Plastic Strain Tensor Research Articles

In situ study of microstructure evolution and α → ω phase transition in annealed and pre-deformed Zr under hydrostatic loading

The detailed study of the effect of the initial microstructure on its evolution under hydrostatic compression before, during, and after the irreversible α→ω phase transformation and during pressure release in Zr using in situ x-ray diffraction is presented. Two samples were studied: one is plastically pre-deformed Zr with saturated hardness and the other is annealed. Phase transformation α→ω initiates at lower pressure for a pre-deformed sample but for a volume fraction of ω Zr, c>0.7, a larger volume fraction is observed for the annealed sample. This implies that the proportionality between the athermal resistance to the transformation and the yield strength in the continuum phase transformation theory is invalid; an advanced version of the theory is outlined. Phenomenological plasticity theory under hydrostatic loading is outlined in terms of microstructural parameters, and plastic strain is estimated. During transformation, the first rule is suggested, i.e., the average domain size, microstrain, and dislocation density in ω Zr for c<0.8 are functions of the volume fraction, c of ω Zr only, which are independent of the plastic strain tensor prior to transformation and pressure. The microstructure is not inherited during phase transformation. Surprisingly, for the annealed sample, the final dislocation density and the average microstrain after pressure release in the ω phase are larger than for the severely pre-deformed sample. The results suggest that an extended experimental basis is required for the predictive models for the combined pressure-induced phase transformations and microstructure evolutions.

Tensorial stress-plastic strain fields in α - ω Zr mixture, transformation kinetics, and friction in diamond-anvil cell

Various phenomena (phase transformations (PTs), chemical reactions, microstructure evolution, strength, and friction) under high pressures in diamond-anvil cell are strongly affected by fields of stress and plastic strain tensors. However, they could not be measured. Here, we suggest coupled experimental-analytical-computational approaches utilizing synchrotron X-ray diffraction, to solve an inverse problem and find fields of all components of stress and plastic strain tensors and friction rules before, during, and after α-ω PT in strongly plastically predeformed Zr. Results are in good correspondence with each other and experiments. Due to advanced characterization, the minimum pressure for the strain-induced α-ω PT is changed from 1.36 to 2.7 GPa. It is independent of the plastic strain before PT and compression-shear path. The theoretically predicted plastic strain-controlled kinetic equation is verified and quantified. Obtained results open opportunities for developing quantitative high-pressure/stress science, including mechanochemistry, synthesis of new nanostructured materials, geophysics, astrogeology, and tribology.

Recent <i>In Situ</i> Experimental and Theoretical Advances in Severe Plastic Deformations, Strain-Induced Phase Transformations, and Microstructure Evolution under High Pressure

Severe plastic deformations (SPD) under high pressure, mostly by high-pressure torsion, are employed for producing nanostructured materials and stable or metastable high-pressure phases. However, they were studied postmortem after pressure release. Here, we review recent in situ experimental and theoretical studies of coupled SPD, strain-induced phase transformations (PTs), and microstructure evolution under high pressure obtained under compression in diamond anvil cell or compression and torsion in rotational diamond anvil cell. The utilization of x-ray diffraction with synchrotron radiation allows one to determine the radial distribution of volume fraction of phases, pressure, dislocation density, and crystallite size in each phase and find the main laws of their evolution and interaction. Coupling with the finite element simulations of the sample behavior allows the determination of fields of all components of the stress and plastic strain tensors and volume fraction of high-pressure phase and provides a better understanding of ways to control occurring processes. Atomistic, nanoscale and scale-free phase-field simulations allow elucidation of the main physical mechanisms of the plastic strain-induced drastic reduction in phase transformation pressure (by one to two orders of magnitude), the appearance of new phases, and strain-controlled PT kinetics in comparison with hydrostatic loading. Combining in situ experiments with multiscale theory potentially leads to the formulation of methods to control strain-induced PT and microstructure evolution and designing economic synthetic paths for the defect-induced synthesis of desired high-pressure phases, nanostructures, and nanocomposites.

Memory surfaces separating the processes of monotonous and cyclic loads

The processes of elastoplastic deformation of structural materials can consist of a sequence of monotonous and cyclic loading regimes, in which peculiar effects and features arise. Mathematical modeling of such processes, as well as resource assessment and forecasting, is a very difficult task. In addition, the analysis of transient processes from cyclic to monotonous and from monotonous to cyclic shows the need to separate these processes. Based on the analysis of the results of experimental studies of samples of stainless steel 12Х18Н10T under a rigid (controlled deformation) deformation process, which is a sequence of monotonic and cyclic loading modes, under conditions of uniaxial tension-compression at normal temperature, the features and differences in the processes of monotonic and cyclic loading are revealed. To describe these features and separate the processes of monotonic and cyclic loading modes in the theories of plastic flow with combined hardening, various variants of memory surfaces are introduced. An analysis of the results of experimental studies of stainless steel showed that in the space of the plastic strain tensor, the size of the memory surface is determined by the range of plastic strains, and the position of the center is determined by the values of average plastic strains under cyclic loading. Various variants of the memory surface are considered, their capabilities and disadvantages are identified, and the most adequate variant of the memory surface is determined. To confirm the operability of this version of the memory surface, together with the equations of the Bondar plasticity model, the calculated and experimental results were compared and a reliable agreement was obtained between these results both in terms of the kinetics of the stress-strain state and in terms of the number of cycles to failure.

ПОСТРОЕНИЕ ПОВЕРХНОСТИ ПАМЯТИ ДЛЯ РАЗДЕЛЕНИЯ ПРОЦЕССОВ МОНОТОННЫХ И ЦИКЛИЧЕСКИХ НАГРУЖЕНИЙ

Non-stationary and asymmetric processes of cyclic deformation are considered, which consist of a sequence of monotonic and cyclic loading modes, in which peculiar effects of landing and ratcheting of plastic hysteresis loops occur. Mathematical modeling of such processes of deformation and damage accumulation is based mainly on variants of plasticity theories belonging to the class of theories of plastic flow with combined (isotropic and anisotropic) hardening. In this paper, mathematical modeling of the processes of deformation and damage accumulation is based on a variant of the theory of plasticity – the Bondar model. Based on the analysis of the results of experimental studies of samples of stainless steel 12Х18Н10T under a rigid (controlled deformation) deformation process, which is a sequence of monotonic and cyclic loading modes, under conditions of uniaxial tension-compression at normal temperature, the features and differences in the processes of monotonic and cyclic loading are revealed. To describe these features and separate the processes of monotonic and cyclic loading modes in the theories of plastic flow with combined hardening, various variants of memory surfaces are introduced. An analysis of the results of experimental studies of stainless steel showed that in the space of the plastic strain tensor, the size of the memory surface is determined by the range of plastic strains, and the position of the center is determined by the values of average plastic strains under cyclic loading. Various variants of the memory surface are considered, their capabilities and disadvantages are identified, and proposed variant of the memory surface.To confirm the operability of proposed version of the memory surface, together with the equations of the Bondar plasticity model, the calculated and experimental results were compared and a reliable agreement was obtained between these results both in terms of the kinetics of the stress-strain state and in terms of the number of cycles to failure.

Formulation and existence of weak solutions for a problem of adhesive contact with elastoplasticity and hardening

This paper presents the weak formulation of a quasi-static evolution model for two deformable bodies with uni-directional adhesive unilateral contact on which external loads act. Small deformations and linearized elastoplasticity with hardening are assumed. The adhesion component is rate-dependent or rate-independent according to the choice of the viscosity coefficient of the glue; elastoplasticity is considered rate-independent. The weak formulation is expressed as a doubly non-linear problem with unbounded multivalued operators, as a function of internal and boundary displacements, plastic and symmetric strain tensors, and the bonding field and its gradient. This paper differs from other formulations by coupling the equations defined inside and on the boundary of the solids in functional form. In addition to this novelty, we verify the existence of solutions by a path other than that displayed in similar articles. The existence of solutions is demonstrated after considering a succession of doubly non-linear problems with an unbounded operator, and verifying that the solution of one of the problems is also a solution to the objective problem. The proof is supported by previous results from non-linear Partial differential equations theory with monotone operators.

Elastoplastic source model for microseismicity and acoustic emission

SUMMARYThe microseismic events can often be characterized by a complex non-double couple source mechanism. Recent laboratory studies recording the acoustic emission during rock deformation help connecting the components of the seismic moment tensor with the failure process. In this complementary contribution, we offer a mathematical model which can further clarify these connections. We derive the seismic moment tensor based on classical continuum mechanics and plasticity theory. The moment tensor density can be represented by the product of elastic stiffness tensor and the plastic strain tensor. This representation of seismic sources has several useful properties: (i) it accounts for incipient faulting as a microseismicity source mechanism, (ii) it does not require a pre-defined fracture geometry, (iii) it accounts for both shear and volumetric source mechanisms, (iv) it is valid for general heterogeneous and anisotropic rocks and (v) it is consistent with elasto-plastic geomechanical simulators. We illustrate the new approach using 2-D numerical examples of seismicity associated with cylindrical openings, analogous to wellbore, tunnel or fluid-rich conduit and provide a simple analytic expression of the moment density tensor. We compare our simulation results with previously published data from laboratory and field experiments. We consider four special cases corresponding to ‘dry’ elastically homogeneous and elastically heterogeneous isotropic rocks, ‘dry’ transversely isotropic rocks and ‘wet’ isotropic rocks. The model highlights theoretical links between stress state, geomechanical parameters and conventional representations of the moment tensor such as Hudson source type parameters.

Modified Theory of Plasticity for Monotonic and Cyclic Deformation Processes

Based on the analysis of the results of experimental studies of 12Х18Н10Т stainless steel specimens under a rigid (controlled deformation) process of deformation, which includes a sequence of monotonic and cyclic loading modes, some features and differences in isotropic and anisotropic hardening processes under monotonic and cyclic loading are revealed. To describe these features within the framework of the theory of plasticity in the space of the plastic strain tensor, a criterion for changing the direction of plastic deformation and a memory surface are introduced, which made it possible to separate the processes of monotonic and cyclic deformation. To describe the transient processes, evolutionary equations are formulated for the parameters of isotropic and anisotropic hardening. The calculated and experimental changes in the stress-strain states are compared for the process of monotonic and cyclic loading.

A variant of the thermoplasticity theory for monotonic and cyclic processes of nonisothermal loads

We revealed some features and differences in isotropic and anisotropic hardening under monotonic and cyclic loads by analyzing the experimental results of the samples made of 12X18H10T stainless steel under a rigid (controlled) deformation process, which includes a sequence of monotonic and cyclic loading modes under uniaxial tension-compression and different temperature levels. To describe these features with the theory of thermoplasticity, which belongs to the class of flow theories for combined hardening, a memory surface is introduced in the space of the plastic strain tensor components that separates the processes of monotonic and cyclic deformations. The main assumptions and equations of the thermoplasticity theory are formulated. To describe the transition from the monotonic to the cyclic and from the cyclic to the monotonic deformations, the evolutionary equations are formulated for the parameters of isotropic and anisotropic hardening. The basic experiment, which determined the material functions, consists of three stages, such as cyclic loading, monotonic loading and the subsequent cyclic up to destruction. The method of identifying the material functions based on the results of the basic experiment is considered. The material functions that close the thermoplasticity theory at different temperature levels are determined for 12X18H10T stainless steel due to the basic experiment and identification method. We considered the results of the computational and experimental studies of the rigid cyclic deformation under isothermal and non-isothermal loadings up to destruction of 12X18H10T stainless steel. The kinetics of the stress range and the average stress during isothermal and non-isothermal cyclic loadings are analyzed. A reliable compliance of the computational and experimental results is obtained.

Study on the influence of the yield surface shape in the hole expansion test

The hole expansion test has become popular since it allows to study the formability of metallic sheets, in particular the onset of necking in stretch flanging areas. The accurate prediction of strain localization and consequent fracture requires a proper description of the plastic behavior, particularly the anisotropic yield function. In the context of strain localization prediction, the yield criterion adopted plays an important role, particularly when using an associated flow rule, since the direction of the plastic strain tensor is modelled by the normal to the yield surface. In this work, the parameters of an advanced yield criterion are calibrated considering a wide set of experimental data, which includes results from uniaxial and biaxial tension tests. This enables establishing yield surfaces with similar shape in the plane defined by the stress components in the rolling and transverse directions and a null shear component in the same plane. However, their shape changes slightly when considering non-null values for that shear component. The numerical simulations of the hole expansion test demonstrate the impact of these slight differences on the thickness strain distribution and, consequently, in the instant and location of the necking.

Mathematical Modeling of the Monotonic and Cyclic Loading Processes

Based on experimental studies of specimens of 12Kh18N10T stainless steel under a rigid (strain-controlled) deformation process, which includes a sequence of monotonic and cyclic loading modes, some features and differences between isotropic and anisotropic hardening under monotonic and cyclic loading are found. To describe these features in terms of the theory of plasticity (the Bondar model), the criterion of change of plastic deformation direction and a memory surface are introduced in the space of the plastic strain tensor, which allow one to separate the processes of monotonic and cyclic loading. To describe the transitional processes, evolutionary equations for isotropic and anisotropic hardening parameters are formulated. The change in the stress-strain state during monotonic and cyclic loading, obtained by calculation and experiment, is compared.

Exact reconstruction formulas for plastic strain distribution in the surface-treated plate and their applications

Surface-treated bodies are considered under the assumption that the plastic strain distribution is uniform along the surface and changes only with depth. The elastic problem of the surface-treated plate is solved. The obtained solution is used to derive the explicit exact reconstruction formulas for distributions of main tangential components of the plastic strain tensor. It has been proved that the deflection of the surface-treated strip does not depend on the width of the strip. These results are intended for use in destructive testing methods and for numerical prediction of strength. The following initial data were used: (a) the dependence of the deflection on the thickness of the removed material layer obtained by the Davidenkov method, (b) the dependence of the tangential residual stress components on the depth, and (c) the dependence of the surface tangential stress components or elastic strain tangential components on the thickness of the removed material layer obtained by the X-ray diffraction method. The obtained formulas were analyzed and approved for all types of initial data, wherefore two experiments were conducted. A way to incorporate the plastic strain distribution into a model of a machine part of complex geometry to take into account the residual stress field in fatigue, wear, cracking, the and corrosion strength predictions is presented.

Computationally efficient necking prediction using neural networks trained on virtual test data

The onset of localized necking under monotonic and non-monotonic loading can be well-predicted by the imperfection-based approach proposed by [1] (MK). However, a large number of virtual imperfections has to be investigated for an accurate necking prediction, making the MK approach computationally expensive and hence preventing the industrial application for full-scale vehicle models. To overcome these issues, a computationally efficient neural network (NN) model is proposed for replacing the MK model in the present work. An extended version of the MK model has been implemented into a User Material for an explicit crash solver. The model continuously computes the “distance to localized necking” as an important engineering quantity. Single shell element simulations are utilized for creating a comprehensive virtual test database for monotonic and non-monotonic loading for a 22MnB5 grade in an as-delivered state. A simple feed-forward NN model, featuring only one hidden layer, is trained and tested against the virtual data, where invariants of the stress and plastic strain tensors represent the input features of the NN and the “distance to necking” represents the output value of the NN. For comparison of the computational cost, the NN architecture has also been implemented in a User Material Routine for shell elements. The predictions of the NN and the MK model are in good agreement, where, due to the simple mathematical structure of the NN, the computational cost of the NN is significantly lower than for the MK implementation.

Tensorial stress\u2212strain fields and large elastoplasticity as well as friction in diamond anvil cell up to 400\u2009GPa

Various phenomena (fracture, phase transformations, and chemical reactions) studied under extreme pressures in diamond anvil cell are strongly affected by fields of all components of stress and plastic strain tensors. However, they could not be measured. Here, we suggest a coupled experimental−theoretical−computational approach that allowed us (using published experimental data) to refine, calibrate, and verify models for elastoplastic behavior and contact friction for tungsten (W) and diamond up to 400 GPa and reconstruct fields of all components of stress and large plastic strain tensors in W and diamond. Despite the generally accepted strain-induced anisotropy, strain hardening, and path-dependent plasticity, here we showed that W after large plastic strains behaves as isotropic and perfectly plastic with path-independent surface of perfect plasticity. Moreover, scale-independence of elastoplastic properties is found even for such large field gradients. Obtained results open opportunities for quantitative extreme stress science and reaching record high pressures.

A fourth-order gauge-invariant gradient plasticity model for polycrystals based on Kröner’s incompatibility tensor

In this paper we derive a novel fourth-order gauge-invariant phenomenological model of infinitesimal rate-independent gradient plasticity with isotropic hardening and Kröner’s incompatibility tensor , where is the symmetric plastic strain tensor. Here, gauge-invariance denotes invariance under diffeomorphic reparametrizations of the reference configuration, suitably adapted to the geometrically linear setting. The model features a defect energy contribution that is quadratic in the tensor and it contains isotropic hardening based on the rate of the plastic strain tensor . We motivate the new model by introducing a novel rotational invariance requirement in gradient plasticity, which we call micro-randomness, suitable for the description of polycrystalline aggregates on a mesoscopic scale and not coinciding with classical isotropy requirements. This new condition effectively reduces the increments of the non-symmetric plastic distortion to their symmetric counterpart . In the polycrystalline case, this condition is a statement about insensitivity to arbitrary superposed grain rotations. We formulate a mathematical existence result for a suitably regularized non-gauge-invariant model. The regularized model is rather invariant under reparametrizations of the reference configuration including infinitesimal conformal mappings.

Учет влияния полей остаточных деформаций в современных физико-механических технологиях обработки конструкционных материалов

AbstractA strict determination of the mechanisms of redistribution of previously accumulated irreversible strains as a result of additional elastic shock actions on the material is given for a nonlinear gradient model of large elastic–plastic strain. It is shown that this redistribution is limited by rigid transport and rotation of the plastic strain tensor. Formulas for a change in the initial components of the plastic strain tensor in elastic waves are derived. It is shown that the preliminary plastic field affects the dynamics of further reversible strain as one of the factors of formation of the initial quasi-static elastic field, which cannot be obtained in a purely elastic process.

Consideration of the Influence of Residual Strain Fields in Modern Physicomechanical Technologies of Treating Structural Materials

A strict determination of the mechanisms of redistribution of previously accumulated irreversible strains as a result of additional elastic shock actions on the material is given for a nonlinear gradient model of large elastic–plastic strain. It is shown that this redistribution is limited by rigid transport and rotation of the plastic strain tensor. Formulas for a change in the initial components of the plastic strain tensor in elastic waves are derived. It is shown that the preliminary plastic field affects the dynamics of further reversible strain as one of the factors of formation of the initial quasi-static elastic field, which cannot be obtained in a purely elastic process.

The currently predominant Taylor principles should be disregarded in the study of plastic deformation of metals

The original Taylor principles offer identical intergranular strain equilibrium without stress equilibrium in metals during deformation. In reality, however, the stress and strain equilibria are maintained individually for different grains. As key points, the principles have become a prerequisite predominantly in the current theories, which unreasonably indicate that strains instead of stresses induce grain deformation despite reaching the stress equilibrium by complicated combinations of the activation of slip systems or other crystallographic mechanism via different approaches. Real intergranular equilibria can be traced if mechanical interactions together with the external loading are considered step by step. In this work, several penetrating and non-penetrating slips were used to obtain the necessary elastic and plastic strain tensors of different grains in a natural manner. Without the Taylor principles, the stress and strain equilibria can be reached naturally, simply, easily, reasonably, and individually without complicated calculations. Results of the experimental observation conformed with the predicted deformation texture when certain important engineering stress conditions are included in the simulation. Therefore, the Taylor principles for plastic deformation of polycrystalline metals should now be disregarded.

FE implementation of HAH model using FDM-based stress update algorithm for springback prediction of AHSS sheets

The homogeneous anisotropic hardening (HAH) model was implemented into a finite element (FE) code in order to predict springback for an advanced high strength steel (AHSS) sheet sample after double-stage U-draw bending. The finite difference method (FDM) was utilized as an alternative way to calculate the derivatives of this advanced distortional plasticity model allowing the update of the equivalent plastic strain and stress tensor at each time step in the user-material subroutines (UMAT and VUMAT). The FDM makes it easier to derive the stress gradient of complex yield surfaces. The proposed FDM-based stress update algorithm was verified by comparing the springback profiles after the single- and double-stage U-draw bending tests for a DP980 sheet sample predicted with analytical and numerical approaches. In addition, the springback measurement parameters and computational efficiencies depending on both approaches were also compared. The results indicate that the computational efficiency and accuracy of the FE simulations with the FDM-based stress update algorithm were similar to those of the analytical method.

К вопросу о реконструкции остаточных напряжений и деформаций пластины после дробеструйной обработки

Предметом исследования является математическое описание формы и напряженно-деформированного состояния стальной пластины, подвергнутой односторонней дробеструйной обработке, его экспериментальное подтверждение и применение результатов для верификации методов реконструкции полей остаточных напряжений и деформаций по экспериментальным данным. Подобная пластина используются на производстве в качестве калибровочного образца для определения времени пневмодробеструйной обработки, необходимого для формирования в поверхностном слое обрабатываемого изделия сжимающих тангенциальных остаточных напряжений заданной величины, а сам метод калибровки оказывается удобным и довольно широко распространенным для различных способов поверхностно упрочняющей обработки. Источником остаточных напряжений в данном случае является пограничный слой пластических деформаций, наводимый рассматриваемым технологическим процессом. Для постановки задачи задается структура поля тензора пластических деформаций. Форма и напряженно-деформированное состояние упругой пластины с пограничным слоем пластических деформаций были рассчитаны численно, в результате чего были выявлены качественные особенности данных полей, ослаблены граничные условия задачи и сформулированы гипотезы о структуре решения соответствующей пространственной задачи теории упругости, которое далее было найдено аналитически. Показано, что в рамках приближения плоского напряженного состояния в поперечных направлениях результат точно соответствует формуле Давиденкова - Биргера, связывающей зависимость тангенциальной компоненты остаточных напряжений от координаты по толщине пластины с функцией прогибов. Получена явная формула для зависимости остаточной (пластической) деформации от координаты по толщине. Проанализированы источники погрешностей полученных выражений и способы их коррекции. Проведен эксперимент по односторонней дробеструйной обработке калибровочной пластины, изготовленной из закаленной стали 65Г, для которой выполнено травление обработанной поверхности с измерением изменения стрелы прогиба (метод Н. Н. Давиденкова). С помощью полученных экспериментальных данных были численно реконструированы профили остаточных напряжений и деформаций с разумной точностью. Результат применим к широкому классу задач для упругих тел с упрочняющими покрытиями, а также имеет определенную методическую ценность для усовершенствования основ экспериментального исследования таких задач, позволяет формулировать и подтверждать экспериментом гипотезы о структуре решения, изучать связь рассматриваемых полей в предельных случаях, верифицировать применение различных способов учета остаточных напряжений и деформаций в численных расчетах. Найденное решение может быть использовано для верификации полей напряжений и перемещений при выборе различных вариантов предварительно напряженных или деформированных поверхностных оболочечных конечных элементов в рамках пакетов прикладных программ для расчета усталостной долговечности деталей машин с поверхностными упрочняющими покрытиями и также представляется опорным для исследования поверхностно упрочненных тел с криволинейной свободной границей, к которым сводится большинство практически важных задач.