Creep Law Research Articles

Investigation on thermal relaxation of residual stresses induced in deep cold rolling of Ti–6Al–4V alloy

In the present study, a finite element model has been developed to simulate the deep-cold rolling process on Ti-6Al-4V specimens and the following short term exposure of the treated components to elevated temperature. The developed model can be effectively used to predict the residual stress profiles induced by the process at room temperature and the following residual stress relaxation at the elevated temperature. In the present study, the thermal relaxation stage is performed using a viscoplastic model which couples the creep and plasticity deformation mechanisms to predict the state of residual stresses at the elevated temperature. For this purpose, a new set of hyperbolic creep law coefficients are identified in order to describe the primary creep at 450 °C. The accuracy of the developed finite element model to predict residual stresses is validated by comparison with the experimental data available in the literature. It has been shown that the finite element predictions correlate well with experimental results with error generally less than 10%. The developed model can be effectively utilized for parametric studies to understand the effect of different process parameters on the induced residual stresses without performing expensive experimental tests.

Non-local structural derivative Maxwell model for characterizing ultra-slow rheology in concrete

Ultra-slow rheological phenomena have widely been observed in engineering materials. The logarithmic law is normally used to describe the slow rheology, but it does not work well for the long-term ultra-slow rheology. In this paper, we devise a new Maxwell-type viscoelastic model to capture the ultra-slow rheology by using the non-local structural derivative, where the inverse Mittag-Leffler (ML) structural function is adopted. The viscoelastic responses of the ultra-slow Maxwell model are analytically derived, including creep and relaxation. The logarithmic creep law can be regarded as a special case of the ultra-slow Maxwell model. In addition, the proposed model is tested by several experimental data of concrete. Compared with the existed models, the present ultra-slow Maxwell model shows the reasonable accuracy. The derived results indicate that the non-local structural derivative involving the inverse ML function is feasible to capture the ultra-slow rheology of concrete.

Prediction of long-term creep behaviour of Grade 91 steel at 873 K in the framework of microstructure-based creep damage mechanics approach

A detailed analysis has been performed for the prediction of long-term creep behaviour of tempered martensitic Grade 91 steel at 873 K using the microstructure-based creep damage mechanics approach. Necessary modifications have been made into the original kinetic creep law proposed by Dyson and McLean in order to account for the influence of microstructural damages arising from the coarsening of M23C6 and conversion of useful MX precipitates into deleterious Z-phase on creep behaviour of the steel. An exponential rate relationship has been introduced for the evolution of number density of MX precipitates with time. It has been shown that the developed model adequately predicts the experimental long-term creep strain–time as well as creep rate-time data. The role of Z-phase on long-term creep behaviour of Grade 91 steel has also been discussed.

Applying Depth Distribution of Seismicity to Determine Thermo-Mechanical Properties of the Seismogenic Crust in Southern California: Comparing Lithotectonic Blocks

We analyze waveform-relocated seismicity (1981–2016) and other geophysical and geological datasets from 16 lithotectonic crustal blocks in southern California. We explore how earthquake depth histograms (EDH) are related to crustal strength, lithology, and temperature of the crust. First, we calculate relative EDHs to quantify the depth distribution of seismicity for each lithotectonic block. Second, we calculate depth profiles of maximum differential stress (“yield strength envelopes”, YSEs) using Byerlee’s law and a non-linear dislocation creep law. We use observed average heat flow values, strain rates, and states of stress to parameterize YSEs for five different crustal candidate lithologies in each lithotectonic block. We assume that seismicity ceases where the mechanical rock strength falls below a critical threshold level, and identify the YSE that best predicts the depth extent of seismicity in each block. The lithologies of the best matching YSEs are found to agree well with expectations from past tectonics: they are mostly quartz-dominated except for the feldspar-rich diorite lithologies in the Great Valley, the southernmost western Sierra Nevada, Inner Continental Borderland, and Rifted crust in the Salton Trough. Similarly, the inferred thermo-mechanical properties, including differential stress, lithology, and geotherms reflect the previously mapped tectonic variability between the 16 lithotectonic blocks. On average, the differential yield stress is smaller and peaks at a shallower depth in hotter and more quartz rich crust but is larger and peaks at greater depths for colder and predominantly diorite crust. The good agreement between the modeled YSEs, the EDHs and tectonic considerations suggests that EDHs indeed reflect long-term geophysical properties of the crust and can be used to infer thermo-mechanical properties at depth. In contrast, shallow seismicity may be more likely to reflect short-term strain transients from fluid flow or recent anthropogenic disturbances.

Creep measurement and choice of creep laws for BGA assemblies' reliability simulation

Since the replacement of hazardous materials by the RoHS directive, the intensive use of lead free solder materials generates countless studies to find new acceleration factors. In that scope, identifying the failure mechanisms involved in accelerated thermal tests has become of great concern for reliability design. Numerical tools are commonly developed for predictive analysis but their accuracy and capacity to be transposed to multiple technologies rely in part on the right choice of material behaviour. This study proposes to fit material coefficients of 4 different available creep laws to a same set of experiments. A unique couple of Finite Element models is defined using global/local calculations. Global calculation gives the assembly behaviour under thermal loads that is used to feed a part of the assembly with a more refined mesh to calculate locally a BGA ball that behaves following the Garofalo's, Anand's, Darveaux's laws of creep. The discrepancies that this comparison highlights, show how much care needs to be taken at that step of modeling for reliability prediction.

Wire edge dependent magnetic domain wall creep

While edge pinning is known to play an important role in sub-$\ensuremath{\mu}\mathrm{m}$ wires, we demonstrate that strong deviations from the universal creep law can occur in 1 to $20\phantom{\rule{0.28em}{0ex}}\ensuremath{\mu}\mathrm{m}$ wide wires. Magnetic imaging shows that edge pinning translates into a marked bending of domain walls at low drive and is found to depend on the wire fabrication process and aging. Edge pinning introduces a reduction of domain wall velocity with respect to full films which increasingly dominates the creep dynamics as the wire width decreases. We show that the deviations from the creep law can be described by a simple model including a counter magnetic field which links the width of the wire to the edge dependent pinning strength. This counter field defines a key nonuniversal contribution to creep motion in patterned structures.

Enhanced double-mechanism creep laws for salt rocks

The double-mechanism creep law (DM model) is an empirical creep constitutive model widely employed in Brazilian salt rock mechanics. This model often presents good performance in steady-state creep prediction. However, transient creep is not accounted for, and whenever early creep estimates are important, the contribution of this phase might be meaningful. This work adds value by presenting two alternatives to account for transient creep in the DM model. The first alternative couples the transient function from Sandia’s multi-mechanism deformation model to the DM model steady-state creep rate (EDMT model). The second alternative couples the DM model response to Norton’s power law when the strain rate given by the latter remains lower than the one from the former (EDMP model). These models can be implemented in numerical simulators at small code extensions of the DM model implementations. Applications from previous works by the authors are revisited to validate the formulations based on experimental data. EDMT and EDMP models differ in the formulation of transient creep and, consequently, in the time of transition between the transient and the steady-state phases. Both methods were successful in treating transient creep and in simulating experimental results.

Numerical and Experimental Investigation on the Hybrid Superplastic Forming of the Conical Mg Alloy Component

Hybrid superplastic forming (SPF) is a novel sheet metal forming technique that combines hot drawing with gas forming process. Compared with the conventional SPF process, the thickness distribution of AZ31B part formed by this hybrid SPF method has been significantly improved. Additionally, the microstructure evolution of AZ31 was examined by electron backscatter diffraction (EBSD). Many subgrains with low misorientation angle were observed in the coarse grains during SPF. Based on the tensile test results, parameters of hyperbolic sine creep law model was determined at 400 oC. The hybrid SPF behavior of non-superplastic grade AZ31B was predicted by ABAQUS using this material forming model. The FEM results of thickness distribution, thinning characteristics and forming height were compared with the experimental results and have shown reasonable agreement with each other.

Deformation of Fine‐Grained Quartz Aggregates by Mixed Diffusion and Dislocation Creep

AbstractHot‐pressed polycrystalline quartz samples with grain sizes of 1.7–12.0 μm and water contents of 3,500 ppm H/Si were deformed in a solid‐pressure‐medium (Griggs‐type) deformation apparatus at temperatures of 600 to 950°C, confining pressures of 0.9 to 1.5 GPa and strain rates of 10−3.3 to 10−5.9/s. Two different flow regimes are distinguished at low and high temperatures. The stress exponent determined at low temperatures (600–750°C) increased from 2.9 to 5.2 with an activation energy of 129 ± 33 kJ/mol, consistent with previous quartz dislocation creep laws indicating operation of dislocation creep. In contrast, the stress exponent determined at high temperatures (800–950°C) is 1.7 ± 0.2 with an activation energy of 183 ± 25 kJ/mol. A fugacity exponent determined at 800°C was 1.0 ± 0.2. All samples show evidence of basal <a> slip. However, flow strengths at high temperatures also depend on grain size with a small grain size exponent of 0.51 ± 0.13. Mechanical, microstructural, and textural results suggest that deformation occurs by a combination of intracrystalline and grain boundary processes. The flow law determined from the high‐temperature data can be fit by with stress, σ in MPa, grain size, d in μm, in MPa, and Tk in Kelvin. At conditions of the middle crust and tectonic strain rates, deformation depends on grain size where the strength is weaker than for pure dislocation creep even for grain sizes >10 μm.

Comparison of brittle- and viscous creep in quartzites: Implications for semi-brittle flow of rocks

The co-existence and interaction between brittle and viscous deformation processes contributes to the integrated strength of the crust and results in a wide range of energy-release mechanisms ranging from earthquakes to creep. Here, we compare flow laws derived for quartz-rich rocks deforming by brittle creep, wet dislocation power-law creep and dissolution-precipitation creep. We investigate theoretically the conditions when both brittle and viscous processes contribute significantly to deformation provided that all processes act independently and in parallel. Utilizing a comprehensive data set for deformation experiments in quartz-rich rocks, we find that the transition between deformation mechanisms is strongly dependent on input variables such as initial flaw size and grain size. The transition can occur abruptly or over hundreds of MPa in differential stress and hundreds of degrees Kelvin at a constant strain rate. The transition is strongly dependent on grain-size and confining pressure. Limitations to this work are first that all three flow laws are poorly constrained by experimental data for conditions relevant for the comparison. Secondly, a need exists for new experiments to infill the knowledge gaps between high-temperature and low-temperature deformation experiments and deriving quantitative flow laws for low-temperature plasticity and high-temperature brittle creep.

Effect of Nonlinear Functions on Prediction of the Creep Behavior in Fibrous Composites Theoretically

This paper presents the analysis of the creep behavior for safe designing the fibrous composite devices. Special theoretical method is presented for analyzing the second stage creep of the short fiber composites subjected to axial load using nonlinear displacement functions. Behavior of creeping matrix is described by an exponential creep law, whilst the fibers behave elastically in general. The major plan of the present research work is prediction of the creep behavior in the fibrous composites for preventing unwanted events, over and above, controlling the creep deformation rate theoretically and numerically. Finally, good compatibilities are seen between finite element method and present theoretical method results. As an important result and finding, in general, the mentioned behaviors are ascending along with soft gradients, and so they are controllable and desired.DOI: http://dx.doi.org/10.5755/j01.mech.24.2.19535

Rheology and stress in subduction zones around the aseismic/seismic transition

Subduction channels are commonly occupied by deformed and metamorphosed basaltic rocks, together with clastic and pelagic sediments, which form a zone up to several kilometers thick to depths of at least 40 km. At temperatures above ~ 350 °C (corresponding to depths of > 25–35 km), the subduction zone undergoes a transition to aseismic behavior, and much of the relative motion is accommodated by ductile deformation in the subduction channel. Microstructures in metagreywacke suggest deformation occurs mainly by solution-redeposition creep in quartz. Interlayered metachert shows evidence for dislocation creep at relatively low stresses (8–13 MPa shear stress). Metachert is likely to be somewhat stronger than metagreywacke, so this value may be an upper limit for the shear stress in the channel as a whole. Metabasaltic rocks deform mainly by transformation-assisted diffusional creep during low-temperature metamorphism and, when dry, are somewhat stronger than metachert. Quartz flow laws for dislocation and solution-redeposition creep suggest strain rates of ~ 10−12 s−1 at 500 °C and 10 MPa shear stress: this is sufficient to accommodate a 100 mm/yr. convergence rate within a 1 km wide ductile shear zone.The up-dip transition into the seismic zone occurs through a region where deformation is still distributed over a thickness of several kilometers, but occurs by a combination of microfolding, dilational microcracking, and solution-redeposition creep. This process requires a high fluid flux, released by dehydration reactions down-dip, and produces a highly differentiated deformational fabric with alternating millimeter-scale quartz and phyllosilicate-rich bands, and very abundant quartz veins. Bursts of dilational microcracking in zones 100–200 m thick may cause cyclic fluctuations in fluid pressure and may be associated with episodic tremor and slow slip events. Shear stress estimates from dislocation creep microstructures in dynamically recrystallized metachert are ~ 10 MPa.

Determination of time-dependent strengths of salt pillars based on strain energy principle

The objective of this study is to determine the time-dependent strengths of salt mine pillars in the Maha Sarakham formation, northeast of Thailand. Strain rate-controlled triaxial compression tests have been performed on salt specimens under confining pressures from 0 MPa to 12 MPa. The strain rates are from 10−7 s−1 to 10−4 s−1. The axial stresses and lateral strains are monitored through the strain-softening region. The results indicate that the strengths and elastic moduli increase exponentially with the strain rates. The power creep law parameters are calibrated with the test results, and hence allows constructing series of strain-time curves for the salt pillars under different depths and extraction ratios. The strain energy density principle is applied to develop a strength criterion for the salt pillars. Combining this criterion with the series of the strain-time curves the time-dependent strengths of the salt pillars for different extraction ratios can be predicted.

Magnetic properties, domain-wall creep motion, and the Dzyaloshinskii-Moriya interaction in Pt/Co/Ir thin films

We study the magnetic properties of perpendicularly magnetised Pt/Co/Ir thin films and investigate the domain wall creep method of determining the interfacial Dzyaloshinskii-Moriya (DM) interaction in ultra-thin films. Measurements of the Co layer thickness dependence of saturation magnetisation, perpendicular magnetic anisotropy, and symmetric and antisymmetric (i.e. DM) exchange energies in Pt/Co/Ir thin films have been made to determine the relationship between these properties. We discuss the measurement of the DM interaction by the expansion of a reverse domain in the domain wall creep regime. We show how the creep parameters behave as a function of in-plane bias field and discuss the effects of domain wall roughness on the measurement of the DM interaction by domain expansion. Whereas modifications to the creep law with DM field and in-plane bias fields have taken into account changes in the energy barrier scaling parameter $\alpha$, we find that both $\alpha$ and the velocity scaling parameter $v_{0}$ change as a function of in-plane bias field.

Current induced domain wall motion and tilting in Pt/Co/Ta structures with perpendicular magnetic anisotropy in the presence of the Dyzaloshinskii–Moriya interaction

Current induced domain wall motion (CIDWM) was studied in Pt/Co/Ta structures with perpendicular magnetic anisotropy and the Dyzaloshinskii–Moriya interaction (DMI) by the spin-orbit torque (SOT). We measured the strength of DMI and SOT efficiency in Pt/Co/Ta with the variation of the thickness of Ta using a current induced hysteresis loop shift method. The results indicate that the DMI stabilizes a chiral Néel-type domain wall (DW), and the DW motion can be driven by the enhanced large SOT generated from Pt and Ta with opposite signs of spin Hall angle in Pt/Co/Ta stacks. The CIDWM velocity, which is 104 times larger than the field driven DW velocity, obeys a creep law, and reaches around tens of meters per second with current density of ~106 A cm−2. We also found that the Joule heating accompanied with current also accelerates the DW motion. Meanwhile, a domain wall tilting was observed, which increases with current density increasing. These results can be explained by the spin Hall effect generated from both heavy metals Pt and Ta, inherent DMI, and the current accompanying Joule heating effect. Our results could provide some new designing prospects to move multiple DWs by SOT for achieving racetrack memories.

Time dependent creep analysis in thick FGM rotating disk with two-dimensional pattern of heterogeneity

Combining a novel modeling algorithm and Generalized Differential Quadrature (GDQ) solution method of differential equations, a time depended creep analysis is performed for a Functionally Graded (FG) rotating thick disk featuring a two-dimensional type of heterogeneity. The disk is made of SiC reinforcement particles distributed functionally both radially and axially in Aluminum based matrix continuously. All mechanical and thermal properties are functions of SiC volume fraction. Primary and secondary creep behaviors are based on Norton's creep law. Creep parameters are depended on temperature, particle size and volume fraction of particles. Using equilibrium, constitutive and strain-displacement equations, nonlinear displacement-based creep equations are obtained. A solution algorithm is developed to solve the nonlinear form of equations. The application of 2-D GDQ method is examined in solution of resulting system of equations. Exemplary case studies confirm that in comparison with one-dimensional models, two-dimensional methodologies are more versatile for the study of FGM structures.

A generalization of the Becker model in linear viscoelasticity: creep, relaxation and internal friction

We present a new rheological model depending on a real parameter $\nu \in [0,1]$ that reduces to the Maxwell body for $\nu=0$ and to the Becker body for $\nu=1$. The corresponding creep law is expressed in an integral form in which the exponential function of the Becker model is replaced and generalized by a Mittag-Leffler function of order $\nu$. Then, the corresponding non-dimensional creep function and its rate are studied as functions of time for different values of $\nu$ in order to visualize the transition from the classical Maxwell body to the Becker body. Based on the hereditary theory of linear viscoelasticity, we also approximate the relaxation function by solving numerically a Volterra integral equation of the second kind. In turn, the relaxation function is shown versus time for different values of $\nu$ to visualize again the transition from the classical Maxwell body to the Becker body. Furthermore, we provide a full characterization of the new model by computing, in addition to the creep and relaxation functions, the so-called specific dissipation $Q^{-1}$ as a function of frequency, which is of particularly relevance for geophysical applications

Modeling Creep Fracture in Rock by Using Kelvin Discretized Virtual Internal Bond

Discretized virtual internal bond (DVIB) is a lattice model, which is composed of bond cells. Each bond cell has a finite number of bonds. The DVIB is used to model the creep fracture. It is done by introducing a viscous bond to the original hyperelastic DVIB. The hyperelastic bond is parallel coupled with a viscous bond together, forming a hybrid hyperelastic-Kelvin bond. The hyperelastic bond reflects the microfracture mechanism, whereas the viscous bond reflects the creep mechanism. Based on this hyperelastic-Kelvin bond, the constitutive relation of a cell is derived. The microbond parameters are calibrated based on the ideal cell approach. The simulation results suggest that this method can represent the typical features of creep and can simulate the creep fracture. The merit of this method lies in that the complicated 3D macrocreep problem is reduced to the 1D microbond creep problem. No creep law is previously derived. The macrocreep fracture behavior is the natural response of the assembly of the micro hyperelastic-Kelvin bonds.

Numerical simulation of creep behaviour of 316LN stainless steel weld joint

The welded structures are prone to fail prematurely when subjected to long time application of high temperature environment. The variation of creep behaviour in 316LN stainless steel weldment and base metal is considered as one of the major parameter to premature failure. The present research elucidates the stress and strain developed in 316LN stainless steel weldment that are fabricated using multi-pass tungsten inert gas welding. Finite element analysis employing Norton’s creep law is carried out at elevated temperature to predict the creep behaviour of weld joint. Parametric studies of various weld groove angles are carried out for a range of stress levels. Based on the analysis, critical locations and induced stress levels are identified and discussed.

Численное и экспериментальное исследование чистого изгиба балок из титанового сплава АБВТ-20 в условиях ползучести с учетом различных свойств на растяжение и сжатие

Рассматривается решение задачи чистого изгиба балки прямоугольного сечения в режиме ползучести с учетом различных свойств ползучести на растяжение и сжатие. Построен алгоритм и разработано программное обеспечение для математического моделирования процесса перераспределения напряжений по высоте балки с учетом накопления повреждений. Моделирование процессов ползучести разупрочняющегося материала происходит на основе уравнений кинетической теории ползучести и повреждаемости. Численное решение задачи проводится на основе метода Рунге - Кутты - Мерсона. Проведено сравнение результатов моделирования с экспериментальными данными чистого изгиба балок прямоугольного сечения из титанового сплава АБВТ-20 при действии знакопеременного изгибающего момента в условиях продолжительного воздействия температуры ($750^\circ$C), которое показало удовлетворительное соответствие результатов расчета экспериментальным данным.