Unified correlation of in‐plane and out‐of‐plane constraints with fracture toughness

In this paper, the specimens with different geometries and loading configurations were used to study the unified correlation of in‐plane and out‐of‐plane constraints with fracture toughness by using numerical simulation method. The results show that the unified constraint parameter A_p which was defined on the basis of the areas surrounded by the equivalent plastic strain isolines ahead of crack tip can characterise both in‐plane and out‐of‐plane constraints induced by different specimen geometries and loading configurations. A sole linear relation between the normalised fracture toughness J_IC/J_ref and was obtained. The J_IC/J_ref ‐ line is a unified correlation line of in‐plane and out‐of‐plane constraints with fracture toughness of a material, and the constraint dependent fracture toughness of a material can be determined from the unified correlation line. The results also demonstrate that the out‐of‐plane constraint effect is related to the in‐plane constraint effect, and there exists interaction between them. The higher in‐plane constraint strengthens the out‐of‐plane constraint effect, whereas the lower in‐plane constraint is not sensitive to the out‐of‐plane constraint effect.     

Three‐dimensional analyses of in‐plane and out‐of‐plane crack‐tip constraint characterization for fracture specimens

Three‐dimensional elastic–plastic finite element analyses have been conducted for 21 experimental specimens with different in‐plane and out‐of‐plane constraints in the literature. The distributions of five constraint parameters (namely T‐stress, Q, h, T_z and A_p) along crack fronts (specimen thickness) for the specimens were calculated. The capability and applicability of the parameters for characterizing in‐plane and out‐of‐plane crack‐tip constraints and establishing unified correlation with fracture toughness of a steel were investigated. The results show that the four constraint parameters (T‐stress, Q, h and T_z) based on crack‐tip stress fields are only sensitive to in‐plane or out‐of‐plane constraints. Therefore, the monotonic unified correlation curves with fracture toughness (toughness loci) cannot obtained by using them. The parameter A_p based on crack‐tip equivalent plastic strain is sensitive to both in‐plane and out‐of‐plane constraints, and may effectively characterize both of them. The monotonic unified correlation curves with fracture toughness can be obtained by using A_p. In structural integrity assessments, the correlation curves may be used in the failure assessment diagram (FAD) methodology for incorporating both in‐plane and out‐of‐plane constraint effects in structures for improving accuracy.     

Three‐dimensional analyses of unified characterization parameter of in‐plane and out‐of‐plane creep constraint

Based on extensive three‐dimensional finite element analyses, the unified characterization parameter A_c of in‐plane and out‐of‐plane creep constraint based on crack‐tip equivalent creep strain for three specimen geometries (C(T), SEN(T) and M(T)) were quantified for 316H steel at 550 °C and steady‐state creep. The distributions of the parameter A_c along crack fronts (specimen thickness) were calculated, and its capability and applicability for characterizing a wide range of in‐plane and out‐of‐plane creep constraints in different specimen geometries have been comparatively analysed with the constraint parameters based on crack‐tip stress fields (namely R*, h and T_Z). The results show that the parameter A_c in the centre region of all specimens appears uniform distribution and lower value (higher constraint), and in the region near free surface it shows protuberant distribution and higher value (lower constraint). The parameter A_c can simultaneously and effectively characterize a wide range of in‐plane and out‐of‐plane creep constraints, while the parameters R*, h and T_Z based on crack‐tip stress fields cannot achieve this. The different capabilities of these parameters for characterizing in‐plane and out‐of‐plane creep constraints originate from their underlying theories. The parameter A_c may be useful for accurately characterizing the overall constraint level composed of in‐plane and out‐of‐plane constraints in actual high‐temperature components, and it may be used in creep life assessments for improving accuracy.     

In‐plane and out‐of‐plane constraint parameters along a three‐dimensional crack‐front stress field under creep loading

Full‐field three‐dimensional (3D) numerical analyses was performed to determine in‐plane and out‐of‐plane constraint effect on crack‐front stress fields under creep conditions of finite thickness boundary layer models and different specimen geometries. Several parameters are used to characterize constraint effects including the non‐singular T‐stresses, the local triaxiality parameter, the T_z ‐factor of the stress‐state in a 3D cracked body and the second‐order‐term amplitude factor. The constraint parameters are determined for centre‐cracked plate, three‐point bend specimen and compact tension specimen. Discrepancies in constraint parameter distribution on the line of crack extension and along crack front depending on the thickness of the specimens have been observed under different loading conditions of creeping power law hardening material for various configurations of specimens.     

In‐plane and out‐of‐plane constraint for single edge notched bending specimen and cruciform specimen under uniaxial and biaxial loading

Considering fracture constraint is an efficient way to describe stress–strain field and fracture toughness more accurately, so it is necessary to realise the relationship with in‐plane and out‐of‐plane constraint for different standard specimens. In this paper, three‐dimensional finite element method is applied to study the in‐plane and out‐of‐plane constraint for both cruciform specimen and single edge notched bending specimen made from commercial pure titanium. Crack length and in‐plane loading as the factors affecting in‐plane constraint, and thickness as the factor affecting the out‐of‐plane constraint are used to study the effect on both in‐plane and out‐of‐plane constraint in this paper. From the results, in‐plane and out‐of‐plane constraint are both related to specimen geometries and loading styles. And there exist relationships with in‐plane and out‐of‐plane constraint because of factors for different specimens. Depending on crack length, out‐of‐plane constraint increases with in‐plane constraint. While depending on transverse loading, out‐of‐plane constraint decreases with in‐plane constraint. In addition, when the in‐plane constraint of a specimen is higher, in‐plane constraint increases with out‐of‐plane constraint (thickness). When the in‐plane constraint is lower, in‐plane constraint almost remains unchanged with out‐of‐plane constraint.     

Induced out‐of‐plane mode at the tip of blunt lateral notches and holes under in‐plane shear loading

As it is well known the Poisson's effect in a cracked plate subjected to anti‐symmetric plane loading leads to the generation of a coupled out‐of‐plane singular mode. Recent theoretical and numerical analyses have shown that this effect is present also in plates weakened by sharp V‐notches and might play a role in failure initiation phenomena of notched plates subjected to Mode II loading, especially in the presence of a large notch opening angle. Dealing with blunt notches with a large notch radius, and not just with sharp notches, the presence or not of an out‐of‐plane mode does not appear to have been systematically investigated in the past. The main aim of this work is to confirm the existence of the stress field associated with the out‐of‐plane mode (Mode O) and to describe its main features in the presence of a notch radius significantly different from zero. The analyses include U‐notches, as well as circular and elliptic holes. The strain energy density in a 3D control volume is utilized to identify the most critical zone (with respect to failure initiation) through the plate thickness at the notch tip.     

On improved crack tip plastic zone estimates based on T‐stress and on complete stress fields

Cracked ductile structures yield locally to form a plastic zone (pz) around their crack tips, which size and shape controls their structural behaviour. Classical pz estimates are based solely on stress intensity factors (SIF), but their precision is limited to very low σ_n/S_Y nominal stress to yield strength ratios. T‐stresses are frequently used to correct SIF‐based pz estimates, but both SIF and SIF plus T‐stress pz estimates are based on truncated linear elastic (LE) stress fields that do not satisfy boundary conditions. Using Griffith's plate complete LE stress field to avoid such truncated pz estimates, the influence of its Williams’ series terms on pz estimation is evaluated, showing that T‐stress improvements are limited to medium σ_n/S_Y values. Then, corrections are proposed to introduce equilibrium requirements into LE pz estimates. Finally, these improved estimates are compared with pz calculated numerically by an elastic–plastic finite element analysis.     

Fatigue crack propagation under in‐phase and out‐of‐phase biaxial loading

Fatigue damage characteristics of aluminium alloy under complex biaxial loads such as in‐phase and out‐of‐phase loading conditions and different biaxiality ratios have been investigated. The effects of microscale phenomena on macroscale crack growth were studied to develop an in‐depth understanding of crack nucleation and growth. Material characterization was conducted to study the microstructure variability. Scanning electron microscopy was used to identify the second phase particles, and energy dispersive X‐ray spectroscopy was performed to analyse their phases and elements. Extensive quasi‐static and fatigue tests were conducted on Al7075‐T651 cruciform specimens over a wide range of load ratios and phases. Detailed fractography analysis was conducted to understand the crack growth behaviour observed during the fatigue tests. Significant differences in crack initiation and propagation behaviour were observed when a phase difference was applied. Primarily, crack retardation and splitting were observed because of the constantly varying mode mixity caused by phase difference. The crack growth behaviour and fatigue lives under out‐of‐phase loading were compared with those under in‐phase loading to understand the effect of mixed‐mode fracture.     

Critical buckling load of paper honeycomb under out‐of‐plane pressure

Two out‐of‐plane buckling criteria for paper honeycomb are proposed by analysing the structure properties and the collapse mechanism of paper honeycomb: these are based on the peeling strength and ring crush strength of the chipboard wall. Taking into account the orthotropic, initial deflection and large deflection properties of the chipboard wall, the two new mechanical models and the calculation methods are developed to represent the out‐of‐plane critical load of paper honeycomb. Theoretical calculations and test results show that the models are suitable for describing the collapse mechanism of paper honeycomb. The peeling strength and ring crush strength determine the critical buckling load of paper honeycomb in different stretch phases. The out‐of‐plane critical buckling load can be predicted when the two models are integrated. Copyright © 2005 John Wiley & Sons, Ltd.     

Reduction of measured toughness due to out‐of‐plane constraint in ductile fracture of aluminium alloy specimens

It is generally believed that a lower bound on the fracture toughness of a material is obtained from a standard test, particularly in metals where yielding occurs prior to fracture. The understanding is that in such a test the material around the crack tip is highly constrained hence reducing the extent of yielding. In this paper, we report the results of fracture tests where a tensile load is applied to a biaxial aluminium alloy specimen in the direction parallel to the crack front in addition to the fracturing load normal to the crack surface. We show that in this case a lower fracture toughness is measured than that obtained from a standard test. Indeed, for the highest value of tensile load used in our tests the J‐integral at fracture was half the value measured in a standard test. It is also shown that the volume of the plastic region can be used to measure the effect of constraint, irrespective of the manner in which the constraint arises. This approach suggests an even lower fracture toughness may be obtained than that measured here in certain loading conditions.     

The effect of T‐stress on crack‐tip plastic zones under mixed‐mode loading conditions

In this paper, the influence of T‐stress on crack‐tip plastic zones under mixed‐mode I and II loading conditions is examined. The crack‐tip stress field is defined in terms of the mixed‐mode stress intensity factors and the T‐stress using William's series expansion. The crack‐tip stress field is incorporated into the Von Mises yield criteria to develop an expression that determines the crack‐tip plastic zone. Using the resultant expression, the plastic zone is plotted for various combinations of mode II to mode I stress intensity factor ratios and levels of T‐stress. The properties of the plastic zone affected by T‐stress and mixed‐mode phase angle are discussed. The observations obtained on plastic zones variations are important for further fatigue and fracture analyses for defects in engineering structures under mixed‐mode loading conditions.     

Structure design of reinforced honeycomb paperboard core and characterization of its out‐of‐plane bearing strength

Honeycomb paperboard's out‐of‐plane bearing performance is one of the important properties in packaging field application. Further improvement of its bearing performance has important value in engineering practice. In this paper, a honeycomb core structure was designed, and the bonding dimension and manufacturing process were designed. The mechanism of out‐of‐plane quasi‐static compression deformation of reinforced honeycomb paperboard was analyzed by experiments. The theoretical model of out‐of‐plane platform stress was constructed by applying the plastic deformation, plastic energy dissipation and energy conservation theory. The results show that the improved structure can be mechanically bonded in a flat state with less technological changes. Under the same honeycomb core material and core size parameters, the bearing strength of the improved structure increases by an average of 3.9 times to conventional structure. In order to meet the same compressive strength requirement, the improved structure can reduce the performance requirements of honeycomb core material or increase the core size compared with the conventional structure. When the honeycomb core cell is larger, the tension on the core layer required for the production process is reduced. The theoretical and experimental data are in good agreement with each other, and the relative errors are all less than 13%.     

Effect of constraint on void growth near a blunt crack tip

The growth of a single cylindrical hole ahead of a blunt crack tip was studied using large deformation finite element analysis in three-point bend specimens with different precrack depth. The effect of small second phase particles was taken into account by incorporating Gurson's constitutive equation. The effects of strain hardening and the initial distance from the hole to the crack tip were also investigated. The results show that the variation of crack tip opening displacement with load is not sensitive to constraint level. The effects of constraint on the growth of hole and ductile initiation toughness are diminished with decreasing initial distance from the hole to the blunt crack tip. This revised version was published online in August 2006 with corrections to the Cover Date.     

Investigation of the seismic out‐of‐plane behaviour of unreinforced masonry walls

The structural stability of unreinforced masonry (URM) walls has to be guaranteed not only under static (permanent and live) loads but also under earthquake loads. Loads transverse to the plane (out‐of‐plane) often have a decisive influence on the load‐bearing capacity. In practical applications, simplified methods from codes, guidelines and literature are often used to analyse and evaluate the out‐of‐plane capacity of load‐bearing and non‐load‐bearing URM walls. The results of these simplified methods can be significantly conservative and inaccurate since essential influencing effects are neglected. For many existing buildings, the simplified methods underestimate the capacity, which leads to cost‐intensive retrofitting and strengthening measures or complete replacement by other wall systems. In order to realistically estimate the out‐of‐plane capacity, parameters such as wall geometry, boundary conditions, vertical loads and especially dynamic effects (e.g. inertia forces) have to be taken into account. In this paper, non‐linear time history simulations are presented to investigate the influence of these effects. The numerically determined maximum acceptable earthquake acceleration is compared with results from simplified analysis models. The comparison shows that the out‐of‐plane capacity is significantly higher than the values predicted by simplified models. Finally, several initial experimental seismic tests conducted on the shaking table of the TU Kaiserslautern are presented, together with the planned extensive experimental test program on the out‐of‐plane capacity of masonry walls.     

Predicting lower bound damage curves for high‐strength low‐alloy steels

Hindered by the distinctive toughness requirements of the current European standards, the high‐strength low‐alloy (HSLA) steels are rarely applied to the pressure vessels industry. The reason is that the design rules specified by the standards define local plastic deformation as limit state. This results in an over‐conservative application of materials. To achieve an effective, economical and energy‐efficient use of HSLA steels, a strain‐based criterion, the damage curve, which considers crack initiation instead of the beginning of plastic deformation as limit state, is proposed in this study for the improved design rules. In the view of the interaction of microstructure and mechanical properties of materials, the new design rule is derived on the basis of the correlation of microstructural features of HSLA steels with the micromechanical damage models. The experimental verification of the result is furthermore investigated with sufficient agreement so that the general applicability of the procedure can be expected. However, further studies for a reliable parameter calibration are necessary.     

Fracture toughness characterisation of a glass fibre‐reinforced plastic composite

By examining the state of the art, it can be realised that few research works have been done on the fracture behaviour of plastic composites reinforced with continuous glass fibres. Therefore, the present paper deals with the fracture toughness of a unidirectional glass fibre‐reinforced plastic (GFRP), such a parameter being analytically determined by means of the modified two‐parameter model (MTPM). The input data of the MTPM are obtained from an experimental campaign related to three‐point bending tests on single edge‐notched specimens characterised by different sizes. The novelty of this research work is that the MTPM, originally proposed for isotropic materials, is here employed to estimate the fracture toughness of GFRPs characterised by orthotropic mechanical properties.     

Higher‐order beam analysis of box beams connected at angled joints subject to out‐of‐plane bending and torsion

The difficulty in the analysis of thin‐walled beams by a beam theory comes from slowly decaying end effects associated with warping and distortion. However, a beam theory without considering such effects yields inaccurate solutions especially near beam ends. Numerical analysis using a higher‐order beam theory capable of representing such effects is now available, but the analysis of a series of box beams connected by angled joints still remains an unsolved problem because of the lack of a matching condition at the joint. The objectives of this investigation are to develop a field‐variable‐matching technique at an angled joint through a higher‐order beam theory and to implement it in the finite element formulation. Thin‐walled box beams in consideration are assumed to be subject to out‐of‐plane bending and torsion. Thus, the minimization of three‐dimensional displacement mismatch is used to relate the field variables at a joint intersection. The minimization condition turns out to represent coupling effects of different deformation kinematics such as torsion, bending, distortion and warping. Point‐wise displacement matching is not possible with a higher‐order beam theory. The validity of the proposed technique was verified by a finite element analysis using two‐node higher‐order beam elements applied to some benchmark problems. Copyright © 2008 John Wiley & Sons, Ltd.     

The effect of T-stress on plane strain dynamic crack growth in elastic–plastic materials

In this paper, the effects of T‐stress on steady, dynamic crack growth in an elastic–plastic material are examined using a modified boundary layer formulation. The analyses are carried out under mode I, plane strain conditions by employing a special finite element procedure based on moving crack tip coordinates. The material is assumed to obey the J₂ flow theory of plasticity with isotropic power law hardening. The results show that the crack opening profile as well as the opening stress at a finite distance from the tip are strongly affected by the magnitude and sign of the T‐stress at any given crack speed. Further, it is found that the fracture toughness predicted by the analyses enhances significantly with negative T‐stress for both ductile and cleavage mode of crack growth.     

Impact of compressive hold on out‐of‐phase thermomechanical fatigue crack growth in IN 718

Accurate characterization and understanding of the fatigue crack growth behaviour of components in jet turbine engines is critical for successfully using a damage tolerant design method to maximise safety and efficiency. The hot section components experience changing loads and temperatures, and hence, fatigue crack growth rates are typically studied under thermomechanical loading. One question that remains unclear is the role of the compressive holds that are often part of an aircraft loading‐temperature spectrum. This experimental study was undertaken to investigate a turbine disk alloy, Inconel 718, subjected to different cycling and temperature profiles considering different lengths of hot compressive holds to determine its effect on the fatigue crack growth rate. It was found that the addition of a compressive hold at temperatures from 650 to 725 °C has no significant impact on the fatigue crack growth rate when compared with a cycle without a compressive hold. Fractographic analysis shows that crack growth is primarily transgranular in all cases studied suggesting that grain boundary oxidation, often observed during hot tensile holds, is insignificant.     

Prediction of crack initiation plane direction in high‐cycle multiaxial fatigue with in‐phase and out‐of‐phase loading

A new method for predicting crack plane direction in high‐cycle multiaxial fatigue is proposed. This method considers material properties and loading conditions. Two situations are considered: (i) in‐phase loading, where the crack plane direction only depends on the loading condition and material properties have little influence on it, and (ii) out‐of‐phase loading, where the crack plane direction is affected by both loading conditions and material properties. The prediction accuracy is assessed by comparison with several experimental results, including different loading conditions and materials. The results show that the proposed method provides a good prediction capability for these experiments.