Plastic Strain-induced Phase Transformations Research Articles

A large deformation constitutive model for plastic strain-induced phase transformation of stainless steels at cryogenic temperatures

In this paper, a new constitutive model for plastic behavior of the metastable austenitic stainless steels at cryogenic temperatures is presented. The constitutive model is a phenomenological hyperelastic-based large deformation model developed in the framework of continuum damage mechanics considering the dissipative phenomena of strain-induced phase transformation and damage growth during plastic deformation. To solve the coupled nonlinear set of equations, incremental equations and the related implicit integration method have been developed. Numerical analysis is performed by implementing the constitutive model in a UMAT subroutine in ABAQUS/STANDARD. In addition, experiments have been conducted to identify the material model parameters for AISI304 steel at the 77K temperature. In addition to calibration, the constitutive model was validated via a tensile test of a notched sample. Also, the model has been compared with the experimental data at 4.2 K available in the literature. The results show that the model is well able to predict the hardening behavior, martensite evolution, and damage growth during the plastic deformation until the fracture with acceptable accuracy.

Strain-induced multivariant martensitic transformations: A scale-independent simulation of interaction between localized shear bands and microstructure

A scale-independent model for the interaction between multivariant phase transformations (PTs) and discrete shear bands is advanced and utilized to simulate plastic strain-induced PTs at high pressure. The model includes a scale-free phase-field theory for martensitic PTs. The localized shear bands are introduced via a contact problem formulation. That is, the continuous distribution of sliding displacements along the prescribed slip surfaces is modeled to reproduce the plastic-strain-induced stress concentrators necessary for nucleation of a high-pressure phase (HPP). The strain-induced PTs in the bi/polycrystalline samples subjected to compression and shear are studied. The simulations show a severe reduction in the PT pressure by the plastic shear in comparison to a hydrostatic condition, even below the phase equilibrium pressure, like in known experiments. Transformation kinetics versus shear strain for each martensitic variant and the volume fraction of the HPP in individual grains and the entire aggregate are determined. The stationary volume fraction of the HPP is the same for polycrystals consisting of 13 and 38 grains, and a further shearing does not cause PT. The local phase equilibrium condition based on the transformation-work criterion is satisfied at almost all stationary phase interfaces. A similar phase equilibrium condition in terms of stresses averaged over the entire polycrystal or HPP is fulfilled. These results are important for the development of the microscale kinetic equations and modeling the sample behavior in traditional and rotational diamond anvils during the high-pressure torsion, ball milling, friction, and other deformation-transformation processes.

In situ quantitative study of plastic strain-induced phase transformations under high pressure: Example for ultra-pure Zr

The first in situ quantitative synchrotron X-ray diffraction (XRD) study of plastic strain-induced phase transformation (PT) has been performed on α−ω PT in ultra-pure, strongly plastically predeformed Zr as an example, under different compression-shear pathways in rotational diamond anvil cell (RDAC). Radial distributions of pressure in each phase and in the mixture, and concentration of ω-Zr, all averaged over the sample thickness, as well as thickness profile were measured. The minimum pressure for the strain-induced α−ω PT, pεd=1.2 GPa, is smaller than under hydrostatic loading by a factor of 4.5 and smaller than the phase equilibrium pressure by a factor of 3; it is independent of the compression-shear straining path. The theoretically predicted plastic strain-controlled kinetic equation was verified and quantified; it is independent of the pressure-plastic strain loading path and plastic deformation at pressures below pεd. Thus, strain-induced PTs under compression in DAC and torsion in RDAC do not fundamentally differ. The yield strength of both phases is estimated using hardness and x-ray peak broadening; the yield strength in shear is not reached by the contact friction stress and cannot be evaluated using the pressure gradient. Obtained results open a new opportunity for quantitative study of strain-induced PTs and reactions with applications to material synthesis and processing, mechanochemistry, and geophysics.

Scale-Free Modeling of Coupled Evolution of Discrete Dislocation Bands and Multivariant Martensitic Microstructure.

A scale-free model for the coupled evolution of discrete dislocation bands and multivariant martensitic microstructure is developed. In contrast to previous phase field models, which are limited to nanoscale specimens, this model allows for treating the nucleation and evolution of martensite at evolving dislocation pileups, twin tips, and shear bands in a sample of an arbitrary size. The model is applied for finite element simulations of plastic strain-induced phase transformations (PTs) in a polycrystalline sample under compression and shear. The solution explains the one to two orders of magnitude reduction in PT pressure by plastic shear, the existence of incompletely transformed stationary state, and optimal shear strain for the strain-induced synthesis of high pressure phases.

Discontinuous plastic flow coupled with strain induced fcc–bcc phase transformation at extremely low temperatures

A popular class of materials massively used at cryogenic temperatures comprises the stainless steels of different grades, such as 304, 304L, 316, 316Ti, 316L, 316LN etc. Such materials are metastable at extremely low temperatures, and usually undergo plastic strain induced phase transformation. In addition, these materials applied in the proximity of absolute zero exhibit the so-called discontinuous (intermittent, serrated) plastic flow (DPF). It consists in frequent, abrupt drops of stress against strain, characterized by increasing amplitude of the stress oscillations. Strong coupling between both phenomena: DPF and phase transformation is observed. Recent experiments performed by means of stainless steel samples tested in liquid helium (4.2 K) clearly indicate strong strain localization during DPF, in the form of shear bands propagating along the sample. However, as soon as the phase transformation process takes place, the motion of shear bands is hindered by formation of secondary phase. A physically based constitutive model developed in the present paper reflects coupling between the discontinuous plastic flow and the plastic strain induced phase transformation in the temperature range 0–T1. The model involves nonlinear mixed hardening, that occurs during the 2nd stage of each serration (stress–strain oscillation). The hardening is based on two mechanisms: interaction of dislocations with the inclusions of secondary phase, evolution of tangent stiffness operator due to changing proportions between the primary and the secondary phases. Nonlinear hardening strongly increases the stress level during each serration, which affects production of the internal lattice barriers, and the amount of the accumulated plastic strain. This, in turn, affects intensity of the phase transformation (full coupling). The constitutive model and its numerical version allow to reproduce the observed serrations, which is crucial for its application in the design of components operating at extremely low temperatures.

Coupled elastoplasticity and plastic strain-induced phase transformation under high pressure and large strains: Formulation and application to BN sample compressed in a diamond anvil cell

In order to study high-pressure phase transformations (PTs), high static pressure is produced by compressing a thin sample within a high strength gasket in a diamond anvil cell (DAC). However, since a PT occurs during plastic flow, it is classified and treated here as a plastic strain-induced PT. A thermodynamically consistent system of equations for combined plastic flow and plastic strain-induced PTs is formulated for large elastic, plastic, and transformation strains. The Murnaghan elasticity law, pressure-dependent J2 plasticity (both dependent of the concentration of a high-pressure phase), and plastic strain-induced and pressure-dependent PT kinetics are utilized. A computational algorithm within finite element method (FEM) is presented and implemented in a user material subroutine (UMAT) in the FEM code ABAQUS. Combined plastic flow and strain-induced PT from the highly-disordered hexagonal boron nitride (hBN) sample to a superhard wurtzitic wBN is simulated within the rhenium gasket for pressures up to 50 GPa. The evolution of the fields of stresses and plastic strains, as well as the concentration of phases in a sample is obtained and discussed in detail. Stress-strain fields in a gasket and diamond are presented as well. An unexpected shape of the deformed sample with almost complete PT in the external part of the sample that penetrated the gasket was found. Obtained results demonstrated the difference between material and system behavior which are often confused by experimentalists. Thus, while plastic strain-induced PT may start (and end) at plastic straining slightly above 6.7 GPa, it is not visible below 12 GPa. It becomes detectable at 21 GPa and is not completed everywhere in a sample even at a maximum pressure of 50 GPa. Due to a strong gasket the gradient of pressure is much smaller than the gradient of plastic strain, and therefore the distribution of the high pressure phase is mostly determined by the plastic strain field instead of the pressure field. Possible misinterpretation of the experimental data and characterization of the PT is discussed. The developed model will allow computational design of experiments for synthesis of high-pressure phases.

Phase field simulations of plastic strain-induced phase transformations under high pressure and large shear

Pressure and shear strain-induced phase transformations (PTs) in a nanograined bicrystal at the evolving dislocations pile-up have been studied utilizing a phase field approach (PFA). The complete system of PFA equations for coupled martensitic PT, dislocation evolution, and mechanics at large strains is presented and solved using the finite element method (FEM). The nucleation pressure for the high-pressure phase (HPP) under hydrostatic conditions near a single dislocation was determined to be 15.9 GPa. Under shear, a dislocation pile-up that appears in the left grain creates strong stress concentration near its tip and significantly increases the local thermodynamic driving force for PT, which causes nucleation of HPP even at zero pressure. At pressures of 1.59 and 5 GPa and shear, a major part of a grain transforms to HPP. When dislocations are considered in the transforming grain as well, they relax stresses and lead to a slightly smaller stationary HPP region than without dislocations. However, they strongly suppress nucleation of HPP and require larger shear. Unexpectedly, the stationary HPP morphology is governed by the simplest thermodynamic equilibrium conditions, which do not contain contributions from plasticity and surface energy. These equilibrium conditions are fulfilled either for the majority of points of phase interfaces or (approximately) in terms of stresses averaged over the HPP region or for the entire grain, despite the strong heterogeneity of stress fields. The major part of the driving force for PT in the stationary state is due to deviatoric stresses rather than pressure. While the least number of dislocations in a pile-up to nucleate HPP linearly decreases with increasing applied pressure, the least corresponding shear strain depends on pressure nonmonotonously. Surprisingly, the ratio of kinetic coefficients for PT and dislocations affect the stationary solution and the nanostructure. Consequently, there are multiple stationary solutions under the same applied load and PT, and deformation processes are path dependent. With an increase in the size of the sample by a factor of two, no effect was found on the average pressure and shear stress and HPP nanostructure, despite the different number of dislocations in a pile-up. The obtained results represent a nanoscale basis for understanding and description of PTs under compression and shear in a rotational diamond anvil cell and high-pressure torsion.

Evaluation of Zero and Sub-zero Temperature Tensile and Impact Properties of Quenched and Tempered Steel Weldments

Combat vehicles using armour grade quenched and tempered (Q&T) steel are needed to be operated under different climatic conditions. However, the properties meeting the room temperature test conditions may not necessarily be required to meet the low temperature operating conditions. The influences of shielded metal arc welding (SMAW) process, flux cored arc welding (FCAW) processes and test temperatures below room temperature up to–60 °C on tensile and impact toughness properties of armour grade Q&T steel welds were reported. The tensile and impact properties of the joints showed incremental trend with respect to the decrement in test temperature up to −40 °C irrespective of the welding processes used. The increment in tensile and impact properties of SMAW and FCAW joints subjected to the low temperature environment is due to plastic strain-induced phase transformation of retained austenite into martensite phase. Among the two processes, SMAW process welded joint showed better tensile and impact properties than its FCAW counterpart joints.

Modeling of Damage Evolution and Martensitic Transformation in Austenitic Steel at Cryogenic Temperature

Abstract In the present work, a constitutive model of materials undergoing the plastic strain induced phase transformation and damage evolution has been developed. The model is based on the linearized transformation kinetics. Moreover, isotropic damage evolution is considered. The constitutive model has been implemented in the finite element software Abaqus/Explicit by means of the external user subroutine VUMAT. A uniaxial tension test was simulated in Abaqus/Explicit to compare experimental and numerical results. Expansion bellows was also modelled and computed as a real structural element, commonly used at cryogenic conditions.

Micromechanics based constitutive modeling of martensitic transformation in metastable materials subjected to torsion at cryogenic temperatures

Special class of multi-phase structural members with graded microstructure can be obtained at cryogenic temperatures as a result of controlled transition from the parent phase to the secondary phase. The required features are obtained via the mechanism of diffusionless plastic strain induced phase transformation from purely austenitic to martensitic lattice (γ→α′). Many ductile materials are known to behave in metastable way when strained at extremely low temperatures. Among them the austenitic stainless steels are often used to construct components of superconducting magnets, cryogenic transfer lines and other cryogenic systems. The multiscale constitutive model developed to describe the plastic strain induced phase transformation at very low temperatures involves plastic hardening where two important effects play fundamental role: (1) interaction of dislocations with martensite inclusions and (2) increase of resultant tangent stiffness due to the evolution of harder martensite within the softer austenite. The micro-mechanism of interaction of dislocations with martensite inclusions is based on the Orowan scenario. The other mechanism takes into account constantly evolving proportion between the primary and the secondary phase and involves the relevant homogenization scheme. Both effects contribute to strong nonlinear hardening that occurs as soon as the phase transformation process begins. The constitutive model has been used in order to obtain a new closed form analytical solution for the case of torsion of round bars at extremely low temperatures. Moreover, numerous experiments involving quasistatic and cyclic loads were carried out in order to trace the profile of phase transformation. An ultimate proof is presented that a functionally graded structural member, characterized by the required profile of volume fraction of both phases, can be obtained. The profile of volume fraction of secondary phase (martensite) is checked by means of 2 independent methods: hardness measurements and verification by using a ferritscope (magnetic induction).

Ductile damage development in two-phase metallic materials applied at cryogenic temperatures

FCC metals and alloys are frequently used in cryogenic applications, nearly down to the temperature of absolute zero, because of their excellent physical and mechanical properties including ductility. These materials, often characterized by the low stacking fault energy (LSFE), undergo at low temperatures three distinct phenomena: dynamic strain ageing (DSA), plastic strain induced transformation from the parent phase ( γ ) to the secondary phase ( α ′ ) and evolution of micro-damage. Especially the third phenomenon leads to irreversible degradation of lattice and can accelerate the process of material failure therefore a suitable constitutive description appears to be fundamental for the correct analysis of structures applied at very low temperatures. The constitutive model presented in the paper takes into account the plastic strain induced phase transformation and the evolution of ductile damage. The FCC–BCC phase transformation results from metastability of LSFE metals and alloys at very low temperatures. The phase transformation process leads to creation of two-phase continuum where the parent phase coexists with the inclusions of secondary phase. Such heterogeneous material structure induces strong strain hardening related to two distinct mechanisms: interaction of dislocations with the inclusions and increase of tangent stiffness as a result of mixture of two phases, each characterized by different parameters. For the micro-damage evolution a generalization of the classical isotropic ductile damage concept ( Chaboche, 1988a,b) to anisotropic model has been introduced. The kinetics of damage evolution is based on the accumulated plastic strain as driving force of ductile damage. The damage rate tensor depends on the strain energy density release rate (conjugate force) and on the tensor of material properties, that reflects the damage anisotropy.

Functionally graded structural members obtained via the low temperature strain induced phase transformation

The notion of functionally graded materials (FGM) covers all domains of discrete and smooth gradation of material microstructure designed in order to obtain macroscopic features suitable for a given application. A special class of multi-phase materials with graded microstructure can be obtained at cryogenic temperatures as a result of smooth transition from the parent phase to the secondary phase. The required continuously graded material features are obtained at low temperatures via the mechanism of controlled strain induced phase transformation from the purely austenitic to the martensitic lattice (γ → α′). Several families of ductile materials are known to behave in a metastable way when strained at very low temperatures. Among them the austenitic stainless steels are extensively used to construct components of the superconducting magnets, cryogenic transfer lines and other structural members loaded in cryogenic conditions. The constitutive model used to describe mathematically the plastic strain induced phase transformation at low temperatures involves strain hardening where two fundamental effects play an important role: interaction of dislocations with the martensite inclusions and increase in material tangent stiffness due to the mixture of harder martensite with softer austenite. The interaction of dislocations with the martensite inclusions is reflected by the hardening modulus that depends on the volume fraction of martensite. Here, a linear approximation, based on the micro-mechanics analysis, is used. On the other hand, evaluation of the material tangent stiffness of two-phase continuum is based on the classical homogenization scheme and takes into account the local tangent moduli of the components, as postulated by Hill [Hill, R., 1965. A self consistent mechanics of composite materials. J. Mech. Phys. Solids 13, 213–222]. In the present paper, the Mori–Tanaka homogenisation scheme is applied. Both effects contribute to strong nonlinear hardening that occurs as soon as the phase transformation process begins. The material model is suitable for a wide range of temperatures, however the best results are obtained at very low temperatures, where the linearized kinetic law of phase transformation is valid [Garion, C., Skoczeń B., 2002. Modeling of plastic strain induced martensitic transformation for cryogenic applications. J. Appl. Mech. 69 (6), 755–762]. As the application field the structural members in the form of rods (cylinders) of circular cross-section, used as parts of the carrying structures, are analyzed. The required graded microstructure of the material is obtained by imposing torsion at cryogenic temperatures. Both the intensity of the phase transformation and the depth of the transformed zone is obtained by suitable kinematic control (angle of twist). The closed form solutions for the stress state and torque as a function of the angle of twist are shown.

Combined Model of Strain-Induced Phase Transformation and Orthotropic Damage in Ductile Materials at Cryogenic Temperatures

Ductile materials (like stainless steel or copper) show at cryogenic temperatures three principal phenomena: serrated yielding (discontinuous in terms of dσ/dε), plastic strain-induced phase transformations and evolution of ductile damage. The present paper deals exclusively with the two latter cases. Thus, it is assumed that the plastic flow is perfectly smooth. Both in the case of damage evolution and for the 0 phase transformation, the principal mechanism is related to the formation of plastic strain fields. In the constitutive modeling of both phenomena, a crucial role is played by the accumulated plastic strain, expressed by the Odqvist parameter p. Following the general trends, both in the literature concerning the phase transformation and the ductile damage, it is assumed that the rate of transformation and the rate of damage are proportional to the accumulated plastic strain rate. The 0 phase transformation converts the initially homogenous material to a two-phase heterogeneous ”composite”. The kinetics of phase transformation is described by the relevant linearized law of evolution of the volume fraction of 0 martensite in the austenitic matrix [Garion, C. and Skoczen, B. (2002a). The evolution of orthotropic damage is characterized by the fact that the principal directions of damage are generally not colinear with the principal directions of stress. The damage rate tensor depends linearly on the strain energy density release rate tensor (conjugate force) and on the material properties tensor C, that reflects the orthotropy level. The relevant kinetic law of damage evolution and the combined constitutive model, including phase transformation, are developed in the present paper. The model is particularly suitable to describe the evolution of highly localized damage fields in thin-walled shells, subjected at cryogenic temperatures to the loads far beyond the yield point. It has been applied to the prediction of the response of the bellows expansion joints (corrugated thin-walled shells) designed for the inter-connections of the Large Hadron Collider at CERN.