Quasi-static Crack Growth Research Articles

An extended/generalized phase‐field finite element method for crack growth with global‐local enrichment

SummaryAn extended/generalized finite element method (XFEM/GFEM) for simulating quasistatic crack growth based on a phase‐field method is presented. The method relies on approximations to solutions associated with two different scales: a global scale, that is, structural and discretized with a coarse mesh, and a local scale encapsulating the fractured region, that is, discretized with a fine mesh. A stable XFEM/GFEM is employed to embed the displacement and damage fields at the global scale. The proposed method accommodates approximation spaces that evolve between load steps, while preserving a fixed background mesh for the structural problem. In addition, a prediction‐correction algorithm is employed to facilitate the dynamic evolution of the confined crack regions within a load step. Several numerical examples of benchmark problems in two‐ and three‐dimensional quasistatic fracture are provided to demonstrate the approach.

On the Crack Quasi-Static Growth

The theoretical model of quasi-static crack growth in the elastic-plastic material under load variation in a wide range. Small-scale yielding is principal assumption and main restriction of proposed theory. The model of crack growth provides for continues and interrelated both the crack propagation and plastic deformation development. The nonlinear first-order differential equation describes the quasi-static process of crack growth. In dimensionless form this equation invariant in respect to geometrical configuration and material. The critical size of the plastic zone is proposed as the characteristics of material resistance which is directly connected with the fracture toughness, but more convenient in practical applications of invariant equation. The demonstration of solution is performed for the double cantilever beam that widely used as the standard (DCB) sample for measurement of the mode-I interlaminar fracture toughness. he short analysis of some properties of solution of the invariant equation and its application is done.

Effect of Point Defects on the Interface Strength of Joint Materials

The paper proposes a nonequilibrium thermodynamic model of quasi-static crack growth at the interface of perfectly elastic materials which are either free of defects or contain nonequilibrium point defects like vacancies or interstitial/substitutional impurity atoms. Considering that the interface of joint materials is a point-defect adsorber from their bulk, the model establishes a crack growth criterion in terms of the defect adsorbability at the interface of two materials and at their free surfaces formed by an open crack, work of reversible interface separation dependent on the defect concentration in the materials, and crack growth rate. The criterion suggests that varying the defect concentration can provide one or another crack mode: growth or healing. The model is supported by a detailed mathematical analysis of the crack growth in the presence of nonequilibrium vacancies. Assuming that only one of the joint materials contains vacancies, the analysis gives an equation for the vacancy concentration which can change the crack mode, shows its analytical solution for two limiting concentrations, and demonstrates how the crack mode is changed as the vacancy concentration increases. The analysis is supplemented with quantitative estimates of the range of external and internal parameters (temperature, tensile stress, parameters of two joint materials, vacancy adsorbability at their interface and free surfaces) that admit of the two limiting cases in actual multilayer structures under typical operating conditions. The research results can be useful for modeling and optimizing the adhesion characteristics and assessing the service life of multilayer structures employed in modern micro- and nanoelectronics.

Fatigue crack growth behavior of thermo-mechanically processed AA 5754: Experiment and extended finite element method simulation

Fatigue and fracture parameters such as sigmoidal curve, fatigue crack growth threshold, stress intensity factor (SIF), and Paris constants are evaluated for thermo-mechanically processed Al-Mg (AA 5754) alloy at different stress-ratio. Experimental and numerical techniques are utilized for fracture and fatigue studies. Cryorolling and metal inert gas (MIG) welding processes are adopted to alter its mechanical and fatigue properties. The effect of cyclic stress ratio, thermo-mechanical processing, and MIG welding has been investigated on crack growth rate of the alloy. Extended finite element method (XFEM)-based program is developed to simulate both two-dimensional (2D) and three-dimensional (3D) crack geometries for quasi-static crack growth.

Full thermo-mechanical coupling using eXtended finite element method in quasi-transient crack propagation

This work aims to present a complete full coupling eXtended finite element formulation of the thermo-mechanical problem of cracked bodies. The basic concept of the extended finite element method is discussed in the context of mechanical and thermal discontinuities. Benchmarks are presented to validate at the same time the implementation of stress intensity factors and numerical mechanical and thermal responses. A quasi-transient crack propagation model, subjected to transient thermal load combined with a quasi-static crack growth was presented and implemented into a home-made object-oriented code. The developed eXtended finite element tool for modeling two-dimensional thermo-mechanical problem involving multiple cracks and defects are confirmed through selected examples by estimating the stress intensity factors with remarkable accuracy and robustness.

The significance of deformation mechanisms on the fracture behavior of phase reversion-induced nanostructured austenitic stainless steel

We describe here the relationship between grain structure, deformation mechanism and fracture characteristics in an austenitic stainless steel. This was accomplished using the novel concept of phase reversion that enabled a wide range of grain size from nanograined/ultrafine grained (NG/UFG) to coarse-grained (CG) regime to be obtained in a single material through change in temperature-time annealing sequence. In the NG/UFG structure, a marked increase in abundance of stacking faults (SFs) and twin density with strain was observed that led to a decrease in the average spacing between adjacent SFs, thus converting stacking faults into twins. Twinning in NG/UFG structure involved partial dislocations and their interaction with the grain boundaries, including SF overlapping and the coordinated nucleation of partial dislocations from the grain boundaries. The plastic zone in the NG/UFG structure resembled a network knitted by the intersecting twins and SFs. With SFE ~30 mJ/m2, the minimum stress for twin nucleation was ~250 MPa for the experiment steel and the corresponding optimal grain size (dop) wa ~120 nm. In contrast, in the CG structure, strain induced martensite formation was the deformation mechanism. The difference in the deformation mechanism led to a clear distinction in the fracture behavior from striated fracture in high strength-high ductility NG/UFG alloy to microvoid coalescence in the low strength-high ductility CG counterpart. The underlying reason for the change in fracture behavior was consistent with change in deformation mechanism from nanoscale twinning in NG/UFG alloy to strain-induced martensite in the CG alloy, which is related to change in the stability of austenite with grain size. An analysis of critical shear stress required to initiate twinning partial dislocations in comparison to that required to nucleate shear bands is presented. The appearance of striated fracture in the NG/UFG alloy suggests a quasi-static step wise crack growth process.

Stressed State of a Plane Wedge-Shaped Specimen with Edge Crack Under Uniaxial Tension

We propose a computational model of plane compact wedge-shaped specimens with stability regions of the stress intensity factor at the tip of an edge crack whose length increases. This model is used for the determination of the crack resistance of materials with regard for the influence of various operating factors. On the basis of the method of singular integral equations, we obtain the numerical solutions of model problems for wedge-shaped specimens subjected to uniaxial eccentric tension. We reveal the influence of the modes of loading of these specimens and their geometric parameters on the behavior of the stress intensity factor depending on the crack length and establish the ranges of the geometric parameters guaranteeing the presence of maximum stability regions for the stress intensity factor in the process of quasistatic crack growth.

Numerical simulation of the interaction between a crack and an inclusion

PurposeThe purpose of this paper is to propose the interaction integral method combing with a XFEM-based local mesh replacement method to evaluate both the stress intensity factors (SIFs) and T-stress at the crack tip near a circular inclusion.Design/methodology/approachSpecial attention is pay to the effect of T-stress on crack initiation angle in 2D composite medium. The generalized maximum tangential stress criterion is employed during the simulation which simultaneously involves the effects of the mixed-mode SIFs, the T-stress and a physical length scale rc (the size of the fracture process zone).FindingsIt is shown that T-stress could affect the crack initiation angle significantly for mixed-mode conditions. Varies types of material mismatch are also considered and their influences on T-stress are given quantitatively.Originality/valueThe proposed numerical method allows a considerable flexibility for such problems and provides a basic framework for quasi-static crack growth in materials containing complex interfaces.

XFEM modeling of multistage hydraulic fracturing in anisotropic shale formations

The aim of the present study is to help understand hydraulic fracture propagation in shale formations through numerical simulations. The hydraulic fracture propagation regime in shale is analyzed considering the anisotropic nature of the shale rock formation and slick-water fracturing fluid. It is determined that the dominant mechanism of hydraulic fracture propagation is the so-called transitional regime that is characterized by a negligibly small fluid lag region and zero fluid front pressure. For the modeling of the hydraulic fracture evolution over time, we assume orthotropic linear-elastic rock media and that the flow of the fracturing fluid is governed by the Reynold's lubrication equation. For the discretization of the coupled solid-fluid equations within the 2D plane-strain context we use the extended finite element method for the rock media and the finite volume method for the lubrication equation. The problem of the hydraulic fracture evolution over time is modeled as stable quasi-static crack growth where time is the result of upholding the mass conservation principle between the fluid inflow and the crack volume. The Picard iterative approach is used to solve the discrete non-linearly coupled solid-fluid equations. Our model is verified against several analytical solutions. Subsequently, a five-stage hydraulic fracturing problem is simulated to study the interactions between the different fractures. Results show that the on-going hydraulic fractures are attracted by the pre-existing hydraulic fractures as a result of the change of the local stress field relative to the initial in-situ stress field. For the cases considered, fracture deflections are found to be most extensive with decreasing fracture spacing and in-situ stress difference, but insensitive with the increasing the ratio of Young's moduli.

Slow Growth of a Crack with Contacting Faces in a Viscoelastic Body

An algorithm for solving the problem of slow growth of a mode I crack with a zone of partial contact of the faces is proposed. The algorithm is based on a crack model with a cohesive zone, an iterative method of finding a solution for the elastic opening displacement, and elasto–viscoelastic analogy, which makes it possible to describe the time-dependent opening displacement in Boltzmann–Volterra form. A deformation criterion with a constant critical opening displacement and cohesive strength during quasistatic crack growth is used. The algorithm was numerically illustrated for tensile loading at infinity and two concentrated forces symmetric about the crack line that cause the crack faces to contact. When the crack propagates, the contact zone disappears and its dynamic growth begins.

Simulation of dynamic and static thermoelastic fracture problems by extended nodal gradient finite elements

In this paper, the recent development of extended consecutive-interpolation 4-node quadrilateral element (XCQ4), which goes beyond certain limitations of conventional methods, is further employed to study dynamic and static thermoelastic fracture problems. In particular, static stress intensity factors (SIFs) and transient dynamic responses of two-dimensional (2-D) stationary cracks in elastic solids under thermal and thermal-mechanical loadings are investigated. In addition, simulation of quasi-static crack propagation under thermal-mechanical loading condition is also performed. To precisely capture the singularity and discontinuity of temperature, heat flux, and displacements due to the presence of crack, the unknown field variables of mechanical displacements and temperature are approximated in terms of the XCQ4 framework by different enrichment strategies, which include either the use of the conventional branch functions or the new ramp function associated with Heaviside function. Only adiabatic condition is considered on the crack surfaces. For quasi-static crack growth analysis, the maximum hoop stress criterion, which is also valid for thermo-elastic fracture mechanics, is adopted. For transient crack analyses, the inertial effect is taken into account for the interaction integral in the consideration of DSIFs. Through eight numerical examples for thermoelastic fracture problems, the validity of the XCQ4 is confirmed by a comparison of the obtained results to those derived from different approaches in previous studies. In addition, the accuracy, performance, and superior advantages of the XCQ4 over standard extended finite element approach are also demonstrated.

Analysis of Semipermeable Crack Growth in Piezoelectric Materials Using Extended Finite Element Method

Piezoelectric materials possess special characteristics of electromechanical coupling behavior and thus have found numerous applications such as transducers, sensors, actuators. Fracture of piezoelectric materials has drawn substantial attention of the research community and is being widely investigated for predicting their failure. Most of the research on piezoelectric materials is based on impermeable crack conditions. In the present study semi-permeable crack boundary conditions has been analyzed using the extended finite element method (XFEM). Combined Mechanical and Electrical loading with quasi-static crack growth has been considered on a pre-cracked rectangular plate with crack at its edge and center. Stress intensity factors have been evaluated by interaction integral approach using the asymptotic crack tip fields. Effect of presence of minor cracks and holes have been analyzed on the intensity factors of semi-permeable major crack.

Influences of different dimensional carbon-based nanofillers on fracture and fatigue resistance of natural rubber composites

The dimensions of reinforcing filler is a key factor in influencing the fracture and fatigue of rubbers. Here, the fracture and fatigue resistance of natural rubber (NR) filled with different dimensional carbon-based fillers including zero-dimensional spherical carbon black (CB), one-dimensional fibrous carbon nanotubes (CNTs) and two-dimensional planar graphene oxide (GO) were explored. To obtain equal hardness, a control indicator in the rubber industry, the amounts of CB, CNTs, and GO were 10.7 vol%, 1.2 vol%, and 1.6 vol%, respectively. J-integral and dynamic fatigue tests revealed that NR filled with CB exhibited the best quasi-static fracture resistance and dynamic crack growth resistance. The much higher hysteresis loss of NR filled with CNTs weakened its fatigue resistance. The planar GO played a limited role in preventing crack growth. Furthermore, digital image correlation revealed that NR filled with CB had the highest strain amplification level and area at the crack tip, which dissipated the most local input energy and then improved the fracture and fatigue performance.

Three-dimensional quasi-static fatigue crack growth analysis in functionally graded materials (FGMs) using coupled FE-XEFG approach

In the present technological scenario, functionally graded materials (FGMs) are being introduced by scientific community for a wide range of effective engineering applications especially in aerospace, nuclear reactor, bio-medical and military uses. Safety and long service life of engineering components made from FGMs are one of the major objectives of engineering design process. The prediction of fracture behaviour is quite essential for safe designing of such critical components. In this work, three dimensional quasi-static fatigue crack growth has been studied in FGMs. Effects of material gradient (due to FGMs nature) have been investigated in detail. Both crack growth contours and fatigue lives are studied with reference of material heterogeneity (FGMs). Detailed analyses are presented in the form of fracture toughness, crack growth contours and fatigue life cycles. Numerical results are presented for cyclic thermal and mechanical loading environments. The variation in the material properties for FGM has been modelled by exponential law. The solution methodology has been implemented in coupled finite element (FE) and extended element free Galerkin (XEFG) approach. Few planar and non-planar crack geometries are simulated to check robustness and accuracy of the proposed technique. It has been shown that employed computational approach provides accurate fracture studies and crack growth predictions for three-dimensional functionally graded materials.

Quasistatic crack growth in 2d-linearized elasticity

In this paper we prove a two-dimensional existence result for a variational model of crack growth for brittle materials in the realm of linearized elasticity. Starting with a time-discretized version of the evolution driven by a prescribed boundary load, we derive a time-continuous quasistatic crack growth in the framework of generalized special functions of bounded deformation (GSBD). As the time-discretization step tends to zero, the major difficulty lies in showing the stability of the static equilibrium condition, which is achieved by means of a Jump Transfer Lemma generalizing the result of [19] to the GSBD setting. Moreover, we present a general compactness theorem for this framework and prove existence of the evolution without imposing a-priori bounds on the displacements or applied body forces.

Quasistatic crack growth based on viscous approximation: a model with branching and kinking

Employing the technique of vanishing viscosity and time rescaling, we show the existence of quasistatic evolutions of cracks in brittle materials in the setting of antiplane shear. The crack path is not prescribed a priori and is chosen in an admissible class of piecewise regular sets that allows for branching and kinking.

Steady-state crack growth in single crystals under Mode I loading

The active plastic zone that surrounds the tip of a sharp crack growing under plane strain Mode I loading conditions at a constant velocity in a single crystal is studied. Both the characteristics of the plastic zone and its effect on the macroscopic toughness is investigated in terms of crack tip shielding due to plasticity (quantified by employing the Suo, Shih, and Varias set-up). Three single crystals (FCC, BCC, HCP) are modelled in a steady-state elastic visco-plastic framework, with emphasis on the influence of rate-sensitivity and crystal structures. Distinct velocity discontinuities at the crack tip predicted by Rice [Rice J.R., 1987. Tensile crack tip fields in elastic-ideally plastic crystals. Mech. Mater. 6, pp. 317–335] for quasi-static crack growth are confirmed through the numerical simulations and highly refined details are revealed. Through a detailed study, it is demonstrated that the largest shielding effect develops in HCP crystals, while the lowest shielding exists for FCC crystals. Rate-sensitivity is found to affect the plastic zone size, but the characteristics overall remain similar for each individual crystal structure. An increasing rate-sensitivity at low crack velocities monotonically increases the crack tip shielding, whereas the opposite behaviour is observed at high velocities. This observation leads to the existence of a characteristic velocity at which the crack tip shielding becomes independent of the rate-sensitivity.

Fatigue crack growth anisotropy in ultrafine-grained iron

Nanocrystalline and ultrafine-grained (UFG) metals produced by severe plastic deformation exhibit often microstructures with elongated grains, which result in orientation dependent mechanical properties. This anisotropy is especially pronounced for the resistance against quasi-static and cyclic crack growth. In order to gain more knowledge about the consequences of anisotropic microstructures in the case of cyclic loading, fatigue crack growth (FCG) experiments were performed on UFG iron processed by high pressure torsion, with a mean grain size of 500 × 400 × 150 nm3. Samples with four different orientations were prepared and tested with two mean stresses to account for crack closure effects. The FCG rate varies by one order of magnitude between cracks propagating parallel to elongated grains and cracks advancing perpendicular to it. This larger difference is discussed in the light of intrinsic and extrinsic toughening mechanisms. It is concluded that crack closure contributions are reduced in UFG Fe, however, geometric shielding due to more frequently occurring crack branching leads to a significantly higher FCG resistance for cracks perpendicular to the grain elongation. Furthermore, it is observed that grain refinement leads to a transition from transgranular to intergranular fracture. However, it can be shown that this intergranular crack growth of UFG iron under cyclic loading is not the result of grain boundary embrittlement, but occurs due to a blunting and re-sharpening process along the grain boundaries.

Quasi-static hydraulic crack growth driven by Darcy’s law

Abstract In the framework of rate independent processes, we present a variational model of quasi-static crack growth in hydraulic fracture. We first introduce the energy functional and study the equilibrium conditions of an unbounded linearly elastic body subject to a remote strain ϵ ∈ ℝ {\epsilon\in\mathbb{R}} and with a sufficiently regular crack Γ filled by a volume V of incompressible fluid. In particular, we are able to find the pressure p of the fluid inside the crack as a function of Γ, V, and ϵ. Then we study the problem of quasi-static evolution for our model, imposing that the fluid volume V and the fluid pressure p are related by Darcy’s law. We show the existence of such an evolution, and we prove that it satisfies a weak notion of the so-called Griffith’s criterion.

Three-dimensional quasi-static interfacial crack growth simulations in thermo-mechanical environment by coupled FE-EFG approach

This paper presents the three-dimensional interfacial crack growth simulations using coupled finite element and element free Galerkin (FE-EFG) approach under mechanical and thermo-elastic loading. Extrinsic partition of unity (PU) enrichment technique has been employed to model a crack in the domain. Crack face and crack front have been modelled by Heaviside and asymptotic branch enrichment functions respectively whereas a material interface has been modelled by a signed distance function. Modified domain based interaction integral approach has been employed to evaluate the individual stress intensity factors. Standard Paris law of fatigue crack growth has been employed for the life prediction of interfacial cracked geometries. Several interfacial fatigue crack growth problems have been solved by the coupled EF-EFG approach.