Positive T-stress Research Articles

Influence of specimen configuration on mode I and mode II fracture toughness of sandstone

In order to investigate the impact of cracked specimen configurations on the apparent fracture toughness of the green sandstone, a series of fracture experiments were conducted utilizing four distinct types of specimens. The experimental results show that the cracked straight-through Brazilian disk (CSTBD) specimens observed the minimum mode I fracture toughness KIC and the maximum mode II fracture toughness KIIC. The KIC measured by notched semi-circular bending (NSCB) specimens is nearly twice that of CSTBD specimens, while the KIIC is only 77% of CSTBD specimens. It indicates that both the specimen configurations and loading conditions are significant factors influencing the apparent fracture toughness, which may be attributed to the different T-stress for different specimens. Furthermore, several fracture criteria were employed to evaluate the experimental results. Simultaneously, a theoretical framework based on the generalized maximum tangential strain (GMTSN) criterion was proposed to investigate the impact of T-stress on apparent fracture toughness. The results indicate that positive T-stress can improve the KIC, while negative T-stress can decrease the KIC. In addition, we also found that the dependence of KIIC on T-stress is exactly opposite to that of KIC. The study provides theoretical references for systematically evaluating experimental results and selecting specimens reasonably.

Fracture of four semi-regular lattices regulated by T-stress in modified boundary layer models

The development of additive manufacturing technology has promoted the rise of various lattice metamaterials, and evaluating their fracture properties is critical for both application and safety. Here, by conducting finite element simulations on four types of elastic-brittle semi-regular lattices (kagome, snub-trihexagonal, elongated-triangular, and snub-square), we indicate that the fracture toughness of lattice metamaterials is not solely determined by the crack-tip singular term, i.e., the critical stress intensity factor, but is also influenced by the non-singular T-stress term. T-stress regulates fracture toughness by modulating geometrical distortions and force-carrying modes of lattice bars around the crack front. This regulation effect depends on the lattice topology and becomes weaker under mixed-mode loading. As the relative density of lattices decreases, T-stress can force elastic buckling of bars around the crack front, which may deteriorate the fracture toughness. Additionally, positive T-stress tends to promote the crack deflection and negative T-stress tends to inhibit. Our results offer new insights into the structure-property relationships in the presence of T-stress, which may contribute to design the geometry of micro/nano-lattice metamaterials.

Development of maximum tangential strain (MTSN) criterion for prediction of mixed-mode I/III brittle fracture

A number of fracture criteria are available for predicting the cracking behavior under combined contribution of opening-tearing damage mechanism (i.e. mixed-mode I/III). But most of them have some limitations or discrepancies with the experimental results. In this research, a new form of maximum tangential strain criterion called 3D-MTSN is developed for predicting mixed-mode I/III fracture phenomenon. This criterion was used originally for analyzing the mixed-mode I/II fracture but we extended its ability for mixed mode I/III as well. Also, the influence of T-stress in addition to modes I and III stress intensity factors was also considered in driving the fracture toughness and fracture angle formulations of the 3D-MTSN criterion. According to this strain-based criterion it is shown that both sign and magnitude of T-stress can affect the fracture behavior of mixed mode I/III. In general, positive T-stress increases the mixed mode I/III fracture envelope and conversely negative sign of T-stress can decrease it. Meanwhile, the Poisson’s ratio (v) has noticeable influence on the mixed mode I/III fracture envelopes and by increasing v, the mixed mode fracture curve predicted by the generalized MTSN criterion shifts down except for those specimens having positive T-stress values and subjected to dominantly pure mode I loading conditions. The practical ability and validity of 3D-MTSN criterion is examined by predicting different sets of mixed mode I/III experimental results reported in the literature for different brittle and quasi-brittle materials tested with different test samples including edge-notch disc bend (ENDB), inclined single edge notch beam (ISENB) and semi-circular bend (SCB) specimens. A comprehensive comparison with the well-known 3D-MTS (stress-based) criterion was also performed. Compared to the other available energy-based fracture models, the presented 3D-MTSN criterion provided better estimations for the experimental results of different materials and specimens under dominantly mode III conditions.

T-stress for the double-edge cracked Brazilian disc under compression

As the nonsingular term in the Williams series solution of the stress field at the crack tip, T-stress has a great influence on the crack propagation behavior. Under the loading condition for pure mode II fracture, negative T-stress (compressive T-stress) reduces the deflection angle for crack initiation and promotes the self-similar propagation of crack; while positive T-stress increases the crack deflection angle, and the crack kinks into tensile wing crack. The closed-form solution of the T-stress for the double-edge cracked Brazilian disc (DCBD) under compression is obtained by the weight function method. The closed-form solution in series form has a good convergence. The normalized T-stresses are calculated for different conditions, then the effects of distribution form, loading angle, contact angle, and relative crack length on T-stresses are analyzed. The results show that the effect of contact angle γ on T-stress should be considered when the loading angle θ and the relative crack length α are both small or great; while the effect of distribution form on T-stress cannot be ignored if and only if θ and α are small. The T-stress in the DCBD specimen is always negative when θ ≤ 20° and α ≥ 0.4. This is advantageous for the realization of true mode II fracture.

Multi-material adhesive joints with thick bond-lines: Crack onset and crack deflection

This study investigates the fracture onset and crack deflection in multi-material adhesive joints with thick bond-lines (≈10 mm) under global mode I loading. The role of adherend-adhesive modulus-mismatch and pre-crack length are scrutinized. The parameters controlling the crack path directional stability are also discussed. Single-material (i.e. steel-steel and GFRP-GFRP) and bi-material (i.e. steel-GFRP) double-cantilever beam joints bonded with a structural epoxy adhesive are tested. The joints are modelled analytically, considering a beam on elastic-plastic foundation, to include characteristic length scales of the problem (e.g. adhesive thickness, plastic zone) and numerically using Finite Element Model. An empirical relation, in terms of geometrical and material properties of the joints, that defines the transition between non-cohesive and cohesive fracture onset is found. Above a specific pre-crack length the stress singularity at pre-crack tip rules over the stress singularity near bi-material corners, resulting in mid-adhesive thickness cohesive fracture onset. However, the cracking direction rapidly deflects out from the adhesive layer centre-line. Positive T-stress along the crack tip is found to be one of the factors for the unstable crack path.

The Effects of Nonproportional Biaxial Loading Paths on Ductile Fracture Initiation: A Void Growth Analysis

Abstract The effects of nonproportional biaxial loading paths on ductile fracture initiation toughness are studied in this article. To this end, the growth of a cylindrical void (hole) located in front of a mode I plane strain crack has been studied using large-deformation finite element analysis. A specific microstructural feature of a steel alloy was thoroughly studied by having a single void positioned at a fixed distance from the crack tip and void that was equal to 10 times the diameter of the void. In particular, the nonproportional biaxial loading path effects on the crack tip blunting, void growth, ligament reduction, and near-tip stress fields are investigated computationally. Under small-scale yielding conditions, one proportional loading and two nonproportional loading paths are applied to the modified boundary layer models, covering both low-constraint (negative T-stress) and high-constraint (positive T-stress) crack tip conditions. We have observed that the nonproportional load paths have a marked effect on the void growth, crack tip blunting, and the resulting ligament reduction. By applying the simple criteria for the coalescence of the crack tip and void, the ductile fracture initiation toughness is estimated. It is shown that the ductile fracture toughness is dependent on loading paths and constraint conditions (the T-stress ratios). Compared with the proportional biaxial loading case, different load sequences of the biaxial loads will result in either an increase or a decrease the fracture initiation toughness. This effect is particularly significant for specimen geometry with low-constraint conditions (i.e., negative T-stress ratios). Results from this study are of relevance to ductile fracture assessment of components or structures, such as pressure vessels that operate under nonproportional biaxial loading conditions.

Crack growth in Fe-Si (2 wt%) single crystals on macroscopic and atomistic level

Abstract This paper is dedicated to experimental and atomistic study of the influence of so called T-stress (acting along the crack plane) on fracture processes in bcc iron. We analyze experimental results from fracture tests performed at room temperature on bcc iron-silicon single crystals with a long edge crack ( 1 ¯ 1 0 ) [1 1 0] (crack plane/crack front). The specimens were loaded in tension mode I under different border conditions inducing different sign of the T-stress. The brittle-ductile behavior at the crack front was monitored on-line via optical microscopy together with external force and prolongation of the specimens. Topology of the specimens has been investigated before and after the fracture tests via the white light interferometer. The microscopic processes produced by the crack itself were studied at 300 K via 3D molecular dynamic (MD) simulations in bcc iron under equivalent boundary conditions and the T-stress was examined by means of stress calculations on the atomistic level. The experimental and atomistic results show that the sign of the T-stress affects the fracture behavior. MD simulations reveal that positive T-stress makes the emission of blunting dislocations 〈1 1 1〉{1 1 2} from the crack front more difficult. As a consequence, higher external loading is needed for crack blunting in the experimental specimens with T > 0 in comparison with the specimen having T

The Influence of Specimen Type on Tensile Fracture Toughness of Rock Materials

Up to now, several methods have been proposed to determine the mode I fracture toughness of rocks. In this research, different cylindrical and disc shape samples, namely: chevron bend (CB), short rod (SR), cracked chevron notched Brazilian disc (CCNBD), and semi-circular bend (SCB) specimens were considered for investigating mode I fracture behavior of a marble rock. It is shown experimentally that the fracture toughness values of the tested rock material obtained from different test specimens are not consistent. Indeed, depending on the geometry and loading type of the specimen, noticeable discrepancies can be observed for the fracture toughness of a same rock material. The difference between the experimental mode I fracture resistance results is related to the magnitude and sign of T-stress that is dependent on the geometry and loading configuration of the specimen. For the chevron-notched samples, the critical value of T-stress corresponding to the critical crack length was determined using the finite element method. The CCNBD and SR specimens had the most negative and positive T-stress values, respectively. The dependency of mode I fracture resistance to the T-stress was shown using the extended maximum tangential strain (EMTSN) criterion and the obtained experimental rock fracture toughness data were predicted successfully with this criterion.

Quasi-static crack tip fields in rate-sensitive FCC single crystals

In this work, the effects of loading rate, material rate sensitivity and constraint level on quasi-static crack tip fields in a FCC single crystal are studied. Finite element simulations are performed within a mode I, plane strain modified boundary layer framework by prescribing the two term (K − T) elastic crack tip field as remote boundary conditions. The material is assumed to obey a rate-dependent crystal plasticity theory. The orientation of the single crystal is chosen so that the crack surface coincides with the crystallographic (010) plane and the crack front lies along \([10\overline 1]\) direction. Solutions corresponding to different stress intensity rates \(\dot{{K}}\), T-stress values and strain rate exponents m are obtained. The results show that the stress levels ahead of the crack tip increase with \(\dot{{K}}\) which is accompanied by gradual shrinking of the plastic zone size. However, the nature of the shear band patterns around the crack tip is not affected by the loading rate. Further, it is found that while positive T-stress enhances the opening and hydrostatic stress levels ahead of crack tip, they are considerably reduced with imposition of negative T-stress. Also, negative T-stress promotes formation of shear bands in the forward sector ahead of the crack tip and suppresses them behind the tip.

Effect of residual stress on cleavage fracture toughness by using cohesive zone model

This study presents the effect of residual stresses on cleavage fracture toughness by using the cohesive zone model under mode I, plane stain conditions. Modified boundary layer simulations were performed with the remote boundary conditions governed by the elastic K-field and T-stress. The eigenstrain method was used to introduce residual stresses into the finite element model. A layer of cohesive elements was deployed ahead of the crack tip to simulate the fracture process zone. A bilinear traction-separation-law was used to characterize the behaviour of the cohesive elements. It was assumed that the initiation of the crack occurs when the opening stress drops to zero at the first integration point of the first cohesive element ahead of the crack tip. Results show that tensile residual stresses can decrease the cleavage fracture toughness significantly. The effect of the weld zone size on cleavage fracture toughness was also investigated, and it has been found that the initiation toughness is the linear function of the size of the geometrically similar weld. Results also show that the effect of the residual stress is stronger for negative T-stress while its effect is relatively smaller for positive T-stress. The influence of damage parameters and material hardening was also studied.

Crack tip plastic zone under Mode I, Mode II and mixed mode (I+II) conditions

The shape and size of the plastic zone around the crack tip are analyzed under pure mode I, pure mode II and mixed mode (I+II) loading for small scale yielding and for both plane stress and plane strain conditions. A new analytical formulation is presented to determine the radius of the plastic zone in a non-dimensional form. In particular, the effect of T-stress on the plastic zone around the crack tip is studied. The results of this investigation indicate that the stress field with a T-stress always yields a larger plastic zone than the field without a T-stress. It is found that under predominantly mode I loading, the effect of a negative T-stress on the size of the plastic zone is more dramatic than a positive T-stress. However, when mode II portion of loading is dominating the effect of both positive and negative T-stresses on the size of the plastic zone is almost equal. For validating the analytical results, several finite element analyses were performed. It is shown that the results obtained by the proposed analytical formulation are in very good agreements with those obtained from the finite element analyses.

The elastic T-stress for slightly curved or kinked cracks

This work presents a solution for the elastic T-stress at the tip of a slightly curved or kinked crack based on a perturbation approach. Compared to other exact or numerical solutions the present solution is accurate for considerable deviations from straightness. The T-stress variation as crack extends along a curved trajectory is subsequently examined. It is predicted that T-stress always keeps negative during crack extension when the crack has an initial negative T-stress. In the case of a positive T-stress and non-zero first and second stress intensity factors initially accompanying the crack, the T-stress is not positive with increasing the extension length until a threshold is exceeded. Based on directional stability criterion with respect to the sign of the T-stress, this result implies that for a straight crack with a positive T-stress, the crack extension path will not turn immediately and instead keep a stable growth until a critical length is reached. This prediction is consistent with experimental observations.

Geometry effects on fracture behaviour of polymethyl methacrylate

The mixed mode I/II fracture resistance of polymethylmethacrylate (PMMA) is studied using four different test samples namely the Brazilian disc (BD), semi-circular bend (SCB), four-point bend (FPB) and diagonally loaded square plate (DLSP) specimens. The noticeable discrepancy observed between the different sets of fracture toughness data reveals the influence of specimen geometry and loading condition on the crack growth behaviour. It is shown that the upper and lower bound mixed mode fracture toughness data obtained from the BD and SCB specimens, respectively are mainly due to the effects of negative T-stress in the BD and positive T-stress in the SCB samples. The FPB and DLSP test data are also close to each other and lie between the BD and SCB test results because the T-stress is nearly zero in both FPB and DLSP specimens.

Effect of T-stress on crack growth under mixed mode I–III loading

For a crack subjected to combined mode I and III loading the influence of a T-stress is analyzed, with focus on crack growth. The solid is a ductile metal modelled as elastic–plastic, and the fracture process is represented in terms of a cohesive zone model. The analyzes are carried out for conditions of small scale yielding, with the elastic solution applied as boundary conditions on the outer edge of the region analyzed. For several combinations of the stress intensity factors K I and K III and the T-stress crack growth resistance curves are calculated numerically in order to determine the fracture toughness. In all situations it is found that a negative T-stress adds to the fracture toughness, whereas a positive T-stress has rather little effect. For given values of K I and T the minimum fracture toughness corresponds to K III = 0.

Cleavage fracture modeling of pressure vessels under transient thermo-mechanical loading

The next generation of fracture assessment procedures for nuclear reactor pressure vessels (RPVs) will combine non-linear analyses of crack front response with stochastic treatments of crack size, shape, orientation, location, material properties and thermal-pressure transients. The projected computational demands needed to support stochastic approaches with detailed 3-D, non-linear stress analyses of vessels containing defects appear well beyond current and near-term capabilities. In the interim, 2-D models become appealing to approximate certain classes of critical flaws in RPVs, and have computational demands within reach for stochastic frameworks. The present work focuses on the capability of 2-D models to provide values for the Weibull stress fracture parameter with accuracy comparable to those from very detailed 3-D models. Weibull stress approaches provide one route to connect non-linear vessel response with fracture toughness values measured using small laboratory specimens. The embedded axial flaw located in the RPV wall near the cladding-vessel interface emerges from current linear-elastic, stochastic investigations as a critical contributor to the conditional probability of initiation. Three different types of 2-D models reflecting this configuration are subjected to a thermal-pressure transient characteristic of a critical pressurized thermal shock event. The plane-strain, 2-D models include: the modified boundary layer (MBL) model, the middle tension (M(T)) model, and the 2-D RPV model. The 2-D MBL model provides a high quality estimate for the Weibull stress but only in crack front regions with a positive T-stress. For crack front locations with low constraint ( T-stress < 0), the M(T) specimen provides very accurate Weibull stress values but only for pressure load acting alone on the RPV. For RPVs under a combined thermal-pressure transient, Weibull stresses computed from the 2-D RPV model demonstrate close agreement with those computed from the corresponding crack front locations in the 3-D RPV model having large negative T-stresses. Applications of this family of 2-D models provide Weibull stress values in excellent agreement with very detailed 3-D models while retaining practical levels of computational effort.

Crack deflection in a biaxial stress state

Cotterell and Rice theory (Int J Fract 16(2):155–169, 1980) on the kinking of a crack submitted to a biaxial loading in a homogeneous material is revisited. Using both an energetic and a stress fracture criteria (Leguillon, Eur J Mech A/Solids 21:61–72, 2002) allows defining a positive threshold of the T-stress T c below which no branching can occur (Selvarathinam and Goree, Eng Fract Mech 60(5–6):543–561, 1998) provided the inhomogeneities size is small compared to the Irwin length. The absence of such a threshold would definitely condemn experimental procedures like the double-cantilever beam (DCB) or compact tension (CT) tests, which result in a positive T-stress at the crack tip. The stress intensity factors K I and T are computed using a contour integral. Calculations provide a very good agreement with the analytical results of the infinite Centrally Notched (CN) plate in tension for instance. An asymptotic analysis makes it possible to define the branching angle as a discontinuous function of T with a jump from 0° to some significant positive value as T reaches T c . Furthermore, for non vanishing K II , a similar analysis is carried out, a positive T-stress increases the kinking angle due to K II alone.

Crack paths and the problem of global directional stability

Crack paths of original Griffith crack and edge crack under biaxial remote mode-I loading after different local disturbances are calculated by using integro-differential equations of first-order perturbation and numerical simulation with FEM respectively. Considering the asymptotic behaviour for large crack lengths the problem of global directional stability is reinvestigated extending the work by Melin. For the Griffith crack, correct power functions for the asymptotic path with one or both crack tips growing have been determined. The well- known critical stress biaxiality ratio R c = 1 for the global directional stability has been obtained independently whether the crack is disturbed by local imperfections in geometry or in loading. For an edge crack the calculated critical stress biaxiality ratio for the global directional stability R c = 0.616, also irrespective of the local disturbances, corresponds to a positive T-stress and is considerably smaller than the value R > 0.95 estimated by Melin (2002). In general, cracks need not propagate asymptotically in the direction perpendicular to the largest principal stress (without crack). This is found to be due to the effect of the boundaries. Considering the initial crack growth exclusively, it is shown that the solution for crack path prediction in series expansion form as derived by Cotterell and Rice (1980) for traction-free crack faces (after correction of a misprint) is exact in the two first terms in all cases. Thus, for small crack growth the Cotterell and Rice solution is universal with respect to all loading and geometrical situations.

Effect of T-stress on fatigue crack closure in 3-D small-scale yielding

This paper investigates the effects of in-plane constraint on 3-D fatigue crack closure in the small-scale yielding regime. The finite element analyses grow a sharp, straight-through crack in a modified boundary layer model under mode I, constant amplitude cyclic loading with prescribed but independent peak values of stress intensity factor, K max, and the T-stress, T max. A purely kinematic hardening law with constant modulus represents the material constitutive behavior. The computational results demonstrate that a two parameter characterization of crack tip fields in terms of K max /σ 0 B and T max/ σ 0, where σ 0 denotes the yield stress of the material, correlates successfully the normalized opening load K op/ K max across variations of thickness ( B), constraint level and material flow properties. Both negative and positive T-stress reduce the through-thickness variation in local opening load levels along the crack front. A negative T-stress increases K op/ K max values, particularly at low peak loads where the plastic zone size remains a fraction of the thickness; a positive T-stress has limited effect on K op/ K max values. The fringe plots of individual plastic strain components reveal (a) in the absence of T-stress ( T max/ σ 0=0), plastic contraction in the thickness direction compensates primarily for permanent stretching in the direction normal to the crack plane required for closure, (b) for negative T-stress ( T max/ σ 0<0), plastic contraction in the in-plane transverse direction contributes the larger share of material flowing into the normal direction, and (c) for positive T-stress ( T max/ σ 0>0), both in-plane directions experience permanent stretching and the thickness direction alone undergoes plastic contraction.

Effect of T-Stress on Crack Growth Along an Interface Between Ductile and Elastic Solids

For crack growth along an interface joining an elastic-plastic solid to an elastic substrate the effect of a non-singular stress component in the crack growth direction in the elastic-plastic solid is investigated. Conditions of small scale yielding are assumed, and due to the mismatch of elastic properties across the interface the corresponding oscillating stress singularity fields are applied as boundary conditions on the outer edge of the region analysed. The fracture process is represented in terms of a cohesive zone model. It is shown that the interface fracture toughness is significantly increased by a negative T-stress in the elastic-plastic solid, while a positive T-stress in the elastic-plastic solid leads to a reduced fracture toughness.

Two parameter approach for elastic-plastic fracture of short cracked specimens under mixed mode loading

For mode-I loading, in order to describe the near-tip stress field in a specimen under large scaled yielding, two parameter approaches such as J-T, J-Q and J-A2 theories have been developed and proved well for their validity and limit. In this work elastic-plastic finite element analysis were performed to investigate the effects of mode mixity and T-stress upon near-tip stress distribution for a small-scale-yield model with the modified boundary layer and CTS (Compact Tension-Shear) configuration under large-scale-yield state. As the results, some peculiar characteristics were found as follows; As the mode mixity increases, normal stresses σrr and σθ θ near the crack tip in the small-scale-yield model get significantly affected by the positive T-stress as well as the negative T-stress, while the shear stress σrθ is little affected by T-stress. Also, the near-tip stress distribution of short cracked CTS specimens under the large-scale-yield state agree fairly well with that of the small-scale-yield model with an appropriate positive T-stress. The two parameters approach with J-integral and T-stress seems to be a good tool for describing the near-tip stress field under a mixed mode loading and large-scale-yield state.