) and the case with infinitely stiff interface with brittle failure (corresponding to linear-elastic fracture mechanics (LEFM) solutions). If the DCB arms are shear-deformable, our solution correctly captures the fact that they will rotate at the crack tip and in front of it even if the interface is infinitely stiff. Expressions defining the distribution of contact tractions at the interface, as well as shear forces, bending moments and cross-sectional rotations of the arms, at and in front of the crack tip, are derived for a linear-elastic interface with brittle failure and in the LEFM limit. For a DCB with prescribed displacement in the LEFM limit we also derive a closed-form expression for the critical energy release rate, \(G_c\) . This formula, compared to the so-called ‘standard beam theory’ formula based on the assumptions that the DCB arms are clamped at the crack tip (and also used in standards for determining fracture toughness in mode-I delamination), has an additional term which takes into account the rotation at the crack tip. Additionally, we provide all the mentioned analytical solutions for the case when the shear stiffness of the arms is infinitely high, which corresponds to Euler–Bernoulli beam theory. In the numerical examples we compare results for Euler–Beronulli and Timoshenko beam theory and analyse the influence of the CZM parameters. PubDate: 2018-11-14 DOI: 10.1007/s10704-018-0324-5

Authors:Y. J. Cui; K. F. Wang; B. L. Wang; P. Wang Abstract: Thermoelectric thin film/substrate structures have many practical applications such as in heat recovery systems. The general problem of thermally induced delamination between a thermoelectric thin film and a substrate is investigated. The temperature varies along the length direction but is constant along the thickness direction of the film. Analytical solutions of the temperature field in the film and the stress intensity factors (SIFs) at the delamination crack tips are obtained. The combined heat convection and heat radiation between the film surfaces and the surrounding medium (i.e., the air) are taken into account. Numerical results show that the SIFs sharply increase as the tips of the delamination crack approach the ends of the film. The combined heat convection and heat radiation can increase or decrease the SIFs. The mechanism for the delamination propagation in the thermoelectric film/substrate system is examined. The critical (permissible) temperature difference across the film governing the delamination propagation is identified. PubDate: 2018-11-12 DOI: 10.1007/s10704-018-0328-1

Authors:Xin Sun; Jian Yao; Guozhong Chai; Yumei Bao Abstract: The integrity of reactor pressure vessel (RPV) is greatly affected by pressurized thermal shock (PTS). Once crack appears in the nozzle region, the stress concentration around the crack tips may lead to crack propagation, and finally cause a serious security problem. When the transient temperature is above the nil-ductility reference temperature, elastic–plastic constitutive relations are considered in the fracture mechanics analysis. The temperature-related properties of the materials are introduced into a 3-D finite element model to establish the temperature field and stress field of a real RPV. Since the test and safety inspection for RPV with defects under PTS loads are quite difficult and dangerous, the process of the ductile crack propagation is simulated by the extended finite element method (XFEM), and the critical crack sizes for different base wall thicknesses are determined. Then, the quantitative analysis of the effect of the crack position on the ultimate bearing capacity is carried out. For the crack tips with different shapes, the crack propagation law and the shape effect on the ultimate bearing capacity of the whole structure are also analyzed. According to their crack propagation paths and damage degrees, a good agreement is obtained. PubDate: 2018-11-08 DOI: 10.1007/s10704-018-0326-3

Authors:Weijie Huang; Zhiping Li Abstract: A numerical study on a new bifurcation phenomenon in multi-void growth is carried out for a 2-D problem in nonlinear elasticity, where the reference configuration is y-axisymmetric with two pre-existing small voids. The numerical experiments show that, under large radially-symmetric displacement boundary conditions, other than the y-axisymmetric equilibrium solution one would normally expect, there are two energetically more favorable non-y-axisymmetric equilibrium solutions in which the growth of one void is overwhelmingly dominant. The relationships between the critical bifurcation boundary displacement and the compressibility of the material as well as some geometric parameters are illustrated by numerical results. The numerical experiments also show that the existence of the secondary bifurcation will effectively expedite the onset of fractures. PubDate: 2018-11-02 DOI: 10.1007/s10704-018-0323-6

Authors:Shengwen Tu; Xiaobo Ren; Jianying He; Zhiliang Zhang Abstract: The increasing demand of energy prompts the petroleum industry exploitation activities to the Arctic region where the low temperature is a strong challenge, both for structural design and material selection. For structural materials exhibiting the Lüders plateau, it has been reported that lowering the temperature will increase the Lüders plateau length. In order to obtain a deep understanding of the Lüders plateau effect on ductile crack growth resistance, we performed numerical analyses with SENT specimens and the Gurson damage model. The Lüders plateau was simplified by keeping the flow stress constant and varying the plateau length. The results show that the existence of the Lüders plateau does not influence the initiation toughness, however, will alter the material’s fracture resistance significantly. It is found that the Lüders plateau effect is in general controlled by the stress triaxiality level in front of the crack tip. Both the strain hardening and the crack depth effects on resistance curves are alleviated due to the Lüders plateau. For materials with very small initial void volume fraction, the Lüders plateau effect is more pronounced. Since the Lüders plateau intensifies the crack driving force and may lower down crack resistance curve, special attention should be paid to the application of materials with the Lüders plateau in the Arctic. PubDate: 2018-11-02 DOI: 10.1007/s10704-018-0327-2

Authors:Ying Zhen; Hongjun Tian; Haijiao Yi; Yuguang Cao; Shihua Zhang Abstract: In engineering design, a difficulty has always existed in those standard laboratory tests that cannot accurately predict the behavior of large structures like pipelines due to the different constraint levels. At present, extensive work has been done to characterize the constraint effects on fracture toughness by introducing a second parameter, while the systematic research on constrained transformation is inadequate. To address this issue, the ductile fracture process of X65 SENB specimen is simulated through the finite-element method coupled with the Gurson–Tvergaard–Needelman model. The parameters crack tip opening displacement (CTOD) and crack tip opening angle (CTOA) are chosen to characterize the fracture behaviors. The effects of specimen thickness on fracture toughness based on CTOD/CTOA and constraints ahead of crack tips in SENB specimen are studied. The results indicate that the critical values of CTOD/CTOA decrease with the increase of specimen thickness, but the constraint parameters are opposite. Furthermore, it finds that there is a near linear relationship between critical values of CTOD/CTOA and the stress constraint ahead of the crack tip. Thus, a constraint-corrected fracture failure criterion based on CTOD/CTOA is proposed, which can be used for the prediction and simulation of stable tearing crack growth in specimens and structures, made of this steel with any thickness value. PubDate: 2018-11-01 DOI: 10.1007/s10704-018-0322-7

Authors:Eric B. Chin; Joseph E. Bishop; Rao V. Garimella; N. Sukumar Abstract: We introduce a framework for modeling dynamic fracture problems using cohesive polygonal finite elements. Random polygonal meshes provide a robust, efficient method for generating an unbiased network of fracture surfaces. Further, these meshes have more facets per element than standard triangle or quadrilateral meshes, providing more possible facets per element to insert cohesive surfaces. This property of polygonal meshes is advantageous for the modeling of pervasive fracture. We use both Wachspress and maximum entropy shape functions to form a finite element basis over the polygons. Fracture surfaces are captured through dynamically inserted cohesive zone elements at facets between the polygons in the mesh. Contact is enforced through a penalty method that is applied to both closed cohesive surfaces and general interpenetration of two polygonal elements. Several numerical examples are presented that illustrate the capabilities of the method and demonstrate convergence of solutions. PubDate: 2018-11-01 DOI: 10.1007/s10704-018-0325-4

Authors:Lufan Long; Zhiping Li Abstract: A dynamic cavity growth problem in an isotropic compressible nonlinear hyper-elastic material subject to a gradually applied traction is numerically studied. The effects of three parameters, namely the material mass density, the maximum traction, and the loading rate, on the evolution of the cavity radius, especially the maximum cavity radius, are investigated. The numerical results show that, while both the inertia and loading rate have only marginal effect on the onset of cavitation, they do significantly affect the maximum cavity radius. Furthermore, the applied traction eventually leads to a periodic oscillatory cavity growth, and a square root power law for the vibration period as a function of the material mass density is found to hold. PubDate: 2018-10-19 DOI: 10.1007/s10704-018-0321-8