Crack Face Tractions Research Articles

Phase-field finite deformation fracture with an effective energy for regularized crack face contact

Phase-field models are a leading approach for realistic fracture problems. They treat the crack as a second phase and use gradient terms to smear out the crack faces, enabling the use of standard numerical methods for simulations. This regularization causes cracks to occupy a finite volume in the reference, and leads to the inability to appropriately model the closing or contacting -- without healing -- of crack faces. Specifically, the classical idealized crack face tractions are that the shear component is zero, and that the normal component is zero when the crack opens and identical to the intact material when the crack closes. Phase-field fracture models do not replicate this behavior. This work addresses this shortcoming by introducing an effective crack energy density that endows the regularized (finite volume) phase-field crack with the effective properties of an idealized sharp crack. The approach is based on applying the QR (upper triangular) decomposition of the deformation gradient tensor in the basis of the crack, enabling a transparent identification of the crack deformation modes. By then relaxing over those modes that do not cost energy, an effective energy is obtained that has the intact response when the crack faces close and zero energy when the crack faces are open. A highlight of this approach is that it lies completely in the setting of finite deformation, enabling potential application to soft materials and other settings with large deformation or rotations. The model is applied to numerically study representative complex loadings, including (1) cyclic loading on a cavity in a soft solid that shows the growth and closing of cracks in complex stress states; and (2) cyclic shear that shows a complex pattern of crack branching driven by the closure of cracks.

Atomic mixed-mode cohesive-zone dual constitutive laws of impurity-embrittled grain boundaries in polycrystalline solids via nanoscale field projection method

Atomic-scale mixed-mode intergranular fracture, featured by non-local non-linear discrete atomic debonding processes near a crack tip along a grain boundary (GB), is modeled with a cohesive zone in a continuum scale controlled by cohesive-zone dual constitutive relations, a balanced traction-separation relationship, i.e., a conventional cohesive-zone law (CZL), and an unbalanced traction (UT)-centerline displacement (CD) relationship. In order to bridge two different scales, we developed a nanoscale field projection method (nano-FPM) based on atomic-scale interaction J and M integrals, for which asymptotic anisotropic elastic fields near an interfacial crack with balanced and unbalanced crack-face tractions are used as probing fields of the CZL and UT-CD relationship, respectively. Cracking phenomena along a GB in nickel with segregated sulfur impurity atoms under mixed-mode loadings are simulated with molecular dynamics. Embrittlement by sulfur impurity atoms is quantitatively estimated with the mixed-mode CZL of the GBs in nickel via the nano-FPM developed in this study. UT-CD relationship, a special feature of atomic fracture, representing the micromechanical change of surface stress between GB and cracked surfaces, is also obtained by the nano-FPM. New functional relationships for CZL, UT-CD relationship, and decohesion potential obtained by the nano-FPM are proposed to facilitate the implementation of them into mesoscale or continuum-scale analysis on mixed-mode intergranular fracture.

A stabilized finite element method for enforcing stiff anisotropic cohesive laws using interface elements

We present a stabilized finite element method that generalizes Nitsche’s method for enforcing stiff anisotropic cohesive laws with different normal and tangential stiffness. For smaller values of cohesive stiffness, the stabilized method resembles the standard method, wherein the traction on the crack surface is enforced as a Neumann boundary condition. Conversely, for larger values of cohesive stiffness, the stabilized method resembles Nitsche’s method, wherein the cohesive law is enforced as a kinematic constraint. We present several numerical examples, in two-dimensions, to compare the performance of the stabilized and standard methods. Our results illustrate that the stabilized method enables accurate recovery of crack-face tractions for stiff isotropic and anisotropic cohesive laws; whereas, the standard method is less accurate due to spurious traction oscillations. Also, the stabilized method could mitigate spurious sensitivity of load–displacement results to displacement increment in mixed-mode fracture simulation, owing to its stability and accuracy.

Quantifying fatigue overload retardation mechanisms by energy dispersive X-ray diffraction

The fatigue crack retardation mechanisms operating after an overload event are investigated for a bainitic steel using high spatial resolution energy dispersive synchrotron X-ray diffraction. The elastic crack-tip strain fields are mapped at mid-thickness of compact tension samples at R-ratios of 0.1 and 0.4. The same overload stress intensity factor (KOL = 60 MPa m1/2) is applied in each case with the cracks then propagating under the same applied stress intensity range, ΔKapp = 27 MPa m1/2. The competing retardation mechanisms are directly quantified and separated, with the associated fatigue crack growth (FCG) rates then being predicted according to a 2-parameter Walker-type assessment and validated against those measured. The stress intensity factor associated with the overload residual stress field is calculated using a weight function approach. For R=0.1, shielding from residual stress controls retardation when crack growth through the overload plastic zone, rpOL, is small (specifically <0.6rpOL). For more extensive crack growth, discontinuous crack closure controls the retardation behaviour, with significant load transfer across opposing crack faces being observed at minimum load (for R=0.1). These crack face tractions are associated with the plastic asperity created during overload. The traction forces holding the crack faces open at minimum load are, for the first time, used to directly quantify the associated stress intensity factor, Kmintract as a function of crack growth. While no crack shielding is expected, nor observed, for R=0.4, the variation in FCG rate after overload is explained by changes in effective R-ratio.

Evaluation of stress intensity factor for a surface cracked butt welded joint based on real welding residual stress

This paper examines the evaluation of stress intensity factor (SIF) for surface cracks in a non-uniform residual stress field. A technique that based on utilizing the crack face traction (CFT) integral implemented in the interaction integral method using an open source code for models with a particular crack-block mesh density was presented. 3-D flat plate models with different crack-block mesh densities were employed under different loading conditions to examine the accuracy, conditions and easiness to use of the present technique. The influence of the crack-block mesh density on the calculated SIF solutions due to the application of non-uniform tractions over the crack face was discussed. A considerable improvement of SIF solutions calculated by crack face tractions was observed when the crack-block mesh density becomes fine enough. The effectiveness of the present technique was then validated for different 3-D numerical examples of welded joints under different loading conditions. The present technique was thereafter employed to evaluate SIF for a surface crack in a butt welded joint using real welding residual stress. The SIF solution calculated due to the application of welding residual stress was verified with that given by mechanical loading (without welding residual stress).

Computing stress intensity factors for curvilinear cracks

SummaryThe use of the interaction integral to compute stress intensity factors around a crack tip requires selecting an auxiliary field and a material variation field. We formulate a family of these fields accounting for the curvilinear nature of cracks that, in conjunction with a discrete formulation of the interaction integral, yield optimally convergent stress intensity factors. In particular, we formulate three pairs of auxiliary and material variation fields chosen to yield a simple expression of the interaction integral for different classes of problems. The formulation accounts for crack face tractions and body forces. Distinct features of the fields are their ease of construction and implementation. The resulting stress intensity factors are observed converging at a rate that doubles that of the stress field. We provide a sketch of the theoretical justification for the observed convergence rates and discuss issues such as quadratures and domain approximations needed to attain such convergent behavior. Through two representative examples, a circular arc crack and a loaded power function crack, we illustrate the convergence rates of the computed stress intensity factors. The numerical results also show the independence of the method from the size of the domain of integration. Copyright © 2015 John Wiley &amp; Sons, Ltd.

Fracture mechanics by three-dimensional crack-tip synchrotron X-ray microscopy.

To better understand the relationship between the nucleation and growth of defects and the local stresses and phase changes that cause them, we need both imaging and stress mapping. Here, we explore how this can be achieved by bringing together synchrotron X-ray diffraction and tomographic imaging. Conventionally, these are undertaken on separate synchrotron beamlines; however, instruments capable of both imaging and diffraction are beginning to emerge, such as ID15 at the European Synchrotron Radiation Facility and JEEP at the Diamond Light Source. This review explores the concept of three-dimensional crack-tip X-ray microscopy, bringing them together to probe the crack-tip behaviour under realistic environmental and loading conditions and to extract quantitative fracture mechanics information about the local crack-tip environment. X-ray diffraction provides information about the crack-tip stress field, phase transformations, plastic zone and crack-face tractions and forces. Time-lapse CT, besides providing information about the three-dimensional nature of the crack and its local growth rate, can also provide information as to the activation of extrinsic toughening mechanisms such as crack deflection, crack-tip zone shielding, crack bridging and crack closure. It is shown how crack-tip microscopy allows a quantitative measure of the crack-tip driving force via the stress intensity factor or the crack-tip opening displacement. Finally, further opportunities for synchrotron X-ray microscopy are explored.

Three-dimensional J-integral evaluation for cracks with arbitrary curvatures and kinks based on domain integral method for quadratic tetrahedral finite element

In this paper, a methodology to compute the three-dimensional J-integral by the domain integral method using the quadratic tetrahedral finite element for the absence of crack face tractions is presented. We carefully examined the properties of the domain integral method and propose a methodology that can deal with cracks with arbitrary curvatures and kinks in three-dimensional space without any crack surface integral terms. Some example problems are presented and the accuracy of present J-integral method is critically discussed.

Evolution of crack-bridging and crack-tip driving force during the growth of a fatigue crack in a Ti/SiC composite

High spatial resolution diffraction and imaging using synchrotron X-rays are combined to monitor the incremental growth of a fatigue crack through the matrix of a Ti-6Al-4V/SCS-6 SiC monofilament metal matrix composite. X-ray tomography is used to quantify the crack opening displacement (COD) and diffraction to measure the crack-tip stress field in each phase, the wear degraded interfacial strengths, as well as the crack face tractions applied by the bridging fibres, at maximum () and minimum () loading as a function of crack length. In this way, it has been possible to quantify the crack-tip driving force (the stress intensity range effective at the crack-tip) in three ways: from the COD, the bridging stresses and the crack-tip stress field. The fibre stresses act to prop open the crack atand shield the crack atsuch that the change in COD is small over the fatigue cycle. Consequently, the effective stress intensity range at the crack tip remains around 10 MPa√m as the crack lengthens, as more and more fibres bridge the crack despite the normally applied stress intensity rising to 60 MPa√m. The implications of the derived fracture mechanics parameters are assessed and the wider potential of X-ray diffraction and imaging for crack-tip microscopy is discussed.

3D Crack‐tip Microscopy: Illuminating Micro‐Scale Effects on Crack‐Tip Behavior

AbstractTraditionally the driving force for crack growth is described in terms of global parameters which, for continua, nevertheless reflect the local conditions at the crack‐tip. Mimicking nature, materials microstructures are now being designed at the microscale, or even the nanoscale, employing interfaces and heterogeneities to shield the propagating crack for improved resistance to sub‐critical crack growth. In such cases global approaches simply will not do. Information is needed about intrinsic damage occurring ahead, and extrinsic shielding mechanisms behind, the crack. Here, high resolution X‐ray imaging and diffraction modes analogous to those in 2D electron microscopy are combined to form a 3D “crack‐tip microscope”. In this way one can quantify the effect of microstructural scale events on the crack‐tip environment. The technique is applied to study overload phenomena under fatigue, to characterize the bridging ligaments under stress corrosion cracking and fiber bridging during fatigue of a metal matrix composite. Besides identifying shielding mechanisms, it provides three methods for calculating the effective stress intensity at the crack tip; namely from the near tip stress field, from crack face tractions and from crack opening displacements. The wider opportunities opened up by this approach to study self‐healing, transformation toughening and other microstructural toughening mechanisms are also discussed.

Weight function based Dugdale model for mixed-mode crack problems with arbitrary crack surface tractions

Weight function theory states crack surface displacements can be found for any arbitrary distribution of mode I, or mixed-mode crack face tractions via that geometry’s weight functions. This statement is validated via finite element analysis of the infinite center-cracked plate for various mixed mode loadings. An elastic-perfectly plastic material is considered using a Dugdale approach and compared to elastic–plastic finite element simulations. The weight function method in all cases agrees well with the finite element simulations for small scale yielding at the crack tip. As the maximum traction value approaches one-half the yield strength discrepancies become larger due to violation of small scale yielding.

Computation of Mixed-Mode Stress Intensity Factors for Cracks in Three-Dimensional Functionally Graded Solids

This work applies a two-state interaction integral to obtain stress intensity factors along cracks in three-dimensional functionally graded materials. The procedures are applicable to planar cracks with curved fronts under mechanical loading, including crack-face tractions. Interaction-integral terms necessary to capture the effects of material nonhomogeneity are identical in form to terms that arise due to crack-front curvature. A discussion reviews the origin and effects of these terms, and an approximate interaction-integral expression that omits terms arising due to curvature is used in this work to compute stress intensity factors. The selection of terms is driven by requirements imposed by material nonhomogeneity in conjunction with appropriate mesh discretization along the crack front. Aspects of the numerical implementation with (isoparametric) graded finite elements are addressed, and examples demonstrate the accuracy of the proposed method.

Interaction integral procedures for 3-D curved cracks including surface tractions

This study examines a two-state interaction integral for the direct computation of mixed-mode stress intensity factors along curved cracks under remote mechanical loads and applied crack-face tractions. We investigate the accuracy of stress intensity factors computed along planar, curved cracks in homogeneous materials using a simplified interaction integral that omits terms to reflect specifically the effects of local crack-front curvature. We examine the significance of the crack-face traction term in the interaction integral, and demonstrate the benefit of a simple, exact numerical integration procedure to evaluate the integral for one class of three-dimensional elements. The work also discusses two approaches to compute auxiliary, interaction integral quantities along cracks discretized by linear and curved elements. Comparisons of numerical results with analytical solutions for stress intensity factors verify the accuracy of the proposed interaction integral procedures.

Intersonic crack growth under time-dependent loading

The problem investigated in this paper is a mode II crack extending at a uniform intersonic speed in an otherwise unbounded elastic solid subjected to time dependent crack-face tractions. The fundamental solution for this problem is obtained analytically, which is then used to construct the general solution for an intersonic crack subjected to arbitrary time-dependent loading. For time-independent loading, this solution reduces to Huang and Gao’s [J. Appl. Mech 68 (2001) 169] fundamental solution. We have also studied two crack-face loadings that are of interest for engineering applications.

An alternating method based on the VNA solution for analysis of damaged solid containing arbitrarily distributed elliptical microcracks

This paper presents the development of an alternating method for the interaction analysis of arbitrary distributed numerous elliptical microcracks. The complete analytical solutions (VNA solutions) for a single elliptical crack in an infinite solid, subject to arbitrary crack-face tractions, are implemented in the present alternating method, together with the coordinate transformations for stress tensors. First, the present method is verified by solving the problems of two interacting cracks for which accurate numerical solutions have been obtained previously. Next, the present method demonstrates obtaining efficient and accurate solutions for the problems of many interacting elliptical cracks, which cannot be solved in a practical sense by the ordinary numerical methods such as the finite element method. Furthermore, damaged solids containing periodically distributed elliptical microcracks are analyzed by the present alternating method. The effective elastic moduli are evaluated for varying microcrack density. Detailed structures of the interactions in the damaged solids are visualized and clarified.

Dynamic crack propagation in piezoelectric materials—Part I. Electrode solution

An analysis is performed for the transient response of a semi-infinite, anti-plane crack propagating in a hexagonal piezoelectric medium. The mixed boundary value problem is solved by transform methods together with the Wiener-Hopf and Cagniard-de Hoop techniques. As a special case, a closed form solution is obtained for constant speed crack propagation under external anti-plane shear loading with the conducting electrode type of electric boundary condition imposed on the crack surface (a second type of boundary condition is considered in Part II of this work). In purely elastic, transversely isotropic elastic solids, there is no antiplane mode surface wave. However, for certain orientations of piezoelectric materials, a surface wave will occur—the BleusteindashGulyaev wave. Since surface wave speeds strongly influence crack propagation, the nature of antiplane dynamic fracture in piezoelectric materials is fundamentally different from that in purely elastic solids, exhibiting many features only associated with the indashplane modes in the elastic case. For a general distribution of crack face tractions, the dynamic stress intensity factor and the dynamic electric displacement intensity factor are derived and discussed in detail for the electrode case. As for inplane elastodynamic fracture, the stress intensity factor and energy release rate go to zero as the crack propagation velocity approaches the surface wave speed. However, the electric displacement intensity does not vanish.

Elastic Analyses of Planar Cracks of Arbitrary Shape

A numerical method is presented for planar cracks of arbitrary shape. The fundamental solution for a dislocation segment is obtained from the point force solution and used to derive three coupled surface integral equations in which the crack-face tractions are expressed in terms of the gradients of the relative crack-surface displacements. Because the singularity of the kernel in the integral equations is one order less for fundamental solutions based on dislocation segments than for those based on dislocation loops or the body force method, no special numerical techniques are required. Most of the integrations over elements are evaluated analytically. The integral equations are solved numerically by covering the crack surface with triangular elements, and taking the relative displacements to vary linearly over the elements. The mesh is generated by optimizing the local aspect ratio, which is related to the difference in the principal stretches of the mapping of a square reference mesh onto the fracture surface. This mesh generator allows cracks of a wide variety of shapes to be analyzed with good accuracy. Comparison with known solutions indicate that accurate numerical solutions are obtained with a relatively coarse mesh.

Toughening mechanisms in mixed-mode interfacial fracture

A series of mixed-mode interfacial crack initiation experiments were conducted using thinner glass/epoxy specimens than had been used in the past [Liechti and Chai (1992, J. Appl. Mech.59, 295–304)]. Three-dimensional effects were eliminated in the sense that the crack front shapes were the same for all mode-mixes. Nonetheless, the thinner specimens were associated with an overall increase in toughness and a noticeable change in the distribution of toughness with mode-mix. Interferometric measurements of normal crack opening displacements (NCOD) revealed values that were lower than predictions based on rate independent and dependent plasticity and even elastic analyses. The lower NCOD were attributed to long range cohesive crack face tractions, possibly due to electrostatic effects. Shielding calculations suggested that the long range tractions were more important than plastic dissipation. Nonetheless, they did not fully explain the shear-induced toughness effect that has been noted for this and other interfaces.

Exact elastodynamic analysis of some fracture specimen models involving strip geometries

A procedure for analysing a class of transient elastodynamic crack problems is presented. These problems model certain experimental situations which can be used to infer fracture toughness values for materials under stress-wave loadings. In particular, the present analysis provides exact expressions for the elastodynamic stress intensity factor at the tip of a long external crack in a striplike body whose lateral (upper and lower) boundaries are parallel to the crack line. Plane stress/strain conditions are assumed to prevail. In this class of problems, the crack may be situated asymmetrically with respect to the mid-strip line, and various dynamic loadings are considered, including crack face tractions and lateral face displacements. The loadings considered will have an arbitrary time dependence, but will be spatially uniform. The problem analysis is based on integral transforms and an asymptotic usage of the Wiener-Hopf technique. Two useful cases are considered in detail as examples; the symmetrically cracked strip with traction-free lateral boundaries under sudden crack face pressure, and the symmetrically cracked strip with suddenly displaced shear-free lateral boundaries.

Crack face bridging mechanisms in monolithic MgAl 2O 4 spinel microstructures

The strongly rising R-curve behavior, typical of coarse grain sized spinels, has been ascribed to crack face tractions in the wake zone which reduce the crack tip stresses. A novel experimental approach is presented which isolates these tractions for direct characterization in tension. With this method, a detailed study of the crack face separation events has identified the toughening contributions from the active microstructural constituents. Critical wake stresses, varying from 3.75 MPa for a small grain size to 13 MPa for a large grain size, are estimated for this high purity spinel containing only minor quantities of a crystalline grain boundary phase. Grain boundary interactions in the process zone are characterized through optical techniques and evidence the load-carrying role of bridging ligaments. The post fracture tensile test results of a medium grain size spinel are validated through a simple model to reconstruct the experimentally determined R-curve.