Inelastic Strain Rate Research Articles

Inelastic Constitutive Equation Based on a New Viscoelastic Material Model.

A new viscoelastic material model was developed to describe deformations under various loads at the high-temperature creep regime. In the model the inelastic strain rate is written in the form of a flow equation of the Norton-Bailey type and the material hardening during deformation under various loads is induced by an internal stress which is subdivided into back stress and friction stress. The back stress represents a conservative part of the creep resistance while the friction stress includes all the dissipative parts in the internal structure such as grain boundary sliding, diffusion and interaction between dislocation and precipitation. An inelastic constitutive equation is established based on the new viscoelastic material model. A case study was done for Hastelloy XR, a Ni-based superalloy. Deformation analyses show a good agreement between calculations and experimental results for creep, relaxation, tensile and stress dip tests.

Isotropic viscoplasticity theory based on overstress (VBO). The influence of the direction of the dynamic recovery term in the growth law of the equilibrium stress

Abstract Two versions of the isotropic, small strain theory of viscoplasticity based on overstress (VBO) are given. They differ only in the dynamic recovery term of the growth law for the equilibrium stress. In the Yao formulation, this term is in the direction of the differences between the equilibrium and the kinematic stress deviators, two state variables of the theory. The inelastic strain rate determines the direction of this term in the Lee formulation. The predictions of the two formulations are compared in numerical experiments simulating proportional and nonproportional cyclic and monotonic loading. The two versions have the same material constants. They give identical results in uniaxial and proportional monotonic and cyclic loadings for a 6061 T6 aluminum alloy. They differ considerably when nonproportional loading is involved. In a strain controlled corner path the strees component corresponding to the strain that is held constant almost reaches zero at large strains in the Yao formulation. In the Lee formulation the stress is very different from zero. Also, the effective stress-strain diagram for the rectangular path ultimately joins the stress-strain diagram obtained in monotonic loading for the Yao model. A permanent difference remains for the Lee formulation. In cyclic 90° out-of-phase loading the hysteresis loops generated by the Lee model are much narrower than the ones predicted by the Yao version. The predictions of the Yao formulation are in good agreement with the results of biaxial experiments on the aluminum alloy. An analysis of the equations shows that differences between the predictions of the two formulations must be expected whenever unloadings are involved. The agreement in cyclic proportional loadings for the aluminum alloy is apparently due to the special material properties of this alloy. When material properties close to that of stainless steel are used a transient difference between the two formulations is found in cyclic proportional loadings. This experience suggests that models that predict the same behavior in monotonic loading may have vastly different responses in other loadings. Biaxial, nonproportional experiments and their numerical simulations are needed for the development of a reliable constitutive equation.

Thermodynamic restrictions on constitutive equations for second-deformation-gradient inelastic behavior

The aim of this paper is to examine the possibility of a model of non-local inelasticity on thermodynamic grounds in the absence of higher-order stresses or additional kinematic entities. On considering a material that cannot support higher-order stresses, it is found that thermodynamics does not allow an explicit non-local dependence in the free-energy and hence the stress and entropy response, but allows it in the strength and internal variable rate response. Rate-independent plasticity presents its own peculiarities in the proof of this result, which are also inherent in deriving the constitutive restrictions for the local material. An account of the basic thermodynamic framework and the reasoning used in these proofs is provided in listing the result for the conventional rate-independent solid. This procedure represents the building blocks for the derivation of the constitutive restrictions of the non-local theory in which second-deformation-gradient-dependent constitutive postulates for inelastic materials are introduced. Such postulates involve a material length scale due to the gradient dependence and achieve uniqueness in constitutive response trivially. The constitutive restrictions are shown not to preclude the existence of a class of second-deformation-gradient-dependent inelastic constitutive postulates for which the corresponding bvp of incremental equilibrium requires no extra boundary conditions above the conventional specification. The general considerations mentioned above are utilized to illustrate the constitutive structure of a rate-independent metal single crystal deforming by multiple slip, whereby contact with earlier work in this area is established. The thermodynamic results are also used to derive conditions for the existence of a gradient-dependent potential for the inelastic strain rate for inelastic rate-dependent materials.

Dynamic failure mechanisms of ceramic bars : Experiments and numerical simulations

Failure mechanisms in ceramics are investigated by means of bar impact experiments and numerical simulations of the wave propagation event. Stress histories are measured by embedding manganin stress gauges in the ceramic bars. The fracture event is examined by high speed photography. A violent radial expansion, in a region close to the impact surface, followed by a cloud of debris is observed. Numerical simulations of the inelastic wave propagation event are performed with a multiple-plane microcracking model. The simulations show that when the impact stress exceeds a material threshold, the stress wave in the bar has a relatively short duration which is controlled by the rate of unconfined compressive damage. A nonzero inelastic strain rate at the wave front is required in the simulations to properly capture the measured stress attenuation with propagation distance. This feature is related to a heterogeneous material microstructure which is a common occurrence in ceramics. Furthermore, the simulations predict a radial expansion of the bar as a result of not only compressive but also tensile damage. The radial velocity histories on the bar surface are functions of wave propagation distance and damage rate. Tensile damage is induced by stress release from the rod surface and is restricted to the bar core, due to wave focusing, and to the bar free end. In the latest case, reflection of the compressive pulse produces bar spallation. The two dimensional distribution of tensile and compressive damage is assessed by means of contour plots of volumetric strain and the second invariant of the inelastic strain tensor.

Creep-Fatigue Interaction of Modified 9Cr-1Mo Steel in a Very High Vacuum Environment and Experimental Analysis of Overstress

Creep-fatigue tests were conducted with Modified 9Cr-lMo steel at 600°C in a very high vacuum of 0.1 μPa in order to investigate a pure creep-fatigue behavior which is free from the environmental effect of the air. On the whole, the time-dependent life reduction occurs when the duration of tension is longer than that of compression. A good correlation was found between the creep-fatigue life and the fracture mode. When the fracture mode is transgranular, no time-dependent life reduction occurs. In contrast, when it is predominantly intergranular, a significant life reduction is observed. The overstress was experimentally analyzed to evaluate the creep-fatigue damage. The relationship between the overstress and the inelastic strain rate can be fitted, being independent of the strain range.

Elasto-Plastic Micro-Macro Modelling of Solid-Solid Phase Transformation ; Application to Transformation Induced Plasticity

Classical plasticity (dislocation motion) coupled to deformation due to lattice changes (phase transformation) occurs in several materials where interfaces are moving under external forces (TRIP (TRansformation Induced Plasticity) steels, even shape memory alloys). In the case of steels (transformation induced plasticity) the internal stress associated with the phase change induces a large additional plastic flow inside austenite and martensite. We propose a micromechanical modelling of such a phenomenon based on a decomposition of strain rate into an elastoplastic part and a given lattice inelastic strain rate field is obtained. The self consistent approximation allows to determine the behavior of the equivalent material and the equivalent transformation strain rate. Applications in the case of cooling under constant applied stresses and isothermal loading give results in good agreement with Finite Element calculations and theoretical results

Some fundamental issues in rate theory of damage-elastoplasticity

Abstract The paper elaborates on some fundamental constitutive issues in the rate theory of damage-elastoplasticity. The analysis combines the constitutive theories of elastoplastic and progressively damaged solids. After defining needed kinematic and kinetic preliminaries, the anisotropic elastic response is analyzed by introducing a set of damage tensors which represent material degradation and induced elastic anisotropy. Decomposition of the rate of stress and deformation tensors into their elastic and inelastic parts is then defined in a manner analogous to the corresponding decomposition used in large-deformation elastoplasdcity theory. The procedure is further developed to partition the inelastic stress and strain rates into the damage and plastic parts, which takes into account the physics of these deformation processes. The energy dissipation rate is derived and the thermodynamic forces conjugate to elastic stiffness and compliance tensors are identified, based on a thermodynamic analysis of isothermal deformation process. The damage potentials for the corresponding fluxes are introduced and the constitutive expressions for the damage stress and strain rates are established. The concept of a damage surface is used to define the onset and evolution of damage. A constitutive analysis for inelastic stress and strain rates is then presented. The inelastic potential function and the yield surface are introduced. A dual formulation is constructed in both the stress and strain spaces. The two limiting cases, one involving plasticity without damage, and the other involving damage without plasticity, are deduced from the developed and more general constitutive framework of damage-elastoplasticity.

THE OVERSTRESS DEPENDENCE OF INELASTIC RATE OF DEFORMATION INFERRED FROM TRANSIENT TESTS

Transient tests are used to demonstrate unusual or even paradoxical deformation behaviors in metallic alloys and polymers at ambient and elevated temperatures. Included are repeated relaxation tests and creep tests of short duration. It is shown that creep rate need not increase with a stress increase and that, at the same stress level, creep rate can be different on loading and unloading. Also, a stress magnitude increase can be found during relaxation. Once steady inelastic flow is reached, the relaxation rate is nearly independent of the stress and strain at which relaxation starts; it depends mainly on the preceding prior inelastic strain rate. These phenomena can be modeled using an overstress (effective stress) dependence of inelastic rate of deformation and a proper evolution law for a single state variable, the equilibrium stress (back stress) within the context of a “unified” state variable theory. The author and his students have developed the viscoplasticity theory based on overstress (VBO) that can model these unusual behaviors.

Modeling Complex Inelastic Deformation Processes in IC Packages’ Solder Joints

Large scale three-dimensional finite element analyses of 68 and 44 pin types Quad Flat I-Leaded packages subjected to −55°C to 150°C temperature cycling are performed to predict the way in which inelastic strains and microstructure evolve in the solder joints. A new set of unified constitutive equations, which correctly accounts for the inelastic strain rate dependency on stress, temperature, and microstructure (internal stresses) and the evolution of the latter with deformation, was used to model the solder joints’ thermo-mechanical behavior. Estimates of the critical inelastic strain ranges in the solder joints were obtained from the calculated inelastic deformation histories. The range of inelastic strain associated with each thermal cycle is small and practically unaffected by the observed ratcheting of the cycles toward positive strains due to the non-zero mean cycle stresses. By combining the numerical results with fatigue life data obtained from leads subjected to the same thermal loading, a fatigue life relation for mean failure probability of the IC packages’ solder joints has been proposed. The analysis correctly predicts the critical locations for crack initiation.

Damage-induced nonassociated inelastic flow in rock salt

The multimechanism deformation coupled fracture model recently developed by Chan et al. [1992], for describing time-dependent, pressure-sensitive inelastic flow and damage evolution in crystalline solids was evaluated against triaxial creep experiments on rock salt. Guided by experimental observations, the kinetic equation and the flow law for damage-induced inelastic flow in the model were modified to account for the development of damage and inelastic dillation in the transient creep regime. The revised model was then utilized to obtain the creep response and damage evolution in rock salt as a function of confining pressure and stress difference. Comparison between model calculation and experiment revealed that damage-induced inelastic flow is nonassociated, dilational, and contributes significantly to the macroscopic strain rate observed in rock salt deformed at low confining pressures. The inelastic strain rate and volumetric strain due to damage decrease with increasing confining pressures, and all are suppressed at sufficiently high confining pressures.

The role of acoustic emission in the study of rock fracture

The development of faults and shear fracture systems over a broad range of temperature and pressure and for a variety of rock types involves the growth and interaction of microcracks. Acoustic emission (AE), which is produced by rapid microcrack growth, is a ubiquitous phenomenon associated with brittle fracture and has provided a wealth of information regarding the failure process in rock. This paper reviews the successes and limitations of AE studies as applied to the fracture process in rock with emphasis on our ability to predict rock failure. Application of laboratory AE studies to larger scale problems related to the understanding of earthquake processes is also discussed. In this context, laboratory studies can be divided into the following categories. 1) Simple counting of the number of AE events prior to sample failure shows a correlation between AE rate and inelastic strain rate. Additional sorting of events by amplitude has shown that AE events obey the power law frequency-magnitude relation observed for earthquakes. These cumulative event count techniques are being used in conjunction with damage mechanics models to determine how damage accumulates during loading and to predict failure. 2) A second area of research involves the location of hypocenters of AE source events. This technique requires precise arrival time data of AE signals recorded over an array of sensors that are essentially a miniature seismic net. Analysis of the spatial and temporal variation of event hypocenters has improved our understanding of the progression of microcrack growth and clustering leading to rock failure. Recently, fracture nucleation and growth have been studied under conditions of quasi-static fault propagation by controlling stress to maintain constant AE rate. 3) A third area of study involves the analysis of full waveform data as recorded at receiver sites. One aspect of this research has been to determine fault plane solutions of AE source events from first motion data. These studies show that in addition to pure tensile and double couple events, a significant number of more complex event types occur in the period leading to fault nucleation. 4) P and S wave velocities (including spatial variations) and attenuation have been obtained by artificially generating acoustic pulses which are modified during passage through the sample.

The Influence of Cool-Down Temperature Histories on the Residual Stresses in Fibrous Metal-Matrix Composites

The vanishing fiber diameter model together with the thermoviscoplastic ity theory based on overstress including a recovery of state formulation is introduced. They are employed to analyze the effects of temperature rate and of annealing at constant temperature on the residual stresses at room temperature when unidirectional fibrous metal-matrix composites are cooled down from 1000°C during the manufacturing process. For the present analysis the fibers are assumed to be transversely isotropic thermoelastic and the matrix constitutive equation is isotropic thermoviscoplastic including recovery of state. All material functions and constants can depend on current temperature. Yield sur faces and loading/unloading conditions are not used in the theory in which the inelastic strain rate is solely a function of the overstress, the difference between stress and the equi librium stress, a state variable of the theory. Assumed but realistic material elastic and vis coplastic properties as a function of temperature which are close to W/9Cr-lMo steel com posite permit the computation of residual stresses. Due to the viscoplasticity of the matrix time-dependent effects such as creep and change of residual stresses with time are found. It is found that the residual stresses at room temperature change considerably with temper ature history. The matrix residual stress, upon reaching room temperature, is highest for the fastest cooling rate, but after 30 days rest the influence of cooling rate is hardly notice able since relaxation takes place. Annealing periods can reduce the residual stresses com pared to continuous cooling.

Thermally and mechanically activated dislocation glide: Experimental results and theoretical analysis

The results of room temperature stress rate change tests conducted on OFE copper are presented. The data exhibit apparent inelastic strain rate discontinuity at the point of stress rate change, indicating the presence of mechanical activation of dislocation glide. However, a new description of deformation that includes thermal and mechanical activation of glide is used to demonstrate that the current testing conditions could not have led to a measurable amount of mechanical activation. Further, a model of deformation that includes only thermal activation is shown to adequately describe the rapid strain rate decay observed in OFE copper following a stress rate decrease. This model indicates that the time resolution of 1 ms required to measure this decay is greater than the experimentally achieved time resolution of 28 ms. Together, these observations demonstrate that mechanical activation of dislocation glide was not detected in the present experiments. The analyses can also be used to predict the conditions under which mechanical activation may be observable.

Extensive modeling of high temperature deformation for cyclically softening material

A unified inelastic constitutive equation is proposed to describe the cyclic inelastic deformation of 2 1 4 Cr-1Mo steel at an elevated temperature of 550°C. In the present study, the evolution of overstress is discussed which is shown to lead to a time-independent term in the inelastic strain rate. This term is proportional to the stress rate. Another significant point proposed in this study is the distinction of the inelastic deformation depending on the evolution of overstress. A mathematical model has been developed based on these assumptions. Many numerical experiments were conducted to examine the validity of the present model, and in comparison with the experimental results, the present model has been shown to give good descriptions for a wide variety of loading conditions, including rate-dependent plasticity, stress relaxation, and cyclic softening.

Mechanical testing using direct control of the inelastic strain rate

A methodology is described for carrying out mechanical tests under direct control of the inelastic strain rate. The method depends on the use of a servohydraulic testing machine and suitable electronic-control circuitry. The technique permits the use of abrupt changes of inelastic strain rate as well as maintenance of constant values of inelastic strain rate. The application of the method to several modes of testing is described.

Extensive Modeling of High-Temperature Deformation Based on Unified Approach

A unified inelastic constitutive equation is proposed to describe the cyclic inelastic deformation of 2 1/4Cr-1Mo steel at 550°C. In the present study, an evolution of the overstress is discussed which leads the time-independent term into the inelastic strain rate. A mathematical model is developed and some examples of the numerical simulation are presented.

Room temperature creep in saturated granite

Cylindrical samples of granite were deformed at 26ºC, constant confining pressure (60 MPa), and constant pore pressure (20 MPa). Axial and volumetric strain were determined from changes in the output of resistance foil strain gauges bonded to the rock surface. In addition, dc electrical resistivity was measured parallel to the sample axis. During each experiment (typically lasting from 1–2 weeks), the deviatoric stress σd applied to the sample was cycled between 70% and 90% of the short–term failure strength. The bulk of the experiments were conducted in the secondary or “steady state” creep regime. Inelastic volumetric strain rate was found to obey the law , where and (compressive stresses are negative). The C coefficient represents a strain‐hardening‐like term. The stress dependence is of the same form as the stress dependence measured for mode I crack growth in double cantilever beam experiments. The observed creep behavior is analyzed in terms of stress corrosion and crack growth models including a formulation based on energy release rate for characteristic microcracks.

Creep-Fatigue Interaction of Modified 9Cr-1 Mo Steel in a Very High Vacuum Environment and Experimental Analysis of Overstress.

Creep-fatigue tests were conducted with Modified 9Cr-1 Mo steel at 600°C in a very high vacuum of 0. 1 μPa in order to observe pure creep-fatigue behavior free from environmental effects. On the whole, time-dependent life reduction occurs when duration of tension is longer than duration of compression. A good correlation was observed between creep-fatigue life and fracture mode. When the fracture mode is transgranular, no time-dependent life reduction occurs. In contrast, when it is predominantly intergranular, a significant life reduction is observed. The overstress was experimentally analyzed to evaluate the creep-fatigue damage. The relationship between the overstress and the inelastic strain rate can be fitted with a bilinear curve independent of the strain range. At a lower strain rate, the line has a gradient ; however, there is no gradient at a higher strain rate.

Kinematic hardening rules with critical state of dynamic recovery, part I: formulation and basic features for ratchetting behavior

Abstract Kinematic hardening rules formulated in a hardening/dynamic recovery format are examined for simulating rachetting behavior. These rules, characterized by decomposition of the kinematic hardening variable into components, are based on the assumption that each component has a critical state for its dynamic recovery to be activated fully. Discussing their basic features, the authors show that they can predict much less accumulation of uniaxial and multiaxial ratchetting strains than the Armstrong and Frederick rule. Comparisons with multilayer and multisurface models are made also, resulting in a finding that the simple one in the present rules is similar to the multilayer model with total strain rate replaced by inelastic (or plastic) strain rate. Part II of this work deals with applications to experiments.

A constitutive model for inelastic flow and damage evolution in solids under triaxial compression

A constitutive model for describing time-dependent, pressure-sensitive inelastic flow and damage evolution in crystalline solids under non-hydrostatic compression has been developed on the basis that the relevant damage and dislocation flow processes both contribute to the overall inelastic strain rate. A damage-based kinetic equation is first formulated using the work-conjugate approach and the continuum damage concept. That relation is then added to the dislocation-based kinetic equation of a multi-mechanism deformation (M-D) model to obtain the macroscopic inelastic strain rate. The proposed kinetic relation for the overall inelastic strain rate is shown to be derivable from a flow potential. The kinetic equation indicates plastic dilatancy under triaxial compression when the damage term is activated, and leads to plastic incompressibility when inelastic straining is primarily provided by dislocation flow mechanisms. The dependence of creep rate and plastic dilatancy on confining pressure shown by model calculations for rock salt is in accordance with experimental observations.