Complete system of equations
Decomposition of the deformation gradient F into elastic and inelastic contributionsDecomposition of the deformation rate into elastic, plastic, and transformation partsThe third-order Murnaghan potential
A diamond anvil cell (DAC), see Fig. 1, allows one an in-situ study of the material's physical and mechanical behavior as well as PTs under high pressure, using advanced diagnostics such as optical, Raman, and X-ray techniques (Lazicki et al., 2012, Nisr et al., 2012, Oganov et al., 2009, Zha et al., 2012). Within a liquid media, the sample in a DAC is under a hydrostatic pressure in which the PTs are pressure-induced. At some pressure, liquid freezes and the sample is subjected to a stress tensor, but most likely still below the yield strength. The highest solidification pressure of 11 GPa at room temperature is for helium. Note that freezing of a liquid transmitting media, even if not reported, exhibits itself in a drastic difference in PT pressures in different media. For example, pressure for the beginning of α → ε and reverse ε → α PTs in iron varied for different transmitting media in the range from 6 to 16 GPa (Bargen and Boehler, 1990). Thus, most PT studies are, in reality, performed under such non-hydrostatic (or quasi-hydrostatic) conditions, and PTs are stress-induced. Without a hydrostatic medium, or after the solidification of the transmitting medium if its yield strength is getting comparable with that for a sample, the sample undergoes plastic deformation. To produce high pressure, the thickness of a sample should be irreversibly reduced by hundreds of percent. Additional large plastic shearing occurs due to friction stress between the sample and diamond. Thus, in this case, PTs occur during large elastoplastic deformations, i.e., they are plastic strain-induced PTs. There are many applications for which plastic strain-induced PT under high pressure is of interest. They include: friction, wear, polishing, and cutting of some strong semiconductors (silicon and germanium) and ceramics (e.g., SiC). Utilizing plastic strain-induced PTs from brittle low-to ductile high-pressure phases, one can introduce a ductile regime of machining of Si, Ge, and SiC, which minimizes microcracking (Patten et al., 2004). Various geophysical applications are based on analysis of combined plastic flow and plasticity. In particular, some of the mechanisms of deep earthquakes are caused by the instability of geological materials due to shear strain-induced PTs (Green and Burnley, 1989). Penetration of a projectile in a target is also accompanied by large plastic deformations and PTs (see, e.g., Chen et al., 2003). While these processes are dynamic, static experiments are still desirable to get initial information since it is easier to extract in situ data in DAC than in shock waves.
Recent discoveries of a new superconducting phase at megabar pressures (Drozdov et al., 2015, Troyan et al., 2016, Dias and Silvera, 2017) are also produced due to plastic strain-induced PTs without pressure transmitting media. The most exciting results are obtained under application of large plastic shear in rotational DAC (Alexandrova et al., 1993, Blank et al., 1984, Blank et al., 1993, Novikov et al., 1999, Levitas et al., 2006, Ji et al., 2012, Blank and Estrin, 2014), where plastic straining significantly reduces the PT pressure by a factor of 2–10 in comparison to hydrostatic conditions. For example, the highly disordered nanocrystalline hBN-to-wBN transformation occurs at 6.7 GPa with plastic straining, whereas wBN has not been obtained even at pressures up to 52.8 GPa under quasi-hydrostatic conditions (Ji et al., 2012). Even without rotation, the irreversible rhombohedral-to-cubic boron nitride (rBN-to-cBN) transformation under compression without hydrostatic media in DAC occurs at 5.6 GPa (Levitas and Shvedov, 2002), whereas it occurs at 55 GPa under the hydrostatic condition (Ueno et al., 1992). Similar results are obtained under lower pressures when the anvil is made of hard alloys (Bridgman, 1935, Zhilyaev et al., 2011; Srinivasarao et al., 2011, Edalati and Horita, 2016), e.g., for PTs in Zr and Ti. This popular processing approach is called high-pressure torsion. In addition, plastic deformation often substitutes a reversible PT with an irreversible PT (Blank and Estrin, 2014, Levitas et al., 2006), which allows one to use high-pressure phases in engineering applications. Thus, the main practical interest in PT under plastic straining is in the possibility to obtain both known and new phases under much lower pressures while keeping them under normal pressure. This illustrates the transformation of discovery into technology. Lastly, plastic straining may lead to new phases that could not be obtained under hydrostatic conditions (Blank and Estrin, 2014, Novikov et al., 1999, Levitas et al., 2012).
Thus, understanding of the interaction between PTs and plasticity is an important fundamental problem of high-pressure material physics and mechanics. DAC and rotational DAC are the only experimental tools that are used for these studies. The main challenges in these studies are the following.
1. The high-pressure community does not clearly recognize and utilize the difference between pressure- or stress-induced PTs and plastic strain-induced PTs under high pressure. Both pressure- and stress-induced PTs start by nucleating at pre-existing defects, e.g., various dislocation configurations or different boundaries (grain, subgrain, and twin). These dislocations and boundaries represent stress tensor concentrators. Plastic strain-induced PTs occur by nucleating at new defects that are continuously generated during the plastic flow. In particular, dislocations are generated and densely piled up against grain boundaries or other obstacles during plastic deformation. They generate a strong concentrator of the stress tensor. Then, the local stresses may satisfy local PT (i.e., lattice instability) criterion and cause barrierless nucleation of the high-pressure phase at a much lower applied pressure than that under hydrostatic conditions (Levitas, 2004a and b). Plastic strain-induced PTs have different mechanisms and require completely different thermodynamic and kinetic descriptions as well as experimental characterization than pressure- and stress-induced PTs. A multiscale theory (Levitas, 2004a and b) for high-pressure mechanochemistry was proposed, which claims that plastic strain-induced PTs could be characterized by a strain-controlled, pressure-dependent kinetic equation of type of Eq. (41). In this equation, the derivative of concentration of the high-pressure phase with respect to the accumulative plastic strain (rather than time) is determined.
2. Due to highly heterogeneous fields of stress and strain tensors, and complex distributions of phases, only the pressure and concentration of high pressure phases along the radius of the sample on a contact surface are experimentally available (Levitas et al., 2006, Blank and Estrin, 2014). In principle, there are currently methods that allow one to measure distribution of elastic lattice strain that may be converted into distribution of stresses (Wenk et al., 2007, Merkel et al., 2013, Nisr et al., 2012). However, the main problem is that distribution of plastic strain cannot be directly measured. The only way toward quantitative high pressure science is to develop models of materials' behavior and FEM procedure. Then we can extract material parameters and functions by comparing FEM modeling with available measurements (Levitas, 2004a, Levitas, 2004b, Levitas and Zarechnyy, 2010a, Feng et al., 2013a, Feng et al., 2013b, Feng et al., 2014). Surprisingly, despite such an interesting, important, and urgent problem, we are the only group which works in this direction. FEM have been developed and applied for investigation of the evolution of stresses, strains, and concentration of phases in the entire sample during plastic flow as well as PTs with external force growth (Feng et al., 2013a, Feng et al., 2013b, Feng et al., 2014, Levitas and Zarechnyy, 2010a). In these papers, coupled problems of elastoplasticity and PT with large plastic deformation were solved. They are quite sophisticated in terms of nonlinear constitutive equations and computational algorithms, as well as convergence of the FEM solutions. In exploratory studies (Feng et al., 2013a, Feng et al., 2013b, Feng et al., 2014, Levitas and Zarechnyy, 2010a), the simplest system of equations was postulated without continuum thermodynamic treatment. In particular, the assumptions of small transformational and elastic strains, linear elasticity, and pressure-independent perfect plasticity have been used. Thus, pressure was limited to 0.1K (K is the bulk modulus), which is relatively low, and diamond anvils were considered as rigid. Transformation strain was pure volumetric and transformation-induced plasticity (TRIP) was neglected. Calculations have been performed for generic material parameters (i.e., without calibration for any specific material), which have been varied in order to understand their effect on PTs and plastic flow.
The major goals of this paper are:
to formulate a continuum thermodynamic framework for coupled elastoplasticity and plastic strain-induced PTs with large elastic, plastic, and tensorial transformation strains, nonlinear elasticity, strain hardening, pressure-dependent yield condition, and TRIP;
to propose a computational algorithm for this theory with emphasis on the stress update and derivation of consistent tangent stiffness;
to specify this model for plastic strain-induced PT in BN;
to model and simulate PT and plastic flow in the BN sample within a rhenium gasket in a DAC, for which all deformations are finite (including deformation of a diamond anvil, and
to interpret some experimental high-pressure phenomena observed in DAC, first of all, difference between material behavior (constitutive equations) and system (sample/gasket/anvil) behavior as well as the difference between pressure-induced PTs and plastic strain-induced PTs at high pressure.
Plastic strain-induced solid-solid PT from highly disordered nanograined hexagonal hBN to superhard wurtzitic wBN will be studied in the paper. This PT does not occur even at 52.8 GPa under quasi-hydrostatic conditions but occurred at 6.7 GPa in rotational DAC with large plastic shear, demonstrating extremely strong effect of plastic straining on PT. During this PT, the bulk and shear moduli increase by a factor of 10 and 19 respectively, and the yield strength increases by a factor of 30. The volume is reduced by 39%, which along with geometrically and physically nonlinear elasticity rules for all materials (BN sample, rhenium gasket, and diamond anvil), as well as large plastic deformations and contact sliding, makes this problem quite challenging for convergence.
As the first step toward this goal, we developed a large deformation model for elastoplasticity without PTs (Feng et al., 2016), and solved the problem for compression of a rhenium sample in DAC with a maximum pressure up to 300 GPa. Results were in good correspondence with experiments in terms of pressure distribution along the sample-diamond surface, and the change in shape (cupping) of this surface.
The paper is organized as follows. General equations for coupled plastic flow and plastic strain-induced PTs are presented in Section 2. Specific models are described in Section 3. Complete system of equations is summarized in Section 4. Algorithmic aspects are discussed in Section 5. Material parameters used in simulations are described in Section 6. Nonlinear elastic equations and material properties for a single-crystal diamond anvil are given in Section 7. Results and discussion on plastic strain-induced PT in the BN sample are described and analyzed in Section 8. Section 9 includes concluding remarks.
Contractions of the second-order tensors and over one and two indices are defined as and , respectively. Similarly, we designate contractions of the forth-order tensor and second-order tensor , over one and two indices as and . The subscripts s and a mean symmetrization and skew-symmetrization, the superscripts t and −1 are the transposition and inverse of a tensor, the subscripts e, p and t designate elastic, plastic and transformational strain or deformation gradient. I is the second-order unit tensor,:= is equal by definition, and subscripts 1 and 2 are for material parameters of the low pressure and high pressure phases, respectively.
The constitutive behavior of two-phase elastoplastic mixture with the variable concentration of phases during PT under large elastic, plastic and transformational strains is extremely complex. While it can be studied numerically for a representative volume using micromechanical approach for some typical loadings, we need analytical expressions to be implemented in the FEM code for the simulation of materials behavior in DAC. It is clear that significant simplifications are required. Also, the
The simplest mixture rule will be used for all elastic and plastic constants. Due to a large ratio of elastic and plastic properties of phases (see Section 6), actual properties strongly depend on morphology of a mixture, which evolves and is unknown. Also, scatter of the pressure dependence of the yield strength (e.g., for Rhenium, see Feng et al. (2016)) is quite large. Under these circumstances, there is no sense to use a more complex model than the linear mixture rule.
Box 1 summarizes all equations in the form used in our simulations. Complete system of equations Decomposition of the deformation gradient F into elastic and inelastic contributionsDecomposition of the deformation rate into elastic, plastic, and transformation partsThe third-order Murnaghan potential
We will formulate the computational algorithm for updating Cauchy stress for elastoplastic materials coupled with plastic strain-induced PTs, which is similar to the radial return algorithm in the book (Simo and Hughes, 1998). However, due to multiple nonlinearities in our constitutive equations, the return direction is not exactly along the radial direction as it is in Simo and Hughes' book for small elastoplastic deformation problems. At time instant (n = 0, 1, 2, …) all state variables
Due to the low yield strength of hBN, the sample gets very thin under high pressure if compressed without gasket. The sample is placed in a gasket made of a strong material to reduce sample radial flow and increase the volume of a high-pressure region in the sample, as well as reduce the radial pressure gradient in the sample, rhenium is often used as a gasket material (Jeanloz et al., 1991, Manghnan et al., 1974, Vohra et al., 1987, Dubrovinsky et al., 2012, Dubrovinsky et al., 2015) due to
The elasticity rule for anisotropic materials (Feng et al., 2016, Feng and Levitas, 2017) has the formwhere is the Kirchhoff stress, is the second Piola-Kirchhoff stress, and since there is no plastic deformation in a diamond, the subscript e is dropped. Under megabar pressures, it is necessary to consider at least the third-order potential with cubic symmetry:
Due to the complexity of the problem, it is impossible to guess the character of the solution. For example, large volumetric transformation strain should lead to pressure drop, but drastic increase of elastic and, especially, plastic properties during PT should lead to pressure growth. Interplay of these effects determines the actual distribution of all fields and PT kinetics. Thus, if the reduction in volume due to PT dominates and the material flows toward the center, then maximum pressure
In this paper, coupled plastic strain-induced PT and plastic flow in a sample compressed in a diamond anvil cell are investigated in the framework of fully geometrically nonlinear formulation and by utilizing FEM. A thermodynamically consistent system of equations is formulated. It includes the Murnaghan elasticity law and pressure-dependent J2 plasticity for both low and high pressure phases, linear mixture rule, and plastic strain-induced and pressure-controlled PT kinetics. Elastic, plastic,
The support of NSF (DMR-1434613), ARO (W911NF-17-1-0225), and Iowa State University (Schafer 2050 Challenge Professorship) is gratefully acknowledged.
A similar two-surface plasticity concept has been developed by Burgold et al. (2020) for TRIP steels in which the first surface is of J2-type and describes plastic yielding, whereas the second surface is of Drucker-Prager type and represents the mechanically induced phase transformation. It should be noted that some large deformation models with appropriate approaches for strain (or stress)-induced phase transformation and TRIP in other materials and states could be found in the literature (Feng & Levitas, 2017; Feng et al., 2019; Mahnken et al., 2012). Another important issue in plastic behavior under the phase transformation (and generally in the multiphase steels) would be the damage growth, decreasing of the load-carrying capacity, and fracture of the material during and after the phase transformation.
Under the same conditions, the deformation-induced anisotropy also does not matter under monotonic loading (if the initial non-deformed material is isotropic). These propositions were confirmed experimentally for a wide variety of materials (Levitas, 1996) and were adopted in the finite element modeling of different high-pressure technics (Feng and Levitas, 2017b; Feng et al., 2019; Pereira and Figueiredo, 2019). The plastic incompressibility of compact materials is usual assumption for severe plastic deformation processes.
The regions of deformation twinning generally span the diameter of the grain in the propagation direction but is only a fraction of the grain size transverse to the twin plane. The work of Morrow et al. (2016) and Feng and Levitas (2017a, b, c) demonstrate that for the single crystal experiments, both primary and secondary twinning occurred. With secondary twinning, additional regions of orientations and interfaces are created with dimensions well below the original grain size of the material.