Dynamic fracture toughness determined from load-point displacement

The paper presents a method to determine dynamic fracture toughness using a notched three-point bend specimen. With dynamic loading of a specimen there is a complex relation between the stress-intensity factor and the force applied to the specimen. This is due to effects of inertia, which have to be accounted for to evaluate a correct value of the stress-intensity factor. However, the stress-intensity factor is proportional to the load-point displacement if the fundamental mode of vibration is predominant in the specimen. The proportionality constant depends only on the geometry and stiffness of the specimen. In the present method we have measured the applied force and load-point displacement by a modified Hopkinson pressure bar, where two-point strain measurement has been used to evaluate force and displacement for times greater than the transit time for elastic waves in the Hopkinson bar. We have compared the method with the stress-intensity factor derived from strain measurement near the notch tip and good agreement was obtained. The method is well suited for high-temperature testing and results from fracture toughness tests of brittle materials at ambient and elevated temperatures are presented.     

Crack-tip dynamic isochromatics in the presence of small-scale yielding

A plasticity correction factor for the dynamic stress-intensity factor,K _I ^dyn , associated with a propagating crack tip in the presence of small-scale yielding, is derived from Kanninen's solution for a constant-velocity Yoffe crack with a Dugdale-strip yield zone. Distortions in the otherwise elastic isochromatics surrounding the constant-velocity crack tip are also studied by the use of this model. This plasticity correction factor is then used to evaluateK _I ^dyn from the dynamic isochromatics of a propagating crack in a 3.2-mm-thick polycarbonate wedge-loaded rectangular double-cantilever-beam specimen. The correctedK _I ^dyn is in good agreement with the corresponding values computed by a dynamic, elastic-plastic finite-element code executed in its generation mode.     

Evaluation of fracture mechanics parameters for free edges in multi-layered structures with weak singularities

This study focuses on the stress intensity factors for free edges in multi-layered structural components. The effects of elastic constants of various material combinations on the weak singularity at free edges are analyzed. Using the H-integral approach, the effects of elastic mismatch parameters, the bond area and the thickness of the thin metal layer on the stress intensity factor are quantified and the results are compared with detailed finite element solutions. A good agreement between numerical predictions obtained from the H-integral and the detailed FE results is achieved, showing the applicability of this approach. Similar to a crack problem, the singular elastic field dominates in an annular region adjacent to the notch tip. The relationship between the valid range of the K-dominated field and the thin-film thickness is then demonstrated. Furthermore, the competition of crack initiation between the free edge interface (180° opening angle) and a 90° notch interface in a generic specimen is investigated, in order to find out which is the prevailing failure mode. Comparison between isotropic Si and anisotropic Si substrate is also illustrated. Anisotropy of the Si substrate has a significant influence on the stress intensity factor when combined with an Au or Al metal layer but not with a Cu layer. Additionally, standardized numerical formulae of the dimensionless stress intensity factor have been derived to guide the engineering application.     

Numerical evaluation of generalized stress-intensity factors in multi-layered composites

The problem of the evaluation of the generalized stress-intensity factors for re-entrant corners in multi-layered structural components is addressed. An approximate analytical model based on the theory of multi-layered beams is presented. This approach provides a simple closed-form solution for the direct computation of the Mode I stress-intensity factor for the general problem of a re-entrant corner symmetrically meeting a bi-material interface. For the self-consistency of the theory, re-entrant corners in homogeneous materials and cracks perpendicular to bi-material interfaces can also be gained as limit cases of this formulation. According to this approach, the effects of the elastic mismatch parameters, the value of the notch angle and the thicknesses of the layers on the stress-intensity factor are carefully quantified and the results are compared with FE solutions. FE results are obtained by applying a combination of analytical and numerical techniques based on the knowledge a priori of the asymptotic stress field for re-entrant corners perpendicular to a bi-material interface and on the use of generalized isoparametric singular finite elements at the notch tip. A good agreement between approximate and analytical/numerical predictions is achieved, showing the effectiveness of this approach.     

The determination of mode I stress-intensity factors by holographic interferometry

The use of linear elastic fracture mechanics generally depends upon the availability of suitable analytical or numerical solutions for the relevant crack-tip stress-intensity factor,K. Convenient experimental verification of such solutions is a valuable aid to their correct application and can provide a practical substitute in real design situations of great complexity. A convenient, new experimental technique for estimating the Mode I stress-intensity factor using holographic interferometry and test pieces cut from thin sheets of commercially available polymethylmethacrylate is described and demonstrated. The test pieces can readily be prepared to model any desired Mode I geometry and boundary conditions. In addition, a prior self-calibration procedure can be employed to enhance both convenience and accuracy. Real-time interference-fringe data from the crack-tip region are easily reduced and plotted to yield a straight line whose slope provides a one-parameter evaluation of the effect of geometry on the stress-intensity factor. This information, together with the crack length and applied stress, completely definesK.     

Crack propagation in concrete: Linear elastic fracture mechanics and boundary element method

Crack propagation in concrete is investigated by application of Linear Elastic Fracture Mechanics (LEFM) and the Boundary Element Method (BEM). Acoustic Emission (AE) measurement is made for detecting the load level at which unstable crack initiation occurs in a notched beam under bending. The corresponding critical stress intensity factor K_Ic does not vary appreciably with notch depth. This result is in contrast to that found by the conventional procedure.The critical stress intensity factor K_Ic is employed to analyze the process of crack propagation in an arbitraty direction. The BEM prediction is in good agreement with the experiments.The detection and assessment of crack initiation in concrete structures is of considerable technical interest.     

Role of elasticity in elastic–plastic fracture: Analytical bi-linear crack-tip fields and finite element analysis

A solution for Model-I plane strain crack tip fields in a bi-linear elastic–plastic material is presented. The elastic–plastic Poisson's ratio is introduced to characterize the influence of elastic deformation on the near tip constraint. Attention is focused on the distribution of elastic/plastic strain energy in the sensitive region of the forward sector ahead of a crack tip. The present study shows that the elastic strain energy can be higher than the plastic strain energy in this sensitive sector while large amount of the plastic strain energy develops outside this sector around the crack tip. The effect of elastic deformation in this sensitive region on the structure of crack-tip fields is considerable and the assumption in some important solutions for crack-tip fields reported in literature that the elastic deformation is small and can be ignored is therefore not physically reasonable. Besides, finite element analysis is carried out to validate the analytical solution and good agreement between them is found. It is seen that the present solution with T-stress can properly describe the crack-tip fields under various constraints for different specimens and an analytical relation is established between the critical value of J-integral, J_c, and T-stress for elastic–plastic fracture.     

Dynamic stress-intensity-factor determination from isopachics

The governing equations for determination of dynamic stress-intensity factor at the tip of a running crack are developed from a dynamic analysis of dynamic isopachicfringe patterns. The equations are applied to investigate dynamic crack propagation in Homalite 100 by means of dynamic holographic interferometry. A simple method based on simultaneous measurement of the widths of two isopachics allows determination of Irwin's additional stress field, and a dynamic correction function for the stress-intensity factor is derived. It was found that dynamicK-values obtained from dynamic isopachic-fringe-pattern analysis are lower than their corresponding static values. This implies a modification of the crack velocity vs. stress-intensity-factor relationship towards lower values ofK for dynamic crack propagation.     

Stress-intensity factors for surface cracks in functionally graded materials under mode-I thermomechanical loading

This paper describes the development and application of a general domain integral method to obtain J-values along crack fronts in three-dimensional configurations of isotropic, functionally graded materials (FGMs). The present work considers mode-I, linear-elastic response of cracked specimens subjected to thermomechanical loading, although the domain integral formulation accommodates elastic–plastic behavior in FGMs. Finite element solutions and domain integral J-values for a two-dimensional edge crack show good agreement with available analytical solutions for both tension loading and temperature gradients. A displacement correlation technique provides pointwise stress-intensity values along semi-elliptical surface cracks in FGMs for comparison with values derived from the proposed domain integral. Numerical implementation and mesh refinement issues to maintain path independent J-values are explored. The paper concludes with a parametric study that provides a set of stress-intensity factors for semi-elliptical surface cracks covering a practical range of crack sizes, aspect ratios and material property gradations under tension, bending and spatially-varying temperature loads.     

Matched asymptotic expansions in nonlinear fracture mechanics—I. longitudinal shear of an elastic perfectly-plastic specimen

A method of analysis based upon matched asymptotic expansions is proposed for a cracked specimen which is subjected to longitudinal shear (mode III) loading. This gives the small-scale yielding estimate of linear fracture mechanics as a first approximation, and provides systematic refinements which take account of the nonlinear interaction between the elastic and the plastic regions. Explicit solutions can be generated for any specimen which is amenable to a linear elastic analysis. Fracture parameters, such as crack opening displacement and the Jintegral, are expressed as power series in the ratio of applied stress to yield stress, and three terms are given explicitly. These are defined from linear elastic solutions alone. The edge-cracked strip and cracking from a semi-circular notch are studied as examples. Comparison with an exact solution for the former geometry suggests that the three-term expansions give useful results up to 75 % of limit load. The latter example is new and shows the effect of a notch on a crack at loads beyond the normal range of validity of linear elastic fracture mechanics.     

Prediction and observation of crack tip microstructure in shape memory CuAlNi single crystals

The formation of martensite at a notch tip in a CuAlNi shape memory alloy loaded in tension is studied. The geometry of the initial martensite plate to form at the notch is predicted theoretically, using the stress field at a crack tip in an anisotropic linearly elastic body together with a listing of all possible austenite-martensite interfaces from the Crystallographic Theory of Martensite (CTM). The stress field and CTM analyses are combined through a selection criterion based on computing the work available from the stress field to transform to each austenite-martensite interface. The resulting predictions are compared to experimentally observed microstructures in notched specimens of single crystal CuAlNi loaded in tension for eight notch orientations. Results show that the available work criterion accurately predicts the orientation, number and order of the austenite-martensite interfaces that initially form near a crack.     

A new fracture-toughness test method for thick-walled cylinder material

A new method for measuring the plane-strain fracture toughness of the material from a thick-walled cylinder is presented. This method utilizes a notched, “C”-shaped test specimen, pin loaded in tension. This specimen has the advantage of most efficiently utilizing the available material to obtain the maximum possible triaxial constraint at the crack tip. Stress-intensity-factor calibrations for this specimen were obtained by two independent experiments. These are a compliance test, as originally proposed by Irwin, and a fatigue-crack-growth test, as suggested by James and Anderson. Very good agreement was obtained between the results of these two experiments. A stress-intensity calibration for a similar geometry was also obtained using a finite-element analysis and a method developed by Kobayashi to determine stress-intensity factors from finite-element results. The results of this method appear to be low by about 10 percent. Comparative fracture-toughness tests of material from a 2-in.-thick plate of special aircraft quality, 4340 steel, were conducted using the proposed new test method and the ASTM standard bend specimen. These results agreed within 2 percent.     

Analysis of crack extension in anisotropic materials based on local normal stress

The normal stress ratio theory is applied to predict crack extension behavior in center-notched unidirectional graphite-epoxy of arbitrary fiber axis orientation, subjected to arbitrary far-field planar loading. The theory is applied within analytical solutions for two infinite plate geometries: a plate with a sharp center crack, and a plate with an elliptical center flaw. A critical analytical case is identified suggesting that application of the theory within a stress solution modelling crack tip shape may increase the accuracy of crack growth direction predictions. Crack extension direction, location of crack extension, and critical stress predictions of the theory are compared to those obtained from experiments on specimens subjected to tensile, shear, and mixed-mode far-field loading. The comparison shows that, applied within each analytical solution, the normal stress ratio theory provides verifiable predictions of crack growth behavior. By modelling actual notch tip shape, the elliptical notch solution is able to provide accurate qualitative predictions of the origin of crack extension along the periphery of a cut notch tip in a way that the sharp crack analysis cannot. The sharp notch solution appears to provide slightly more accurate crack growth direction predictions, however. Also, in predicting critical applied far-field stresses, the sharp crack solution appears to exhibit a stronger ability to model subtle experimental trends.     

A coupled kinetic-constitutive approach to the study of high temperature crack initiation in single crystal nickel-base superalloys

A rate-dependent crystallographic constitutive theory coupled with a mass diffusion model has been used to study crack initiation in single crystal nickel-base superalloys, exposed to an oxidising environment and subjected to mechanical loading. The time to crack initiation under constant load has been predicted using a strain-based failure criterion. A notched compact tension (CT) specimen containing a single casting defect, idealised as a cylindrical void close to the notch surface, has been studied. Finite element analysis of the CT specimen revealed that, due to the strong localisation of inelastic strain at the void, a microcrack will initiate in the vicinity of the void rather than at the notch surface. The numerical results have also shown that the time to crack initiation depends strongly on the void location. The coupled diffusion-deformation studies have revealed that environmental effects reduce the time to crack initiation due to the oxidation-induced material softening in the vicinity of the notch and void. The applicability of a failure assessment approach, based on the linear elastic stress intensity factor, K, to predict the crack initiation time under creep loading is examined and a probabilistic framework for prediction of component lifetime is proposed.     

Modeling of fatigue crack growth from a notch

When a fatigue crack is nucleated and propagates into the vicinity of the notch, the crack growth rate is generally higher than that can be expected by using the stress intensity factor concept. The current study attempted to describe the crack growth at notches quantitatively with a detailed consideration of the cyclic plasticity of the material. An elastic–plastic finite element analysis was conducted to obtain the stress and strain histories of the notched component. A single multiaxial fatigue criterion was used to determine the crack initiation from the notch and the subsequent crack growth. Round compact specimens made of 1070 steel were subjected to Mode I cyclic loading with different R-ratios at room temperature. The approach developed was able to quantitatively capture the crack growth behavior near the notch. When the R-ratio was positive, the crack growth near a notch was mainly influenced by the plasticity created by the notch and the resulted fatigue damage during crack initiation. When the R-ratio was negative, the contact of the cracked surfaces during a part of a loading cycle reduced the cyclic plasticity of the material near the crack tip. The combined effect of notch plasticity and possible contact of cracked surface were responsible for the observed crack growth phenomenon near a notch.     

Three-dimensional speckle interferometric investigation of the stress-intensity factor along a crack front

The variation in Mode I stress-intensity factor throughout the thickness of an ASTM standard compact tension specimen was determined using scattered-light speckle interferometry. Two very thin sheets of coincident coherent light traveling in opposite directions were passed through a Plexiglas specimen normal to the crack faces. A double-exposed photograph of the scattered-light speckle pattern was taken while the specimen was subjected to a small load increment. From this double-exposed photograph, the change in the crack-opening displacement could be determined. From the information about the crack-opening displacement in the region of the crack tip, the stress-intensity factor was calculated for various interior planes and on the surface of the specimen. For the compact tension specimen tested, the stress-intensity factor did not vary throughout the specimen's thickness. The method of scattered-light speckle interferometry proved to be very powerful in solving this complex three-dimensional problem.     

An improved experimental method for the determination of the critical stress intensity factor KIc of brittle materials

The critical stress intensity factor K_Ic is determined by a simple and accurate method, using small test specimens and a simple procedure in this paper.Single edge V-notched tension specimens made of PMMA are subjected to a load which is slowly increased until the crack begins to move from the notch tip. During the crack propagation event shadow patterns at the tip of the crack are recorded in a video recorder. Under these loading conditions, the creating real crack propagate slowly until the crack propagation velocity take an abrupt increase and the entire fracture of the specimen takes place. The stress intensity factor which correspond to the transition from the slow to fast crack speed, is the critical stress intensity factor K_Ic and it can be the fracture toughness of the material.The results are accurate and in good agreement with those values of K_Ic which are calculated by approximate theoretical expressions.The purpose of this paper is to introduce an improved, simple and accurate experimental method for the determination of fracture toughness of brittle materials.     

Effect of initial flaw shape on crack extension in orthotropic composite materials

An analytical study is presented showing the effects of the notch tip geometry on the location and direction of crack growth from an existing notch in a unidirectional fibrous composite modelled as a homogeneous, anisotropic, elastic material. Anisotropic elasticity and the normal stress ratio theory are used to study crack growth from elliptical notches in unidirectional composites. Sharp cracks, circular holes, and ellipses are studied under far-field tension, and shear loading. Limited comparisons are made showing good correlation with experimental results.     

Creep deformation at crack tips in elastic-viscoplastic solids

The evaluation of crack growth tests under creep conditions must be based on the stress analysis of a cracked body taking into account elastic, plastic and creep deformation. In addition to the well-known analysis of a cracked body creeping in secondary (steady-state) creep, the stress field at the tip of a stationary crack is calculated for primary (strain-hardening) or tertiary (strain-softening) creep of the whole specimen. For the special hardening creep-law considered, a path-independent integral C¹_h, can be defined which correlates the near-tip field to the applied load.It is also shown how, after sudden load application, creep strains develop in the initially elastic or, for a higher load level, plastic body. Characteristic times are derived to distinguish between short times when the creep-zones, in which creep strains are concentrated, are still small, and long times when the whole specimen creeps extensively in primary and finally in secondary and tertiary creep. Comparing the creep-zone sizes with the specimen dimensions or comparing the characteristic times with the test duration, one can decide which deformation mechanism prevails in the bulk of the specimen and which load parameter enters into the near-tip stress field and determines crack growth behavior. The governing load parameter is the stress intensity factor K₁ if the bulk of the specimen is predominantly elastic and it is the J-integral in a fully-plastic situation when large creep strains are still confined to a small zone. The C¹_h-integral applies if the bulk of the specimen deforms in primary or tertiary creep, and C¹ is the relevant load parameter for predominantly secondary creep of the whole specimen.     

A technique for studying interaction between a screw dislocation dipole or a concentrated load and a mode III crack crossing an interface

A method of potentially wide application is developed for deriving analytical expressions of the elastic interaction between a screw dislocation dipole or a concentrated force and a crack cutting perpendicularly across the interface of a bimaterial. The cross line composed of the interface and the crack is mapped into a line, and then the complex potentials are educed. The Muskhelishvili method is extended by creating a Plemelj function that matches the singularity of the real crack tips, and eliminates the pseudo tips’ singularity induced by the conformal mapping. The stress field is obtained after solving the Riemann–Hilbert boundary value problem. Based on the stress field expressions, crack tip stress intensity factors, dislocation dipole image forces and image torque are formulated. Numerical curves show that both the translation and rotation must be considered in the static equilibrium of the dipole system. The crack tip stress intensity factor induced by the dipole may rise or drop and the crack may attract or reject the dipole. These trends depend not only on the crack length, but also on the dipole location, the length and the angle of the dipole span. Generally, the horizontal image force exerted at the center of the dislocation dipole is much smaller than the vertical one. Whether the dipole subjected to clockwise torque or anticlockwise torque is determined by whether the Burgers vector of the crack-nearby dislocation of the dipole is positive or negative. A concentrated load induces no singularity to crack tip stress fields as the load is located at the crack line. However, as the concentrated force is not located on the crack line but approaches the crack tip, the nearby crack tip stress intensity factor K_IIIu increases steeply to infinity.