Unified characterisation of in‐plane and out‐of‐plane constraint based on crack‐tip equivalent plastic strain

Constraint can be divided into two conditions of in‐plane and out‐of‐plane, and each of them has its own parameter to characterize. However, in most cases, there exists a compound change of both in‐plane and out‐of‐plane constraint in structures, a unified measure that can reflect both of them is needed. In this paper, the finite element method (FEM) was used to calculate the equivalent plastic strain (ɛ_p) distribution ahead of crack tips for specimens with different in‐plane and out‐of‐plane constraints, and the FEM simulations based on Gurson–Tvergaard–Needleman (GTN) damage model and a small number of tests were used to obtain fracture toughness for the specimens with different constraints. Unified measure and characterisation parameter of in‐plane and out‐of‐plane constraints based on crack‐tip equivalent plastic strain has been investigated. The results show that the area A_PEEQ surrounded by the ɛ_p isoline ahead of crack tips can characterize both in‐plane and out‐of‐plane constraints. Based on the area A_PEEQ, a unified constraint characterisation parameter A_p was defined. It was found that there exists a sole linear relation between the normalised fracture toughness J_IC/J_ref and regardless of the in‐plane constraint, out‐of‐plane constraint and the selection of the ɛ_p isolines. The unified J_IC/J_ref−reference line can be used to determine constraint‐dependent fracture toughness of materials. The FEM simulations with the GTN damage model (local approach) can be used in obtaining the unified J_IC/J_ref−reference line for materials with ductile fracture.     

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.     

A three‐dimensional stress field solution for pointed and sharply radiused V‐notches in plates of finite thickness

By making use of the generalized plane strain hypothesis, an approximate stress field theory has been developed according to which the three‐dimensional governing equations lead to a system where a bi‐harmonic equation and a harmonic equation should be simultaneously satisfied. The former provides the solution of the corresponding plane notch problem, and the latter provides the solution of the corresponding out‐of‐plane shear notch problem. The system can be applied not only to pointed three‐dimensional V‐notches but also to sharply radiused V‐notches characterized by a notch tip radius small enough. Limits and degree of accuracy of the analytical frame are discussed comparing theoretical results and numerical data from FE models.     

Generalized approach to estimation of strains and stresses at blunt V‐notches under non‐localized creep

Geometrical discontinuities such as notches need to be carefully analysed by engineers because of the stress concentration generated by them. Notches become even more important when the component is subjected, in service, to very severe conditions, such as high‐temperature fatigue and imposed viscoplastic behaviour such as creep. The knowledge of strains and stresses in such stress concentration zones is essential for an efficient and safe design process. The aim of the paper is to present an improvement and extension of the existing notch‐tip creep stress–strain analysis method developed by Nuñez and Glinka, validated for U‐notches only, to a wide variety of blunt V‐notches. The key in obtaining the extension to blunt V‐notches is the substitution of the Creager–Paris equations with the more generalized Lazzarin–Tovo solution, allowing a unified approach to the evaluation of linear elastic stress fields in the neighbourhood of both cracks and notches. Numerous examples have been analysed to date, and the stress fields obtained according to the proposed method were compared with appropriate finite element data, resulting in a very good agreement. In view of the promising results discussed in the paper, authors are considering possible further extension to sharp V‐notches and cracks introducing the concept of the strain energy density.     

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.     

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.     

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.     

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%.     

Fatigue thresholds of holed components under in‐phase and out‐of‐phase torsional and axial loadings

The resistance‐curve (R‐curve) method was applied to the prediction of the fatigue thresholds of notched components under in‐phase and out‐of‐phase combinations of cyclic torsion and axial loadings. The prediction was compared with the experimental data obtained from thin‐walled tubular specimen of medium‐carbon steel with a hole. The stress was completely reversed and the mean stress was zero. The crack was nucleated at the position of the maximum range of the circumferential stress on the periphery of a hole, and propagated almost straight for all cases examined. The experimental data of the thresholds for crack initiation and fracture agreed well with the predictions for in‐phase and for out‐of‐phase loadings with 45° phase difference. For out‐of‐phase loading with 90°, the threshold for fracture was close to the crack initiation limit, because of the reduction of crack closure due to crack face rubbing by mode II shear cycling.     

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.     

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.     

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.     

Stress singularity analysis for orthotropic V‐notches in the generalised plane strain state

The singularity for the V‐notch under the generalised plane deformation is investigated by the combination of the asymptotic analysis with the interpolating matrix method developed by part of the authors before. The displacement asymptotic expansions at the vicinity of the V‐notch vertex are introduced into the equilibrium equations, which are transformed into a set of characteristic ordinary differential equations with respect to the notch singularity orders. The boundary conditions and interfacial compatibility conditions are also represented by the combination of the singularity orders and characteristic angular functions. The determination of the singularity orders and characteristic angular functions are transformed into solving the ordinary differential equations with variable coefficients, which are solved by the interpolating matrix method. The present method is suitable for the singularity analysis for isotropic and orthotropic V‐notches. It is versatile for analysing the stress singularity of single material V‐notches, bi‐material V‐notches, interface edges and cracks. The correctness of the results by the proposed method is ensured by the comparison with the published ones.     

Analytical study of the elastic–plastic stress fields ahead of parabolic notches under antiplane shear loading

An analytical study is carried out on the elastic–plastic stress and strain distributions and on the shape of the plastic zone ahead of parabolic notches under antiplane shear loading and small scale yielding. The material is thought of as obeying an elastic-perfectly-plastic or a strain hardening law. When the notch root radius becomes zero, the analytical frame matches the solutions for the crack case due to Hult–McClintock (elastic-perfectly-plastic material) and Rice (strain hardening material). The analytical frame provides an explicit link between the plastic stress and the elastic stress at the notch tip. Neuber’solution for blunt notches under antiplane shear is also obtained and the conditions under which such a solution is valid are discussed in detail by using elastic and plastic notch stress intensity factors. Finally, revisiting Glinka and Molski’s equivalent strain energy density (ESED), these factors are used also to give, under antiplane shear loading, the increment of the strain energy at the notch tip with respect to the linear elastic case.     

Fracture analysis of functionally graded materials with U‐ and V‐notches under mode I loading using the averaged strain‐energy density criterion

One of the most powerful criteria to predict the critical fracture load in plates with notches is the strain‐energy density averaged over a well‐defined control volume ahead of the notch tip. Although the averaged strain‐energy density (ASED) criterion has been proposed for homogeneous materials, it has been shown in this paper that this criterion can also be applied for non‐homogeneous materials, especially for functionally graded materials (FGMs). A numerical method has been used to evaluate the control volume boundary, the averaged strain‐energy density over the control volume, and also the critical fracture load in FGMs under mode I loading. A new set of experimental results on fracture of blunt V‐notched samples made of austenitic–martensitic functionally graded steel under mode I loading have been provided.     

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.     

Experimental Analysis of the folding force in thin wall extruded cells under quasi‐static in‐plane loading

In most of the engineering structures, specially moving ones and generally in structures under dynamic and static loadings, energy absorber systems are implemented for preventing or reducing damages. These systems are employed in all on‐road vehicles, train wagons, airplanes and ships. Energy absorbers are subjected to two forms of in‐plane and out‐of‐plane loadings. In this paper, the effects of geometrical parameters such as thickness and height of structure and mechanical parameters such as yield stress in the cell structure are investigated. The effect of changing boundary conditions on the folding force is also investigated. According to the results, the energy absorbed by the cell, directly relates to the increase in the number of rows; the main reason of this increase is the increase in shared walls. Moreover, doubling the wall thickness has resulted in a 4.7 times higher energy absorption. In addition, the accuracy of different analytical models is compared, and the model with most precise predictions is introduced.