The effects of triaxial stress on void growth and yield equations of power-hardening porous materials

In engineering applications, especially for ductile fracture of materials, nucleation, growth and coalescence of voids have often been observed. Currently there is an increase in interest for the effects of voids on the behaviour of engineering materials. In this paper, by the method of combining micro- and macro-parameters, the effects of triaxial stress on the rates of void growth and yield equations are presented for porous materials with power-hardening. The relations between triaxial stress and the rates of void growth for different n-values and yield equations with different n-values and void volume fractions are discussed. Following results have been obtained: For a porous material with power-hardening, the yield equation can be approximately expressed by an elliptical equation in equivalent stress and triaxial stress. Both the long half-axis and the short half-axis of the elliptical equation are functions of the void volume fraction for a given hardening exponent. The triaxial stress has a strong effect on the growth rates of voids. For linear hardening materials, the relation between the growth rate of voids and the triaxial stress is linear. For elastic/perfectly plastic materials with a small void volume fraction, the growth rate of voids can be described in relation to the triaxial stress with an exponential function. The results from this paper are compared with theoretical results from other researchers for elastic/perfectly plastic materials. A good agreement is shown.     

Numerical Simulation of Ductile Fracture Based on Local Approach Concept

In this paper a finite-element analysis on ductile fracture in two-dimensional quasi-static state is performed by using the local approach concept in continuum damage mechanics. An isotropic damage model based on the generalized concept of effective stress is proposed. Crack propagation is achieved by removing critically damaged elements. The finite-element approximation of a largely deforming body based on the incremental total Lagrangian concept is carried out. As numerical examples, the mesh size sensitivity analysis and the simulation of the shearing mode failure in plane strain state are carried out to verify the present formulation qualitatively. For an edge cracked plate under plane stress state, load-displacement curves and successively fractured shapes are shown. It can be concluded that the proposed model may be stated as a reasonable tool to explain ductile fracture initiation and crack propagation.     

Explicit Finite-Element Analysis of 2024-T3/T351 Aluminum Material under Impact Loading for Airplane Engine Containment and Fragment Shielding

Uncontained aircraft engine failure can cause catastrophic damaging effects to aircraft systems if not addressed in the aircraft design. Mitigating the damaging effects of uncontained engine failure and improving the numerical modeling capability of these uncontained engine events are crucial. In this paper, high strain rate material behavior of one of the most extensively used materials in the aircraft industry is simulated and the results are compared against ballistic impact tests. Ballistic limits are evaluated by utilizing explicit finite-element (FE) simulations based on the corresponding ballistic impact experiments performed at different material thicknesses. LS-DYNA is used as a nonlinear explicit dynamics FE code for the simulations. A Johnson–Cook material model with different sets of parameters is employed as a thermo-viscoplastic material model coupled with a nonlinear equation of state and an accumulated damage evaluation algorithm for the numerical simulations. Predictive performance of the numerical models is discussed in terms of material characterization efforts, material model parameters, mesh sensitivities, and effects of stress triaxiality. It is shown that mesh refinement does not necessarily provide better results for ballistic limit simulations without considering and calibrating these interrelated factors. Moreover, it is shown that current models that can only fit a specific function for damage evaluation as a function of stress triaxiality are not always successful in predicting failure, especially if the state of stress changes significantly.     

The effect of constraint-induced normal stress on the failure of notched TiAl components

Titanium aluminides are candidates for replacing nickel superalloys in some aircraft engine components. In uniaxial tension tests, these materials experience plastic strains at failure that place them in-between traditional definitions for ductile and brittle materials. This study considers the appropriate continuum mechanics failure criterion for these materials under multiaxial loading conditions (based on either the maximum equivalent plastic strain or the maximum normal stress). The material tested is a Ti-47.0Al-2.0Cr-1.9Nb alloy having a predominantly lamellar γ/α ₂ microstructure. Cylindrical notched tensile specimens that experience elevated normal stresses in their interiors due to circumferential constraint during plastic deformation have been investigated. Results are presented that quantify reductions in failure loads due to elevated normal stress, compared to those predicted by finite element models using a maximum equivalent plastic strain criterion. To properly interpret the experimental results, the effects of notch strengthening must be included in the model predictions. Model and experimental results suggest that this TiAl alloy has some sensitivity to normal stress and that a combined failure criterion is needed to accurately predict failure under multiaxial loading conditions. A fracture initiation and failure mechanism requiring a combination of normal stress and plastic straining is suggested that is consistent with observed features at fracture initiation sites.     

Workability Studies in Forming of Sintered Fe-0.35C Powder Metallurgy Preform During Cold Upsetting

 An experimental investigation on the workability behaviour of sintered Fe-035C steel preforms under cold upsetting, have been studied in order to understand the influence of aspect ratio and lubrication condition on the workability process. The above mentioned powder metallurgy sintered preform with constant initial theoretical density of 84% of different aspect ratios, namely, 04 and 06 respectively were prepared using a suitable die-set assembly on a 1 MN capacity hydraulic press and sintered for 90 min at 1200 ℃. Each sintered preform was cold upset under nil/no and graphite frictional constraint, respectively. Under the condition of triaxial stress densification state, axial stress, hoop stress, hydrostatic stress, effective stress and formability stress index against axial strain relationship was established and presented in this work. Further more, attained density was considered to establish formability stress index and various stress ratio parameters behaviour.     

Plasticity Model for Concrete under Triaxial Compression

Using the experimental background of 130 triaxial tests conducted on cylindrical specimens, a plasticity-based constitutive model of concrete behavior is developed. Parameters of the reference experimental database include the water:cement ratio (i.e., f′c), degree of saturation at testing, and load path used in the tests. In the model, damage is quantified by the volumetric expansion that builds up progressively in the material as it approaches failure and is caused by propagation of microcracks. This behavioral index is calibrated with reference to the available tests and subsequently used as the primary state variable in the model, determining for any stress state the degree of stiffness and strength degradation and the ductility in the response. Because failure is modeled as a damage-driven continuous process rather than a distinct event, the characteristic failure envelope is expanding (hardening) or contracting (softening) as a function of a scalar measure of plastic deformation. A nonassociated plastic flow rule calibrated against the experiments is used to describe the direction of plastic deformation. The model was tested against published triaxial test series and empirical confinement models. It was also used in the context of a finite-element formulation to study the mechanical behavior of reinforced-concrete circular columns. This particular test problem was selected because it is a real-life example of the experimental conditions used to derive the model.     

Shakedown Approaches to Rut Depth Prediction in Low-Volume Roads

Rutting, due to permanent deformations of unbound materials, is one of the principal damage modes of low traffic pavements. Flexible pavement design methods remain empirical; they do not take into account the inelastic behavior of pavement materials and do not predict the rutting under cyclic loading. A finite-element program, based on the concept of the shakedown theory developed by Zarka for metallic structures under cyclic loadings, has been used to estimate the permanent deformations of unbound granular materials subjected to traffic loading. Based on repeated load triaxial tests, a general procedure has been developed for the determination of the material parameters of the constitutive model. Finally, the results of a finite-element modeling of the long-term behavior of a flexible pavement with the simplified method are presented and compared to the results of a full-scale flexible pavement experiment performed by Laboratoire Central des Ponts et Chaussées. Finally, the calculation of the rut depth evolution with time is carried out.     

Characterization of Damage in Triaxial Braided Composites under Tensile Loading

Carbon fiber composites that utilize flattened, large tow yarns in woven or braided forms are being used in many aerospace applications. The complex fiber architecture and large unit cell size in these materials present challenges for both understanding the deformation process and measuring reliable material properties. In this paper composites made using flattened 12k and 24k (referring to the number of fibers in the fiber tow) standard modulus carbon fiber yarns in a 0°/+60°/?60° triaxial braided architecture are examined. Standard straight-sided tensile coupons were tested with the 0° axial braid fibers either parallel to (axial tensile test) or perpendicular to (transverse tensile test) the applied tensile load. The nonuniform surface strain resulting from the triaxial braided architecture was examined using photogrammetry. Local regions of high strain concentration were examined to identify where failure initiates and to determine the local strain at the time of failure initiation. Splitting within fiber bundles was the first failure mode observed at low to intermediate strains. For axial tensile tests the splitting was primarily in the ±60° bias fibers, which were oriented 60° to the applied load. At higher strains in the axial tensile test, out-of-plane deformation associated with localized delamination between fiber bundles or damage within fiber bundles was observed. For transverse tensile tests, the splitting was primarily in the 0° axial fibers, which were oriented transverse to the applied load. The initiation and accumulation of local damage caused the global transverse stress-strain curves to become nonlinear and caused failure to occur at a reduced ultimate strain for both the axial and transverse tensile tests. Extensive delamination at the specimen edges was also observed. Modifications to the standard straight-sided coupon geometry are needed to minimize these edge effects when testing the large unit cell type of material examined in this work.     

Methodology for Impact Modeling of Triaxial Braided Composites Using Shell Elements

In this paper, a two-dimensional triaxial braided composite model has been studied using the nonlinear explicit finite-element code LSDYNA. The unit cell consists of six subcells and material properties associated with shell element integration point simulate braiding architecture. The local material properties were selected by correlation of the global behavior of a coupon model with static specimen tests. By changing subcell size and orientation angle at integration points, different braids architectures were obtained. Panel ballistic models were performed with benefits of computation efficiency of shell elements. Mechanical properties, panel impact threshold velocities, and failure initiations for braids with bias angles of 75, 60, 45, and 30° were studied. Boundary effects were also investigated.     

Effect of Constraint on Creep Behavior of 9Cr-1Mo Steel

The effect of constraint on creep rupture behavior of 9Cr-1Mo steel has been investigated. The constraint was introduced by incorporating a circumferential U-notch in a plain cylindrical creep specimen of 5 mm diameter. The degree of constraint was increased by decreasing the notch root radius from 5 to 0.25 mm. Creep tests were conducted on plain and notched specimens at stresses in the range of 110 to 210 MPa at 873 K (600 °C). The creep rupture life of the steel was found to increase under constrained conditions, which increased with the increase in degree of constraint and applied stress, and tended to saturate at a higher degree of constraint. The creep rupture ductility (pct reduction in area) of the steel was found to be lower under constrained conditions. The decrease in creep ductility was more pronounced at a higher degree of constraint and lower applied stresses. Scanning electron microscopic studies revealed a change in fracture behavior with stress and degree of constraint. The fracture surface appearance for relatively lower constrained specimens at higher stresses was predominantly transgranular dimple. Creep cavitation-induced intergranular brittle fracture near the notch root was observed for specimens having a higher degree of constraint at relatively lower stresses. The creep rupture life of the steel under constrained conditions has been predicted based on the estimation of damage evolution by continuum damage mechanics coupled with finite element analysis of the triaxial state of stress across the notch. It was found that the creep rupture life of the steel under constrained conditions was predominantly governed by the von-Mises stress and the principal stress became progressively important with increase in the degree of constraint and decrease in applied stress.     

Test and Analysis for Ultimate Load-Carrying Capacity of Existing Reinforced Concrete Arch Ribs

The structural performance of reinforced concrete bridges gradually deteriorates due to material aging and concrete cracking. Reported here are the experimental investigations and the nonlinear finite-element analysis of two arch ribs removed from a decommissioned bridge and reinstalled in the laboratory. The old bridge had been in service for 28 years. The full-scale static tests for two arch ribs were performed. The load–displacement and load–strain relationships, the residual load-carrying capacity, and the failure form are explored in detail. The structural analysis software Marc is invoked in the theoretical computations. Both geometrical and material nonlinearities are considered. Moreover, the material aging and the structural damage are introduced in the finite-element model. For comparison, the undamaged and geometrically perfect arch rib is analyzed at the same time. A comparison between the experimental and theoretical results is made. It can be concluded that the initial cracks, the reinforcement corrosion, and the variation of the arch axial line shape are the crucial effects for the structural ultimate load-carrying capacity and failure mode.     

Statistical Damage Constitutive Model of Quasi-Brittle Materials

Recent studies have shown that statistical damage mechanics is one effective method to study the failure process of quasi-brittle materials. There are two key problems in setting up the statistical damage constitutive model of quasi-brittle materials, namely, determining the microunit strength and the parameters of statistical distribution that the microunit strength obeys. The four-parameter criterion is a failure criterion consisting of four unknown parameters. When the four parameters equal appropriate values, it may become the Drucker–Prager criterion (for rock), Mohr–Coulomb criterion (for rock), and Hsieh–Ting–Chen criterion (for concrete), so the four-parameter criterion may be used to simulate the elastoplastic behavior of rock and concrete quasi-brittle materials. In the paper, microunit strength is determined with the four-parameter criterion, thus the statistical damage constitutive model suits rock and concrete. The deficiencies of existing methods in determining the distribution parameters are investigated, and a new method for determining the distribution parameters is proposed. First, the theoretical relationships between the parameters and the strain and stress at the peak point of material failure curve are derived; second, the approximate relations between the strain and stress at the peak point of material failure curve and confining pressure are established through the curve fitting method; finally, the relations between the parameters and confining pressure are established. The proposed statistical damage softening constitutive model of quasi-brittle materials has universal meaning, the determination of distribution parameters has strict theoretical basis, and the distribution parameters can be conveniently obtained with general triaxial tests. Numerical examples are also presented to validate the model.     

Fracture initiation at hydrides in zirconium

The effect of hydride size and stress state on fracture initiation at hydrides in a reactor grade Zr material has been studied. Uniaxial and triaxial states of stress were imposed by using smooth and notched tensile specimens, respectively. Crack initiation at hydrides was monitored using acoustic emission (AE). The specimens contained, nominally, either 0.18 or 0.90 at. pct hydrogen. Plate-or needle-shaped hydrides having different lengths were produced by varying the cooling rate to room temperature from the hydrogenation temperature. Initial orientation of the plate normals of the hydrides with respect to the tensile axis was mainly random. After deformation, the hydrides near the fracture surface were all oriented with their plate normals perpendicular to the tensile axis direction. Regardless of the hydride size, fracture at hydrides commenced at stress levels just above the proportional limit under uniaxial deformation. Average plastic strain values at initiation were ~0.2 pct. Slightly lower values of plastic strain were needed to initiate fracture at hydrides under triaxial loading. Fracture of the hydrides was always through-thickness and specimen fracture ductile. This is in contrast to previous results on hydride fracture obtained using the pressure tube alloys. In these materials, the fracture of hydrides with their plate normal oriented parallel to the tensile axis became less ductile as the hydride length increased.     

Advanced test methods of material property characterization:high strain-rate testing and experimental simulation of multiaxial stress states

Optimum utilization of the loading capability of engineering materials is an important and active contribution to protect nature’s limited resources,and it is the key for economic design methods.In order to make use of the materials’ resources,those must be known very well;but conventional test methods will offer only limited informational value.The range of questions raised is as wide as the application of engineering materials,and partially they are very specific.The development of huge computer powers enables numeric modelling to simulate structural behaviour in rather complex loading environments-so the real material behaviour is known under the given loading conditions.Here the art of material testing design starts.To study the material behaviour under very distinct and specific loading conditions makes it necessary to simulate different temperature ranges,loading speeds, environments etc.and mostly there doesn’t exist any commonly agreed test standard.In this contribution two popular,non-standard test procedures and test systems will be discussed on the base of their application background,special design features as well as test results and typically gained information:The demand for highspeed tests up to 1000 s^-1 of strain rate is very specific and originates primarily in the automotive industry and the answers enable CAE analysis of crashworthiness of vehicle structures under crash conditions.The information on the material behaviour under multiaxial loading conditions is a more general one.Multiaxial stress states can be reduced to an equivalent stress,which allows the evaluation of the material’s constraint and criticality of stress state.Both discussed examples shall show that the open dialogue between the user and the producer of testing machines allows custom-tailored test solutions.     

Simulation of shear plugging through thin plates using the GRIM Eulerian hydrocode

Ballistic experiments have been performed using aluminum spheres against 10-mm rolled homogenous armour (RHA), MARS270, MARS300, and titanium alloy plates to investigate the influence of the plugging mechanism on material properties. The experiments have measured the threshold for plug mass and velocity as well as the recovered aluminum sphere mass over a range of velocities. Some of the experiments have been simulated using the in-house second generation Eulerian hydrocode GRIM. The calculations feature advanced material algorithms derived from interrupted tensile testing techniques and a triaxial failure model derived from notched tensile tests over a range of strain rates and temperatures. The effect of mesh resolution on the results has been investigated and understood. The simulation results illustrate the importance of the constitutive model in the shear localization process and the subsequent plugging phenomena. The stress triaxiality is seen as the dominant feature in controlling the onset and subsequent propagation of the crack leading to the shear plug. The simulations have demonstrated that accurate numerics coupled with accurate constitutive and fracture algorithms can successfully reproduce the observed experimental features. However, extrapolation of the fracture data leads to the simulations overpredicting the plug damage. The reasons for this are discussed. This article is based on a presentation given in the symposium entitled “Dynamic Behavior of Materials—Part II”, held during the 1998 Fall TMS/ASM Meeting and Materials Week, October 11–15, 1998, in Rosemont, Illinois, under the auspices of the TMS Mechanical Metallurgy and the ASM Flow and Fracture Committees.     

Simulation of shear plugging through thin plates using the GRIM eulerian hydrocode

Ballistic experiments have been performed using aluminum spheres against 10-mm rolled homogenous armour (RHA), MARS270, MARS300, and titanium alloy plates to investigate the influence of the plugging mechanism on material properties. The experiments have measured the threshold for plug mass and velocity as well as the recovered aluminum sphere mass over a range of velocities. Some of the experiments have been simulated using the in-house second generation Eulerian hydrocode GRIM. The calculations feature advanced material algorithms derived from interrupted tensile testing techniques and a triaxial failure model derived from notched tensile tests over a range of strain rates and temperatures. The effect of mesh resolution on the results has been investigated and understood. The simulation results illustrate the importance of the constitutive model in the shear localization process and the subsequent plugging phenomena. The stress triaxiality is seen as the dominant feature in controlling the onset and subsequent propagation of the crack leading to the shear plug. The simulations have demonstrated that accurate numerics coupled with accurate constitutive and fracture algorithms can successfully reproduce the observed experimental features. However, extrapolation of the fracture data leads to the simulations overpredicting the plug damage. The reasons for this are discussed. This article is based on a presentation given in the symposium entitled “Dynamic Behavior of Materials—Part II,” held during the 1998 Fall TMS/ASM Meeting and Materials Week, October 11–15, 1998, in Rosemont, Illinois, under the auspices of the TMS Mechanical Metallurgy and the ASM Flow and Fracture Committees.     

Effects of Cross Anisotropy on Three-Dimensional Behavior of Sand. II: Volume Change Behavior and Failure

The volume change behavior of cross-anisotropic sand is studied using results of a series of cubical triaxial tests. The relationships between the volumetric response, failure, and shear localization are addressed. Rates of dilation under various three-dimensional stress conditions are evaluated in conjunction with the peak shear resistance and initiation of shear banding in specimens of dense Santa Monica beach sand. The location of the line in principal stress space along which the tendency to deform changes from compressive to dilative (the characteristic line) is determined using two different methods. The uniqueness of this characteristic line for cross-anisotropic materials is analyzed.     

冲击荷载下岩石裂纹扩展研究进展

为了给深部资源开采和大型地下空间工程中围岩体的变形机理及稳定性控制提供理论基础,通过查阅大量关于表征岩石裂纹扩展的裂纹扩展模型、应力强度因子和断裂韧性的国内外文献,总结了前人的研究成果。依据现有研究,提出了动荷载作用下岩石裂纹扩展的几点建议：（1）综合考虑弹性力学、断裂力学和损伤力学建立岩石材料从微观断裂到宏观破坏这一演变过程的理论模型,使理论模型更加适应岩石材料的非线性特征;（2）采用分形、自组织和混沌等非线性理论表征动荷载作用下岩石内部以及表面裂纹的扩展演化特征;（3）采用颗粒离散元和有限差分模拟岩石材料裂纹扩展演化特征。     

The influence of hydride size and matrix strength on fracture initiation at hydrides in zirconium alloys

The effects of matrix strength (yield stress) on hydride fracture and alloy ductility have been studied as a function of stress state, hydride content, hydride size, and precipitation stress. Uniaxial and triaxial states of stress were investigated by using smooth and notched tensile specimens, respectively, containing 0.18 or 0.90 at. pct H, with the longest hydride platelet dimension varying from 5 to 400 μm. The majority of the hydrides in the specimens had their plate normals oriented parallel to the tensile axis direction. Crack initiation at hydrides was monitored using acoustic emission, finiteelement calculations were employed to determine the stresses and strains in the notched specimens, and metallographic and fractographic analyses were carried out to determine the state of fractured hydrides/voids near and on the fracture surface. These techniques showed that, up to a hydride platelet length of ∼50 to 100 μm and regardless of the stress state, a critical plastic strain, independent of matrix strength, controls the initiation of fracture in hydrides. The amount of plastic strain needed to fracture hydrides decreases as (a) the average hydride length increases and (b) the axiality of stress increases. The equivalent plastic strain to fracture small hydrides is ∼ 1 pct under a triaxial as opposed to ∼5 pct under a uniaxial state of stress. When the average hydride platelet lengths are longer than ∼50 to 100 μm, negligible plastic deformation is required to fracture hydrides. A critical applied stress then is the governing factor in all three materials, ranging from 750 to 850 MPa, depending on the stress state.