Remote Tensile Stress Research Articles

Strength-based regularization length in phase field fracture

We propose a length-free (LF) implementation of phase field (PF) approach to fracture. The input parameters of the LF-PF model are the same as those of the coupled criterion (CC), namely the critical energy release rate Gc and the material tensile strength σc. This formulation relies on the previously determined correlation between the internal length in classical PF fracture models and the material tensile strength. Similar failure loads are obtained using either AT1 and AT2 versions of the LF-PF, a larger diffused damaged zone being observed using AT1. The LF-PF formulation allows retrieving the size effect for a crack under remote tensile stress in infinite medium and shows square hole specimen failure stresses close to the predictions obtained using the CC, which allows using both models in a complementary manner. Finally, the proposed formulation overcomes the problem of identifying different internal lengths for different specimen configurations.

Material index for strain localization susceptibility in cracked bodies: Examination of single‐crystal Cu using crystal plasticity finite element method

AbstractOur previous study demonstrated that, in a cracked single‐crystal subjected to remote tensile stress, the interactions among material behavior, strain localization, mechanical behavior, and strain concentration caused the formation of a strain field was quite different from the HRR singularity introduced in the homogeneous continuum. Moreover, to quantify strain localization ability for metals, a material index for strain localization susceptibility (SLS) was proposed. However, the effectiveness and availability of that index were not examined in detail. In this study, a Cu‐based single crystal was examined using the crystal‐plasticity finite element method (CPFEM) model. This model incorporates the strain localization mechanism from the non‐crystallographic shear band. Additionally, a new material index for SLS was introduced. The results showed that the new material index could reflect the SLS of the material. Meanwhile, a critical material index value based on strain localization was determined for predicting the crack behavior. This study also discusses the plastic deformation behavior near the crack tip with different material indices for SLS.

Early crack initiation mode during tensile fracture process in a punch-processed precipitation-hardened steel plates and evaluation method to treat the crack as an equivalent pre-crack

For partially worked metals such as punching-processed steel plates, a process-affected zone designated “shear-affected zone” (SAZ) is formed. In this study, tensile tests of punched specimens were conducted to determine the strength of partially worked metals. In the fracture surface of the specimen, a semi-elliptical crack originating in the SAZ, or “shear crack,” was observed, inclined at 45° to the remote tensile stress direction. This shear crack changed the behavior of the metal during tensile loading, e.g., ductile-brittle transition. In specimens exhibiting macroscopic brittle fracture, microscopic brittle fractures cause shear cracks before the specimen reaches its macroscopic yield point. The shear cracks in ductile fractured specimens include a similar brittle-fractured region that can be considered a pre-crack; the punched material can be considered a pre-cracked material. This study highlights the requirement for a novel mechanical model based on linear fracture mechanics to evaluate and predict the fracture mode of punched materials.

Hydrogen induced fast-fracture

One of the recurring anomalies in the hydrogen induced fracture of high strength steels is the apparent disconnect between their toughness and uniaxial tensile strength in identical hydrogen environments. Here we propose, supported by detailed atomistic and continuum calculations, that unlike macroscopic toughness, hydrogen-mediated tensile failure is a result of a fast-fracture mechanism. Specifically, we show that failure originates from the fast propagation of cleavage cracks that initiate from cavities that form around inclusions such as carbide particles. The failure process occurs in two stages. In stage-A, hydrides rapidly form around the roots of stressed notches on the cavity surfaces with hydrogen fed from the hydrogen gas within the cavity. These hydrides promote cleavage fracture with the cracks propagating at >100ms−1 until the hydrogen gas in the cavity is exhausted. Predictions of this hydrogen-assisted crack growth mechanism are supported by atomistic calculations of binding energies, mobility barriers and molecular dynamics calculations of the fracture process. Typically, cracks grow by less than 1 μm via this hydrogen-assisted mechanism and thus insufficient to cause macroscopic fracture of the specimen. However, this stage is then followed by a stage-B process where these fast propagating cracks can continue to grow, now in the absence of hydrogen supply, given an appropriate level of remote tensile stress. This is surprising because the fracture energy is now that of Fe in the absence of H and cleavage fracture requires opening tractions on the order of 15 GPa to be generated. Thus, fracture is usually precluded due to plasticity around the crack-tip. Here we show via macroscopic continuum crack growth calculations in a rate dependent elastic-plastic solid with fracture modelled using a cohesive zone that cleavage is possible if the crack propagates fast enough. This is because strain-rates at the tips of fast propagating cracks are sufficiently high for the drag on the motion of dislocations resulting from phonon scattering to limit plasticity. This combined atomistic/continuum model is used to explain a host of well-established experimental observations including (but not limited to): (i) insensitivity of the strength to the concentration of trapped hydrogen; (ii) the extensive microcracking in addition to the final cleavage fracture event and (iii) the higher susceptibility of high strength steels to hydrogen embrittlement. Importantly, we also show that the stage-A hydrogen-assisted fracture process only occurs in certain crystallographic orientations with crack-tip plasticity processes, such as twinning, blunting cracks in other orientations. This inhibits the fast-fracture mechanism in a macroscopic toughness on a polycrystalline material and thus explains the apparent contradiction between the hydrogen-assisted macroscopic toughness and tensile strength of steels.

Tensile fracture of an adhesive joint: the role of crack length and of material mismatch

The tensile strength of an adhesive joint is predicted for a centre-cracked elastic layer, sandwiched between elastic substrates, and subjected to remote tensile stress. A tensile cohesive plastic zone, of Dugdale type, is placed at each crack tip, and the cohesive zone is characterised by a finite strength and a finite toughness. An analytical theory of the fracture strength is developed (and validated by finite element simulations). The macroscopic strength of the adhesive joint is determined as a function of the relative magnitude of crack length, layer thickness, plastic zone size, specimen width and elastic modulus mismatch between layer and substrates. Fracture maps are constructed to reveal competing regimes of behaviour. The maps span the full range of behaviour from a perfectly brittle response (with no crack tip plasticity) to full plastic collapse. When the sum of crack length and cohesive zone length is less than 0.3 times the layer height, the effect of elastic mismatch between substrate and adhesive layer has only a minor influence upon the macroscopic fracture strength. For this case, the cracked adhesive layer behaves as a centre-crack in an infinite solid made from adhesive, and a transition from toughness control to strength control occurs when the crack length is comparable to that of the cohesive zone length. Alternatively, when the sum of crack length and cohesive zone length exceeds 0.3 times the layer height, the elastic mismatch plays a major role; again there is a transition from toughness control to strength control, but it occurs at a ratio of crack length to layer thickness that depends upon both the elastic mismatch and the ratio of cohesive zone length to layer height. The study also highlights the importance of a structural length scale in the form of layer height times modulus mismatch: this scale is on the order of 1 metre when the layer height equals one millimetre and the elastic modulus of the substrate is one thousand times that of the adhesive layer. The in-plane structural dimensions (including crack length) must exceed this structural dimension in order for a remote K-field to exist within the substrate. Experimental validation of the cohesive zone approach is achieved by measuring the sensitivity of fracture strength to crack length and layer height for a centre-cracked strip made from cellulose acetate layer, sandwiched between aluminium alloy substrates.

Application of the cyclic R-curve method to notch fatigue analysis

The paper deals with the prediction of the fatigue limit of notched specimens. Some authors in the literature have shown that this issue can be solved by employing an approach based on critical distances, others used elastic fracture mechanics. In this paper the fatigue limit of notched specimens is given by the non-propagating condition of mechanically small surface cracks. According to the so-called cyclic R-curve method, the crack driving force of a growing small crack is compared to its resistance force, which incorporates the gradual build-up of crack closure. The fatigue limit is determined by that applied nominal stress, for which the tangency condition of crack driving and resistance force is satisfied. The approach has been modified to incorporate plasticity effects in the mechanically short crack regime and it has been applied to a mild and a high-strength steel, in case of an infinite plate with circular hole under remote tensile stress. The method has been successfully applied to the evaluation of fatigue notch-sensitivity in case of surface roughness.

An experimental investigation of dyke injection under regional extensional stress

AbstractDyke injection is a fundamental process of magma transport in the crust, occurring in all tectonic settings. The effect of extensional stress regimes on dyke injections is particularly important to understanding a wide spectrum of processes including continental rifting and volcanic activity. Yet dyke injection in extensional regimes has been relatively understudied. In addition, the effect of dyke‐dyke interaction modifying the surrounding stress field and leading to dyke rotation about the vertical axis has not been addressed. We present the results from 23 laboratory analogue experiments investigating lateral dyke injections in a remote extensional stress field. This study is unique in that it addresses the effect of both extension and dyke‐dyke interaction on the lateral propagation and rotation of dykes. The experiments study the interrelationship between successive lateral dyke injections by examining dyke injection thickness, injection spacing, injection orientation, extension, and structural relationship. A relationship between the rotation angle between two successive intrusions and the distance separating them under given extensional stress conditions is established. The rotation angle depends on two dimensionless numbers: the ratio of fluid overpressure of the first injection and remote tensile stress, and the ratio of the spacing between injections and the height of the first intrusion. The experiments show how the stress field is perturbed by an intrusion and how the remote stress field is locally relieved by this intrusion. The results show furthermore that measuring or estimating the rotation angles between successive intrusions within rift zones allows the spatial distribution of these intrusions to be estimated. In the case of the actively spreading Red Sea rift in Afar, Ethiopia, we find that the vast majority of the dykes are predicted to intrude within 10 km of each other and most frequently between 4 and 5 km, in good agreement with independent geophysical observations.

Fatigue life analysis of internal circumferential crack of tubular structures

In this paper, the fatigue life of cracked tubular structures subjected to remote cyclic tensile stress is studied by using the fracture mechanics and Paris’ law. To evaluate the flooded member detection, a reassessed leak-before-break approach is implemented, in which a more realistic crack development, by taking the partly through thickness crack stage into consideration, has been investigated, and that has never been reported in literature for tubular member with internal crack. Parametric study demonstrates that the time available to detect the presence of the crack in its flooded life depends particularly on the rate of accumulation of fatigue damage and on the initial length of surface crack presented. The results of the analysis have been applied by a case study of a particular offshore structure and loading situation, obtained to support inspection intervals of three years for flooded member detection. By comparing the current results with the conventional leak-before-break procedure in flooded member detection, it is found that for the situation studied in this paper, the flooded life predicted from conventional leak-before-break method for the crack is always on the safe side. Therefore, it is too conservative for setting up the inspection interval for flooded member detection.

Tensile strength and toughness of partially sintered ceramics using discrete element simulations

The fracture behavior of highly porous ceramics is simulated using the discrete element method. Spherical particles model the powder used to obtain the partially sintered ceramic material. Three-dimensional representative volume elements of porous microstructures made of several tens of thousands of particles were first generated by numerical sintering. Elastic force–displacement laws model the solid bonds formed between particles during sintering. A realistic fracture criterion, based on the local stress intensity factor associated with the bond between two particles, is also introduced. The simulation of tensile tests allows the assessment of the effective strength of these microstructures as a function of the residual porosity. Furthermore, the introduction of a pre-crack in a sample subjected to a remote tensile stress allows the critical stress intensity factor to be calculated. Porous electrodes for electrochemical applications represent an important application field for these ceramics. Those discrete element simulations should be an effective tool for optimizing their microstructure at the micron length scale.

A strip yield model for two collinear cracks

In this paper two alternative approaches for analysing the non-linear interaction between two equal-length collinear cracks subjected to remote tensile stress on infinity are developed. The approaches are based on the classical Dugdale strip yield model and utilise the distributed dislocation formulation to represent two through-the-thickness cracks in an infinite plate. In the first approach the solution is obtained analytically by solving Föppl integral equation; in the second a numerical procedure based on the Gauss–Chebyshev quadrature method is applied. The latter approach can be extended to other multi-crack problems or more complicated types of loading. The developed approaches are validated against previously published studies and new results for crack tip opening displacement are presented.

Fracture mechanics of porous ceramics using discrete element simulations

The fracture behavior of highly porous ceramics is simulated using the discrete element method. A representative volume element made of spherical particles models the powder used to obtain a partially sintered ceramic material. Three-dimensional porous microstructures made of several tens of thousands of particles are then generated. Elastic force-displacement laws model the bonds formed between particles during sintering. A realistic fracture criterion, based on the local stress intensity factor associated with the bond between two particles, is also introduced. Based on these simulations, we compute the effective strength of these microstructures in tension as a function of the residual porosity. Furthermore, the introduction of a pre-crack in a sample subjected to a remote tensile stress allows the critical stress intensity factor to be calculated. Porous electrodes for electrochemical applications represent an important application field for these ceramics. Those discrete element simulations should be an effective tool for optimizing their microstructure at the submicronic length scale.

Effects of triaxiality on the growth of crack-like cavities in soft incompressible elastic solids

We study the finite deformation of an isolated circular or penny-shaped crack in an infinite block of soft incompressible hyper-elastic solid. The crack is subjected to remote tensile true stresses that are parallel (S) and normal (T) to the undeformed crack faces. Two material models are considered, an ideal rubber (neo-Hookean solid) and a hyperelastic material that hardens exponentially. The energy release rates for different triaxiality ratios S/T from uniaxial tension (S/T = 0) to hydrostatic tension (S/T = 1) are determined using a finite element method (FEM). For small deformations, our results agree with linear elastic theory where the energy release rate is independent of S. This is not the case at finite strains; our results show that the energy release rate increases rapidly with triaxiality. For the special case of pure hydrostatic tension, the energy release rate approaches infinity for the neo-Hookean solid at a finite tension, consistent with a previous result by Lin and Hui (Y.Y. Lin and C.Y. Hui, Int. J. Fract., 2004, 126, 205–221). Our result shows that strain hardening significantly reduces the energy release rate for the same remote loading. In particular, for the case of hydrostatic tension, the energy release rate remains bounded for the exponentially hardening solid.

Segmented lineaments on Europa: Implications for the formation of ridge complexes and bright bands

We describe several segmented lineaments on Europa’s surface. These lineaments are extensive, stretching for 100s–1000s of km, and have ridge complex or bright band morphologies. The geometries of the segmented portions of these features are diagnostic of the remote normal and shear stress environment in which they formed and, therefore, constrain ridge complex and bright band formation mechanisms. Analysis of four ridge complexes indicates that they formed in a remote normal stress environment that was tensile and isotropic (or nearly so) and that these lineaments may have formed in a manner more analogous to bands on Europa than to ridges. The stress environment associated with these ridge complexes may also explain the anastomosing nature of their interior morphology. Analysis of two bright bands indicate that one formed in a remote normal stress environment that was tensile and the other was reactivated under a combination of remote tensile normal stress and remote sinistral shear stress. Aspects of the morphologies of these features also indicate that bright bands likely have complex deformation histories that can include multiple episodes of reactivation.

Three-dimensional stress concentrations at elliptic holes in elastic isotropic plates subjected to tensile stress

The through-thickness variations of stress concentration factors along the wall of elliptic holes in finite thickness plates of isotropic materials subjected to remote tensile stress have been systematically analyzed using the finite element method. The three-dimensional stress concentration factor K t is found to be a function of the thickness to root radius ratio B/ ρ and the aspect ratio t (short to long axial length) of the elliptic hole under tensile loading. It is found that the maximum stress concentration factor through the thickness, ( K t ) max, is 24–123% higher than the value on the free surface ( K t ) surf when t changes from 1 to 0.01, and the ratio of the surface value ( K t ) surf to the corresponding planar solution ( K t ) p− σ at the root is only 0.82–0.48 when t ranges from 1 to 0.01 if B/ ρ is large enough. When B/ ρ is smaller than 1, both of the ratios will approach 1. Simple and efficient empirical formulae for the relationships between the three values were obtained by fitting the numerical results with good engineering accuracy for large range of B/ ρ (from 1 to 100,000) and t (from 0.01 to 1). Combining with the plentiful known two-dimensional stress concentration factors in handbooks, the proposed formulae can be used to solve the maximum three-dimensional stress concentration factors of holes, notches and be useful for strength and fatigue designs of engineering structures with similar defects.

Numerical investigations of maximum stress concentration at elliptic holes in finite thickness piezoelectric plates

The through-thickness variations of stress-concentration factors along the wall of elliptic holes in finite thickness plates of transversely isotropic piezoelectric materials subjected to uniaxial remote tensile stress and applied electric field have been systematically analyzed using the finite element method. The three-dimensional stress concentration factor K t is found to be a function of the thickness to root radius ratio B/ ρ and the aspect ratio t (short to long axial length) of the elliptic holes under tensile loading. It is found that the maximum stress-concentration factor through the thickness, ( K t ) max, is 20–150% higher than the value on the free surface ( K t ) surf when t changes from 1 to 0.01, and the ratio of the surface value ( K t ) surf to the corresponding planar solution ( K t ) p− σ at the roots is only 0.84–0.44 when t ranges from 1 to 0.01 if B/ ρ is large enough. When B/ ρ is decreasing to 1, both the ratios are approaching unity. Simple empirical formulae for the relationships between the three values were obtained by fitting the numerical results with good engineering accuracy for large range of B/ ρ (from 1 to 100,000) and t (from 0.01 to 1). The proposed formulae will be useful for strength and fatigue designs of engineering structures with notches and holes. Additional applied electric field can cause higher opening stress in the interior and lower opening stress at the free surface of the plate near the hole, and enhance the out-of-plane constraint significantly. Therefore, the three-dimensional effects can be much stronger in piezoelectric ceramics than in metallic materials.

Experimental and theoretical fracture mechanics applied to fracture of the crust of Venus

Mapping of closely spaced, parallel extensional fractures in the Guinevere and Sedna Planitia regions of Venus reveals a concentric pattern of fractures around the edge of the large topographic rise of Western Eistla Regio. We have constructed 13 transects through these closely spaced parallel fractures (CSPF) and find a mean spacing of between 0.8 and 1.2 km. A two‐dimensional, nonlayered, fracture mechanics computer model for the formation of CSPF is described. For cracks extended by a remote tensile stress the stress intensity factor controls the depth of penetration, which in turn, governs the spacing between adjacent cracks based upon the stress‐shadow principle. The stress required to initiate cracks depends upon the strength of the material; thus spacing of CSPF depends only on the physical properties of the preexisting rock mass. Using a new fracture mechanics apparatus designed to simulate Venusian conditions (90 bar CO2, 450°C) the fracture toughness of basalt was measured for confining pressures up to 20 MPa and for temperature up to 600°C. Fracture toughness was found to increase from ∼2.4 MPam1/2 at ambient pressure to ∼3.0 MPam1/2 at 10 MPa confining pressure. Fracture toughness showed no clear trend with temperature. The experimental results for fracture toughness suggest that the preexisting rock mass that fits best with observations contains an inverse square distribution of flaws with maximum sizes of only ∼20–80 cm and that a remote tensile stress of ∼3–6 MPa is required to form the observed CSPF.

Viscoplastic BEM fracture analysis of creeping metallic cracked structures in plane stress using complex variable techniques

This paper extends a boundary element approach developed earlier by the present author to the solution of viscoplastic fracture problems arising in creeping cracked metallic structural components undergoing plane stress deformation conditions. By using a plane stress viscoplastic formulation this work adopts the procedure of Lyness and Moler to express the derivative of displacement rate as the ratio of the imaginary part of the displacement rate to the imaginary part of a complex variable representing the coordinates of a collocation point. In this manner the evaluation of creep stresses at internal points is greatly simplified. The results thus obtained are highly accurate in comparison with conventional BEM. The application of the present BEM approach is illustrated by obtaining creep stress and strain distribution for center cracked and standard CT fracture specimens subjected to remote tensile and shear stress loading under plane stress deformation conditions.

701 片側き裂を有する帯板の応力拡大係数

A series of finite element analyses were performed for a single edge cracked plate with finite width subjected to point load at a point on the crack surface. The ratio of crack length to plate width a/W was varied in a range from 0.1 to 0.9. Stress intensity factor K was evaluated by means of the virtual crack extension method. Approximate equations for K were made in the same manner as those for a single edge cracked semi-infinite plate proposed by Hartranft et al. K values of a single edge cracked plate with uniform remote tensile stress were evaluated using these approximate equations for various a/W conditions, and compared with those calculated using the existing formula. It was clarified that the estimation error of these approximate equations was less than 1%.

Use of interferometric techniques for measuring the displacement field in the plane of a part-through crack existing in a plate

An experimental procedure based on holographic interferometry coupled with computer-aided image processing techniques was used to estimate the stress intensity factor (SIF) of a part-through crack in a plate. The loading consisted of a uniform remote tensile stress, leading to an opening and edge rotation displacement field along the crack plane. The evaluation of the crack opening displacements and edge rotations, carried out after the numerical interpretation of the fringe pattern, made possible an estimation of the SIF along the crack line, following the theoretical basis of the Line-Spring Model developed by Rice and Levy.

An energy approach to the link-up of multiple cracks in thin aluminum alloy sheets

An energy approach has been used in the study of the coalescence or linkage of multiple cracks in aluminum alloy sheets. The study was motivated by concern for the structural integrity of aging aircraft. Forty reported tests for 2024-T3 aluminum panels with a major crack and several multiple-site damage (MSD) cracks have been analyzed via a simple computational model with a Dugdale–Barenblatt [D.S. Dugdale, J. Mech. Phys. Solids 8 (1960) 100–104; G.I. Barenblatt, in: H.L. Dryden, Th. VonKarman (Eds.), Advances in Applied Mechanics, vol. II, 1962, pp. 55–130] type of plastic or inelastic deformation. For simplicity, the computational model considers only the plastic interaction between the major crack and two symmetrically adjacent MSD cracks in an infinite sheet under remote tensile stress. By following the approach given in [B. Cotterell, J. K. Reddel, Int. J. Fract. 13 (1977) 267–277], the specific work to cause ligament failure is found to be a linear function of the normal extent of the confined plastic region for most tests considered. A few exceptions to this linear relation are attributed to the limitation of the employed computational model. A new criterion and an engineering method to predict crack link-up in an MSD sheet are proposed based on this specific work concept, and they have been demonstrated through application to stiffened panels.