Mixed-mode crack growth in ductile thin-sheet materials under combined in-plane and out-of-plane loading

Ductile thin-sheet structures, such as fuselage skin or automobile panels, are widely used in engineering applications. These structures often-times are subjected to mixed mode (I/II/III) loading, with stable crack growth observed prior to final fracture. To characterize specific specimen deformations during stable tearing, a series of mixed-mode I/III stable tearing experiments with highly ductile thin-sheet aluminum alloy and steel specimens have been measured by using three-dimensional digital image correlation (3D-DIC). Measurements include (a) specimen’s deformed shape and 3D full-field surface displacement fields, (b) load-crack extension response and (c) crack path during stable tearing, (d) angular and radial distributions of strains and (e) the mixed mode crack-opening displacement (COD, measured at 1-mm from crack tip along crack surface) variation as a function of crack extension. Results indicate that for both aluminum alloy and steel at all mixed-mode I/III loading conditions (Φ = 30°, 60° and 90°), the crack tip fields have almost identical angular and radial polar strain distributions. The mixed mode I/III fields were different from those observed for the nominal Mode I loading case (Φ = 0°). The effect of the Mode III loading component is that it lowers the magnitude of the dominant strain component ε _θθ ahead of the growing crack tip and increases the singularity of the strain as compared with that in the mode I case. In addition, measurements indicate that the average mixed mode I/III stable COD for AL6061-T6 (GM6208 steel) is 4×(3×) greater than the average Mode I stable COD.     

Numerical modeling of ductile crack growth in 3-D using computational cell elements

This study describes a 3-D computational framework to model stable extension of a macroscopic crack under mode I conditions in ductile metals. The Gurson-Tvergaard dilatant plasticity model for voided materials describes the degradation of material stress capacity. Fixed-size, computational cell elements defined over a thin layer at the crack plane provide an explicit length scale for the continuum damage process. Outside this layer, the material remains undamaged by void growth, consistent with metallurgical observations. An element vanish procedure removes highly voided cells from further consideration in the analysis, thereby creating new tractionfree surfaces which extend the macroscopic crack. The key micro-mechanics parameters are D, the thickness of the computational cell layer, and f ₀ , the initial cell porosity. Calibration of these parameters proceeds through analyses of ductile tearing to match R-curves obtained from testing of deep-notch, through-crack bend specimens. The resulting computational model, coupled with refined 3-D meshes, enables the detailed study of non-uniform growth along the crack front and predictions of specimen size, geometry and loading mode effects on tearing resistance, here described by J-a curves. Computational and experimental studies are described for shallow and deep-notch SE(B) specimens having side grooves and for a conventional C(T) specimen without side grooves. The computational models prove capable of predicting the measured R-curves, post-test measured crack profiles, and measured load-displacement records.     

Automatic construction of 3-D models in multiple scale analysis

This paper discusses techniques for the automatic construction of numerical analysis models for multiple scale analyses which employ interacting models at two, or more, physical scales. Consideration is given to the methods to define the geometric representations and generate the discretizations needed by the numerical analysis procedures. The application of the techniques to multichip modules and composite structures, with interacting macromechanical and micromechanical level analyses, is demonstrated. In the multichip module analyses both heat conduction and thermomechanical analysis are performed using different numerical analysis techniques, and the two interaction of the analyses at the through levels is through a basic global/local methodology. The composite structure analysis considers crack propagation at the micromechanical level interacting with the macromechanical analysis through finite element based adaptive multiscale analysis. In both example applications the focus of the discussion is on the automatic construction of the required geometric models and their automatic discretization.     

Simulation of ductile crack growth in thin aluminum panels using 3-D surface cohesive elements

This work describes the formulation and application of a 3-D, interface-cohesive finite element model to predict quasi-static, ductile crack extension in thin aluminum panels for mode I loading and growth. The fracture model comprises an initially zero thickness, interface element with constitutive response described by a nonlinear traction-separation relationship. Conventional volumetric finite elements model the nonlinear (elastic-plastic) response of background (bulk) material. The interface-cohesive elements undergo gradual decohesion between faces of the volumetric elements to create new traction free crack faces. The paper describes applications of the computational model to simulate crack extension in C(T) and M(T) panels made of a 2.3 mm thick, Al 2024-T3 alloy tested as part of the NASA-Langley Aging Aircraft program. Parameters of the cohesive fracture model (peak opening traction and local work of separation) are calibrated using measured load vs. outside surface crack extensions of high constraint (T-stress > 0) C(T) specimens. Analyses of low constraint M(T) specimens, having widths of 300 and 600 mm and various a/W ratios, demonstrate the capabilities of the calibrated model to predict measured loads and outside surface crack extensions. The models capture accurately the strong 3-D effects leading to various degrees of crack front tunneling in the C(T) and M(T) specimens. The predicted crack growth response shows rapid convergence with through-thickness mesh refinement. Adaptive load increment procedures to control the rate of decohesion in the interface elements leads to stable, rapidly converging iterations in the globally implicit solution procedures.     

A coupled FE–EFG approach for modelling crack growth in ductile materials

In this work, a coupled finite element–element free Galerkin approach has been used to model crack growth in ductile materials under monotonic and cyclic loads. In this approach, a small discontinuous domain near crack is modelled by EFG method, whereas the rest of the domain is modelled by FEM to exploit the advantages of both the methods. A ramp function has been used in the transition region to maintain the continuity between FE and EFG domains. Two plasticity models (GTN and von‐Mises) and three hardening rules (isotropic, kinematic and mixed) have been used to model the nonlinear material behaviour. Four different problems, i.e. single edge notched tension specimen, double edge notched tension specimen, compact tension specimen and three‐point bend specimen, are solved under plane strain condition using J–R curve approach. Finally, a CT specimen problem is also solved by coupled approach using three hardening rules and two plasticity models under cyclic loading.     

A plasticity-corrected stress intensity factor for fatigue crack growth in ductile materials under cyclic compression

For prediction of the fatigue crack growth (FCG) behavior under cyclic compression, a plasticity-corrected stress intensity factor (PC-SIF) range ΔK_pc is proposed on the basis of plastic zone toughening theory. The FCG behaviors in cyclic compression, and the effects of load ratio, preloading and mean load, are well predicted by this new mechanical driving force parameter. Comparisons with experimental data showed that the proposed PC-SIF range ΔK_pc is an effective single mechanical parameter capable of describing the FCG behavior under different cyclic compressive loading conditions.     

Polycrystal models for the analysis of intergranular crack growth in metallic materials

In polycrystal materials the intergranular decohesion is one important damage phenomena that leads to microcrack initiation. The paper presents a mesoscale model, which is focused on the brittle intergranular damage process in metallic polycrystals. The model reproduces the crack initiation and propagation along cohesive grain boundaries between brittle grains. An advanced Voronoi algorithm is applied to generate polycrystal material structures based on arbitrary distribution functions of grain size. Therewith, the authors are more flexible to represent realistic grain size distributions. The polycrystal model is applied to analyze the crack initiation and propagation in statically loaded samples of aluminium on the mesoscale without the necessity of initial damage definition.     

Some observations on the viability of crack tip opening angle as a characterising parameter for plane strain crack growth in ductile materials

Wnuk's application of the Dugdale-Bilby-Cottrell-Swinden model to stable crack growth in ductile materials, is extended to incorporate a simplified idealization of the fracture process zone. The results, when considered with Rice and Sorensen's recent analytical and finite element results, provide further support for the use of crack tip opening angle as a viable characterising parameter for plane strain stable crack growth.

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Résumé L'application faite par Wnuk du modèle de Dugdale-Bilby-Cottrell et Swinden relatif à la croissance stable d'une fissure dans un matériau ductile est étendue de manière à incorporer dans une idéalisation simplifiée la zône où se produit la rupture. Les résultats, lorsqu'ils sont examinés à la lueur des résultats analytiques récents ainsi que des résultats par éléments finis obtenus par Rice et Sorensen, fournissent un support complémentaire à l'utilisation de l'angle d'ouverture de l'extrémité d'une fissure en tant que paramètre caractéristique pour décrire la croissance stable d'une fissure en état plan de déformation.

     

An inclusion model for crack arrest in fiber reinforced materials

A tensile crack bridged by strong and tough fibers is modeled in such a manner that it can be analyzed based on concepts from the mechanics of inclusion. Analytical expressions for the energy release rate, stress intensity factor and crack opening are derived. It is shown that the presence of the supporting fibers reduces the values of those quantities which saturate at a certain level determined by the size and spacing of the fibers.     

An energy approach to account for crack initiation in nanocrystalline materials

An energy balance method to calculate the initiation of crack at triple junctions in nanocrystalline materials with the finest grains is developed. In the steady state of crack initiation, work done by an applied stress is considered to be dissipated as heat by specific rotational deformation, grain boundary sliding and diffusion. The stress field at crack tips, the energies of rotational deformation, grain boundary sliding and grain boundary diffusion are calculated. The analysis demonstrates that the existence of finest grains will lead to enhanced local fracture toughness.     

An XFEM method for modeling geometrically elaborate crack propagation in brittle materials

We present a method for simulating quasistatic crack propagation in 2‐D which combines the extended finite element method (XFEM) with a general algorithm for cutting triangulated domains, and introduce a simple yet general and flexible quadrature rule based on the same geometric algorithm. The combination of these methods gives several advantages. First, the cutting algorithm provides a flexible and systematic way of determining material connectivity, which is required by the XFEM enrichment functions. Also, our integration scheme is straightforward to implement and accurate, without requiring a triangulation that incorporates the new crack edges or the addition of new degrees of freedom to the system. The use of this cutting algorithm and integration rule allows for geometrically complicated domains and complex crack patterns. Copyright © 2011 John Wiley & Sons, Ltd.     

An energy release rate criterion for crack growth in unidirectional composite materials

A criterion for crack initiation in unidirectional composite materials is presented. It includes two assumptions:

1. (1) crack initiation occurs in the direction of fibers;

2. (2) rapid crack growth occurs when the energy release rate reaches a critical value, the resistance to crack growth in the direction of fibers.

Then a model of energy release rate for crack growth in unidirectional composites is presented. The model is applied to characterize the fracture of off-axis, unidirectional glass-epoxy and graphite-epoxy composites. The results indicate that the loads corresponding to crack growth predicted by the present model agree well with the experimental results.     

A novel approach to obviousness: An algorithm for identifying prior art concerning 3-D printing materials

With the development and commercialization of the recyclebot (plastic extruders that fabricate 3-D printing filament from recycled or virgin materials) and various syringe pump designs for self-replicating rapid prototypers (RepRaps), the material selection available for consumers who produce products using 3-D printers is expanding rapidly. This paper provides an open-source algorithm for identifying prior art for 3-D printing materials. Specifically this paper provides a new approach for determining obviousness in this technology area. The potential ramifications on both innovation and patent law in the 3-D printing technological space are discussed.     

An augmented cohesive zone element for arbitrary crack coalescence and bifurcation in heterogeneous materials

We demonstrate that traditional cohesive zone (CZ) elements cannot be accurate when used in conjunction with solid elements with arbitrary intra‐element cracking capability, because they cannot capture the load transfer between cohesive interfaces and the solid elements when crack bifurcation or coalescence occurs. An augmented cohesive zone (ACZ) element based on the augmented finite element method formulation is therefore proposed. The new element allows for arbitrary separation of the cohesive element in accordance with the crack configuration of the abutting solid elements, thus correctly maintaining the non‐linear coupling between merging or bifurcating cracks. Numerical accuracy and effectiveness of the proposed ACZ element are demonstrated through several examples. Copyright © 2011 John Wiley & Sons, Ltd.     

An approximate 2-D solution for the shear-induced strain fields in eigenstrained cubic materials

Summary An approximate analytical 2-D solution for the strain field components ₁₁, ₁₂ and ₂₂ occurring in a cubic material due to a coherently bonded shear eigenstrained inclusion of cylindrical geometry was obtained by means of Continous Fourier Transforms (CFT). A Discrete Fourier Transform (DFT) based numerical model was used in order to test the validity of the results. For the case where the cylindrical inclusion and the surrounding media are elastically homogeneous and the orientation of their principal crystal axes are the same, a correlation between the analytical and numerical models is demonstrated, both for strongly and weakly anisotropic materials. Moreover, the strain fields within the inclusion are shown to be of homogeneous isotropic type. Finally, an expression for the closed-form strain energy of two cylindrical inclusions at arbitrary radius and angle was derived, and then used to determine the minimum energy configuration for the system.Dedicated to Professor Krzysztof Wilmanski on the occasion of his 60th birthday     

An analysis of the energy release rate for non-coplanar crack growth in fiber reinforced composite materials

In view of the fact that non-coplaner crack growth is a common characteristic of crack propagation in fiber reinforced composite materials, an attempt is made to derive the equation of the energy release rateG for non-coplaner crack extension in orthotropic linear elastic solids. Combining the idea developed in our previous paper and the analysis by Sihet al., the equation ofG is obtained for the cases of plane symmetric loading, plane skew-symmetric loading, antiplane shear loading, and a combination of these. The result will serve as a basic tool for developing a fracture mechanics approach to composites.     

An alternating technique for evaluating weight functions for 3-D surface flaws in finite solid bodies

An efficient method, based on the Schwarz–Neumann alternating technique, is presented for computing weight functions of a general solid (3-D as well as 2-D), with embedded or surface-flaw configurations. The total rate of change of the crack-opening displacements, due to simple perturbations of crack-dimension characteristics, is conveniently decomposed into the infinite-domain and boundary-correction parts. The former is determined from available analytical solutions of ideal-shaped cracks, whereas the latter is computed numerically by imposing nil boundary-traction requirements for the displacement field corresponding to the weight functions. Numerical examples, with solutions for 3-D weighted-average and local stress intensity factors, indicate that the proposed method is very accurate and efficient.     

An integral-equation/singularity-method approach for 3-D electromagnetic field determination in the end region of a turbine-generator

A mathematical model using an integral-equation/singularity-method approach is derived for determining the magnetic field and electromagnetic forces induced by current-carrying conductors in a region bounded by 3-D material-body surfaces which have complex configurations. Special analytical and numerical techniques that eliminate near-field computational difficulties and bypass the cumbersome matrix manipulations required by other integral-equation approaches are described. A comprehensive computer program package has been developed using this approach for obtaining the 3-D solutions in the end region of a turbine-generator due to armature end windings. Special computational techniques for handling the complex end-winding and surface geometry are described and detailed numerical results are presented for the 3-D field solution and the forces acting on the conductors.     

Short and long crack data for fatigue of 2024‐T3 Al sheets: binariness of scale segmentation in space and time

Pedagogically speaking, crack initiation–growth–termination (IGT) belongs to the process of fracture, the modelling of which entails multiscaling in space and time. This applies to loadings that are increased monotonically or repeated cyclically. Short and long crack data are required to describe IGT for scale ranges from nano to macro, segmented by the SI system of measurement. Unless the data at the nano scale can be connected with the macro, IGT remains disintegrated. The diversity of non‐homogeneity of the physical properties at the different scale ranges results in non‐equilibrium. These effects dubbed as non‐equilibrium and non‐homogeneous are hidden in the test specimens and must be realized. They can be locked into the reference state of measurement at the mi‐ma scale range by application of the transitional functions and transferred to the nano‐micro and macro‐large scale ranges. The aim of this work is to convert the ordinary crack length data to those referred to as short cracks that are not directly measurable. All test data are material, loading and geometry (MLG) specific. The results obtained for the 2024‐T3 aluminium sheets hold only for the MLG tested. The differences are more pronounced for the short cracks. These effects can be revealed by comparing the incremental crack driving force (CDF) for the ma‐mi range the ma‐large range and the na‐mi range The CDF is equivalent to the incremental volume energy density factor (VEDF). The incremental mi‐ma CDF is found to be 10–10⁵ kg mm^?1 for cracks 3–55 mm long travelling at an average velocity of 10^?5 mm s^?1. The crack velocity rises to 10^?3 mm s^?1 when the incremental CDF is increased to 10⁵–10⁶ kg mm^?1, while the crack lengths are 49–260 mm. The crack velocity for the na‐mi range of 0.040–0.043 mm slowed down to 10^?8 mm s^?1, and the incremental CDF reduces further to 10^?8–10^?2 kg mm^?1. Note that changed several orders of magnitude while the crack advanced from 0.040 to 0.044 mm. Such behaviour is indicative of the highly unstable nature of nanocracks. All results are based on using the transitionalized crack length (TCL). The TCL fatigue crack growth increment Δa is postulated to depend on the incremental CDF ΔS or ΔVEDF. The form invariance of , and is invoked by scale segmentation to reveal the multiscale nature of IGT that is inherent to fatigue crack growth. While the choice of directionality from micro to macro is not the same as that from macro to micro, this difference will not be addressed in this work.     

An element-based formulation for ES-FEM and FS-FEM models for implementation in standard solid mechanics finite element codes for 2D and 3D static analysis

Edge-based and face-based smoothed finite element methods (ES-FEM and FS-FEM, respectively) are modified versions of the finite element method allowing to achieve more accurate results and to reduce sensitivity to mesh distortion, at least for linear elements. These properties make the two methods very attractive. However, their implementation in a standard finite element code is nontrivial because it requires heavy and extensive modifications to the code architecture. In this article, we present an element-based formulation of ES-FEM and FS-FEM methods allowing to implement the two methods in a standard finite element code with no modifications to its architecture. Moreover, the element-based formulation permits to easily manage any type of element, especially in 3D models where, to the best of the authors' knowledge, only tetrahedral elements are used in FS-FEM applications found in the literature. Shape functions for non-simplex 3D elements are proposed in order to apply FS-FEM to any standard finite element.