Uniaxial Stress-strain Response Research Articles

On the Constraint Factor and Tabor Coefficient Pertinent to Spherical Indentation

Measuring the uniaxial stress–strain response from indentation testing has been of great interest to the materials community ever since the seminal work on spherical indentation by David Tabor. In this regard, spherical indentation is the primary choice due to the ability to access a wide range of strains in a single test. While indentation testing is fairly simple to perform, the conversion factors required to calculate the uniaxial flow stress from hardness, which is commonly referred to as constraint factor and uniaxial strain from indentation contact radius and ball radius, which is called the Tabor coefficient, are not necessarily constant and most of the prior work involves assumptions about one of these conversion factors to calculate the other. In this work, we present a finite element analysis-based approach to independently determine the constraint factor and Tabor coefficient in the fully plastic indentation regime which is a pre-requisite for this analysis. The criteria to determine whether fully plastic indentation regime is satisfied has also been presented. The proposed approach has been validated by comparing the uniaxial stress–strain response from spherical indentation tests on OFHC copper and the data obtained by conventional uniaxial testing. Excellent agreement has been found between the two approaches which can be readily applied for measuring the uniaxial stress–strain response of coatings which is otherwise difficult to determine.

Mechanical response of two polyimides through coarse-grained molecular dynamics simulations

Coarse-grained molecular dynamics (MD) simulations allow us to predict the mechanical responses of polymers, starting merely with a description of their molecular architectures. It is interesting to ask whether, given two competing molecular architectures, coarse-grained MD simulations can predict the differences that can be expected in their mechanical responses. We have studied two crosslinked polyimides PMR15 and HFPE52—both used in high- temperature applications—to assess whether the subtle differences in their uniaxial stress–strain responses, revealed by experiments, can be reproduced by carefully coarse-grained MD models. The coarse graining procedure for PMR15 is outlined in this work, while the coarse grain forcefields for HFPE52 are borrowed from an earlier one (Pandiyan et al 2015 Macromol. Theory Simul. 24 513–20). We show that the stress–strain responses of both these polyimides are qualitatively reproduced, and important insights into their deformation and failure mechanisms are obtained. More importantly, the differences in the molecular architecture between the polyimides carry over to the differences in the stress–strain responses in a manner that parallels the experimental results. A critical assessment of the successes and shortcomings of predicting mechanical responses through coarse-grained MD simulations has been made.

Micromechanical modelling of constitutive behavior of austempered ductile iron (ADI) at high strain rate

In this work, the uniaxial stress-strain response of austenitic ductile iron at high strain rate has been investigated. Traction tests were performed at low (0.001/s) and high rates (740/s and 1200/s) at RT and low temperature. Results showed that the strain rate has a considerable effect on work hardening and ductility. The work hardening seems to be the result of the competition of ferrite and retained austenite behavior under varying combinations of temperature and strain rate. The strain rate was found to increase the material failure strain. Numerical simulations performed with unit cell model indicated that this effect could due to strain localization along inter-spheroids ligaments supported by thermal softening.

Determining the tensile response of materials at high temperature using DIC and the Virtual Fields Method

An experimental approach based on Digital Image Correlation (DIC) is successfully applied to predict the uniaxial stress-strain response of 304 stainless steel specimens subjected to nominally uniform temperatures ranging from room temperature to 900°C. A portable induction heating device equipped with custom made water-cooled copper coils is used to heat the specimen. The induction heater is used in conjunction with a conventional tensile frame to enable high temperature tension experiments. A stereovision camera system equipped with appropriate band pass filters is employed to facilitate the study of full-field deformation response of the material at elevated temperatures. Using the temperature and load histories along with the full-field strain data, a Virtual Fields Method (VFM) based approach is implemented to identify constitutive parameters governing the plastic deformation of the material at high temperature conditions. Results from these experiments confirm that the proposed method can be used to measure the full field deformation of materials subjected to thermo-mechanical loading.

Modelling of uniaxially pre-strained and annealed AZ31 magnesium sheet based on microstructural considerations

Material deformation at room temperature and subsequent thermal aspects are analyzed at the microstructural level to understand the uniaxial deformation behavior of AZ31 magnesium sheet from pre-straining and annealing (PSA) process. The sheet is pre-strained in uniaxial tension to various strains by conducting interrupted tests and its uniaxial stress-strain responses are obtained. The pre-strained specimens are then annealed at different temperatures and times and then reloaded in uniaxial tension to obtain their post-anneal stress-strain response up to large strains. Pre and post-annealing mechanical responses are analyzed and correlated with static recovery, grain recrystallization and growth kinetics from annealing. Experimental results are utilized to develop and validate a microstructure-based model to capture the room temperature mechanical and grain-level responses of AZ31 sheet from the PSA process. The model incorporates the effect of recovery on the recrystallization kinetics, as well as non-constant nucleation and growth behavior in order to predict the grain size at the end of recrystallization and within the grain growth stage. In addition, the mechanical properties of PSA materials are predicted by considering the resulting microstructure as a composite of un-recrystallized, recrystallized and coarsened grain regions and by employing the rule of mixture.

Correlation of spherical nanoindentation stress-strain curves to simple compression stress-strain curves for elastic-plastic isotropic materials using finite element models

The stress-strain fields realized in spherical indentation tests are highly heterogeneous, and present a significant challenge to the recovery of bulk stress-strain responses such as those measured in simple compression tests performed on samples with a uniform cross section in the gauge section. In this paper, we critically explore the correlations between indentation stress-strain curves and the simple compression stress-strain curves using the finite element model of indentation as a surrogate for the actual experiment. The central advantage of using a finite element model is that it allows us to critically explore the sensitivity of various assumptions or values of parameters or other choices made in the analyses protocols on the extracted results. Based on this study, a general protocol has been established to reliably recover the uniaxial stress-strain response directly from the indentation stress-strain curve for isotropic elastic-plastic materials. The protocols developed are validated for a range of hardening behaviors.

A molecular dynamics study of water transport inside an epoxy polymer matrix

Classical all-atom molecular dynamics simulations were used to build and study a polymer network model of EPON-828 as an epoxy and diethylenetriamine as a cross-linker. A cut-off based cross-linking algorithm was adopted to make the cross-linking bonds of the epoxy network. A step-wise cross-linking process was implemented to achieve an epoxy polymer network with a maximum conversion of 0.8. Based on the uniaxial stress–strain response the elastic, bulk and strain moduli were determined and found to match with previous experimental and simulation studies. Water transport inside epoxy networks was analyzed by preparing the epoxy polymer network models containing different amounts of water. The water–epoxy hydrogen bonding interactions strongly influence the diffusion of water molecules in the polymer. Radial distribution functions, volume swelling analysis and mean squared displacements indicated two different types of water molecules in the polymer matrix. Water-matrix hydrogen bonding dominates during the initial sorption process and then the water–water interactions gradually increase their influence on the diffusion process by forming bigger water clusters.

Small Scale Mechanical Characterization of Thin Foil Materials via Pin Load Microtesting

In situ scanning electron microscope (SEM) experiments, where small-scale mechanical tests are conducted on micro- and nanosized specimens, allow direct visualization of elastic and plastic responses over the entirety of the volume being deformed. This enables precise spatial and temporal correlation of slip events contributing to the plastic flow evidenced in a stress–strain curve. A new pin-loading methodology has been employed, in situ within the SEM, to conduct microtensile tests on thin polycrystalline metal foils. This approach can be tailored to a specific foil whose particular grain size may range from microns to tens of microns. Manufacture of the specialized pin grip was accomplished via silicon photolithography-based processing followed by subsequent focused ion beam finishing. Microtensile specimen preparation was achieved by combining a stencil mask methodology employing broad ion beam sputtering along with focused ion beam milling in the study of several metallic foil materials. Finite-element analyses were performed to characterize the stress and strain distributions in the pin grip and micro-specimen under load. Under appropriately conceived test conditions, uniaxial stress–strain responses measured within these foils by pin-load microtensile testing exhibit properties consistent with larger scale tests.

A comprehensive energy approach to predict fatigue life in CuAlBe shape memory alloy

Stabilized dissipated energy is an effective parameter on the fatigue life of shape memory alloys (SMAs). In this study, a formula is proposed to directly evaluate the stabilized dissipated energy for different values of the maximum and minimum applied stresses, as well as the loading frequency, under cyclic tensile loadings. To this aim, a one-dimensional fully coupled thermomechanical constitutive model and a cycle-dependent phase diagram are employed to predict the uniaxial stress-strain response of an SMA in a specified cycle, including the stabilized one, with no need of obtaining the responses of the previous cycles. An enhanced phase diagram in which different slopes are defined for the start and finish of a backward transformation strip is also proposed to enable the capture of gradual transformations in a CuAlBe shape memory alloy. It is shown that the present approach is capable of reproducing the experimental responses of CuAlBe specimens under cyclic tensile loadings. An explicit formula is further presented to predict the fatigue life of CuAlBe as a function of the maximum and minimum applied stresses as well as the loading frequency. Fatigue tests are also carried out, and this formula is verified against the empirically predicted number of cycles for failure.

Direct numerical determination of stabilized dissipated energy of shape memory alloys under cyclic tensile loadings

When shape memory alloys are subjected to cyclic loadings, the stabilized dissipated energy is an effective parameter in studying their performance, for instance, the fatigue life. However, thermomechanical coupling in the behavior of shape memory alloys under cyclic loadings causes the amount of stabilized dissipated energy to be obtainable once the responses of all transient cycles are determined. In this article, direct formulae are proposed to numerically evaluate stabilized dissipated energy of a shape memory alloy under cyclic tensile loadings as a function of maximum and minimum applied stresses as well as the loading frequency. A one-dimensional fully coupled thermomechanical constitutive model with a cycle-dependent phase diagram is utilized to be able to directly predict the uniaxial stress–strain response of a shape memory alloy in a specified cycle with no need of solving the previous cycles. The results are experimentally assessed for NiTi and CuAlBe specimens. Since the backward transformation in CuAlBe is realized to more gradually occur than that in NiTi, an enhanced phase diagram is proposed in which different slopes are considered for the start and finish of backward transformation strip. The numerical predictions of the present approach are shown to be in a good agreement with the experimental findings for cyclic tensile loadings.

Emergence of localized plasticity and failure through shear banding during microcompression of a nanocrystalline alloy

Microcompression testing is used to probe the uniaxial stress–strain response of a nanocrystalline alloy, with an emphasis on exploring how grain size and grain boundary relaxation state impact the complete flow curve and failure behavior. The yield strength, strain hardening, strain-to-failure and failure mode of nanocrystalline Ni–W films with mean grain sizes of 5, 15 and 90nm are studied using taper-free micropillars that are large enough to avoid extrinsic size effects. Strengthening is observed with grain refinement, but catastrophic failure through strain localization is found as well. Shear banding is found to cause failure, resembling the deformation of metallic glasses. Finally, we study the influence of grain boundary state by employing heat treatments that relax nonequilibrium boundary structure but leave grain size unchanged. A pronounced strengthening effect and increased strain localization are observed after relaxation in the finer grained samples.

Deformation Behavior and Formability of Sheet Metal Laminate Consisting of Perforated Core Sheet and Thin Skin Sheets

This study presents a new type of sheet metal laminate for lightweight products, and investigates its plastic behavior and formability. The sheet metal laminate consists of three layers, i.e. two thin skin sheets and a perforated core sheet with round holes, which are bonded together by diffusion bonding. Pure copper sheets are used for both of the core and the skins. Plastic deformations of the laminate and its component layers under uniaxial and biaxial tension are examined experimentally and analytically. Results of uniaxial stress-strain responses and yield loci (contours of plastic work) show that the perforated core sheet exhibits anisotropic behavior induced by the hole array but the laminated sheet becomes rather isotropic. Forming limit diagrams of the laminate and its component layers are also obtained by performing stretch forming test. Forming limit of the perforated core sheet is markedly lower than that of the monolithic sheet, and that of the thin skin is in between. It is found that forming limit of the laminate is comparable to that of the thin skin.

Micromechanics Modeling of Composites Subjected to Multiaxial Progressive Damage in the Constituents

The high-fidelity generalized method of cells composite micromechanics model is extended to include constituent-scale progressive damage via a proposed damage model. The damage model assumes that all material nonlinearity is due to damage in the form of reduced stiffness, and it uses six scalar damage variables (three for tension and three for compression) to track the damage. Damage strains are introduced that account for interaction among the strain components and that also allow the development of the damage evolution equations based on the constituent material uniaxial stress strain response. Local final-failure criteria are also proposed based on mode-specific strain energy release rates and total dissipated strain energy. The coupled micromechanics-damage model described herein is applied to a unidirectional E-glass/epoxy composite and a proprietary polymer matrix composite. Results illustrate the capability of the coupled model to capture the vastly different character of the monolithic (neat) resin matrix and the composite in response to far-field tension, compression, and shear loading.

Internal state variable plasticity-damage modeling of the copper tee-shaped tube hydroforming process

This paper presents a parametric finite element analysis using a history-dependent internal state variable model for a hydroforming process. Experiments were performed for the internal state variable model correlation and for validating a 2-in. copper tee hydroforming process simulation. The material model constants were determined from uniaxial stress–strain responses obtained from tensile tests on the tube's material. In the finite element simulations, the mesh and boundary conditions were integrated with the geometry and process parameters currently used in industry. The study provides insights for the variation of different process parameters (velocity and pressure profiles, and bucking system characteristics) related to the finished product.

Molecular dynamics simulations of void growth and coalescence in single crystal magnesium

The growth and coalescence of voids in magnesium single crystals at the nanoscale have been investigated using molecular dynamics simulations and the embedded atom method. One void and two void specimens with identical initial void volume fractions were utilized to study the mechanism of void growth and coalescence. In order to study the influences of material length scale on void evolution in single crystals four specimen sizes with the same initial volume fraction of voids were considered. Investigations of the effects of temperature and strain rate were also performed. Uniaxial stress–strain curves were monitored during increasing employed strain. The simulation results show that the specimen size, loading strain rate and temperature had apparent influences on the twin or dislocation pattern, void evolution shape and uniaxial stress–strain responses, but negligible effects on the initial slopes of the uniaxial stress–strain curves. Furthermore, the nucleation stress of twin bands in orientation A – x[0 0 0 1]– y[ 1 2 ¯ 1 0 ]– z[ 1 0 1 ¯ 0 ] was much higher than that of plastic deformation in orientation B – x[ 1 2 ¯ 1 0 ]– y[ 1 0 1 ¯ 0 ]– z[0 0 0 1].

Coarse-graining scheme for simulating uniaxial stress-strain response of glassy polymers through molecular dynamics

Simulation of the deformation of polymers below their glass transition through molecular dynamics provides an useful route to correlate their molecular architecture to deformation behavior. However, present computational capabilities severely restrict the time and length scales that can be simulated when detailed models of these macromolecules are used. Coarse-graining techniques for macromolecular structures intend to make bigger and longer simulations possible by grouping atoms into superatoms and devising ways of determining reasonable force fields for the superatoms in a manner that retains essential macromolecular features relevant to the process under study but jettisons unnecessary details. In this work we systematically develop a coarse-graining scheme aimed at simulating uniaxial stress-strain behavior of polymers below their glass transition. The scheme involves a two step process of obtaining the coarse grained force field parameters above glass transition. This seems to be enough to obtain "faithful" stress-strain responses after quenching to below the glass transition temperature. We apply the scheme developed to a commercially important polymer polystyrene, derive its complete force field parameters and thus demonstrate the effectiveness of the technique.

On measuring the strength of metals at ultrahigh strain rates

The strain rate sensitivity of materials is normally measured through a combination of quasistatic, Hopkinson bar, and pressure-shear experiments. Recent advances in uniaxial strain ramp loading provide a new means to reach strain rates significantly higher than achievable in pressure-shear experiments. One way to determine strength in ramp loading is by comparing the uniaxial stress-strain response to an appropriate pressure-density response obtained from an equation of state for the material. Using this approach, strengths for aluminum are obtained for strain rates of 105–108 s−1. Two issues arise in this calculation: heating due to plastic work and the effect of the superimposed hydrostatic stress on the strength. Heating due to plastic work is calculated and accounted for within the context of the equation of state for the material in a straightforward manner, but neglecting this heating can lead to significant errors in the calculated strength at higher compression levels. A simple scaling of strength with the pressure-dependent shear modulus is utilized to estimate the strength at zero pressure for ramp loading and pressure-shear experiments. When examined in this manner, the strain rate dependence of aluminum is found to be less than previously reported, with little increase in strength below strain rates of about 107s−1. The effects on ramp loading strength measurements of heating due to plastic work and of hydrostatic pressure are also examined for copper and tantalum using simple equation of state and strength models. The effect of plastic heating is similar for the three materials for a given strain level but quite different for a constant stress, with aluminum showing greater effects than the other materials. The effect of hydrostatic pressure in ramp loading experiments is similar for all three materials, but the effect is likely to be much greater in pressure-shear experiments for aluminum than the other materials.

Inelastic Second-Order Analysis of Frame Structures Using the Stability-Based Beam-Column Element

In this study, the stability-based beam-column element has been proposed to consider the geometric second-order effects and material inelasticity, and to take advantage of the computational efficiency of the second-order analysis. A new force interpolation function matrix is developed to consider a moment magnification due to the axial force and lateral displacement. The constitutive behaviour of the cross-section is formulated by integration of the uniaxial stress-strain response of fibres to model the material inelasticity. The generalized displacement control (GDC) method incorporating the incremental secant stiffness method is applied to solve the nonlinear equilibrium equations with multiple limit points. Evaluation of the results of this study shows that the proposed method consistently predicts well the nonlinear inelastic behaviour of steel and composite structures, and gives good efficiency.

Uni-axial stress–strain response and thermal conductivity degradation of ceramic matrix composite fibre tows

A physical model, previously developed by one of the authors, has been extended to cover thermal conductivity degradation owing to the uni-axial stress–strain response of aligned groups of fibres or tows found in ceramic matrix composites. Both the stress–strain and thermal models, together with their coupling, have been shown to predict known composite behaviour qualitatively. The degradation of longitudinal thermal properties is shown to be driven by strain-controlled fibre failure; while the degradation of transverse thermal properties is because of the growth of fibre–matrix interface wake-debonded cracks, coupled with strain-driven fibre failure.

Relation between mechanical properties and microstructure of human fetal membranes: An attempt towards a quantitative analysis

Objective We sought to measure the mechanical baseline behavior of fetal membranes in order to determine constitutive mechanical model parameters for fetal membranes, and to examine their relation to molecular correlates for mechanical function, i.e. collagen and elastin. Study design The uniaxial stress–strain response of nine human term fetal membranes was measured. Methods of nonlinear continuum mechanics were applied for the analysis of the stress–strain curves. Thickness of amnion and chorion were determined from histologic sections for each fetal membrane sample. Complementary biochemical analysis was performed to quantify the soluble collagen and soluble elastin components for each sample. Results We report a straightforward histologic modality for measurements of amnion and chorion thickness. Average thickness of the amnion and chorion layers were 111 ± 78 μm, and 431 ± 113 μm, respectively, which are about twice larger than previously reported. The average content of acid-soluble elastin was 2.1% of wet weight and the one of pepsin/acetic acid-soluble collagen was 10.5% of dry weight. Our data show an inverse proportionality between soluble elastin and soluble collagen content. The low strain elastic modulus ranged between 10 and 25 kPa. Correlations were found between biochemical data and mechanical parameters: there is clearly a direct proportionality between small strain elastic modulus and elastin content. Further, a (less pronounced) direct correlation was observed also between soluble collagen content and the parameter governing the increase in stiffness at larger strains in the nonlinear mechanical model. The mechanical tests revealed a relatively low variability for samples from the same membrane but a large variation between donors. The proposed nonlinear model provides a good fit of the experimental data, with a coefficient of determination, R 2, typically in the range of 0.94. Membranes failure originated at the clamping points thus impairing the quantification of ultimate stress and strain. Thus, no correlation was found between maximum stress and collagen or elastin content. Conclusions This study provides a starting point for comprehensive quantitative analysis of the relationship between fetal membranes microstructure and their nonlinear deformation behavior. These insights could become useful in identifying potential medical interventions to prevent membranes rupture.