Elastic-plastic Stress-strain Response Research Articles

Analytical and Experimental Investigation of Elastic–Plastic Strain Distributions at 2-D Notches

Abstract Both elastic and elastic–plastic stress–strain response were studied for a series of part-circular and V-shaped notches in flat specimens. Both the peak strain at the notch root and the strain distribution along the notch bisector were studied. These quantities were determined analytically using detailed finite element analysis (FEA), and it was shown that significant differences can arise between detailed FEA results and two of the commonly used approximations for the peak notch strain response: Neuber’s rule and Glinka’s equivalent strain energy density theorem. These differences can become significant for notches with low acuity. The strain response for these configurations was also studied experimentally using two measurement techniques: optical fibers and surface differential displacement mapping. Agreement was shown between computed response strains and measured strains for some of the eight notch configurations and two aerospace alloys studied, while for others, experimental difficulties prevented agreement. These difficulties are described in detail.

The mechanical and electrical properties of direct-spun carbon nanotube mat-epoxy composites

Composites of direct-spun carbon nanotube (CNT) mats and epoxy are manufactured and tested in order to determine their mechanical and electrical properties. The mats are spun directly from a floating catalyst, chemical vapour deposition reactor. The volume fraction of epoxy is varied widely by suitable dilution of the epoxy resin with acetone. Subsequent evaporation of the acetone, followed by a cure cycle, leads to composites of varying volume fraction of CNT, epoxy and air. The modulus, strength, electrical conductivity and piezoresistivity of the composites are measured. The CNT mats and their composites exhibit an elastic-plastic stress-strain response under uniaxial tensile loading, and the degree of anisotropy is assessed by testing specimens in 0°, 45° and 90° directions with respect to the draw direction of mat manufacture. The electrical conductivity scales linearly with CNT volume fraction, irrespective of epoxy volume fraction. In contrast, the modulus and strength depend upon both CNT and epoxy volume fractions in a non-linear manner. The macroscopic moduli of the CNT mat-epoxy composites are far below the Voigt bound based on the modulus of CNT walls and epoxy. A micromechanical model is proposed to relate the macroscopic modulus and yield strength of a CNT mat-epoxy composite to the microstructure.

New pseudo stress correction method for estimating local strains at notch under multiaxial cyclic loading

A notch correction method is proposed under multiaxial cyclic loading, in which the real stress increments can be directly obtained by correcting the pseudo stress increments at the notch. Moreover, the proposed method does not inherently restrict the direction and magnitude of real backstress increment in the real stress space. Additionally, on the basis of Kanazawa et al.’s model, the proposed method utilizes pseudo notch strains in replacement of measured strains to take account of the additional hardening due to the non-proportionality of external loadings. Based on the proposed correction method integrated with constitutive equations using Garud’s hardening model, a notch stress-strain estimation methodology is developed to calculate the local elastic-plastic stress-strain responses at the notch area. The presented methodology does not require iteration. The applicability of the presented methodology was verified by the experimental data of TC21 titanium alloy notched tubular specimens in this study and SAE 1070 steel notched shaft specimens from the literature. The results showed that the calculated notch strains correlate well with the measured strains under proportional and non-proportional loadings.

A finite element study on residual stress stability and fatigue damage in high-frequency mechanical impact (HFMI)-treated welded joint

Fatigue improvement from high-frequency mechanical impact (HFMI) is considered to rely on compressive residual stresses, improved weld toe geometry and localized strain hardening at the weld toe. Of these, the existence of compressive residual stresses is generally assumed critical for the effectiveness of the method. In this study, the influence of stress ratio and peak loads on residual stress relaxation and fatigue damage in as-welded and HFMI-treated S700 transverse attachments was investigated. Elastic-plastic stress-strain response for as-welded and HFMI-treated conditions was simulated considering the effects of initial residual stress distribution, local geometry and local material properties. Relative fatigue damage was estimated using the Smith-Watson-Topper parameter. Full residual stress relaxation was observed for a stress ratio of 0.5 and a compressive overload of 0.6 times the nominal yield strength. The fatigue damage assessment showed benefit from HFMI with respect to the as-welded state for all simulated load conditions. The results are consistent with experimentally observed behaviour. The stepwise analysis indicated that the remaining benefit after residual stress relaxation was due to geometry improvement and strain hardening.

Numerical validation of computational stress and strain analysis model for notched components subject to non-proportional loadings

In this paper, a previously developed computational model for the multiaxial stress–strain analysis of the notched bodies is further discussed and validated. The model calculates elastic–plastic stress–strain responses using linear elastic finite element solution of notched components. The validity of the model was presented by comparing results calculated from the proposed model against Finite Element Analysis (FEA) results of SAE 1070 steel notched bar subject to monotonic and cyclic non-proportional loading paths. The comparison showed that the model predicted notch-root elasto-plastic stresses and strains with good accuracy using linear-elastic stresses as a load input. The model becomes a significant advantageous tool and provides more efficient and simple analysis method in comparison to expensive experimental component tests and more complex and time consuming incremental non-linear FEA. The model is particularly efficient and practical to perform the elasto-plastic stress–strain analysis of notched components subjected to complex non-proportional loading paths.

Nonlinear Buckling Analysis of Slender Piles with Geometric Imperfections

AbstractThe buckling response of straight and initially bent, i.e., geometrically imperfect, end-bearing piles in soil subjected to axial load has been examined in the current study through FEM, considering geometric and material nonlinearities exclusively. Nonlinear buckling analysis is performed herein using the Static/Riks method. Material nonlinearity has been considered by assuming elastic-plastic stress-strain response of the steel pile and the surrounding soil. Parametric sensitivity studies have been performed by changing the slenderness ratios and the initial curvature of the piles. The analyses results show that the load-carrying capacity of the piles decreases with increasing pile curvature and increasing slenderness ratio of the initially bent piles. It was also found that the piles with a half-sine bent carry smaller loads than the piles with a quarter-sine bent.

Evaluation of Cyclic Hardening Parameters for Type 304LN Stainless Steel

Engineering components are often subjected to cyclic load excursions beyond elastic limit and hence cyclic plastic deformation of engineering materials becomes inevitable. Since the resultant elastic-plastic stress-strain response of the material plays a pivotal role in analysis, design and failure of the component, it becomes important to understand the cyclic plastic deformation behaviour of engineering materials. Also, cyclic hardening parameters are required in the design of structural components subjected to large plastic deformation. Constitutive equations were proposed by Prager, Armstrong and Frederick, Chaboche, and Ohno-Wang based on the stabilized strain-controlled hysteresis curve to evaluate the hardening parameters. In the present study, cyclic hardening parameters for SA 312 Type 304LN stainless steel have been determined based on the results of constant amplitude strain-controlled fatigue tests carried out earlier at CSIR-SERC under five different strain amplitude values, viz, 0.20%, 0.35%, 0.65%, 0.80% and 0.95%. It is observed that in isotropic hardening, the values of Q decreased with increase in strain amplitude. In kinematic hardening, the values of C1and γ1 are constant for all values of strain amplitude.

Micromechanics-based constitutive modeling of plastic yielding and damage mechanisms in polymer–clay nanocomposites: Application to polyamide-6 and polypropylene-based nanocomposites

Abstract The present work focuses on the continuum-based micromechanical modeling of the elastic–plastic stress–strain response including damage mechanisms of polymer–clay nanocomposites. The micromechanical elastic–plastic-damage model includes both the actual microstructure of the said composites using a multi-scale approach and the microstructural evolution related to damage accumulation under applied macroscopic deformation. The interfacial debonding between clay nanoparticles and polymer matrix, and the polymer matrix voiding are the two prevalent damage events considered. The tensile stress–strain response and micromechanical deformation processes of polyamide-6 and polypropylene-based systems reinforced with modified montmorillonite clay at various concentrations are experimentally investigated by a video-controlled technique. The usual shear yielding deformation mode of neat polyamide-6 is altered by the presence of clay platelets which induce a dilatational process due to interfacial debonding. In addition to matrix shear yielding, a dual-dilatational deformation mechanism by crazing and interfacial debonding is revealed in polypropylene–clay nanocomposites. Using the intrinsic deformation micromechanisms and elastic–plastic properties of the polymer matrix, and the nanocomposite structural characteristics, the micromechanical model is found to successfully describe the experimental results of the two nanocomposite materials in terms of tensile stress–strain response and inelastic volumetric strain.

Modeling and characterization of the moisture-dependent bilinear behavior of regenerated cellulose composites

In this study, we investigate the tensile mechanical properties of unidirectional Lyocell/epoxy composites under wet and dry conditions using a multi-scale analysis approach. Characterization of the samples shows a bilinear stress–strain behavior of the fibers and composites under tension loading. The bilinear elastic–plastic stress–strain response of the Lyocell fibers is incorporated into a p-version finite element model for presenting a methodology for structural analysis of this composite system. The proposed finite element models were successfully able to relate the micro to macro-mechanical behavior enabling an approach for determining the 3D orthotropic elastic–plastic constants of regenerated cellulose/epoxy composites.

Modeling of Fatigue Crack Growth under High-Low Sequence Loading

The fatigue crack growth behavior of one compact tension specimen of 16MnR steel under high-low sequence loading was investigated. The symmetric half finite element model under plane-stress state was used to calculate the elastic-plastic stress-strain responses, in which the Armstrong-Frederick type cyclic plasticity model was implemented as a user material subroutine UMAT of ABAQUS. A recently developed dynamic crack growth model was used to simulate the effects of high loading step on the successive low loading step. The detailed evolution process of the crack closure and cyclic plastic zone within the retardation region of fatigue crack growth was obtained. The extend of the crack closure, the size of cyclic plastic zone and the stress gradient have significant influence on the fatigue crack growth rate. The predicted fatigue crack growth rate is in good agreement with the experimental data.

Simulation of Fatigue Initiation and Non-Self-Similar Fatigue Crack Growth under Mixed Mode I/II Loading

The fatigue initiation and non-self-similar fatigue crack growth behavior of three notched compact tension and shear specimens of 16MnR steel under mixed mode I/II loading were investigated. The plane-stress finite element model with the implemented Armstrong-Frederick type cyclic plasticity model was used to calculate the elastic-plastic stress-strain responses. A recently developed dynamic crack growth model was used to simulate the effects of loading history on the successive crack growth. With the outputted numerical results, a multiaxial fatigue damage criterion based on the critical plane was used to determine the location of fatigue initiation. A formula of fatigue crack growth rate, which is based on the postulation that the fatigue initiation and crack growth have the same damage mechanism, was then used to calculate the transient crack growth rate and determine the non-self-similar crack growth path. The predicted fatigue initiation position, crack path and crack growth rate are in excellent agreement with the experimental data.

Study of the effect of size and clay structural parameters on the yield and post-yield response of polymer/clay nanocomposites via a multiscale micromechanical modelling

In order to predict the nanostructure as well as the particle size dependence of the elastic–plastic stress–strain response of polymer/clay nanocomposites, a micromechanical model based upon a multiscale approach starting from the nanostructure is proposed. The multiscale micromechanical model takes into account the interphase between the polymeric matrix and the inorganic reinforcement, and the intercalated nanostructure. Considering the interphase thickness as a characteristic length scale, the nanoparticle size effect is explicitly introduced in the present model. The intercalated nanostructure is taken into account according to an equivalent stiffness method in which the clay stacks are replaced by homogeneous nanoparticles with predetermined equivalent anisotropic stiffness. The physical and mechanical properties of nylon-6/montmorillonite nanocomposites (with clay weight fractions ranging from 1 up to 7.5%) are investigated by means of differential scanning calorimetry, dynamic mechanical analysis, thermogravimetric analysis and video-controlled tensile mechanical tests. The microstructure was characterized by transmission electron microscopy. The amount of interphase was estimated from the thermal analysis. The reinforcing effect of clay is discussed with respect to the multiscale micromechanical model. A parametric study is carried out to investigate the effect of nanoparticle shape and size and nanoparticle structural parameters (i.e. number of clay layers in the nanoparticle and interlayer spacing) on the elastic–plastic stress–strain response of polymer/clay nanocomposites. Comparison of the model results with the experimental data demonstrates that the evaluation of the reinforcing effect of clay involves considering the elastic stiffness and yield stress simultaneously. It was further found that the model correctly predicts the elastic–plastic stress–strain data.

Material characterization of SS 316 in low-cycle fatigue loading

This study deals with simulation of low-cycle fatigue (LCF), followed by evaluation of fatigue parameters, which would be suitable for estimating fatigue lives under uniaxial loading. The cyclic elastic–plastic stress–strain responses were analyzed using the incremental plasticity procedures. Finite-element (FE) simulation in elastic–plastic regime was carried out in FE package ABAQUS. Emphasis has been laid on calibration of SS 316 stainless steel for LCF behavior. For experimental verifications, a series of low-cycle fatigue tests were conducted using smooth, cylindrical specimens under strain-controlled, fully reversed condition in INSTRON UTM (Universal Testing Machine) with 8,800 controller at room temperature. The comparisons between numerical simulations and experimental observations reveal the matching to be satisfactory in engineering sense. Based on the cyclic elastic–plastic stress–strain response, both from experiments and simulation, loop areas, computed for various strain amplitude, have been identified as fatigue damage parameter. Fatigue strain life curves are generated for fatigue life prediction using Coffin–Manson relation, Smith–Watson–Topper model, and plastic energy dissipated per cycle (loop area). Life prediction for LCF has been found out to be almost identical for all these three criteria and correlations between predicted and experimental results are shown. It is concluded that the improvement of fatigue life prediction depends not only on the fatigue damage models, but also on the accurate evaluations of the cyclic elastic–plastic stress/strain responses.

Ductile damage micromodeling by particles’ debonding in metal matrix composites

The elastic–plastic behaviour of particle-reinforced metal matrix composites undergoing ductile damage is modelled using a two-level micro-structural approach. The considered heterogeneous material is a polycrystal containing intra-crystalline elastic particles. Ductile damage is initiated by the matrix/particle interface debonding and the subsequent voids growth with plastic straining of the crystalline matrix. Homogenization techniques are used twice: first at mesoscale to derive the equivalent grain behaviour and then to obtain the macroscopic behaviour of the material. Plastic deformation of the crystalline matrix is due to crystallographic gliding on geometrically well-defined slip systems. The associative plastic flow rule and the hardening law are described on the slip system level. The evolution of micro-voids volume fraction is related to the plastic strain. The elastic–plastic stress–strain response of particle composite is investigated. Predictions of the proposed model are compared to experimental data to illustrate the capability of the suggested method to represent material behaviour. Furthermore, specific aspects such as the stress triaxiality and yield surfaces are discussed.

Simulations of Cyclic Plasticity and Fatigue Behavior of Structural Steels under Multiaxial Loading

The local cyclic elastic-plastic stress-strain responses were simulated using the incremental plasticity procedures of ABAQUS finite element code. It is shown a better understanding on the evolutions of the local cyclic stress-strain and the strong interactions between the most stressed material element and its neighboring material elements in the plastic deformations and stress redistributions. Based on the stress/strain states of the stabilized cycle, a new damage parameter, proposed on modification of the ASME code parameter, is applied and improved correlations between the predicted and the experimental fatigue lives are shown. It is concluded that the improvement of fatigue life prediction depends not only on the fatigue damage models, but also on the accurate evaluations of the cyclic elasto-plastic stress/strain responses.

Simulation of cyclic stress/strain evolutions for multiaxial fatigue life prediction

This study deals with simulation for cyclic stress/strain evolutions and redistributions, and evaluation of fatigue parameters suitable for estimating fatigue lives under multiaxial loadings. The local cyclic elastic–plastic stress–strain responses were analyzed using the incremental plasticity procedures of ABAQUS finite element code for both smooth and notched specimens made of three materials: a medium carbon steel in the normalized condition, an alloy steel quenched and tempered and a stainless steel, respectively. Emphasis is on the studying of ‘intelligent’ material behaviors to resist fracture, such as stress redistribution and relaxation through plastic deformations, etc. For experimental verifications, a series of tests of biaxial low cycle fatigue composed of tension/compression with static and cyclic torsion were carried out on a biaxial servo-hydraulic testing machine (Instron 8800). Different multiaxial loading paths were used to verify their effects on the additional cyclic hardening. The comparisons between numerical simulations and experimental observations show that the FEM simulations allow better understanding on the evolutions of the local cyclic stress–strain and it is shown that strong interactions exist between the most stressed material element and its neighboring material elements in the plastic deformations and stress redistributions. Based on the local cyclic elastic–plastic stress–strain responses, the energy-based multiaxial fatigue damage parameters are applied to correlating the experimentally obtained lives. Improved correlations between the predicted and the experimental results are shown. It is concluded that the improvement of fatigue life prediction depends not only on the fatigue damage models, but also on the accurate evaluations of the cyclic elasto-plastic stress/strain responses.

A Model for the Inclusion of Notch Plasticity Effects in Fatigue Crack Growth Analysis

Abstract Notches or other stress concentrations are by far the most common sites for the initiation and growth of fatigue cracks in aircraft structures. The growth of these cracks is directly influenced by the material stress-strain response in the vicinity of the notch. Specifically, when the applied (remote) stress is sufficient to cause local plastic deformation at the notch, the response (local) stresses can no longer be found using elastic stress concentration factors, and they become dependent on the prior loading history. This is to say that the response stresses can no longer be treated as state variables. The occurrence of fatigue crack growth at notches which experience local yielding one or more times during their design lifetime is, in fact, quite common in many cyclically loaded structures. Some of the assumptions inherent in “traditional” Linear Elastic Fracture Mechanics (LEFM) based fatigue crack growth analysis may be inappropriate for such problems. In particular, the assumption that the stress distribution on a critical plane remains proportional to the elastic distribution throughout the loading history becomes incorrect when one or more of the applied loads causes plastic deformation and introduces or alters a residual stress field in this region. This paper first describes an elastic-plastic stress-strain response algorithm which may be used to estimate response stress distributions on a critical plane on a cycle-by-cycle basis. This is followed by a discussion of the manner by which stress intensity factors may be calculated based on these response stress distributions using Green's functions. Finally, the use of these stress intensity factors for the calculation of crack growth rate and, ultimately, crack growth life, is demonstrated.

Biaxial fatigue for proportional and non‐proportional loading paths

ABSTRACTIn order to study the use of a local approach to predict crack‐initiation life on notches in mechanical components under multiaxial fatigue conditions, the study of the local cyclic elasto‐plastic behaviour and the selection of an appropriate multiaxial fatigue model are essential steps in fatigue‐life prediction. The evolution of stress–strain fields from the initial state to the stabilized state depends on the material type, loading amplitude and loading paths. A series of biaxial tension–compression tests with static or cyclic torsion were carried out on a biaxial servo‐hydraulic testing machine. Specimens were made of an alloy steel 42CrMo4 quenched and tempered. The shear stress relaxations of the cyclic tension–compression with a steady torsion angle were observed for various loading levels. Finite element analyses were used to simulate the cyclic behaviour and good agreement was found. Based on the local stabilized cyclic elastic–plastic stress–strain responses, the strain‐based multiaxial fatigue damage parameters were applied and correlated with the experimentally obtained lives. As a comparison, a stress‐invariant‐based approach with the minimum circumscribed ellipse (MCE) approach for evaluating the effective shear stress amplitude was also applied for fatigue life prediction. The comparison showed that both the equivalent strain range and the stress‐invariant parameter with non‐proportional factors correlated well with the experimental results obtained in this study.

Stress-Strain Response of Free-Standing Nano-Crystalline Gold Thin-Films

ABSTRACTFree-standing gold thin-film tension specimens of thicknesses in the range 500-700 nm were fabricated by e-beam deposition methods. The films have grains which have thicknesses essentially equal to the film thickness, and an in-plane grain-size distribution which is bi-modal, with a collection of small grains with an average size of 130 nm, and another collection of larger grains with an average size of 380 nm. The films possess a strong texture with the {111} crystallographic planes aligned with the plane of the film. The thin-film specimens have been tested in simple tension using a novel testing machine which enables measurement of the elastic-plastic stress-strain response of the films with high resolution. The experiments show that the gold thin-films have an initial yield strength of around 100 MPa, which is followed by rapid strain hardening to a stress level of 360 MPa at a strain of only ~1.2%, at which point the specimens fail abruptly. Transmission electron microscopy (TEM) investigations of the deformed films shows that the rapid strain hardening is attributable to extensive dislocation activity in the larger grains. The reason behind the low ductility in our gold specimens is still unclear, but it may be attributable to the observed lack of dislocation activity in the smaller grains, whose boundaries then serve as easier sites for alternative inelastic mechanisms such as grain boundary sliding and decohesion leading to low macroscopic strains to failure.

Multiaxial Stress Concentration in an Externally Pressurized Cylinder With an External Circumferential Groove

Abstract A thick-walled cylindrical specimen containing an external circumferential groove was subjected to external pressure. To investigate the maximum pressure sustainable by the reduced wall thickness, strain gage measurements were taken during external pressurization tests. For comparison to experimental results, an elastic-plastic notch stress-strain analysis was conducted based on Neuber’s rule. The analysis utilized multiaxial elastic finite element results along with elastic-plastic tensile test data for the cylinder material. Based on experimental observations, it was necessary to supplement the approach with an additional relation between elastic and elastic-plastic multiaxial strains for the axisymmetric geometry under investigation. Assuming an invariant hoop to radial strain ratio rather than an invariant hoop to axial strain ratio provided better agreement with experimental results. It is demonstrated that the boundary conditions used to model the specimen had a substantial effect on the finite element results, even though the boundary was somewhat removed from the region of concentrated stress. Biaxial strain measurements are presented versus pressure over the elastic and into the plastic regime, and deformation plasticity theory was used to compute stress and radial strain components corresponding to measured strains. It is demonstrated that in order to apply a multiaxial Neuber’s rule to accurately estimate the elastic-plastic stress-strain response (using elastic stress concentration information and elastic-plastic material data), it is necessary to utilize an experimental observation that the ratio of the hoop to radial strain remains invariant from the elastic to the elastic-plastic regime. This differs from published assumptions about invariant hoop-to-axial strain ratios based on analysis of circumferentially grooved solid shafts. The predictions are accurate for moderate plastic strains, but correlation breaks down for bulk plastic deformation.