Steady-state Deformation Research Articles

Crystallographic orientation dependence in creep deformation of micron-thick silicon films for 3-D microstructures

In this study, the steady-state creep deformation properties of 5 μm-thick single crystal silicon (Si) films at elevated temperatures were investigated using punch creep-forming tests for the design of three-dimensional (3-D) microstructured MEMS. The relationship between the creep strain rate and stress of the Si films with {100}, {110}, and {111} planes at temperatures of 1223–1323 K was derived using finite element analysis following punch creep-forming tests, and a power-law creep constitutive equation for Si films was determined. The steady-state creep properties of the Si film samples varied clearly with crystallographic orientations. Among the three crystallographic orientations, the creep strain rate of the micron-thick Si films was the fastest for the {011} plane, followed by the {111} and {001} planes, in the applied stress range of 110 MPa to 220 MPa. This is consistent with the increasing order of magnitude of the thermal activation energy. The crystallographic orientation dependence of the steady-state creep properties of Si films is discussed based on the mobility of dislocation gliding per unit lattice and scanning electron microscope observations. The sample-size dependence of the creep strain rate of Si with the {001} plane was clearly observed between the micron- and millimeter-thick samples, which was attributed to the difference in the frequency factor of the power-law creep. However, the thermal-activation energy remained constant. This study successfully revealed the steady-state creep properties of Si films and provided useful engineering data for the design of 3-D Si-MEMS fabrication by the plastic processing of Si films.

Coupled thermo-elastic dynamics of rotating pre-twisted blades exposed to thermal shock including nonlinear rotational effects

Rotating blades operating under high-temperature experience intense mechanical and thermal stresses induced by centrifugal forces and thermal shock. In this research, to address this critical yet understudied topic, coupled thermo-elastic dynamics of rotating blades incorporating precise rotational effects is investigated. A comprehensive modeling approach, characterizing blade geometry in terms of pre-twist and pre-set angles is employed based on the theory of surfaces. Thermal strain–displacement field is defined based on third-order shear deformation theory in the model integrated with a third-order expansion of temperature change distribution within the blade. Rotational factors related to Coriolis, centrifugal stiffening, and rotational softening effects are considered. In the case of stiffening effects, two different modeling approaches including direct integration of centrifugal forces (DICFs), and pre-stressed dynamics about steady-state equilibrium deformations (SSEDs) are employed. When incorporating large-amplitude SSEDs in the latter, nonlinear rotational effects are incorporated into the model. To overcome the complexity arising from coupled thermo-elasticity, and rotational motion, the spectral Chebyshev technique is applied to the derived integral boundary value problems and coupled energy equations. Finally, natural frequencies, and thermal and centrifugal deformations are investigated comprehensively by including DICFs and pre-stressed dynamics. Results show that thermo-elastic dynamics of the system is highly affected by the modeling approaches used for centrifugal stiffening effects.

Construction of meta-dynamic recrystallization dynamic model of 316L stainless steel based on grain orientation spread during continuous variable rate thermal deformation

Because the deformation parameters (temperature, strain rate) in the actual manufacturing process are non-constant. We investigate meta-dynamic recrystallization behavior under the transient mutation state. First, based on a single-pass thermal deformation experiment and electron backscatter diffraction (EBSD) technique, this study establishes a quantitative calculation method for the volume fraction of meta-dynamic recrystallization under different deformation conditions, and find the GOS threshold basis for judging the occurrence of meta-dynamic recrystallization. Comparing the experimental results with the numerical simulation results, it is found that the MDRX behavior of the material could be accurately described. The meta-dynamic recrystallization evolution law of the material in the thermal deformation insulation stage under the variable strain rate state is investigated by continuous variable rate thermal deformation experiment and numerical simulation. It is observed that an increase of the average strain rate in the continuous variable strain rate state would promote the meta-dynamic recrystallization softening behavior of the material in comparison with the steady state deformation stage, and vice versa.

Elastic particle model for coil-stretch transition of dilute polymers in an elongational flow

The phenomenon of the ‘coil-stretch’ (C-S) transition, wherein a long-chain polymer initially in a coiled state undergoes a sudden configuration change to become fully stretched under steady elongational flows, has been widely recognized. This transition can display intricate hysteresis behaviours under specific critical conditions, giving rise to unique rheological characteristics in dilute polymer solutions. Historically, microscopic stochastic models and Brownian dynamics simulations have shed light on the underlying mechanisms of the transition by uncovering bistable configurations of polymer chains. Following the initial work by Cerf (J. Chem. Phys., vol. 20, 1952, pp. 395–402), we introduce a continuum model in this study to investigate the C-S transition in a constant uniaxial elongational flow. Our approach involves approximating the unfolding process of the polymer chain as an axisymmetric deformation of an elastic particle. We make the assumption that the particle possesses uniform material properties, which can be represented by a nonlinear, strain-hardening constitutive equation to replicate the finite extensibility of the polymer chain. Subsequently, we analytically solve for the steady-state deformation using a polarization method. By employing this reduced model, we effectively capture the C-S transition and establish its specific correlations with material and geometric properties. The hysteresis phenomena can be comprehended through a force-balance analysis, which involves comparing the externally applied viscous forces with the intrinsic elastic responsive forces. We demonstrate that our model, while simple, unveils rich elastohydrodynamics of the C-S transition.

An isochronous stress ratio logarithmic strain curve based clay creep model considering the effects of hardening and damage

Creep is one of the typical mechanical properties of clay, and studying the creep mechanical properties of clay is of great significance to construction projects in clay sites. This study conducted creep tests on Chengdu clay and found that the soil mass underwent elastic deformation, decay creep deformation, steady-state creep deformation, and accelerated creep deformation. The isochronous stress ratio-logarithmic strain curves and their mathematical models were proposed to thoroughly analyze clay creep mechanical properties. Creep automatic feature points, such as linear elastic extreme point, initial yield point, long-term strength point, and plastic point, were identified on the curve. Considering the hardening and damage effects during creep loading, linear elastic and viscoelastic elements considering the time-dependent damage, a viscoplastic element considering the load hardening effect, and viscoplastic and plastic elements considering the load damage effect were established based on the element model and the Riemann–Liouville fractional derivative. Based on the mechanical properties of the whole clay creep process, the creep mechanical feature points, and the established element model, a clay creep model was proposed considering the hardening and damage effects. The rationality and regularity of the creep model were verified using the creep test data. This research accurately revealed the creep mechanical properties of clay and facilitated soil deformation prediction, thus providing technical guidance and references for construction projects in clay sites.

Decoding stress-strain parameters in FCC metals using digital constitutive analyses to devolve dynamic obstacle-strength factor and diffuse necking

Simulation codes use constitutive relations of work hardening to virtually predict shape change during metal forming. Recent analysis has shown that the modified Hollomon-type relation correlates to stress-aided thermal activation at obstacles for dislocation movement. The sweeping action of dislocation gives rise to strain and the dislocation intersections to strain rate sensitivity during work-hardening. The constitutive relation analyses (CRA) encompass fitting parameters which remain constant with strain and its validation is the precision to replicate the measured stress-strain diagram. In this study, digital constitutive analyses (DCA) are examined whereby the modelled-fit parameters are simultaneously numerically adjusted as strain proceeds. For bulk properties such as expended work, volume fraction of vacancy creation and mean slip velocity, CRA predictions have been validated. However, DCA enables the identification of defects being created with strain using derived obstacle-strength factor (αγ). The changes in mechanisms during work-hardening can be decoded using the αγ – γ plot whereby constant αγ indicate steady-state deformation and its rapid decrease, the start of diffuse necking. Thus, αγ is a composite factor of defects being continuously created whereas conventional α is a measure of the stored work up to that strain. The tensile data from polycrystalline super-pure aluminum tested at 78 K were used to validate the derived relations which were applied to the DCA of age-hardenable aluminum alloys tested at 298 K. The present work shows that integral replication of stress-strain diagram is essential, but a differential analysis is required to devolve the creation of crystal defects with strain.

Quasi-static and dynamic response of AA-2219-T87 aluminium alloy

Quasi-static and high-strain rate uniaxial compression testing on AA-2219-T87 aluminium alloy were carried out under varying strain rates (ranging from 1×10−3 to 7×103 s−1) and temperatures. Quasi-static testing showcased no strain-rate sensitivity (SRS) at room temperature, while positive SRS was observed at elevated temperatures (180, 230, and 300°C), showcasing higher flow stress with an increase in strain rate. This temperature-dependent SRS is attributed to the typical activation of thermally activated processes, greater dynamic recovery, and reduced effectiveness of traditional strengthening mechanisms at higher temperatures. High-strain rate testing, performed using a Split Hopkinson pressure bar (SHPB), revealed no SRS across all tested temperatures (25, 180, and 230°C). At high-strain rates (>103), once the flow stress reaches a common flow stress, it undergoes either a steady state deformation or minor softening. This behaviour is explained by the interplay between strain hardening and thermal softening phenomena. Specifically, the formation of intense shear flow zones (including dynamic recrystallization) during impact loading at strain rates of 4×103 and 7×103 s−1 promoted thermal softening over strain hardening, resulting in flow stress levels similar to those observed at a strain rate of 1×103 s−1. The intense shear flow volume fraction increased with an increase in strain rate, which is linked to no SRS and higher strain at 4×103 s−1 and 7×103 s−1. The SHPB testing data was utilized to develop the constitutive Johnson-Cook model, enabling the determination of the plastic behaviour of the AA-2219-T87 alloy at high-strain rates and temperatures. These findings enhance our understanding of the alloy's mechanical response and provide valuable insights for optimizing its performance in high-strain-rate applications.

Suspensions of viscoelastic capsules: effect of membrane viscosity on transient dynamics

Membrane viscosity is known to play a central role in the transient dynamics of isolated viscoelastic capsules by decreasing their deformation, inducing shape oscillations and reducing the loading time, that is, the time required to reach the steady-state deformation. However, for dense suspensions of capsules, our understanding of the influence of the membrane viscosity is minimal. In this work, we perform a systematic numerical investigation based on coupled immersed boundary–lattice Boltzmann (IB-LB) simulations of viscoelastic spherical capsule suspensions in the non-inertial regime. We show the effect of the membrane viscosity on the transient dynamics as a function of volume fraction and capillary number. Our results indicate that the influence of membrane viscosity on both deformation and loading time strongly depends on the volume fraction in a non-trivial manner: dense suspensions with large surface viscosity are more resistant to deformation but attain loading times that are characteristic of capsules with no surface viscosity, thus opening the possibility to obtain richer combinations of mechanical features.

Deformation transients of confined droplets within interacting electric and magnetic field environment

A theoretical exploration and an analytical model for the electro-magneto-hydrodynamics (EMHD) of leaky dielectric liquid droplets suspended in an immiscible confined fluid domain have been presented. The analytical solution for the system, under small deformation approximation in the creeping flow regime, has been put forward. The study of droplet deformation suggests that its temporal evolution is exponential and dependent on the electric and magnetic field interaction. Furthermore, the direction of the applied magnetic field relative to the electric field determines whether the magnetic force opposes or facilitates the interfacial net electrical force caused by the electric field. Validation of the proposed model at the asymptotic limits of vanishing magnetic field shows that the model accurately reduces to the case of the transient electrohydrodynamic model. We also propose a magnetic discriminating function (ϕM) to quantify the steady-state droplet deformation in the presence of interacting electric and magnetic fields. The change of droplets from a spherical shape to prolate and oblate spheroids corresponds to ϕM> 0 and < 0 regimes, respectively. It is shown that a substantial augmentation in the deformation parameter and the associated EMHD circulation within and around the droplet is achieved when aided by a low-magnitude magnetic field. The analysis also reveals the deformation lag and specific critical parameters that aid or suppress this lag behavior, discussed in terms of relevant non-dimensional parameters.

Deformation and breakup behaviors of a Giesekus viscoelastic droplet in Newtonian shear flow

The deformation and breakup of a Giesekus viscoelastic droplet suspended in a Newtonian matrix under simple shear flow is investigated through a lattice Boltzmann method, which uses the color-gradient model for immiscible two-phase flows while coupling a lattice advection-diffusion scheme for the Giesekus constitutive equation. The influence of the Deborah number (De) and shear-thinning mobility parameter (α) of the droplet is first explored for relatively low capillary numbers (Ca). The transient overshoot is observed before the deformation reaches the steady state, and the overshoot generally increases with De but decreases with α. With the increase of De, the steady-state deformation parameter first decreases and then increases, which is attributed to two opposite effects on the elastic force, namely the promoting effect from the increased polymer extension and the inhibiting effect caused by the deviation of the location where the maximum elastic force occurs from the droplet tips. The droplet inclination angle varies little with α but increases with De, where the latter can be explained as an increased vertical component of viscous and elastic forces in the tip regions. Upon increasing α, the steady-state deformation parameter becomes higher due to the reduced elastic force at the droplet tips. In addition, the phase diagrams of Ca-De and Ca-α are presented in which the critical capillary number Cacr characterizing the droplet breakup can be identified. It is found that Cacr varies non-monotonically with De, while Cacr is not very sensitive to α.

Viscoelastic effects on the deformation and breakup of a droplet on a solid wall in Couette flow

The deformation, movement and breakup of a wall-attached droplet subject to Couette flow are systematically investigated using an enhanced lattice Boltzmann colour-gradient model, which accounts for not only the viscoelasticity (described by the Oldroyd-B constitutive equation) of either droplet (V/N) or matrix fluid (N/V) but also the surface wettability. We first focus on the steady-state deformation of a sliding droplet for varying values of capillary number ( $Ca$ ), Weissenberg number ( $Wi$ ) and solvent viscosity ratio ( $\beta$ ). Results show that the relative wetting area $A_r$ in the N/V system is increased by either increasing $Ca$ , or by increasing $Wi$ or decreasing $\beta$ , where the former is attributed to the increased viscous force and the latter to the enhanced elastic effects. In the V/N system, however, $A_r$ is restrained by the droplet elasticity, especially at higher $Wi$ or lower $\beta$ , and the inhibiting effect strengthens with an increase of $Ca$ . Decreasing $\beta$ always reduces droplet deformation when either fluid is viscoelastic. The steady-state droplet motion is quantified by the contact-line capillary number $Ca_{cl}$ , and a force balance is established to successfully predict the variations of $Ca_{cl}/Ca$ with $\beta$ for each two-phase viscosity ratio in both N/V and V/N systems. The droplet breakup is then studied for varying $Wi$ . The critical capillary number of droplet breakup monotonically increases with $Wi$ in the N/V system, while it first increases, then decreases and finally reaches a plateau in the V/N system.

Flow behavior and dynamic transformation of titanium alloy Ti62A during deformation at different temperatures and strain rates

The effect of deformation temperature and strain rate on the hot deformation behavior and dual-phase microstructure evolution of the titanium alloy Ti62A was examined using electron backscatter diffraction. In general, the activation energy of Ti62A during steady-state deformation in the (α + β) phase is 295 kJ mol−1. The primary recovery mechanisms of the β phase during hot deformation are dynamic recovery and dynamic recrystallization (DRX). Moreover, discontinuous DRX occurs at low temperatures and high strain rates, whereas continuous DRX occurs at high temperatures and low strain rates. Furthermore, high strain rates in the (α + β) phase and high deformation temperatures are advantageous to dynamic phase changes during dynamic transformation (DT). The β phase penetrates the lamellar α s phase, causing fragmentation and spheroidization of the α s phase. Finally, DT begins more easily in the fine α s phase than in the coarse α p phase.

A weak-form spectral Chebyshev technique for nonlinear vibrations of rotating functionally graded beams

This study presents the spectral Chebyshev technique (SCT) for nonlinear vibrations of rotating beams based on a weak formulation. In addition to providing a fast-converging and precise solution for linear vibrations of structures with complex geometry, material, and physics, this method is further advanced to be able to analyze the nonlinear vibration behavior of continuous systems. Rotational motion and material gradation further complicate this nonlinear behavior. Accordingly, the beam is considered to be axially functionally graded (FG) and a model representing the forced nonlinear vibrations of the beam about steady-state equilibrium deformations (SSEDs) is developed. The model includes Coriolis, centrifugal softening, and nonlinear stiffening effects caused by coupling of the axial, chordwise, and flapwise motions, and large amplitude deformations. The integral boundary value problem for the rotating structure is discretized using the SCT and element-wise multiplication definition. As a result, mass, damping, and stiffness matrices, as well as internal nonlinear forcing functions and external forcing vectors, are obtained for a given rotating beam. This formulation provides a general representation of nonlinear strain relations in matrix form and circumvents the complexity rising from obtaining and solving the partial differential equations directly. In addition, nonlinear forcing functions are obtained in matrix form which facilitates the application of harmonic balance method easier to obtain the forced nonlinear response.

Dislocation structures after creep in an Al-3.85 %Mg alloy studied using EBSD-KAM technique

The electron backscatter diffraction (EBSD) technique is used to investigate the dislocation structures formed after steady-state creep deformation of an Al-3.85%Mg alloy. This material is crept at two different stress levels, corresponding to the so-called power-law and power-law breakdown regimes. The results show that, regardless of the creep stress level, the strain tends to localize, leading to the formation of intragranular bands. The thickness of such bands is larger when the material is tested at loads corresponding to the power-law breakdown. This suggests enhanced diffusion by dislocation pipes.

Forging Temperature Effects on Crack Tip Creep Behaviour of Hot Hammer Forged Ti-6Al-4V Alloy

The creep behaviour of hot hammer forged Ti-6Al-4V alloy under different forging temperatures has an essential influence on their high-temperature service performance. In this article, the high-temperature mechanical property and creep behaviour of Ti-6Al-4V alloy forging at different forging temperatures were first investigated experimentally. Then, in view of the critical effect of microcracks and other defects on the service life of components in forging, the creep characteristics at the tip of a 2 mm crack were characterized by the finite element method. The results show that when the Ti-6Al-4V alloy was forged at 1000°C, the high-temperature mechanical property was improved, and the steady-state creep rate and creep residual deformation were significantly reduced. In addition, the creep strain and creep rate at the crack tip were decreased apparently during creep, and the creep strain gradient in the crack tip region was also significantly reduced. This indicates that the creep deformation of Ti-6Al-4V alloy can be controlled by an appropriate forging temperature during hot hammer forging process and might delay the crack propagation to improve the service life of components.

Shape of a tethered filament in various low-Reynolds-number flows

Starting from a slender-body formulation, we numerically and asymptotically characterize the steady-state deformation of a tethered, elastic filament in viscous flows with uniform, shear, and parabolic profiles.

INFLUENCE OF MINOR ALUMINUM ADDITION ON THE SUPERPLASTIC DEFORMATION OF A MICRODUPLEX CU-ZN ALLOY

High residual porosity in superplastically deformed brass carries the risk of reducing the mechanical properties. Multicomponent brasses demonstrate lower residual porosity, associated with a lower grain size and more effective accommodation mechanisms of grain boundary sliding. In this paper, a comparison is made of microstructural evolution in the prepolished surface and in the bulk of duplex brass samples and brass samples with minor aluminum addition during steady-state superplastic deformation. After superplastic deformation, dislocation clusters and dislocation walls are revealed in the α-grains of both alloys, indicating the activation of the dislocation creep mechanism. It is shown that aluminum reduces the contribution of grain boundary sliding along the phase boundaries from ~75 to ~30% and causes strain localization in the ß-phase region with the formation of less than ~300-nm grains as a result of dynamic recrystallization. Alloying with 0.4% Al reduces the flow stress by 20%, increases the relative elongation by a factor of 1.5, and decreases the amount of residual porosity by a factor of 3. This leads to a much lower loss of strength in superplastically deformed specimens at room temperature.

A new hyperelastic lookup table for RT-DC.

Real-time deformability cytometry (RT-DC) is an established method that quantifies features like size, shape, and stiffness for whole cell populations on a single-cell level in real-time. A lookup table (LUT) disentangles the experimentally derived steady-state cell deformation and the projected area to extract the cell stiffness in the form of the Young's modulus. So far, two lookup tables exist but are limited to simple linear material models and cylindrical channel geometries. Here, we present two new lookup tables for RT-DC based on a neo-Hookean hyperelastic material numerically derived by simulations based on the finite element method in square and cylindrical channel geometries. At the same time, we quantify the influence of the shear-thinning behavior of the surrounding medium on the stationary deformation of cells in RT-DC and discuss the applicability and impact of the proposed LUTs regarding past and future RT-DC data analysis. Additionally, we provide insights about the cell strain and stresses, as well as the influence resulting from the rotational symmetric assumption on the cell deformation and volume estimation. The new lookup tables and the numerical cell shapes are made freely available.

Deformation dynamics of spherical red blood cells in viscous fluid driven by ultrasound

In this paper, the deformation dynamics of spherical red blood cells in viscous fluid driven by ultrasound are theoretically and numerically studied. Red blood cells are considered to be composed of a homogenous cytoplasm enclosed by a biological membrane. The developed theoretical framework consists of two sets of equations, which describe the acoustic wave propagation and the time-averaged mean dynamics of the fluid-cell system, respectively. Specifically, the separated acoustics and mean responses of viscous fluid are formulated based on the acoustic perturbation method in a generalized Lagrangian framework. Considering the viscoelasticity of the membrane, the cell deformation is coupled to the fluid mean motion through the time-averaged fluid–membrane coupling boundary conditions. A computational model is established by deriving weak form formulations of the final governing equations and implementing them by the finite element method. The computational model is verified by comparing the steady-state deformation of the numerical results with previous experimental results. This model can accurately characterize the deformation of cells over time, which helps to extract the viscoelastic properties of cells.

Investigation on a lightweight type broad band-gap metamaterial beam for low-frequency vibration control

Phononic crystals (PCs) and acoustics/elastic metamaterials (AMs/EMs) have aroused increasing attentions for potential applications in low-frequency vibration control due to their outstanding mechanical and physical performances. So far, most reported metamaterials aim to obtain low-frequency broad band-gaps as required to satisfy the practical application. In this paper, we propose a new lightweight low-frequency broadband metamaterial beam. The low-frequency broadband from 390 to 920 Hz of the proposed metamaterial beam is confirmed through vibration transmittance frequency response functions (FRFs) obtained numerically and experimentally. Finally, the steady-state vibration deformations are acquired to illustrate the attenuating effect of the metamaterial beam band-gaps. The research in this paper offers a novel and advantageous method for designing of structures that both broadband damping and lightweight required, such as space-arm and framework of antenna in the field of aerospace.