Shape Of Plastic Zone Research Articles

A Numerical Estimation of Fatigue Crack Tip Plastic Zone Size and Shape under Mixed-Mode Loading Condition

A 2D elastic-plastic FEM simulation of growing fatigue crack under combined mode I and II loading was performed. An inclined fatigue crack propagated in a sheet of an Al-alloy D16CT1. The effect of increasing mode-mixity on cyclic zone size, shape and the amount of dissipated energy was investigated.

3D effects around notch and crack tips

Notches and cracks are usually treated as two-dimensional problems in most structural design and analysis applications, employing 2D limit solutions from plane elasticity theories to evaluate highly localized stress/strain concentration effects around their tips. However, such effects are also associated to high stress gradients, which cause three-dimensional stress/strain fields around those tips that can severely restrict local Poisson-induced transversal strains. Modeling of fatigue crack initiation and propagation, estimation of plastic zone sizes and shapes, and localized constraint effects are typical problems affected by such 3D effects, which may lead to non-conservative damage and life predictions if neglected. To quantify how important they can be in practical applications, first traditional finite element techniques are used to simulate effects of component thickness and notch tip radius in the stress and strain fields along and around such tips, and to evaluate their importance from the structural design point of view. Then, versatile sub-modeling techniques are used to study similar effects along the fronts of short and long cracks. Finally, a stepwise remeshing routine is used to show how an initially straight crack must slightly curve its front during its propagation by fatigue, due to the unavoidable 3D effects that always surround real crack tips.

Numerical Investigation on LEFM Limits under LSY Conditions

In this work, a parametric numerical investigation on the plastic zone size and shape at the crack front of a through crack in a plate is presented. The thickness of the plate, the size of the crack and the applied remote load values are considered as parameters. The obtained results allow assessing the limits of the linear elastic fracture mechanics in presence of an extensive material yielding at the crack front and the relationship between the plastic zone size and other parameters of the elastic-plastic fracture mechanics theory is highlighted.

Solutions to the problem of thermo-mechanical yielding in functionally graded material cylindrical vessels

Generally, in functionally graded material structures, a ceramic material as reinforcing part is distributed in a metallic matrix. These structures benefit the merits of both their metallic and ceramic parts such as the mechanical strength and thermal resistance. An important aspect in the function and study of functionally graded material structures is the effect of temperature. Usually, the resulted thermal stresses are so high that partial failure of structure is inevitable. In this paper, by using of different exact and approximate mathematical models and solution techniques, the yielding of a functionally graded material cylindrical pressure vessel is studied. Different solution techniques such as closed form solution, FE solution of ABAQUS and semi-analytical solution of variable material properties are used. Based on the von Mises and Tresca’s yield criteria, the shape of plastic zone are obtained and compared. The influences of different structural parameters as well as the effect of failure criterion upon the results are studied. Accordingly, some justifications are made regarding the selection of different material parameters or failure models.

Theoretical prediction of the potential plastic zone of shallow tunneling in vicinity of pile foundation in soils

It is often unavoidable for metro tunnel constructions to disturb and be affected by nearby existing structure foundations. In order to predict and control the interactions between the tunnel construction and the mechanical behavior of the existent structures, it is necessary to evaluate the degree and extent of tunneling disturbance to the ground, especially the shape and range of the potential plastic zone in tunneling-induced secondary stress field. In this regard, existing theoretical solutions are in general all based upon the premise that the ground is free from structure foundation loads. The present study, assuming homogeneous and isotropic ideal elasto-plastic soil, presents a theoretical procedure for prediction of the secondary stress field and the related potential plastic zone caused by tunneling in the vicinity of pile foundations. The procedure involves superimposing the analytical solution for tunneling in greenfield with the analytical solution for pile foundation loads. Calculations are conducted as examples to compare the theoretical results with corresponding numerical simulations and to demonstrate the effects of the position, magnitude and pattern of pile foundation loads on tunneling induced plastic zones. The calculations reveal that the proposed theoretical calculation procedure produces similar results as from numerical simulations, and that pile foundation loads of sufficient magnitude, if close enough to the tunnel, may significantly alter the potential plastic zones caused by nearby tunneling.

A NUMERICAL ESTIMATION OF THE EFFECT OF A BI-MATERIAL INTERFACE ON CRACK TIP PLASTIC ZONE SIZE AND SHAPE

Fatigue crack propagation in ferrite (bainite) near the austenitic cladding was modelled by means of the elastic-plastic finite element method and the effect of the bi-material interface on the cyclic plastic zone size and shape was investigated. It was demonstrated that cyclic plastic strains in softer austenite are already present at the moment when the cyclic plastic zone around the crack tip in harder ferrite reaches the interface. Crack tip plastic zones under plane stress and plane strain conditions vary in size and shape and are affected by the approaching material interface. The level of dissipated energy in front of the fatigue crack and, therefore, the crack rate increased as the boundary of the cyclic plastic zone reached the material interface. th�� lyp�|��umposed of a-Mg solid solution and interdendritic intermetallic phases. Hardness and tensile strength increased with incresing the total amount of alloying elements. Moreover, rare earths elements showed strong strengthening effects. The best corrosion resistance in the simulated physiological solution was observed in the case of alloys containing rare earts elements. n lang]G>0}`�x/or across banded distribution of eutectic particles and across bands of solid solution matrix.

Crack Tip Parameters Under Large Scale Yielding Condition

In recent years, the study of the behaviour of damaged structures has been focusing on cracked components in presence of an extensive material yielding at the crack tip; under this condition, linear elastic fracture mechanics theory is not able to describe the real stress-strain state at the crack tip and consequently either the static or the fatigue behaviour of the component. In this work, an extensive parametric numerical analysis of the plastic zone size and shape at the crack tip for a through cracked plate under Mode I loading condition is presented. The obtained results allow assessing the limits of the linear elastic fracture mechanics in presence of an extensive material yielding at the crack tip and the relationships between the plastic zone size and other parameters of the elastic plastic fracture mechanics theory are pointed out.

Cohesive zone modeling of ductile tearing process in brazed joints

Ductile tearing process in low carbon steel brazed joints with copper filler metal is studied using the Cohesive Zone Model. The Cohesive Zone Model is characterized by two parameters related to the cohesive strength and cohesive energy. A new method is developed to estimate the Cohesive Zone Model parameters through direct experimental measurements and numerical modeling. The cohesive energy of the brazed joints is obtained from the fracture test results performed on the single-edge notched bend specimens. Based on the obtained cohesive energy parameter, the fracture test is simulated using a cohesive zone finite element model. A unique value for the joint cohesive strength is determined by best fitting the finite element results to the experimental load-crack mouth opening displacement curves. The merit of the characterized Cohesive Zone Model is explored by finite element modeling of the tensile test performed on the single-edge notched tension specimens. The load-crack mouth opening displacement curve obtained from the finite element modeling conforms to the corresponding tensile test results. Furthermore, the effect of the crack tip stress constraint on the plastic zone shape and size is successfully captured by the model. The good agreement between the finite element simulation results and the experimental data demonstrates the applicability of the Cohesive Zone Model for fracture analysis of the brazed joints.

Triangular dislocation loop model for indented material

We present triangular dislocation loop (TDL) model to describe the pile-up and the residual stress field caused by Berkovich indent according to Mura’s method. Various regular polygonal loops are derived from analytical solution of angular dislocation, and correction terms necessary for the exact displacement are provided. The elastic field by different numbers of TDLs is similar, but influenced by the plastic zone shape when the slope of indenter is ideally constant. The size dependent hardening behavior among the similar internal stress field is discussed from the view points of the dislocation density and the geometry of plastic zone. Prediction of pileup displacement shows good agreement with reported values when hemi-spherical type dislocation arrangement is assumed for plastic zone.

Quantitative measurement of plastic strain field at a fatigue crack tip

A novel approach is introduced to map the mesoscale plastic strain distribution resulting from heterogeneous plastic deformation in complex loading and component geometries, by applying the discrete Fourier transform (DFT) to backscattered electron (BSE) images of polycrystalline patches. These DFTs are then calibrated against the full width at half the maximum of the central peak of the DFTs collected from the same material tested under in situ scanning electron microscopy uniaxial tensile conditions, which indicates a close relationship with the global tensile strain. In this work, the technique is demonstrated by measuring the residual strain distribution and plastic zone size around a fatigue crack tip in a commercially pure titanium compact tension specimen, by collecting BSE images in a 15×15 array of 115 μm square images around the fatigue crack tip. The measurement results show good agreement with the plastic zone size and shape measured using thermoelastic stress analysis.

Crack tip fields in rubber‐toughened polymers

AbstractNumerical simulations of the formation and the characteristic shape of plastic zones at crack tips in rubber‐toughened polymers are performed in order to get a better understanding of the underlying micromechanisms. Complementing previous work, the present study focuses on the contribution of distributed crazing. A macroscopic constitutive model therefore has been developed that describes the overall effect of distributed crazing as an anisotropic flow process. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the plastic zone size and crack tip opening displacement of a sub-interface crack in an infinite bi-material plate

Elastic–plastic stress analysis has been carried out for the plastic zone size and crack tip opening displacement of a sub-interface crack with small scale yielding. In our study, the shape of plastic zone is assumed as a long, slim strip at both crack tips. In the plastic zone, both normal stress and shear stress exist and are considered due to the bi-material interface. The values of the plastic zone size, normal stress and shear stress are determined by satisfying the conditions where both Modes I and II stress intensity factors vanish and Von Mises yield criterion is met. In the present paper, the sub-interface crack is simulated by continuously distributed dislocations which will result in singular integral equations. Those singular integral equations can be solved by reducing them to a set of linear equations. The values of the plastic zone size and crack tip opening displacement are obtained through an iterative procedure. Finally, the effect of normalized loading, normalized crack depth (distance to the interface) and Dundurs’ parameters on the normalized plastic zone size and the normalized crack tip opening displacement is discussed.

Crack retardation mechanism due to overload in base material and laser welds of Al alloys

To determine the retardation mechanisms due to overload and to predict the subsequent evolution of crack growth rate, investigations are conducted on crack retardation caused by single tensile overloads in base material and laser-welded sheets of AA6056-T6 Al alloy. The effect of the overload ratio on the fatigue crack propagation behaviour of the C(T) 100 specimens was analysed by using experimental and Finite Element (FE) methods. The crack growth rate and fracture surface features were investigated for both base material and laser-welded sheets. The retardation due to overload is described in terms of the affected regions in front of the crack tip. The size and shape of the crack-tip plastic zone and the damage profile induced during the application of the overload in the base material are predicted by FE analysis in conjunction with a porous-metal plasticity model. The results show that the mechanisms of retardation in under-matched welds are substantially different from that of the homogenous base material. More significant crack retardation due to overload has been observed in the laser weld of AA6056-T6. Based on SEM observations of the fracture surfaces and the damage profiles predicted by the proposed FE model, the shape of the crack front formed during the overload application can be predicted. During the overload, the crack front extends into a new shape, which can be predicted by the ductile damage model; a higher load results in a more curved crack front. These outcomes are used to determine the dominant retardation mechanisms and the significance of retardation observed in each region ahead of the crack tip and finally to define the minimum crack growth rate after overload.

Effect of secondary orientation on notch-tip plasticity in superalloy single crystals

Numerical and experimental evolutions of slip fields in notched Ni-Base Single Crystal superalloy tensile specimens are presented as a function of secondary crystallographic orientation. The numerical predictions based on three-dimensional anisotropic elasticity and crystal plasticity are compared with experimental observations. The results illustrate the strong dependence of the slip patterns and the plastic zone size and shape on the secondary orientation of notches, which can have important consequences on crack initiation. Specific orientations or non-symmetric notch geometries lead to non-symmetric patterns on both sides of the sample. The computations show that strongly different plastic zones are expected in the core of the sample and at free surfaces. The ability of the anisotropic elastic model to anticipate the plastic domains, based on identifying dominant slip systems, is confirmed by the crystal plasticity computations, at low load levels. An important observation is that kink shear banding is a real deformation mode operating at crack tips and notches in high strength nickel-based single crystal superalloys for specific orientations.

Mixed-mode (I/II) crack initiation direction for elastic-plastic materials based on crack-tip plastic zone

In this paper, crack-initiation direction for an elastic-plastic material (interstitial free steel) under mixed-mode (I/II) load is studied based on crack-tip plastic zone under monotonic load. The feasibility of using the minimum plastic zone radius (MPZR) criterion for elastic-plastic materials is also examined. The shape and size of crack-tip plastic zones have been estimated by elastic-plastic finite element analysis in a plane stress single edge notch tensile (SENT) specimen under mixed-mode (I/II) loading according to von Mises yield criteria. The results show that the minimum plastic zone sizes were not clearly identified; indicating MPZR criterion is not suitable for the analysis of elastic-plastic materials. It is shown that the crack initiation direction for elastic-plastic materials depends on the path of the maximum plastic zone size. The variation of crack initiation angle is also studied with respect to mode mixity factor and the results are found to be in good agreement with the experimental results available in the literature.

Continuum crystal plasticity analyses of the plastic flow features underneath single-crystal indentations

Continuum crystal plasticity finite element simulations are performed for archetypal pure and alloyed fcc crystals to investigate the role of the crystalline orientation, hardening response and dislocation interactions on the plastic flow patterns developing underneath spherical and pyramidal indenter tips. Following our prior analyses, the orientation of plastic features such as subsurface lobes and surface rosettes goes along that of specific in-plane and out-of-plane slip systems. Interestingly, however, we currently show that the activity of the closely oriented slip systems in such lobes and rosettes is, in general, unaccountable to their development. In highly symmetric (001), (011) and (111) indentations, it is found that the slip systems with a net out-of-plane slip direction may contribute to rosette formation at the surface, whereas in-plane slip directions lead to lobe formation in the subsurface. The present results also show that while the isocontours of maximum shear stress τmax from anisotropic elasticity analyses indeed provide an indication of the indentation-induced elastic field, it is the projection of the stress tensor in all slip systems that drives lobe formation. The isocontours of τmax may not therefore dictate the plastic zone shape, even though they are useful in explaining some of its features. Finally, a conical shear band shape is found to develop immediately underneath the imprint, dictating accumulation of shear strains and their spreading towards the thickness of the crystal. This feature varies depending on crystal orientation, hardening response and on whether or not the cross-section under analysis contains normal slip directions.

3D finite element analysis on crack-tip plastic zone

This paper presents an effect of specimen thickness on crack-tip plastic zone shape and size. The shape of the plastic zone is estimated in 3D with the help of elastic and elastic-plastic finite element analyses. The results indicate that the shape of plastic zone does not correspond to classical “dog-bone shape” and the maximum plastic zone size occurs beneath the free surface, contrary to conventional shape. It is observed that for specimens with B/W=0.5, the ASTM requirement of specimen thickness to estimate KIC, the plastic zone shape and size is much larger than the one obtained in 2D plane strain analysis. The results also show a significant change in the shape and size of plastic zone in elastic and elastic plastic conditions. International Journal of Engineering, Science and Technology, Vol. 2, No. 6, 2010, pp. 47-58

Fatigue crack propagation law measured from integrated digital image correlation: the example of Ti35 thin sheets

In this study, a crack in commercially-pure Titanium (CP-Ti) is observed with optical means during its propagation in fatigue. Digital image correlation is then used to extract directly, with no post-processing step, the crack tip location and the corresponding stress intensity factor amplitude to evaluate the parameters of Paris’ law, the T-stress component and an estimate of the plastic zone shape.

Analysis of crack-tip plastic zone in a Compact Tensile Shear (CTS) Specimen

The minimum plastic zone radius (MPZR) criterion is one of the recent criterions to estimate crack initiation direction under mixed mode loading. The MPZR theory is based on the theoretical computations of plastic zone size (PZS). In this investigation, the shape and size of crack-tip plastic zones have been estimated by analytical and the elastic finite element computations in a Compact Tensile Shear (CTS) specimen under mixed mode (I/II) loading according to von Mises yield criteria. The theoretical and the finite element analysis results are compared to analyze the minimum plastic zone radius (MPZR) criterion for crack-initiation angle with reference to the loading angle and stress intensity factor.

Analytic solutions to crack tip plastic zone under various loading conditions

Based on stress field equations and Hill yield criterion, the crack tip plastic zone is determined for orthotropic materials and isotropic materials under small-scale yielding condition. An analytical solution to calculating the crack tip plastic zone in plane stress states is presented. The shape and size of the plastic zone are analyzed under different loading conditions. The obtained results show that the crack tip plastic zones present “butterfly-like” shapes, and the elastic–plastic boundary is smooth. The size of the plastic zone for orthotropic composites is less at the crack tip for various loading conditions, compared with the case of isotropic materials. Crack inclination angle and loading conditions affect greatly the size and shape of crack tip plastic zone. The mode I crack has a crucial effect on the plastic zone for mixed mode case in plane stress state. The plastic zone for pure mode I crack and pure mode II crack have a symmetrical distribution to the initial crack plane.