Time-dependent Inelastic Deformation Research Articles

Evaluation of reservoir subsidence due to hydrocarbon production based on seismic data

Environmental problems associated with depleted oil and gas reservoirs upon long-term production from them are likely to become important challenges in future decades. With the increasing trend of production from hydrocarbon reservoirs, more and more reservoirs across the world are reaching the second half of their life—a fact that places an emphasis on the necessity of investigating what is known as reservoir subsidence. Different analytical and numerical approaches have been introduced for analyzing the subsidence on the basis of the elasticity theory but in the form of case studies, leaving a comprehensive model yet to be proposed. In this work, a formulation was introduced for estimating reservoir subsidence by integrating the rock physics, rock mechanics, and thermo-poroelasticity theories. Then, a modified version of this formulation was developed to calculate compaction in an actively producing reservoir that is suspect of subsidence, as a case study. For this purpose, triaxial hydrostatic tests were carried out on core plugs obtained from the considered reservoir, and then, compaction parameters (i.e., compression index and coefficient of deformation) were obtained at a laboratory scale. In order to evaluate the subsidence at a reservoir scale, the laboratory-scale results and in situ reservoir properties were integrated with well-logging and 3D seismic data at well location to come up with 3D cubes of compaction information. Continuing with the research, time-dependent inelastic deformation was modeled considering continued production for different future periods. The field observations showed that the estimated compaction is not visible at the surface in the form of subsidence due to the high depth and stiffness of the studied reservoir. However, collapse of casing at some of wells drilled into the studied reservoir could be attributed to the reservoir subsidence. Finally, variations of compaction with pore pressure were investigated to propose a model for predicting the subsidence in future periods. Findings of this research can be used to forecast subsidence at well location to take the required measures for avoiding possible casing collapse and/or relevant environmental issues.

Evaluation of local stress state due to grain-boundary sliding during creep within a crystal plasticity finite element multi-scale framework

Previous studies demonstrate that grain-boundary sliding could accelerate creep rate and give rise to large internal stresses that can lead to damage development, e.g. formation of wedge cracks. The present study provides more insight into the effects of grain-boundary sliding (GBS) on the deformation behaviour of realistic polycrystalline aggregates during creep, through the development of a computational framework which combines: i) the use of interface elements for sliding at grain boundaries, and ii) special triple point (in 2D) or triple line (in 3D) elements to prevent artificial dilation at these locations in the microstructure with iii) a physically-based crystal plasticity constitutive model for time-dependent inelastic deformation of the individual grains. Experimental data at various scales is used to calibrate the framework and compare with model predictions. We pay particular consideration to effects of grain boundary sliding during creep of Type 316 stainless steel, which is used extensively in structural components of the UK fleet of Advanced Gas Cooled Nuclear Reactors (AGRs). It is found that the anisotropic deformation of the grains and the mismatch in crystallographic orientation between neighbouring grains play a significant role in determining the extent of sliding on a given boundary. Their effect on the development of stress within the grains, particularly at triple grain junctions, and the increase in axial stress along transverse boundaries are quantified. The article demonstrates that the magnitude of the stress along the facets is highly-dependent on the crystallographic orientations of the neighbouring grains and the relative amount of sliding. Boundaries, transverse to the applied load tend to carry higher normal stresses of the order of 100-180 MPa, in cases where the neighbouring grains consist of plastically-harder crystallographic orientations. An approach to estimate the amount of sliding as a function of GB particle morphology and diffusive accommodation is developed and employed. The possible implications of these findings on the initiation of creep damage are commented on.

Strength and deformation of shocked diamond single crystals: Orientation dependence

Understanding and quantifying the strength or elastic limit of diamond single crystals is of considerable scientific and technological importance, and has been a subject of long standing theoretical and experimental interest. To examine the effect of crystalline anisotropy on strength and deformation of shocked diamond single crystals, plate impact experiments were conducted to measure wave profiles at various elastic impact stresses up to \ensuremath{\sim}120 GPa along [110] and [111] crystal orientations. Using laser interferometry, particle velocity histories and shock velocities in the diamond samples were measured and were compared with similar measurements published previously for shock compression along the [100] direction. Wave profiles for all three orientations showed large elastic wave amplitudes followed by time-dependent inelastic deformation. From the measured wave profiles, the elastic limits were determined under well characterized uniaxial strain loading conditions. The measured elastic wave amplitudes for the [110] and [111] orientations were lower for higher elastic impact stress (stress attained for an elastic diamond response), consistent with the result reported previously for [100] diamond. The maximum resolved shear stress (MRSS) on the {111}⟨110⟩ slip systems was determined for each orientation, revealing significant orientation dependence. The MRSS values for the [100] and [110] orientations (\ensuremath{\sim}33 GPa) are 25%--30% of theoretical estimates; the MRSS value for the [111] orientation is significantly lower (\ensuremath{\sim}23 GPa). Our results demonstrate that the MRSS depends strongly on the stress component normal to the {111} planes or the resolved normal stress (RNS), suggesting that the RNS plays a key role in inhibiting the onset of inelastic deformation. Lower elastic wave amplitudes at higher peak stress and the effect of the RNS are inconsistent with typical dislocation slip mechanisms of inelastic deformation, suggesting instead an inelastic response characteristic of shocked brittle solids. The present results show that the elastic limit (or material strength) of diamond single crystals cannot be described using traditional isotropic approaches, and typical plasticity models cannot be used to describe the inelastic deformation of diamond. Analysis of the measured wave profiles beyond the elastic limit, including characterization of the peak state, requires numerical simulations that incorporate a time-dependent, anisotropic, inelastic deformation response. Development of such a material description for diamond is an important need.

Evaluation and Modeling of time-dependent inelastic deformation in small strain range for thermosetting polymer

Evaluation and Modeling of time-dependent inelastic deformation in small strain range for thermosetting polymer

Shock compression and release of a-axis magnesium single crystals: Anisotropy and time dependent inelastic response

To gain insight into inelastic deformation mechanisms for shocked hexagonal close-packed (hcp) metals, particularly the role of crystal anisotropy, magnesium (Mg) single crystals were subjected to shock compression and release along the a-axis to 3.0 and 4.8 GPa elastic impact stresses. Wave profiles measured at several thicknesses, using laser interferometry, show a sharply peaked elastic wave followed by the plastic wave. Additionally, a smooth and featureless release wave is observed following peak compression. When compared with wave profiles measured previously for c-axis Mg [Winey et al., J. Appl. Phys. 117, 105903 (2015)], the elastic wave amplitudes for a-axis Mg are lower for the same propagation distance, and less attenuation of elastic wave amplitude is observed for a given peak stress. The featureless release wave for a-axis Mg is in marked contrast to the structured features observed for c-axis unloading. Numerical simulations, using a time-dependent anisotropic modeling framework, showed that the wave profiles calculated using prismatic slip or (101¯2) twinning, individually, do not match the measured compression profiles for a-axis Mg. However, a combination of slip and twinning provides a good overall match to the measured compression profiles. In contrast to compression, prismatic slip alone provides a reasonable match to the measured release wave profiles; (101¯2) twinning due to its uni-directionality is not activated during release. The experimental results and wave profile simulations for a-axis Mg presented here are quite different from the previously published c-axis results, demonstrating the important role of crystal anisotropy in the time-dependent inelastic deformation of Mg single crystals under shock compression and release.

Effect of atomic number and atomic weight on time-dependent inelastic deformation in metals

Effect of atomic number and atomic weight on time-dependent inelastic deformation in metals

Unloading and reloading response of shocked aluminum single crystals: Time-dependent anisotropic material description

The measured unloading and reloading wave profiles in shocked metals have long been known to show deviations from the usual elastic-plastic response used to model shocked metals. To gain insight into inelastic deformation mechanisms during unloading and reloading from the shocked state and to avoid complexities associated with material heterogeneities present in polycrystalline metals, this work has focused on the shock wave response of single crystals. Here, we report on wave propagation simulations to examine the unloading and reloading response of aluminum single crystals shocked along [100], [110], and [111] orientations to 21 GPa peak stresses. The simulations utilized a dislocation dynamics based, time-dependent anisotropic material description to model shock-induced elastic-plastic deformation. The calculated profiles provide good overall agreement with measured unloading and reloading wave profiles [H. Huang and J. R. Asay, J. Appl. Phys. 101, 063550 (2007)] for all three crystal orientations and for two different peak stresses. Our single crystal simulations show that differences between the profiles calculated using simpler elastic-plastic models and the measured profiles can be understood in terms of time-dependent material inelastic response. Therefore, the general applicability of previous time-independent analysis methods to infer shear strength from unloading and reloading wave profiles is questionable. Our simulations also show that dislocation mechanisms governing the unloading response of shocked Al single crystals differ from those governing the reloading response. Overall, our findings show that a dislocation-based approach provides a useful physical description to explain unloading and reloading of shocked Al single crystals. More generally, the present work demonstrates that understanding and modeling the unloading and reloading response of shocked metals may require the incorporation of time-dependent inelastic deformation response.

Design of functionally graded thermal barrier coatings based on a nonlinear micromechanical approach

This study presents the methodology of analysis and design of functionally graded thermal barrier coatings (FG TBCs) consisting of ceramics and metals based on a mean-field micromechanical approach taking into account the time-independent and dependent inelastic deformation, such as plasticity of metals, creep of metals and ceramics, and diffusional mass flow at the ceramic/metal interface. Ni–ZrO 2 systems were numerically studied and the effect of the compositional gradation patterns and time-dependent inelastic deformation on the micro- and macro-stress states in the FG TBCs were examined. The suitable compositional gradation patterns were proposed for typical thermo-mechanical boundary conditions. It was shown that the creep deformation near the ceramic surface can introduce large tensile stresses, potentially leading to thermal fracture, which is possibly reduced by controlling the ability of creep of the ceramics.

Fractal Geometry-Based Hypergeometric Time Series Solution to the Hereditary Thermal Creep Model for the Contact of Rough Surfaces Using the Kelvin-Voigt Medium

This paper aims at constructing a continuous hereditary creep model for the thermoviscoelastic contact of a rough punch and a smooth surface of a rigid half-space. The used model considers the rough surface as a function of the applied load and temperatures. The material of the rough punch surface is assumed to behave as Kelvin-Voigt viscoelastic material. Such a model uses elastic springs and viscous dashpots in parallel. The fractal-based punch surface is modelled using a deterministic Cantor structure. An asymptotic power law, deduced using approximate iterative relations, is used to express the punch surface creep which is a time-dependent inelastic deformation. The suggested law utilized the hypergeometric time series to relate the variables of creep as a function of remote forces, body temperatures, and time. The model is valid when the approach of punch surface and half space is in the order of the size of the surface roughness. The closed-form results are obtained for selected values of the system parameters; the fractal surface roughness and various material properties. The obtained results show good agreement with published experimental results, and the methodology can be further extended to other structures such as the Kelvin-Voigt medium within electronic circuits and systems.

Edward Walter Hart

Edward Walter Hart, a theoretical and experimental physicist who was well known for developing the universal equation of state for mechanical deformation of solids, died on 22 December 2004 in Lancaster, Pennsylvania.Born in Easton, Pennsylvania, on 14 January 1918, Ed graduated from the City College of New York in 1938 and then received MS and PhD degrees from the University of California, Berkeley. During World War II, he worked for the US Navy in Washington, DC, where he invented magnetic compass correctors and a better ship compass—work for which he received a Meritorious Civilian Award. He next spent 25 years at the General Electric Research and Development Laboratory in Schenectady, New York, before becoming a joint professor in the materials science and engineering department and in the theoretical and applied mechanics department at Cornell University in 1976. His research years at Cornell, from 1976 to 1988 when he became emeritus, were particularly productive.Ed invested four decades of research on micromechanics of materials and, specifically, on the microscopic nature of failure and inelasticity. Beginning with his early research in 1952 on state-variable modeling of inelastic deformation, a tremendous amount of effort has been invested in linking a material’s microstructure (including grain-boundary and dislocation motion) and its mechanical properties. He is rightfully famous for his theoretical work in the modeling of inelasticity, fracture, precipitation hardening, dislocation motion, self-diffusion, grain-boundary motion, and the mechanics of superplasticity. The typical senior-level textbook on the mechanical properties of materials has a detailed section on the generalized equation of state, which can be obtained from a tension test and then applied to plastic deformation processes for all time-independent combinations of stress and strain. The last two decades of Ed’s work were dedicated to adding time dependence to the equation-of-state approach. The input again was the tension test, but Ed added time as a variable along with the results from stress-relaxation tests at points of fixed generalized strain. His highly cited 1967 Acta Metallica paper “Theory of the Tensile Test” and numerous papers on load relaxation and creep led to a generalized time-dependent equation of state that could be obtained from experiments. A complement and challenge to those theorists who attempt to use atomic terms to describe point defects, dislocations, planar defects, second phase particles, and so on, the universal equation of state has led to enormous elucidation of time-dependent inelastic and permanent deformation. Ed was awarded an Alexander von Humboldt Senior Scientist Award by West Germany in 1982.Ed was a Renaissance man. As a young man, he studied composition under Aaron Copland. He played guitar, viola, and piano and directed a choir and a chorus. He studied modern dance with Welland Lathrop, and he was a founder of the Schenectady Civic Ballet Co and served as its president from 1960 to 1963. A lover of nature, mountain climbing, and camping, he was a long-time member of the Adirondack Mountain Club. Ed enriched the lives of many, and he is sorely missed by his family and friends. Edward Walter Hart PPT|High resolution© 2005 American Institute of Physics.

Time-dependent inelastic deformation of shocked soda-lime glass

Plate impact experiments were carried out to understand inelastic deformation in soda-lime glass shocked between 3 and 10.8GPa. In-material, wave profile measurements were obtained using longitudinal and lateral stress gauges (4.6–10.8GPa), and electromagnetic particle velocity gauges (2.9–6GPa) at comparable sample thicknesses. The 4.6 and 6GPa experiments revealed time-dependent material inelastic response along with time-dependent loss of material strength. Because of the unsteady, two wave structure observed in the longitudinal wave profiles in conjunction with the time-dependent changes in the lateral stress data, previous interpretations of the shocked soda-lime glass response in terms of a propagating failure wave are not valid. At higher peak stresses (∼10GPa), the experimental results do not display time-dependent strength loss. The shock wave response of soda-lime glass over the 4–10GPa range is complex, and material strength and inelastic deformation features depend significantly on the peak stress. Using the experimental results, a phenomenological continuum model incorporating the various material phenomena was developed. Wave profile simulations using the continuum model show reasonable overall agreement with the experimental profiles at different stress levels. Because of the approximate nature of the continuum model, all of the experimental details were not reproduced in the wave propagation simulations. It is likely that around and above 10GPa, other material phenomena not included in our model may need to be considered.

Suppression of inelastic deformation of nanocoated thin film microstructures

We study the suppression of time-dependent inelastic deformation due to creep, stress relaxation, and microstructural evolution in multilayer thin film microstructures by the use of alumina nanocoatings realized by atomic layer deposition (ALD). Gold (0.5 μm thick)/polysilicon (1.5 or 3.5 μm thick) beam and plate microstructures were fabricated using surface micromachining. The microstructures were then coated on each side with a 40-nm-thick amorphous Al2O3 layer by ALD. The beam and plate microstructures were initially thermal cycled between room temperature and 190 °C to partially stabilize the gold microstructure. After the initial thermal cycles, the microstructures were cooled from 190 to 120 °C and held at 120 °C for about 700 h (4 weeks). We measured, using an interferometric microscope with a custom-built temperature chamber, full-field deformed shapes (and from these determined the average curvatures in the x and y directions) of the microstructures during the initial thermal cycles, during the cooling process from 190 to 120 °C, and during the isothermal hold. Measurements were made on both coated and uncoated microstructures to assess the influence of the coating. We find that while the 40-nm-thick coating has a small effect on the thermoelastic response of the microstructure, it significantly reduces the extent of inelastic deformation during the isothermal hold. We modeled the curvature evolution with time assuming the inelastic deformation mechanism can be described by power-law creep in the gold, ε̇=Aσn, and that the polysilicon and alumina deform elastically. The simple model describes the observed behavior reasonably well for the uncoated microstructures (when the power-law parameters are fit using the measured curvature), however, for the coated microstructures, the model predicts a decrease in the inelastic deformation, but nowhere near the magnitude observed. This suggests not only an alternation of the stress state in the gold film by the nanoscale coating, but also a change in the fundamental inelastic deformation mechanisms.

Staggering error control of a class of inelastic processes in random microheterogeneous solids

In this work a staggering solution strategy for the simulation of the time-dependent inelastic mechanical deformation of a class of solids, possessing irregular heterogeneous microstructure, is developed. The system of coupled equations involved consists of (1) a dynamic equation of momentum balance, where the primary field variable is the displacement, (2) an evolution equation for material degradation, where the primary field variable is a state damage function, and (3) an evolution equation for the inelastic strains in the solid where the primary field variable is a plastic strain field. Clearly, the damage and plasticity variables are implicit functions of the displacement, however, for the staggering scheme strategy, it is convenient to formulate them as individual fields during the solution process. The key concept for the strategy to operate efficiently is to estimate and control the so-called staggering error, i.e. the error due to incompletely resolving the coupling between the field equations in a staggering process. This error is a function of the time step size. However, because the coupling is temporally variable, possibly becoming stronger, weaker, or oscillatory, it is extremely difficult to ascertain a priori the time step size needed for prespecified error control. In the present work, to induce desired staggering rates of convergence within each time step, thus controlling the staggering error, an adaptive strategy is developed whereby the time step size is manipulated, enlarged or reduced, to control the intrinsic contraction mapping constant of the staggering system operator. The overall goal is to deliver accurate solutions where temporal discretization error control dictates the upper limits on the time step size, while the iterative staggering strategy refines the step size further to control the staggering error. Three-dimensional numerical experiments are performed to illustrate the solution strategy.

Analysis of Thermal Stresses in a Functionally Graded Material Plate Considering Time-Dependent Inelastic Deformation by Micromechanical Approach.

The purpose of this paper is to present a numerical analysis method on thermal stresses in a functionally graded material (FGM) plate considering time-dependent inelastic deformation. This FGM plate has a ceramic-to-metal graded microstructure with spherical particles, which is subjected to a unidirectional heat flow through the thickness. The procedure of this analysis is composed of micromechanics and macromechanics. Micromechanics is based on Eshelby's equivalent inclusion method and Mori-Tanaka's mean field approximation, and according to this theory, the constitutive equation for two-phase composite materials subjected to equally biaxial loadings is formulated. This equation deals with inelastic deformation, that is plastic deformation, creep deformation and interfacial diffusion between ceramic phase and metal one. On the other hand, for macromechanics, the classical laminated plate theory is applied and the above constitutive equation is put in this theory. As an example, some numerical results for Al2O3/Ni-FGM and shown.

Micromechanical Analysis of Inelastic Thermal Stress Relaxation in a Functionally Graded Laminate. Effects of Graded Compositional Distributions and Number of Layers.

In this paper, transient thermal stresses in an Al2O3-Ni functionally graded material (FGM) plate subjected to through-thickness heat flow have been numerically analyzed under three different mechanical boundary conditions by taking account of time dependent inelastic deformation due to interfacial diffusion between ceramic and metal phases as well as creep of both phases. First, an outline of the analytical method, which consists of micromechanical and macromechanical approaches, is briefly represented. The micromechanical approach stems from Eshelby's equivalent inclusion method and Mori-Tanaka's mean-field approximation, and the macromechanical one is based on the classical laminated plate theory. Second, we have systematically examined the effects of graded compositional distributions and the number of layers on the thermal stresses in the FGM plate in detail.

An Inelastic Constitutive Equation of Fiber Reinforced Plastic Laminates

A Constitutive model for describing the time dependent inelastic deformation of unidirectional and symmetric angle-ply CFRP (Carbon Fiber Reinforced Plastics) laminates is developed. The kinematic hardening creep law of Malinin and Khadjinsky and the evolution equation of Armstrong and Frederick are extended to describe the creep deformation of initially anisotropic materials. In particular, the evolution equations of the back stresses of the anisotropic material were formulated by introducing a transformed strain tensor, by which the expression of the equivalent strain rate of the an isotropic material has the identical form as that of the isotropic materials. The resulting model is applied to analyze the time dependent inelastic deformation of symmetric angle ply laminates. Comparison between the predictions and the experimental observations shows that the present model can describe well the time dependent inelastic behavior under different loadings.

Fatigue Life Predictions for Thermally Loaded Solder Joints Using a Volume-Weighted Averaging Technique

Fatigue lives of thermally loaded solder joints are predicted using the finite element method. An appropriate constitutive relation to model the time-dependent inelastic deformation of the near-eutectic solder is implemented into a commercial finite element code, and the stress-strain responses of different electronic assemblies under the applied temperature cycles are calculated. The finite element analysis results are coupled with a newly developed approach for fatigue life predictions by using a volume-weighted averaging technique instead of an approach based on the maximum stress and strain locations in the solder joint. Volume-weighted average stress and strain results of three electronic assemblies are related to the corresponding experimental fatigue data through least-squares curve-fitting analyses for determination of the empirical coefficients of two fatigue life prediction criteria. The coefficients thus determined predict the mean cycles-to-failure value of the solder joints. Among the two prediction criteria, the strain range criterion uses the inelastic shear strain range and the total strain energy criterion uses the total inelastic strain energy calculated over a stabilized loading cycle. The obtained coefficients of the two fatigue criteria are applied to the finite element analysis results of two additional cases obtained from the literature. Good predictions are achieved using the total strain energy criterion, however, the strain range criterion underestimated the fatigue life. It is concluded that the strain information alone is not sufficient to model the fatigue behavior but a combination of stress and strain information is required, as in the case of total inelastic strain energy. The superiority of the volume-weighted averaging technique over the maximum stress and strain location approach is discussed.

ＣＦＲＰ積層材に対する時間依存非弾性変形のモデリング

A constitutive model for describing the inelastic deformation of unidirectional and symmetric angle-ply CFRP (Carbon Fiber Reinforced Plastics) laminates is developed. The kinematic hardening creep flow law of Malinin and Khadjinsky and the evolution equation of Armstrong and Frederick are extended to describe the creep deformation of anisotropic materials. In order to express anisotropic hardening, back stress taking account of anisotropic inelastic strain sensitivity is incorporated into the creep constitutive equation. Then, the resulting model is applied to analyze the time-dependent inelastic deformation of symmetric angle-ply laminates. Comparison between the prediction and the experimental observation shows that the present model can describe well the time-dependent inelastic behavior under different loadings.

Thermo-elastoviscoplastic snapthrough behavior of cylindrical panels

The thermo-elastoviscoplastic snapthrough behavior of simply supported cylindrical panels is investigated. The analysis is based on nonlinear kinematic relations and nonlinear rate-dependent unified constitutive equations which include both Bodner-Partom's and Walker's material models. A finite element approach is employed to predict the inelastic buckling behavior. Numerical examples are given to demonstrate the effects of several parameters which include the temperature, thickness and flatness of the panel. Comparisons of buckling responses betweeen Bodner-Partom's model and Walker's model are given. The creep buckling behavior, as an example of time-dependent inelastic deformation, is also presented.

Back stress and stress relaxation behavior under biaxial loading in the high temperature environment.

Tension-torsion biaxial creep-fatigue tests have been conducted with 304 stainless steel at 650°C. An overstress and back stress were analyzed under biaxial stress conditions. Under proportional biaxial loading, peak stress components were well arranged by using Von Mises equivalent stress. However, the overstress and the back stress showed a dominant anisotropy. A long-term stress relaxation behaviour was examined under proportional loading. The result suggests that a time-dependent inelastic deformation process is composed of at least two origins. One is caused from a relaxation of the overstress and the other is caused from that of the back stress. The former is much faster than the latter. The relaxation of the back stress indicates a recovery of the anisotropy.