Elastic Strain Components Research Articles

Effect of Microstructure on Electromigration-Induced Stress

In this paper, a finite element based simulation approach for predicting the effect of microstructure on the stresses resulting from electromigration-induced diffusion is described. The electromigration and stress-driven diffusion equation is solved coupled to the mechanical equilibrium and elastic constitutive equation, where a diffusional inelastic strain is introduced. Here, the focus is on the steady state, infinite life case, when the current-driven diffusion is balanced by the resulting stress gradient. The effect of the crystal orientation in Sn-based solder joints on the limiting current density for an infinite life is investigated and compared to experimental observations in the literature. The effect of the grain structure for Al interconnect lines on the dominant diffusion path and estimates for the effective charge number for two different diffusion paths in Al interconnects determined by matching simulations to experimental measurements of elastic strain components in the literature are also presented.

Photo‐Thermo‐Mechanical Behaviour Under Quasi‐Static Tensile Conditions of a PMMA‐Core Optical Fibre

AbstractAs the optical power transmitted by an optical fibre under tensile stress varies with strain, it can be used as a sensor for strain monitoring in structural elements. In the present work, quasi‐static tensile tests of step index polymer optical fibres (POF) with simultaneous measurement of surface temperature and optical power are described. Young's modulus, yield stress and tensile strength are derived from experimental tests. Morphological characterization of the POF fibres using scanning electron microscope images and differential calorimetry technique is performed. The contributions of both elastic and plastic strain components to the variation of temperature and optical power loss are also estimated. The evolution of the POF mechanical properties as well as that of temperature and optical power loss is explained in terms of the progressive relative movement and alignment of the molecular chains in the direction of the applied load. Strain, temperature and optical power loss are then correlated.

A phenomenological stress–strain model for wrought magnesium alloys under elastoplastic strain-controlled variable amplitude loading

Wrought magnesium alloys typically reveal strong basal textures and thus, non-symmetric sigmoidal shaped hysteresis loops within the elastoplastic load range. A detailed description of those hysteresis loops is necessary for numerical fatigue analyses. Therefore, a one-dimensional phenomenological model was developed for elastoplastic strain-controlled constant and variable amplitude loading. The phenomenological model consists of a three-component equation, which considers elastic, plastic, and pseudoelastic strain components with a set of eight material constants. Experimentally and numerically determined hysteresis loops of four different magnesium alloys were compared by means of different examples with constant and variable amplitude. Good correlation is reached and the relevant fatigue parameters like strain energy density were estimated with good accuracy. Applying an energy based fatigue parameter on modelled hysteresis loops, the fatigue life is predicted adequately for constant and variable amplitude loading including mean strain and mean stress effects.

Effect of the configuration of a wrinklon on the distributions of the energy and elastic strain in a graphene nanoribbon

The formation of wrinkles in thin membranes is a widespread phenomenon. In particular, wrinkles can appear in graphene, which is the thinnest natural membrane, and affect its properties. A region where wrinkles with different wavelengths are linked is called wrinklon. Conditions of the fixing of an elastically deformed graphene sheet dictate a certain wavelength of wrinkles near the fixed edge. Wrinkles with a longer wavelength become more energetically favorable with an increase in the distance from the edge. As a result, wrinklons appear and reduce the potential energy of the system by uniting wrinkles into larger wrinkles with an increase in the distance from the edge. The possibility of implementing various equilibrium configurations of wrinklons at given plane strains in graphene has been demonstrated by the molecular quasistatic method. The distributions of the energy and elastic strain components in wrinklons with various configurations for nanoribbons with different widths have been calculated.

Improved methodology for determining tensile elastic and plastic strain components of alloy 617

This paper presents an improved methodology for calculating tensile elastic and plastic strain components in total creep strain, which are needed in the generation of the isochronous stress-strain curves (ISSC). The procedures for the RCC-MR code and Blackburn’s methods to determine the total strain are analyzed in detail, and a new method is proposed to determine the tensile elastic and plastic strain components. To practically obtain the tensile elastic and plastic strain components for Alloy 617, high-temperature tensile tests were conducted at 800, 850, 900, and 950°C. Using the tested data, three methods of the RCC-MR code, Blackburn, and the newly proposed method were applied and compared. In the elastic regime, the three methods were identical in the tensile curves, but in the plastic regime, Blackburn’s method was higher in the stress-strain curves than the RCC-MR code. The difference gap for the two methods was larger with an increase in the plastic strain. In addition, the proposed new method revealed almost midway values for the two methods. It was suggested that the proposed method is useful to determine tensile and plastic strain components.

Low cycle fatigue behaviour of HAYNES 230 alloy

HAYNES® 230® alloy is a commercial solid–solution strengthened Ni-base superalloy that has found wide spread use in the hot sections of gas turbines. Recently, the alloy has been considered for other applications, e.g. in concentrated solar power plants. One common key performance criterion for all these applications is the ability of the material to withstand large numbers of thermal cycles due to start-ups, operatings, and shut-downs. Such thermal cycles may trigger low-cycle fatigue (LCF) due to differential thermal expansion and contraction. This paper reports isothermal LCF performance data of 230 alloy standard mill-annealed plate product for a wide range of temperatures 427–982°C and strain ranges (up to 1·5%), relevant to many of the aforementioned applications. Low-cycle fatigue lives are analysed in terms of the contributions of cyclic elastic and plastic strain components to fatigue damage. The fit parameters are discussed in the context of the alloy’s tensile properties.

On the Evaluation of the Stress State in Rail Head for Assessing Fatigue Resistance

To search for a single parameter to evaluate the stress state in rail head during wheel/rail rolling contact situations, the stress-based and the strain based phenomenological approaches for multiaxial fatigue analysis can be considered as the candidates. Following the stress-based approach, the maximum von Mises stress range can be applied as a single parameter to evaluate the stress state in the rail head. However, the von Mises stress range only relies on the stress field in the rail head for the fatigue analysis, which is not sufficient for assessing the fatigue resistance of the rail steel. The Smith-Watson-Topper (SWT) method, the strain-based phenomenological approach for multiaxial fatigue analysis which considers stress, elastic strain and plastic strain components, is then adopted to study rolling contact fatigue in the rail head. Combining with the three-dimensional finite element modelling of a steady-state wheel/rail rolling contact, the numerical procedure to calculate the SWT parameter in the rail head is presented. The capability of the SWT method to predict the initiation of fatigue cracks in the rail head is confirmed in a case study. Consequently, the maximum SWT parameter is proposed as a single parameter to effectively evaluate the stress state in the rail head.

Russeting in apple and pear: a plastic periderm replaces a stiff cuticle

Russeting in apples (Malus × domestica Borkh.) and pears (Pyrus communis L.) is a disorder of the fruit skin that results from microscopic cracks in the cuticle and the subsequent formation of a periderm. To better understand russeting, rheological properties of cuticular membranes (CM) and periderm membranes (PM) were studied from the russet-sensitive apple 'Karmijn de Sonnaville' and from 'Conference' pear. The CM and PM were isolated enzymatically, investigated by microscopy and subjected to tensile tests, creep/relaxation tests and to stepwise creep tests using a material testing machine. The isolated CM formed a continuous polymer, whereas the PM represented a cellular structure of stacked cork cells. Tensile tests revealed higher plasticity of the hydrated PM compared with the CM, as indicated by a higher strain at the maximum force (ɛ(max)) and a lower modulus of elasticity (E). In apple, the maximum force (F(max)) was higher in the CM than in the PM but in pear the higher F(max) value was found for the PM. In specimens obtained from the CM : PM transition zone, the weak point in apple was found to be at the CM : PM borderline but in pear it was within the CM. In both apple and pear, creep/relaxation tests revealed elastic strain, creep strain, viscoelastic strain and viscous strain components in both the PM and CM. For any particular force, strains were always greater in the PM than in the CM and were also greater in pear than in apple. The ɛ(max) and F(max) values of the CM and PM were lower than those of non-russeted and russeted whole-fruit skin segments, which included adhering tissue. In russeting, stiff CM are replaced by more plastic PM. Further, the cell layers underlying the CM and PM represent the load-bearing structure in the fruit skin in apple and pear.

Nonlinear elastic behaviors of low and high strength steels in unloading and reloading

The elastic moduli of four sheet steels were measured in continuous loading–unloading–loading (LUL) tests after different amounts of plastic pre-strain. The four materials were a low-strength steel (Mild270), a bake-hardenable steel (BH340) and two advanced high-strength steels (AHSS), DP490 and DP590. Hysteresis loops were observed due to a non-linearity of the elastic modulus during the unloading and reloading cycles. The total strain recovery during unloading could be separated into a linear part and a non-linear part. The latter was found to be proportional to the unloading stress for all materials. The unloading and reloading elastic moduli decreased as the plastic pre-strain increased and reached saturation towards specific constant values. They could well be approximated as a function of the plastic strain with an empirical exponential-type model, although there were some deviations for BH340. The baking heat treatment and different strain rate had little influence on the unloading and reloading elastic moduli. The increase of the non-linear elastic strain component as a function of the unloading stress could be explained by the Kocks–Mecking (KM) model in which the micro-plastic strain is related to the mobile dislocation density.

Stress and Strain in the Sweet Cherry Skin

Rain-cracking of sweet cherry (Prunus avium L.) fruit involves failure of the exocarp caused by excessive stress and strain. The objective of our study was to quantify exocarp strain in developing cherries. The release of linear elastic strain was followed in vivo using a gaping assay, whereas the release of biaxial elastic strain was followed in vitro after excision of small exocarp segments (ESs) that were submerged in silicone oil and strain release quantified by image analysis. When mature sweet cherry fruit were cut (by making two or more deep, longitudinal incisions parallel to the stylar/pedicel axis and on opposing sides of the fruit down to the pit), the incisions rapidly “gaped.” The gaping wounds continued to widen as they progressively released the linear elastic strain in the skin. By 24 hours the combined widths of two gapes represented 8.8% ± 0.1% of the fruit circumference. Increasing the number of cuts from two to 12 increased the cumulative gape widths to 14.9% ± 0.2%. In ES, monitoring the time course of relaxation after excision revealed a rapid release of biaxial strain, having a half-time of ≈2.7 minutes. Relaxation continued, but at a decreasing rate, for up to 48 hours. Across eight cherry cultivars, the biaxial strain in the exocarp at maturity ranged from 18.7% ± 1.9% in ‘Lapins’ to 36.0% ± 1.8% in ‘Katalin’. Elastic strain in the ES was always lower than that measured in an isolated cuticular membrane (CM). Increasing the temperature from 2 to 35 °C increased the rate of strain release and also the total percent strain released at 96 hours. In developing ‘Hedelfinger’ sweet cherry fruit, there was essentially no elastic strain in the exocarp at 45 days after full bloom (DAFB). Thereafter, significant elastic strain developed, reaching a maximum of 47.6% ± 2.5% at 87 DAFB. The effect of exocarp cell turgor on strain in the ES (evidenced by the difference in the reversible strain between ES with and without turgor) was closely and positively related to the relative area growth rate of the skin (r2 = 0.957). Strain release peaked at ≈59 DAFB, and there was no effect of turgor on strain release in mature fruit. Our data demonstrated the following: 1) the exocarp is a viscoelastic material composite; 2) at maturity, plastic and elastic strain components make up 66% and 34% of the total percent strain, respectively; 3) elastic strain in the exocarp increases during Stage III development; and 4) the strain in the exocarp is unaffected by strain in the CM. Thus, the epidermis and hypodermis layers must represent the main, load-bearing structure in sweet cherry fruit with the cuticle making a mechanically insignificant contribution.

Design of thin-film polyvinylidene fluoride sensor rosettes for isolation of various strain components

Thin-film polyvinylidene fluoride piezoelectric sensors have long been recognized as a promising alternative to traditional metal foil strain gauges in applications where only dynamic or quasistatic signals are of interest. Compared to metal foil strain gauges, polyvinylidene fluoride sensors feature high sensitivity, high dynamic range, and broad frequency bandwidth. However, transverse sensitivity of the polyvinylidene fluoride sensor is higher than that of a metal foil strain gauge, making it more difficult to isolate a particular strain component or a deformation mode when the host structure is under complex loading. In addition, polyvinylidene fluoride films are sensitive to changes in ambient temperature due to the pyroelectric effect. In this article, three temperature-compensated polyvinylidene fluoride sensor rosette designs are proposed for isolating specific strain component(s) and deformation mode(s) of interest. First-principles based models are derived to relate the polyvinylidene fluoride sensor rosette output to the actual elastic strain component of interest. Experimental validation is conducted to verify the proposed models and to compare the performance of the polyvinylidene fluoride sensor rosettes with their metal foil strain gauge counterparts.

Thermomechanical analysis of residual stresses in brazed diamond metal joints using Raman spectroscopy and finite element simulation

Thermal residual stresses are one of the crucial parameters in engineered grinding tool (EGT) life and its consistency. Predicting failure of brazed diamond metal joints in EGTs is related to analyzing the thermal residual stresses during the cooling process. Thus thermal residual stresses have been simulated in a model with realistic materials properties, for instance isotropic hardening and a hyperbolic-sine creep law for SS316L and the silver–copper–titanium active filler alloy, named Cusil ABA™. Also, special modeling techniques such as tie constraint and sub-modeling have been used to model an intermetallic layer titanium-carbide (TiC) with dimensions in nanometers, where the rest of the model’s dimensions are in millimeters. To verify the simulated stress state of the diamond, Raman-active optical phonon modes at three different paths in the diamond were measured. As the experiments with Raman spectroscopy (RS) do not deliver stress components, the solution is to directly compute the peak shift of Raman spectrum. The splitting in phonon frequencies and the mixing of phonon modes contain information about the thermal residual stresses in the diamond. Finally the shift in the phonon frequencies was calculated from the different numerical residual elastic strain components and compared to the experimental results.

The Modulation of Jahn–Teller Coupling by Elastic and Binding Strain Perturbations—A Novel View on an Old Phenomenon and Examples from Solid-State Chemistry

Cations in 6-coordination with orbitally degenerate E(g) ground states, such as Cu(2+) and low-spin Co(2+), play an important role in coordination chemistry-in particular, in modern complex biochemistry. The stereochemistry and the binding properties within the basic polyhedra are the subject of pronounced modifications due to vibronic coupling in such cases, but may be also significantly influenced by what is usually called an imposed strain. The latter effect makes allowance for the general observation that the host sites into which the Jahn-Teller unstable centers are substituted are seldom of O(h) symmetry and built from six equal ligands. Hence, the finally observed molecular and binding structure of the pseudo-octahedral complex is the result of the combined action of vibronic coupling and strain. The closer analysis of host-site strain effects demands to distinguish between elastic strain components, which modify the force constant of the vibronically active (here, ε(g)) vibration, and binding strain perturbations, which take account of possibly present ligands with different binding properties. A symmetry-met semiempirical strain model on such a basis is presented and a corresponding formulation within the vibronic coupling formalism is given, on the molecular level. Well-established model examples of Cu(2+) in octahedral fluoride coordination in various host solids, where a great variety of experimental results is available, are given. The derived parameters allow a detailed characterization of the structural and energy qualities of the Jahn-Teller centers, and might help to steer these properties in cases where synthesis strategies are needed. The proposed strain concept is more complex than that of Ham [F. S. Ham, Electron Paramagnetic Resonance; Plenum Press: New York, 1972; F. S. Ham, Phys. Rev. 1965, A138, 1727]; the advantage is that it is directly tied to the structure and energy of the Jahn-Teller complex in focus, although more data (experimental and possibly computed) are needed in such a model.

Influence of high-cycle fatigue on the tension stiffening behavior of flexural reinforced lightweight aggregate concrete beams

The objective of this study was to experimentally investigate the bond-related tension stiffening behavior of flexural reinforced concrete (RC) beams made with lightweight aggregate concrete (LWAC) under various high-cycle fatigue loading conditions. Based on strain measurements of tensile steel in the RC beams, fatigue-induced degradation of tension stiffening effects was evaluated and was, compared to reinforced normal weight concrete (NWC) beams with equal concrete compressive strengths (40 MPa). According to applied load-mean steel strain relationships, the mean steel strain that developed under loading cycles was divided into elastic and plastic strain components. The experimental results showed that, in the high-cycle fatigue regime, the tension stiffening behavior of LWAC beams was different from that of NWC beams; LWAC beams had a lesser reduction in tension stiffening due to a better bond between steel and concrete. This was reflected in the stability of the elastic mean steel strains and in the higher degree of local plasticity that developed at the primary flexural cracks.

High-temperature yield strength and its dependence on primary creep and recovery

Elasto-Delayed-Elastic-Viscous (EDEV), a predictive material model for ‘quasi constant-structure’ primary creep is used in developing an algorithm for predicting stress–strain diagrams at constant strain rates. The strain rate dependence of the 0.2% offset yield-strength ( σ y), time to yield, and the amounts of permanent (viscous) and the recoverable (anelastic or delayed elastic) strain components of the 0.2% offset strain can also be evaluated. Six material constants required for calculations can be obtained from short-term (few seconds to a few hours), constant-stress Strain Relaxation and Recovery Tests (SRRTs). The method is illustrated by calculations made for Ni-based fcc alloy, Waspaloy at 732 °C (0.62 T m) and Ti-based, alpha (hexagonal)-beta(bcc) alloy Ti-6246 at 600 °C (0.45 T m). In the ASTM recommended strain-rate range, 5 × 10 −5 s −1 to 1.2 × 10 −4 s −1, σ y increases by 6% for Waspaloy and 15% for Ti-6246. In this range, the 0.2% offset strain consists of 50% delayed elastic (anelastic) strain for Waspaloy and 70% for Ti-6246. Proportional limit is governed entirely by the delayed elastic effect or the Elasto-Delayed-Elastic (EDE) regime. Intragranular dislocation creep does not play the dominant role up to yield for the materials and temperatures considered, suggesting a shift in paradigm from conventional ideas. The EDEV model, in conjunction with SRRTs (which can be realized using a single specimen), can be used as a tool for characterizing and developing new alloys and ceramics or optimizing processing including grain-boundary engineering.

Closed-Form Solutions for a Circular Tunnel in Elastic-Brittle-Plastic Ground with the Original and Generalized Hoek–Brown Failure Criteria

The paper presents a closed-form solution for the convergence curve of a circular tunnel in an elasto-brittle-plastic rock mass with both the Hoek–Brown and generalized Hoek–Brown failure criteria, and a linear flow rule, i.e., the ratio between the minor and major plastic strain increments is constant. The improvement over the original solution of Brown et al. (J Geotech Eng ASCE 109(1):15–39, 1983) consists of taking into account the elastic strain variation in the plastic annulus, which was assumed to be fixed in the original solution by Brown et al. The improvement over Carranza-Torres’ solution (Int J Rock Mech Min Sci 41(Suppl 1):629–639, 2004) consists of providing a closed-form solution, rather than resorting to numerical integration of an ordinary differential equation. The presented solution, by rigorously following the theory of plasticity, takes into account that the elastic strain components change with radial and circumferential stress changes within the plastic annulus. For the original Hoek–Brown failure criterion, disregarding the elastic strain change leads to underestimate the convergence by up to 55%. For a rock mass failing according to the generalized Hoek–Brown failure criterion, using the original failure criterion leads to a high probability (97%) of underestimating the convergence by up to 100%. As a consequence, the onset or degree of squeezing may be underestimated, and the loading on the support/reinforcement calculated with the convergence/confinement method may be largely underestimated.

Analysis of Constant and Variable Amplitude Strain-Life Data Using a Novel Probabilistic Weibull Regression Model

The relation between the total strain amplitude and the fatigue life measured in cycles is usually given as strain-life curves based on the former proposals of Basquin, for the elastic strain-life, and Coffin–Manson, for the plastic strain-life. In this paper, a novel Weibull regression model, based on an existing well established Weibull model for the statistical assessment of stress-life fatigue data, is proposed for the probabilistic definition of the strain-life field. This approach arises from sound statistical and physical assumptions and not from an empirical proposal insufficiently supported. It provides an analytical probabilistic definition of the whole strain-life field as quantile curves, both in the low-cycle and high-cycle fatigue regions. The proposed model deals directly with the total strain, without the need of separating its elastic and plastic strain components, permit dealing with run-outs, and can be applied for probabilistic lifetime prediction using damage accumulation. The parameters of the model can be estimated using different well established methods proposed in the fatigue literature, in particular, the maximum likelihood and the two-stage methods. In this work, the proposed model is applied to analyze fatigue data, available for a pressure vessel material—the P355NL1 steel—consisting of constant amplitude, block, and spectrum loading, applied to smooth specimens, previously obtained and published by authors. A new scheme to deal with variable amplitude loading in the background of the proposed regression strain-life Weibull model is described. The possibility to identify the model constants using both constant amplitude and two-block loading data is discussed. It is demonstrated that the proposed probabilistic model is able to correlate the constant amplitude strain-life data. Furthermore, it can be used to correlate the variable amplitude fatigue data if the model constants are derived from two-block loading data. The proposed probabilistic regression model is suitable for reliability analysis of notched details in the framework of the local approaches.

A DFT Study of the Energetical and Structural Landscape of the Tetrahedral to Square‐Planar Conversion of Tetrahalide Complexes of Copper(II)

AbstractDFT results and the analysis of the energetic and structural landscape of the title complexes are reported. It is shown that the minima of the ground state potential energy surface occur at positions corresponding to tetragonally compressed structures of D2d symmetry, with rather distinct electronic stabilization energies in respect to the tetrahedral parent complexes. The structural distortion and the accompanying energetic effects are found to increase with increasing ligand hardness, i.e. from Br– to Cl– to F–. The results are discussed and compared by conventional vibronic coupling and angular overlap model calculations. The Jahn–Teller interaction between the 2T2 ground state and the ϵ‐vibrational mode of tetrahedral CuX42– splits the 2T2 ground state; this coupling is supported by 3d–4s mixing, which softens the ground state potential energy surface along the ϵ normal mode [formally a T2(d9) ⊕ ϵ ⊕ T2 (d8s1) pseudo Jahn–Teller coupling interaction in Td]. The vibronic activity may also occur along the trigonal τ2 modes, which leads to C3v‐ or, if both types of pathway are involved, C2v‐ [2T2 ⊕ (ϵ + τ2) coupling] distorted structures, which are very weak as compared to D2d deformations. Theoretical results are validated and used to explain the structural diversity and electronic spectra reported for CuCl42– in various crystal structures. A strain model, which differentiates between an elastic and a binding component, is qualified for getting an understanding for the appearance of structures from tetrahedral to square planar. In the considered cases, it is generally the elastic strain component, which governs the structural and energetic scene, though the binding strain increment is not negligible sometimes (e.g. Cs2CuCl4). Predictions in respect to corresponding compounds with F– and Br– ligands are made.

Modified Schapery’s Model for Asphalt Sand

Asphalt sand is described as a viscoelastoplastic materials. To involve elastic, viscoelastic, and viscoplastic strain components, a modified Schapery’s model is proposed through adding a viscoplastic term to the Schapery’s model with elastic and viscoelastic terms, and a strain-hardening model is used to describe viscoplastic behavior. A set of uniaxial compression tests, repeated and simple creep-recovery tests in different stress levels are performed to determine the stress-dependent parameters in the model. Finally, the model is validated by comparison with the results from the uniaxial creep experiments. Better agreement than the Schapery’s model indicates that the modified model is practicable to describe the mechanical behavior of asphalt sand in uniaxial compression.

Analytical solution of stress-strain relationship of modified Cam clay in undrained shear

The modified Cam clay (MCC) model is used to study the response of virgin compressed clay in undrained compression. The MCC deviatoric stress-strain relationship is obtained in closed form. Elastic and plastic deviatoric strains are taken into account in the analysis. For the determination of the elastic strain components, both a variable shear modulus and constant shear modulus are considered. Constitutive relationships are applied to the well-known London and Weald clays sheared in undrained compression.