Local Stress-strain Response Research Articles

Unveiling the quantitative relationship between microstructural features and quasi-static tensile properties in dual-phase titanium alloys based on data-driven neural networks

The quasi-static mechanical properties of α+β dual-phase titanium alloys are susceptible to their microstructural features, presenting a complex, high-dimensional nonlinear relationship, which hinders the rapid development of high-performance materials. In this work, 4065 micro-representative models were virtually constructed with varying volume fractions of α and β phases and characteristic dimensions via high-throughput finite element simulation, incorporating a cohesive zone model to simulate the interfaces between the two phases. Especially, the established representative models were experimentally verified by two groups of real material microstructures, and the results showed that the relative errors were not more than 9.5 % in microstructural characteristics and quasi-static mechanical properties. Afterward, a neural network model was developed to correlate the quasi-static tensile properties with the microstructural features of the dual-phase TC6 titanium alloys, achieving an 88.2 % accuracy in predicting overall mechanical performance. Utilizing the Shaply Additive Explanation method, it was found that the primary α phase's volume fraction and the secondary α phase's width were the most significant microstructural features affecting quasi-static strength. Specifically, the volume fraction of the primary α phase and the width of the secondary α phase negatively affected strength, while the width of the secondary α phase positively influenced plasticity. Notably, the primary α phase's volume fraction had a quadratic curve pattern of influence on plasticity. The intrinsic mechanisms behind these laws were further revealed based on local stress-strain responses and crack propagation analysis. Ultimately, the optimal microstructural features with strength-plasticity balance were identified through the lower threshold method: a secondary α phase width of about 1 μm and a primary α phase volume fraction ranging from 0.1 to 0.2, effectively facilitating microstructure design.

A Combined Numerical–Analytical Study for Notched Fatigue Crack Initiation Assessment in TRIP Steel: A Local Strain and a Fracture Mechanics Approach

In the fatigue design of metallic components using the safe-life approach, fatigue crack initiation as a development of slip systems at the nanoscale, followed by microstructurally short crack growth, is critical for the onset of structural failure. The development of reliable analytical tools for the prediction of crack initiation, although very complex due to the inherent multiscale fatigue damage processes involved, is important for promoting a more sophisticated design but, more importantly, enhancing the safety in regard to fatigue. The assessment of fatigue crack initiation life at the root of a V-shaped notch is performed by implementing a local strain and a fracture mechanics concept. In the low cycle fatigue analysis, the finite element method is used to determine the local stress–strain response at the notch root, which takes into account elastoplastic material behavior. Fatigue crack initiation is treated as the onset of a short corner crack by incremental damage accumulation and failure of a material element volume at the notch root. The finite element results are compared against established methodologies such as the Neuber and strain energy density methods. In the fracture mechanics approach, fatigue crack initiation is treated as the onset and propagation of a corner crack to a finite short crack. Fatigue experiments in two different transformation-induced plasticity (TRIP) steels were conducted to evaluate the analytical predictions and to determine the physical parameters for the definition of crack initiation. The analytical results show that the finite element method may be successfully implemented with existing fatigue models for a more accurate determination of the local stress–strain behavior at the notch tip in order to improve the assessment of fatigue crack initiation life compared to the established analytical methodologies.

A novel technique to measure the local mechanical properties of third generation advanced high strength steel resistance spot welds

Distinct microstructures and mechanical properties are produced in resistance spot welds (RSW) of advanced high-strength steels (AHSS). The local mechanical properties within the fusion zone (FZ) and heat-affected zones (HAZ) govern the global failure response of RSW automotive structural assemblies in crash events. In this study, a novel experimental procedure was developed to characterize the local properties of RSW sub-regions using sub-size tensile and simple shear specimens extracted from the spot weld, with local strain measurement utilizing digital image correlation (DIC) techniques. The FZ properties were obtained using a simple shear test, while the HAZ properties were characterized using a tensile test containing a central RSW. A range of AHSS grades were considered to evaluate the proposed methodology, including 3rd Gen-980, 3rd Gen-1180, and press-hardened steel (PHS1500) with respective minimum ultimate tensile strengths of 980 MPa, 1180 MPa, and 1500 MPa. Strain localization occurred wherever the hardness was lowest such as in the subcritical heat-affected zone, with limited deformation in the harder regions, such as the upper-critical HAZ and FZ. The local stress-strain responses were then compared to a hardness scaling approach to estimate the tensile properties of RSW based on the local hardness in each zone. The hardness scaling method only appears to be a viable method if the microstructures across the weld have similar microconstituents in all sub-zones across the weld.

Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)

The mechanical behavior of inelastic materials with microstructure is very complex and hard to grasp with heuristic, empirical constitutive models. For this purpose, multiscale, homogenization approaches are often used for performing reliable, accurate predictions of the macroscopic mechanical behavior of solids and structures. Nevertheless, the calculation cost of such approaches is extremely high and prohibitive for real-scale applications involving inelastic materials.Recently, data-driven approaches based on machine learning and, in particular, deep learning have risen as a promising alternative to replace ad-hoc constitutive laws and speed-up multiscale numerical methods. However, such approaches require huge amounts of high quality data, fail to give reliable predictions outside the training range and they lack a rigorous frame based on the laws of physics and thermodynamics. As a result, their application to model materials with complex microstructure in inelasticity is not yet established.Here, we propose the so-called Thermodynamics-based Artificial Neural Networks (TANN) for the constitutive modeling of materials with inelastic and complex microstructure. Our approach integrates thermodynamics-aware dimensionality reduction techniques and thermodynamics-based deep neural networks to identify, in an autonomous way, the constitutive laws and discover the internal state variables of complex inelastic materials. The ability of TANN in delivering high-fidelity, physically consistent predictions is demonstrated through several examples both at the microscopic and macroscopic scale.The efficiency and accuracy of TANN in predicting the average and local stress–strain response, the free-energy and the dissipation rate is demonstrated for both regular and perturbed two- and three-dimensional lattice microstructures in inelasticity. TANN manage to identify the internal state variables that characterize the inelastic deformation of the complex microstructural fields. These internal state variables are then used to reconstruct the microdeformation fields of the microstructure at a given state. Finally, a double-scale homogenization scheme (FEM×TANN) is used to solve a large scale boundary value problem. The high performance of the homogenized model using TANN is illustrated through detailed comparisons with microstructural calculations at large scale. An excellent agreement is shown for a variety of monotonous and cyclic stress–strain paths.

Effect of Internal Stirrups on the Eccentric Compression Behavior of FRP-Confined RC Columns Based on Finite-Element Analysis

Fiber-reinforced polymers (FRPs) have been widely used in the retrofitting and rehabilitation of reinforced concrete (RC) columns because they can provide substantial lateral confinement to concrete. Conventionally, the presence of internal stirrups is neglected in the retrofitting design due to a lack of understanding on the effect of internal hoops when both external FRP and internal steel confinement exist, particularly under eccentric loading. In this study, a finite-element (FE) method is developed to investigate the eccentric compression behavior of FRP-confined RC columns. A reliable concrete plastic-damage model that accurately considers the confining stiffness ratio between the FRP and steel is proposed for the first time, which facilitates the study of a difficult problem: the behavior of RC columns with the dual confinement system under eccentric loading. The accuracy of the model is verified by test results. An extensive parametric study of various affecting factors is carried out to investigate the confinement mechanisms of the concrete columns in terms of the axial stress and confining pressure distributions, lateral principal stress ratio, local stress–strain response, and interaction between the FRP/steel hoops and concrete. It is found that the nonuniform confining pressures under eccentric compression are significantly affected by the internal stirrups. With the existence of discrete transverse steel reinforcement, the axial resistance of the concrete core is increased, thereby improving the ultimate load capacity of the columns. An ultimate axial load model for RC columns that considered combined FRP–steel confinement is subsequently proposed. More accurate results are obtained using the proposed model compared with those calculated from existing design codes.

Transitions in the morphology and critical stresses of gliding dislocations in multiprincipal element alloys

Refractory multiprincipal element alloys (MPEAs) are promising material candidates for high-temperature, high-strength applications. However, their deformation mechanisms, particularly at the dislocation scale, are unusual compared to those of conventional alloys, making it challenging to understand the origin of their high strength. Here, using an atomistically informed phase-field dislocation dynamics model, we study transitions in the morphology and critical stresses of long gliding screw dislocations over extended distances in a refractory MPEA. The model MPEA crystal accounts for the atomic-scale fluctuations in chemical composition across the glide planes via spatially correlated lattice energies and for the differences in glide resistances between screw and edge dislocations of unit length. We show that the dislocation moves in a stop-start motion, alternating between wavy morphology in free flight and nearly recovered straight screw orientation in full arrest. The periods of wavy glide are due to variable kink-pair formation and migration rates along the length, where portions with higher rates glide more quickly. The critical stress to initiate motion corresponds to the stress required to form and migrate a kink pair at the weakest region along the length of the dislocation. Heterogeneity in lattice energy leads to variability in the local stress-strain response and to a strain hardening-like response, in which the critical stress to reactivate glide increases with glide distance. Statistical assessment of hundreds of realizations of dislocations indicates that the amount of hardening directly scales with the dispersion in underlying lattice energy.

A phase-field formulation for dynamic cohesive fracture

We extend a phase-field/gradient damage formulation for cohesive fracture to the dynamic case. The model is characterized by a regularized fracture energy that is linear in the damage field, as well as non-polynomial degradation functions. Two categories of degradation functions are examined, and a process to derive a given degradation function based on a local stress–strain response in the cohesive zone is presented. The resulting model is characterized by a linear elastic regime prior to the onset of damage, and controlled strain-softening thereafter. The governing equations are derived according to macro- and microforce balance theories, naturally accounting for the irreversible nature of the fracture process by introducing suitable constraints for the kinetics of the underlying microstructural changes. The model is complemented by an efficient staggered solution scheme based on an augmented Lagrangian method. Numerical examples demonstrate that the proposed model is a robust and effective method for simulating cohesive crack propagation, with particular emphasis on dynamic fracture.

On the nature of twin boundary-associated strengthening in Fe-Mn-C steel

We unravel the nature of twin boundary-associated strengthening in Fe-Mn-C twinning-induced plasticity steel (TWIPs) by micro-pillar compression tests. Dislocation interactions with a coherent twin boundary and their role on strain hardening were investigated. The results indicate that twin-matrix bundles dynamically introduced by deformation twinning and their interaction with dislocations are required for strengthening Fe-Mn-C TWIPs, while single coherent twin boundaries enable dislocation transmission. Correlative studies on orientation dependent deformation mechanisms, detailed dislocation-twin boundary interactions, and the resulting local stress-strain responses suggest that twin boundary-associated strengthening is primarily caused by the reduction of the mean free dislocation path in nano-twinned microstructures.

Numerical simulation of progressive damage and failure in composite laminates using XFEM/CZM coupled approach

This study seeks to establish a high-fidelity mesoscale simulation methodology that can predict the progressive damage and resultant failure of carbon fiber reinforced plastic laminates (CFRPs). In the proposed scheme, the plastic behavior (i.e., pre-peak nonlinear hardening in the local stress-strain response) is characterized through the pressure-dependent elasto-plastic constitutive law. The evolution of matrix cracking and delamination, which result in post-peak softening in the local stress-strain response, is modeled through cohesive zone models (CZM). The CZM for delamination is introduced through an interface element, but the CZM for matrix cracking is introduced through an extended finite element method (XFEM). Additionally, longitudinal failure, which is dominated by fiber breakage and typically depends on the specimen size, is modeled by the Weibull criterion. The validity of the proposed methodology was tested against an off-axis compression (OAC) test of unidirectional (UD) laminates and an open-hole tensile (OHT) test of quasi-isotropic (QI) laminates. Finally, sensitivity studies were performed to investigate the effect of plasticity and thermal residual stress against the prediction accuracy in the OHT simulation.

Allowable stresses in high-frequency mechanical impact (HFMI)-treated joints subjected to variable amplitude loading

The effectiveness of high-frequency mechanical impact (HFMI) is considered to rely on the existence of compressive residual stresses. To determine when residual stress relaxation occurs, and what the resulting influence on fatigue improvement is, local stress-strain response in as-welded and HFMI-treated weld toes was modelled under different peak stress conditions. Then, effective notch stress analysis was used to correlate these results with available experimental observations. The simulations showed that high stress ratios and compressive peak stresses were critical with respect to residual stress relaxation, as expected. A compressive peak stress of 0.6f y (nominal yield strength) resulted in full residual stress relaxation. The relative fatigue damage calculations and the notch stress analysis indicated, however, that fatigue improvement could be expected even after significant residual stress relaxation. Based on this and previously observed benefit for high stress ratios, an increase in maximum allowable stresses for HFMI-treated welded steel joints is suggested. The maximum stress ratio is proposed to be increased from R = 0.52 to R = 0.7, and the maximum stress range to limit compressive stresses is proposed to be increased from ΔS max = 0.9f y to ΔS max = 1.2f y , which corresponds to S min = −0.6f y for stress ratio R = −1.

A Non-Contact Test Method on Local Mechanical Properties of Weld Joint

A comprehensive study was carried out on the determination of mechanical parameters in the weld zone by using the digital image correlation (DIC) method. The DIC method provided quantitative two-dimensional strain maps of the deformation field across the weld joint throughout the tensile test. Then the local stress-strain response was extracted based on these strain maps on the uniform stress hypothesis. The strain distribution evolution during loading performed a good consistency with the indentation test results and the cracking position of the specimen. The deformation related parameters of local areas in weld joint were successfully extracted by DIC method. These parameters are very valuable for the structure design of the weld components, especially in the case of the structure in which the allowable stress is controlled by the deformation feature but not the strength of the material such as the high temperature applications.

Crystal Plasticity Finite-Element Simulation of Single Phase Titanium Alloy with 3D Polycrystalline Models

Based on the rate-dependent crystal plasticity theory, a finite element code which considers crystallographic slip as deformation mechanism of material was developed to investigate the stress–strain response of the β phase of Ti-5553 during uniaxial tension. Three dimensional models with random grain shapes generated by Voronoi tessellation were used for simulations, and two discretization methods were used to disperse the models. Firstly, the parameters of material were identified by fitting simulation stress-strain curves with experimental data. Then the global stress-strain curves were calculated, and effects of mesh type and mesh density were discussed. Results show that mesh type has a relatively significant influence on overall responses, whereas the influence of mesh density is slight. Investigate of local stress-strain response in each grain was also conducted, and obvious inter-granular heterogeneities were observed. Quantitative analysis indicates that the range of stress and strain variations is affected by mesh type.

Nucleation of Recrystallization in Mg Alloys during and after Hot Working

Recrystallization nucleates at heterogeneities. The impact of this on local texture and stress-strain response in hot worked magnesium is considered in the present paper. Two aspects of bulge nucleation during dynamic recrystallization are considered.

In Situ Evaluation of Tensile Properties of Heat‐Affected Zones from Welded Steel Pipes

Abstract: Currently, measuring the local tensile strength in inhomogeneous materials is not standardised, nor accepted techniques are available despite such technique would be beneficial in a variety of technological applications. Thus, this work introduces an innovative method for assessing stress–strain properties at a sub‐millimeter scale and illustrates the potential of the technique by evaluating the strength of a sub‐region in the HAZ from welded steels pipes. The method employs a fully instrumented stage inside a scanning electron microscope that stretches small tensile specimens (2.0 mm × 0.5 mm cross‐section, 12.5 mm gage length) while registering detailed images of the deformed region. The specimens, cut from full‐scale welds, include in their gage length weld metal, base metal and HAZ and have an 85 μm period grid of evaporated lead on their surface to visualise the deformation. Upon straining, local strain is determined by correlating sequential images of the specimen surface with an open source code for particle image velocimetry. The calculated local strain within the HAZ and the load values recorded during testing are converted into a local stress–strain response. The results for two different heat inputs agree with usual, but indirect and less accurate assessments procedures, including local hardness measurements and notched bar testing.

Elastoplastic finite element analysis and fatigue life prediction for notched specimens under biaxial cyclic loading

An elastoplastic finite element analysis is adopted for thin tubular and notched specimens under biaxial proportional and nonproportional tension-torsion loading at high temperature. Elastoplastic material property is described by the Von Mises yield criterion, the multilinear kinematic hardening rule and uniaxial cyclic stress-strain data under high temperature. Tension-torsion loading with strain controlling is carried out by applying the axial and circular displacement on one end of the specimen in the cylindrical coordinate system. The method is proved suitable by comparison between FEA and experimental results. Then the elastoplastic finite element method is used to the notched specimens. The local cyclic stress-strain responses at notch root are obtained. Fatigue crack initiation life is predicted on the basis of the local stress-strain data with Kandil-Brown-Miller and Smith-Watson-Topper models.

Fatigue Analysis of Notched Shafts under Multiaxial Synchronous Cyclic Loading

Parametrical elastic-plastic finite element analyses of a circumferentially notched shaft subjected to multiaxial synchronous fatigue loading are performed considering two load combinations: (1) constant tension with cyclic torsion and (2) constant torsion with cyclic tensioncompression. The load amplitudes and the mean loads are varied to investigate their influences on the local stress-strain responses. The Multilayer Plasticity Model of Besseling in conjunction with the von Mises yield criterion is applied to describe the elastic-plastic material behavior. Coarse and fine meshes as well as three different types of multilinear approximations (twenty-, five- and threesegments) of the material stress-strain curve are used. Numerical results are presented to reveal the mutual interactions between the applied normal and torsional loads and the stress-strain response at the notch-root.

The constitutive behavior of laser welds in 304L stainless steel determined by digital image correlation

A digital image correlation (DIC) method has been used to characterize the constitutive tensile stress-strain response in 304L austenitic stainless steel weldments produced by both continuous-wave (CW) and pulsed-wave (PW) laser welding. The method provides quantitative two-dimensional (2-D) strain maps of the deformation field across the transverse weld samples throughout the tensile test. Local stress-strain response was extracted from regions within the fusion zone and compared to base metal response. The weldments were found to have a higher yield strength than the base metal. The metallurgical origin for the fusion zone strengthening was largely attributed to Hall-Petch and ferrite content effects. While failures localized in the fusion zone with little appreciable necking, the material within the fusion zone retained considerable local ductility: more than 45 pct strain at failure. Significant weld root porosity found in the PW condition and absent in the CW condition appeared to have no deleterious effect on the mechanical performance under the present test conditions in this very ductile, flaw-tolerant alloy.

Fourier series for computing the response of periodic structures with arbitrary stiffness distribution

A method based on Fourier series is presented, which allows to calculate the local stress–strain response of a three-dimensional periodic structure subjected to a spatial average of strain. The periodicity allows the reduction of the problem to that of a Representative Volume Element (RVE). The solution operator (which can easily be calculated in Fourier space) is defined for a simplified problem, and it is shown that this operator may also be used for the original problem. In order to illustrate the use of this procedure, an example problem is presented. A global error is defined and calculated for the example problem.

Characterisation of the creep-fatigue behaviour of a 1CrMoV turbine steel

This paper concerns the creep-fatigue lifetime assessment of a 1CrMoV steel. A procedure based on the calculation of fatigue and creep damage fractions and their summation according to a non-linear interaction rule is applied for the endurance evaluation of isothermal fatigue and service-like thermo-mechanical fatigue tests performed with plain (uniaxial) and notched testpieces. In order to avoid the uncertainties associated with heat-to-heat variability, testpieces were manufactured from a single heat, for which the fatigue and creep deformation/endurance properties were specifically characterised. With the availability of a reliable constitutive model, finite element calculations could be used to determine the local stress-strain response at the position of crack initiation in each test. The predictive capabilities of the proposed approach for lifetime calculation as well as the suitability of uniaxial fatigue and creep data for the damage assessment of notched components are evaluated.

Utility of micro-indentation technique for characterization of the constitutive behavior of skin and interior microstructures of die-cast magnesium alloys

There has been increasing thrust lately on the development of lightweight cast magnesium alloy components for structural automotive and other applications. The microstructure of the high-pressure die-cast Mg alloys usually contains a fine-grained “skin” having a microstructure significantly different from that of the bulk material. Characterization of the local constitutive behavior of the skin microstructure is of interest as it can affect the overall mechanical response of the component. However, the standard mechanical tests on the macro-specimens are not useful for characterization of the local stress–strain response of the skin microstructure. In this contribution, we present a novel methodology based on a combination of micro-indentation experiments and three-dimensional (3D) finite elements based simulations that permits computation of the local stress–strain (constitutive) behavior of the skin and the interior microstructures at the length scales of 100 μm in a cast high-pressure die-cast AM60 Mg-alloy. The methodology involves development of a numerical solution to the inverse problem. The computed constitutive equations are then utilized to simulate the effect of skin thickness on the overall global mechanical response of the alloy under uniaxial compression.