Elastic Strain Evolution Research Articles

Residual elastic strain evolution due to thermal cycling of a ceramic-metal composite (WC-Cu) via high energy X-ray diffraction and analytical modeling

Residual stress, when superimposed with in-service loading, can significantly reduce the lifetime and performance of a component. Ceramic-metal composites are susceptible to residual stresses due to the thermal expansion mismatch of the ceramic and metallic phases. The WC-Cu composite explored in the present study provides a promising combination of thermal conductivity and strength properties, while exhibiting counterintuitive improvements in strength and ductility after thermal cycling. This work quantifies the evolution of the residual elastic strains as a result of processing and cyclic thermal loading in a co-continuous WC-Cu composite through experimental high energy X-ray diffraction and kinetics-based modeling. Both analyses indicate that processing-induced residual tensile stress in the copper phase is relieved upon subsequent thermal cycling, with kinetics modeling revealing the cyclic-dependent nature of the active power-law creep mechanisms. The results indicate that, through stress relaxation, this material system maintains structural stability during thermal cycling. The illustrated kinetics of relaxation can inform general material processors and designers of ceramic-metal composites to minimize detrimental residual stress and improve performance of these material systems.

An elastic‐inelastic model and embedded bounce‐back control for layered printing with cementitious materials

AbstractThis paper presents a finite‐deformation model for extrusion‐based layered printing with cementitious materials. The evolution of mechanical properties as the printed material cures and stiffens results in nonphysical reduction in the magnitude of elastic strains when standard constitutive models are employed. This elastic recovery of the printing induced deformation contradicts the experimentally observed behavior of the printed cementitious materials that harden at a nearly‐frozen deformed state. A thermodynamically motivated constraint on the evolution of elastic strains is imposed on the constitutive model to remedy the nonphysical bounce‐back effect. An algorithm that is based on a strain‐projection technique for the elastic part of deformation is developed that complements the inelastic response given by the Drucker–Prager model. It is then embedded in a finite strain finite element framework for the modeling and simulation of cure hardening and inelastic response of the early age cementitious materials. A ghost mesh method is proposed for continuous layer‐wise printing of the material without the need for intermittent mesh generation technique or adaptive remeshing methods. The model is validated via comparison with experimental data and representative test cases are presented that investigate the mathematical and computational attributes of the proposed model.

Interplay of dislocation substructure and elastic strain evolution in additively manufactured Inconel 625

The unique thermal history of direct metal laser sintering (DMLS) leads to complex microstructures and local elastic residual strains which accumulate to affect the global residual stress and mechanical properties of components. Characterization of residual stress using neutron and x-ray diffraction result in bulk and grain level residual stress measurements; however, the contribution of the microscale residual strain remains largely unaccounted for. High-resolution electron backscatter diffraction (HR-EBSD) has emerged as a promising tool for the characterization of such micron-level elastic strains. This work presents an early effort in analysis of microscale elastic strain in conjunction with subgrain dislocation structures to further the understanding of microstructural evolution during laser additive manufacturing (AM) techniques. Elastic strain in DMLS fabricated IN625 is analyzed using open source cross-correlation software OpenXY while the geometrically necessary dislocation (GND) density is calculated from EBSD data using the Nye tensor. Dislocation structures previously seen in similar materials are shown here to contain low elastic strain gradients, supporting the assertion that these structures occur as strain minimization mechanisms during solidification.

Crystal plasticity modeling and neutron diffraction measurements of a magnesium AZ31B plate: Effects of plastic anisotropy and surrounding grains

This study used in-situ neutron diffraction measurements and Elastic ViscoPlastic Self-Consistent polycrystal plasticity model, which incorporates a Twinning and DeTwinning scheme (denoted by EVPSC-TDT), to examine the macro-and micro-mechanical behaviors of a rolled AZ31B plate subjected to uniaxial tension. Three specimens were specifically designed for minimum, maximum and intermediate twinning: (1) loading along the rolling direction, (2) loading along the plate normal, and (3) loading along the direction 45° with respect to the plate normal. Apart from the macroscopic stress strain response, the measured diffraction intensities and internal elastic strains were obtained to examine the activities of the deformation modes at the grain level. The diffraction intensity evolution signaled the volume fraction change of twinning, while the internal elastic strain evolution designated the stress partitioning among the grain orientations. The effect of the surrounding grains on the development of the internal elastic strain was investigated by identifying the corresponding deformation mechanisms. Notably, the corresponding modeling work revealed that the EVPSC-TDT model permitted the prediction of the strain hardening and anisotropic behavior along the directions with minimum, maximum and intermediate twinning at the macroscale, and the evolution of the diffraction intensities and internal strains at the microscale. The results provide a physical understanding of the effects of the load direction, texture and surrounding grains on the role of the deformation modes in hexagonal close-packed polycrystalline materials.

Extension of the visco-plastic self-consistent model to account for elasto-visco-plastic behavior using a perturbed visco-plastic approach

We propose a numerically stable and computationally efficient elasto-visco-plastic self-consistent (VPSC) crystal plasticity model, where the effect of elasticity on visco-plasticity is introduced as a perturbation to the visco-plastic regime. The perturbation enters as a fictitious visco-plastic eigen-strain rate and is derived from the elastic inclusion problem. As a consequence, the numerical efficiency of the VPSC model (Lebensohn and Tomé 1993 Acta Metall. Mater. 41 2611–2624) is retained, with the added capability of predicting both, elastic and visco-plastic strain components in the grains. The proposed model (VPSC+) is validated by comparing with the self-consistent crystal plasticity model of Lebensohn and Tomé (VPSC) (Lebensohn and Tomé 1993 Acta Metall. Mater. 41 2611–2624), the elastic-rigid-plastic model (EPSC) of Turner and Tomé (1994 Acta Metall. Mater. 42 4143–4153), and the elasto-visco-plastic model (EVPSC) developed by Wang et al (2010 J. Mech. Phys. Solids 58 594–612). The prediction of texture evolution, stress–strain response, strain rate sensitivity, and internal elastic strain evolution were consistent with those of the other models, in simulations of 316 L stainless steel and Mg AZ31 alloy. VPSC+ showed a superior numerical stability and computational speed when compared to EVPSC, and convergence was successfully achieved with relatively large increments. It is claimed that VPSC+ provides an elasto-VPSC constitutive law for polycrystalline metals applicable to simulation of large deformation, and which is numerically efficient for interfacing with finite element codes.

Effect of Applied Compressive Stress and Impregnation Material on Internal Strain and Stress State in Nb3Sn Rutherford Cable Stacks

The Nb3Sn superconductor in accelerator magnets must resist high mechanical stresses. In order to better understand the effect of the coil impregnation system on the stresses exerted on the strain-sensitive Nb3Sn superconductor, we have measured the elastic strain evolution in the conductor constituents under externally applied loads. For this purpose, a dedicated load frame that enables rotation of the sample load axis with respect to the neutron scattering geometry was installed in the Stress-Spec beamline at the neutron source Heinz Maier-Leibnitz FRM II. The Nb3Sn- and Cu-loading strains were measured in situ by neutron diffraction under monotonic and cyclic compressive loading. So-called ten-stack samples composed of Nb3Sn Rutherford type cables with different impregnation and coil blocks extracted from an 11 T dipole short model coil were investigated.

Strain rate sensitivities of deformation mechanisms in magnesium alloys

Strain rate sensitivity (SRS) is an important material property that governs the rate dependent mechanical behaviors associated with deformation rate changes, creep, stress relaxation, formability, etc. The variety of activated deformation mechanisms of magnesium alloys under different loading paths, e.g. tension vs. compression, implies that SRS of magnesium alloys obviously depends on loading paths, and each deformation mechanism has its own SRS. However, a single SRS scheme is commonly employed in numerical modeling to describe the rate dependent behaviors of magnesium alloys, which disregards the distinction of SRSs among different deformation mechanisms. The implementation of the constitutive model that works for a wide range of values of SRSs has been a challenge to crystal plasticity modeling for metals with multiple deformation mechanisms like magnesium. Especially, very small values of SRS, corresponding to low rate-sensitivity, generally lead to high nonlinearity involved in the governing equations, and then computational failure. In this paper, the elasto-viscoplastic self-consistent (EVPSC) crystal plasticity model is improved to enhance its numerical robustness for very small SRS values. Taking advantage of this improvement, different SRSs for various deformation mechanisms are employed to investigate the strain rate dependent behaviors of magnesium alloys at room temperature. First, the SRSs for various deformation mechanisms are determined based on the compressive stress relaxation tests on an AZ31 alloy plate; secondly, the obtained SRSs are applied to interpret internal elastic strain evolution of the same magnesium alloy under in-plane compression; finally, the determined SRSs are applied to investigate the deformation of another AZ31 alloy under various deformation paths and strain rates. The present work is the first effort on studying effects of strain rate-sensitivity on mechanical behavior of Mg alloys under wide range of applied strain rates by using an improved self-consistent polycrystal plasticity model. Good agreement between the experiments and simulations reveals the importance and necessity of using different SRSs for the deformation mechanisms involved. The rate dependent behaviors of magnesium alloys can be better described by using multiple SRSs associated to each operative deformation mechanism.

Bright x-rays reveal shifting deformation states and effects of the microstructure on the plastic deformation of crystalline materials

The plastic deformation of crystalline materials is usually modeled as smoothly progressing in space and time, yet modern studies show intermittency in the deformation dynamics of single-crystals arising from avalanche behavior of dislocation ensembles under uniform applied loads. However, once the prism of the microstructure in polycrystalline materials disperses and redistributes the load on a grain-by-grain basis, additional length and time scales are involved. Thus, the question is open as to how deformation intermittency manifests for the nonuniform grain-scale internal driving forces interacting with the finer-scale dislocation ensemble behavior. In this work we track the evolution of elastic strain within individual grains of a creep-loaded titanium alloy, revealing widely varying internal strains that fluctuate over time. The findings provide direct evidence of how flow intermittency proceeds for an aggregate of $\ensuremath{\sim}700$ grains while showing the influences of multiscale ensemble interactions and opening new avenues for advancing plasticity modeling.

High Temperature Deformation Mechanism in Hierarchical and Single Precipitate Strengthened Ferritic Alloys by In Situ Neutron Diffraction Studies

The ferritic Fe-Cr-Ni-Al-Ti alloys strengthened by hierarchical-Ni2TiAl/NiAl or single-Ni2TiAl precipitates have been developed and received great attentions due to their superior creep resistance, as compared to conventional ferritic steels. Although the significant improvement of the creep resistance is achieved in the hierarchical-precipitate-strengthened ferritic alloy, the in-depth understanding of its high-temperature deformation mechanisms is essential to further optimize the microstructure and mechanical properties, and advance the development of the creep resistant materials. In the present study, in-situ neutron diffraction has been used to investigate the evolution of elastic strain of constitutive phases and their interactions, such as load-transfer/load-relaxation behavior between the precipitate and matrix, during tensile deformation and stress relaxation at 973 K, which provide the key features in understanding the governing deformation mechanisms. Crystal-plasticity finite-element simulations were employed to qualitatively compare the experimental evolution of the elastic strain during tensile deformation at 973 K. It was found that the coherent elastic strain field in the matrix, created by the lattice misfit between the matrix and precipitate phases for the hierarchical-precipitate-strengthened ferritic alloy, is effective in reducing the diffusional relaxation along the interface between the precipitate and matrix phases, which leads to the strong load-transfer capability from the matrix to precipitate.

Effect of martensitic phase transformation on the behavior of 304 austenitic stainless steel under tension

The present work integrates in-situ neutron diffraction, electron backscatter diffraction and crystal plasticity modeling to investigate the effect of martensitic phase transformation on the behavior of 304 stainless steel under uniaxial tension. The macroscopic stress strain response, evolution of the martensitic phase fraction, texture evolution of each individual phase, and internal elastic strains were measured at room temperature and at 75°C. Because no martensitic transformation was observed at 75°C, the experimental results at 75°C were used as a reference to quantify the effect of formed martensitic phase on the behavior of 304 stainless steel at room temperature. A crystallographic phase transformation model was implemented into an elastic–viscoplastic self-consistent framework. The phase transformation model captured the macroscopic stress strain response, plus the texture and volume fraction evolution of austenite and martensite. The model also predicts the internal elastic strain evolution with loading in the austenite, but not in the martensite. The results of this work highlight the mechanisms that control phase transformation and the sensitivity of modeling results to them, and point out to critical elements that still need to be incorporated into crystallographic phase transformation models to accurately describe the internal strain evolution during phase transformation.

Elastic strain evolution in nanocomposite structure of YBa2Cu3O7+BaZrO3 superconducting films

Lattice parameters of YBa2Cu3O7 (YBCO) and BaZrO3 (BZO) were evaluated in YBCO+BZO nanocomposite superconducting films using X-ray diffraction (XRD), and stress-strain calculation was performed using finite element method (FEM). FEM calculation well described BZO content dependence of lattice parameters in XRD using three fitting parameters; BZO volume fraction ratio between film and target, BZO lattice parameter without strain, and misfit. The fitting parameter values were consistent with results of transmission electron microscopy and energy dispersive X-ray spectroscopy (EDX), which showed that Y was included in nanorods. These results show that elastic strain determined the lattice parameters in YBCO+BZO nanocomposite films and that the FEM calculation is useful for analysis and design of elastic strain in the nanocomposite films.

Stress evaluation in thin films: Micro-focus synchrotron X-ray diffraction combined with focused ion beam patterning for do evaluation

Nanocrystalline metallic coatings of sub-micron thickness are widely used in modern microelectronic applications. In X-ray diffraction experiments to determine both the residual and applied stresses in nanocrystalline coatings, one difficult challenge that comes up invariably is the determination of the strain-free lattice spacing do. The present study addresses this challenge by using the focused ion beam (FIB) to generate a built-in strain-free reference by patterning (milling) a 50×50μm2 region of the coating to produce an array of small stress-relieved “islands” ~0.8×0.8μm2 each. Transmission X-ray diffraction setup was used for data collection at DIFFABS beamline (Synchrotron SOLEIL, France). A 150nm-thick multi-layered W–Cu nano-composite thin film on polyimide (Kapton®) substrate was studied. The samples were loaded incrementally using a compact uniaxial loading device, and micro-beam diffraction data were collected on and away from the reference array. It was shown experimentally that the “island” array remained approximately strain free throughout the experiment, providing an on-board do lattice spacing reference. The changing lattice spacing d in the coating was also monitored away from the array, to deduce the elastic strain evolution during deformation. The results and their implications are presented and discussed.

Multiscale modelling and diffraction-based characterization of elastic behaviour of human dentine

Human dentine is a hierarchical mineralized tissue with a two-level composite structure, with tubules being the prominent structural feature at a microlevel, and collagen fibres decorated with hydroxyapatite (HAp) crystallite platelets dominating the nanoscale. Few studies have focused on this two-level structure of human dentine, where the response to mechanical loading is thought to be affected not only by the tubule volume fraction at the microscale, but also by the shape and orientation distribution of mineral crystallites, and their nanoscale spatial arrangement and alignment. In this paper, in situ elastic strain evolution within HAp in dentine subjected to uniaxial compressive loading along both longitudinal and transverse directions was characterized simultaneously by two synchrotron X-ray scattering techniques: small- and wide-angle X-ray scattering (SAXS and WAXS, respectively). WAXS allows the evaluation of the apparent modulus linking the external load to the internal HAp crystallite strain, while the nanoscale HAp distribution and arrangement can be quantified by SAXS. We proposed an improved multiscale Eshelby inclusion model that takes into account the two-level hierarchical structure, and validated it with a multidirectional experimental strain evaluation. The agreement between the simulation and measurement indicates that the multiscale hierarchical model developed here accurately reflects the structural arrangement and mechanical response of human dentine. This study benefits the comprehensive understanding of the mechanical behaviour of hierarchical biomaterials. The knowledge of the mechanical properties related to the hierarchical structure is essential for the understanding and predicting the effects of structural alterations that may occur due to disease or treatment on the performance of dental tissues and their artificial replacements.

Effect of X-ray irradiation on the elastic strain evolution in the mineral phase of bovine bone under creep and load-free conditions

Both the load partitioning between hydroxyapatite (HAP) and collagen during compressive creep deformation of bone and the HAP residual strain in unloaded bone have been shown in previous synchrotron X-ray diffraction studies to be affected by the X-ray irradiation dose. Here, through detailed analysis of the X-ray diffraction patterns of bovine bone, the effect of X-ray dose on (i) the rate of HAP elastic strain accumulation/shedding under creep conditions and (ii) the HAP lattice spacing and average root mean square (RMS) strain under load-free conditions are examined. These strain measurements exhibit three stages in response to increasing X-ray dose. Up to ∼75kGy (stage I) no effect of dose is observed, indicating a threshold behavior. Between ∼75 and ∼300kGy (stage II) in unloaded bone the HAP d-spacing increases and the RMS strain decreases with dose, indicating strain relaxation of HAP. Furthermore, under constant compressive load creep conditions, the rate of compressive elastic strain accumulation in HAP decreases with increasing dose until, at ∼115kGy, it changes sign, indicating that the HAP phase is shedding load during creep deformation. These stage II behaviors are consistent with HAP–collagen interfacial damage, which allows the HAP elastic strain to relax within both the loaded and unloaded samples. Finally, for doses in excess of ∼300kGy (stage III, measured up to 7771kGy) the HAP lattice spacing and RMS strain for load-free samples and the rate of HAP elastic strain shedding for crept samples remain independent of dose, suggesting a saturation of damage and/or stiffening of the collagen matrix due to intermolecular cross-linking.

Evolution of load transfer between hydroxyapatite and collagen during creep deformation of bone

While the matrix/reinforcement load-transfer occurring at the micro- and nanoscale in nonbiological composites subjected to creep deformation is well understood, this topic has been little studied in biological composites such as bone. Here, for the first time in bone, the mechanisms of time-dependent load transfer occurring at the nanoscale between the collagen phase and the hydroxyapatite (HAP) platelets are studied. Bovine cortical bone samples are subjected to synchrotron X-ray diffraction to measure in situ the evolution of elastic strains in the crystalline HAP phase and the evolution of viscoelastic strains accumulating in the mineralized collagen fibrils under creep conditions at body temperature. For a constant compressive stress, both types of strains increase linearly with time. This suggests that bone, as it deforms macroscopically, is behaving as a traditional composite, shedding load from the more compliant, viscoelastic collagen matrix to the reinforcing elastic HAP platelets. This behavior is modeled by finite-element simulation carried out at the fibrillar level.

On the complex behavior of strain relaxation in (In,Ga)As/GaAs(001) quantum dot molecules

A detailed growth scenario of surface quantum dot molecules (QDM) in the system (In,Ga)As/GaAs(001) has been investigated in terms of shape and elastic strain evolution. QDMs are grown by a combined approach using droplet epitaxy for initial homoepitaxial GaAs mounds, which subsequently serve as nucleation spots for surrounding (In,Ga)As surface quantum dots. Atomic force micrographs trace a detailed pathway toward the final QDM containing up to six quantum dots with perfect inherent symmetry. Synchrotron-based grazing incidence diffraction together with grazing incidence small angle x-ray scattering reveal a relaxation behavior, which for all growth stages comprises a strained lattice along [1¯10] and partial elastic relaxation along [110]. Numerical finite element calculations on the three-dimensional strain profile support the experimental findings.

A new criterion for elasto-plastic transition in nanomaterials: Application to size and composite effects on Cu–Nb nanocomposite wires

Nanocomposite wires composed of a multi-scale Cu matrix embedding Nb nanotubes are cyclically deformed in tension under synchrotron radiation in order to follow the X-ray peak profiles (position and width) during mechanical testing. The evolution of elastic strains vs. applied stress suggests the presence of phase-specific elasto-plastic regimes. The nature of the elasto-plastic transition is uncovered by the “tangent modulus” analysis and correlated to the microstructure of the Cu channels and the Nb nanotubes. Finally, a new criterion for the determination of the macroyield stress is given as the stress to which the macroscopic work hardening, θ a = d σ a/d ε 0, becomes smaller than one third of the macroscopic elastic modulus.

Elastic strain evolution and ε-martensite formation in individual austenite grains during in situ loading of a metastable stainless steel

The (hcp) ε-martensite formation and the elastic strain evolution of individual (fcc) austenite grains in metastable austenitic stainless steel AISI 301 has been investigated during in situ tensile loading up to 5% applied strain. The experiment was conducted using high-energy X-rays and the 3DXRD technique, enabling studies of individual grains embedded in the bulk of the steel. Out of the 47 probed austenite grains, one could be coupled with the formation of ε-martensite, using the reported orientation relationship between the two phases. The formation of ε-martensite occurred in the austenite grain with the highest Schmid factor for the active {111}<12¯1> slip system.

Strain and texture evolution of ED-rotated cubes during quasi-static and dynamic tensile testing of Al–Mg–Si-profiles in the as-extruded T1-temper condition

High-energy synchrotron X-ray diffraction has been used to study through-thickness deformation response in extruded Al–Mg–Si-profiles during tensile testing, in terms of micro- and mesoscopic distributions and dynamical evolution of elastic strains and grain rotations. Local averaging with analysis at intermediate length scales reveals strongly inhomogeneous through-profile elastic strains, caused by the presence of three distinct microstructure regions and the compatibility relations that apply at their interfaces. Variations in elastic strains at characteristic microstructure lengths are found to be large; typically 1 σ Gaussian spreads for the different ɛ ij -components of the elastic strain tensor are minimal and of the order 1.0 × 10 −3 in the central profile region at low stresses. The spread increases with the tensile loads, but even more dramatically with decreasing distance to the surfaces where maximum 1 σ spreads up to 6–7 × 10 −3 are encountered. The evolution and distribution of certain texture components have been analysed, showing grain rotations to be a non-negligible part of the deformation response that activates at quite modest plastic deformations. Inhomogeneous strain response at local and intermediate length scales together with the strain and texture component relations that apply across the microstructure region boundaries are found to be decisive to surface roughening. All together, the results point in the direction that strain and texture evolution should be considered together in order to provide a more complete description of microstructure mechanics in metals.

Plasticity of multiscale nanofilamentary Cu∕Nb composite wires during in situ neutron diffraction: Codeformation and size effect

In situ neutron diffraction was performed on Cu∕Nb nanocomposite wires composed of a multiscale Cu matrix embedding Nb nanofilaments with a diameter of 267nm and spacing of 45nm. The evolution of elastic strains and peak profiles versus applied stress evidenced the codeformation behavior with different elastic-plastic regimes: the Cu matrix exhibit size effect in the finest channels while the Nb nanowhiskers remain elastic up to the macroscopic failure, with a strong load transfer from the Cu matrix onto the Nb filaments. The measured yield stress in the finest Cu channels is in agreement with calculations based on a single dislocation regime.