skip to main content

Title: Phase field modeling of coupled crystal plasticity and deformation twinning in polycrystals with monolithic and splitting solvers

For some polycrystalline materials such as austenitic stainless steel, magnesium, TATB, and HMX, twinning is a crucial deformation mechanism when the dislocation slip alone is not enough to accommodate the applied strain. To predict this coupling effect between crystal plasticity and deformation twinning, we introduce a mathematical model and the corresponding monolithic and operator splitting solvers that couple the crystal plasticity material model with a phase field twining model such that the twinning nucleation and propagation can be captured via an implicit function. While a phase field order parameter is introduced to quantify the twinning induced shear strain and corresponding crystal reorientation, the evolution of the order parameter is driven by the resolved shear stress on the twinning system. To avoid introducing an additional set of slip systems for dislocation slip within the twinning region, we introduce a Lie algebra averaging technique to determine the Schmid tensor throughout the twinning transformation. Three different numerical schemes are proposed to solve the coupled problem, including a monolithic scheme, an alternating minimization scheme, and an operator splitting scheme. Three numerical examples are utilized to demonstrate the capability of the proposed model, as well as the accuracy and computational cost of the solvers.

more » « less
Award ID(s):
Author(s) / Creator(s):
Publisher / Repository:
Wiley Blackwell (John Wiley & Sons)
Date Published:
Journal Name:
International Journal for Numerical Methods in Engineering
Page Range / eLocation ID:
p. 1167-1189
Medium: X
Sponsoring Org:
National Science Foundation
More Like this
  1. Abstract Magnesium, the lightest structural metal, usually exhibits limited ambient plasticity when compressed along its crystallographic c -axis (the “hard” orientation of magnesium). Here we report large plasticity in c -axis compression of submicron magnesium single crystal achieved by a dual-stage deformation. We show that when the plastic flow gradually strain-hardens the magnesium crystal to gigapascal level, at which point dislocation mediated plasticity is nearly exhausted, the sample instantly pancakes without fracture, accompanying a conversion of the initial single crystal into multiple grains that roughly share a common rotation axis. Atomic-scale characterization, crystallographic analyses and molecular dynamics simulations indicate that the new grains can form via transformation of pyramidal to basal planes. We categorize this grain formation as “deformation graining”. The formation of new grains rejuvenates massive dislocation slip and deformation twinning to enable large plastic strains. 
    more » « less
  2. null (Ed.)
    Continuum dislocation dynamics models of mesoscale plasticity consist of dislocation transport-reaction equations coupled with crystal mechanics equations. The coupling between these two sets of equations is such that dislocation transport gives rise to the evolution of plastic distortion (strain), while the evolution of the latter fixes the stress from which the dislocation velocity field is found via a mobility law. Earlier solutions of these equations employed a staggered solution scheme for the two sets of equations in which the plastic distortion was updated via time integration of its rate, as found from Orowan’s law. In this work, we show that such a direct time integration scheme can suffer from accumulation of numerical errors. We introduce an alternative scheme based on field dislocation mechanics that ensures consistency between the plastic distortion and the dislocation content in the crystal. The new scheme is based on calculating the compatible and incompatible parts of the plastic distortion separately, and the incompatible part is calculated from the current dislocation density field. Stress field and dislocation transport calculations were implemented within a finite element based discretization of the governing equations, with the crystal mechanics part solved by a conventional Galerkin method and the dislocation transport equations by the least squares method. A simple test is first performed to show the accuracy of the two schemes for updating the plastic distortion, which shows that the solution method based on field dislocation mechanics is more accurate. This method then was used to simulate an austenitic steel crystal under uniaxial loading and multiple slip conditions. By considering dislocation interactions caused by junctions, a hardening rate similar to discrete dislocation dynamics simulation results was obtained. The simulations show that dislocations exhibit some self-organized structures as the strain is increased. 
    more » « less
  3. Abstract

    The deformation of crystalline materials by dislocation motion takes place in discrete amounts determined by the Burgers vector. Dislocations may move individually or in bundles, potentially giving rise to intermittent slip. This confers plastic deformation with a certain degree of variability that can be interpreted as being caused by stochastic fluctuations in dislocation behavior. However, crystal plasticity (CP) models are almost always formulated in a continuum sense, assuming that fluctuations average out over large material volumes and/or cancel out due to multi-slip contributions. Nevertheless, plastic fluctuations are known to be important in confined volumes at or below the micron scale, at high temperatures, and under low strain rate/stress deformation conditions. Here, we develop a stochastic solver for CP models based on the residence-time algorithm that naturally captures plastic fluctuations by sampling among the set of active slip systems in the crystal. The method solves the evolution equations of explicit CP formulations, which are recast as stochastic ordinary differential equations and integrated discretely in time. The stochastic CP model is numerically stable by design and naturally breaks the symmetry of plastic slip by sampling among the active plastic shear rates with the correct probability. This can lead to phenomena such as intermittent slip or plastic localization without adding external symmetry-breaking operations to the model. The method is applied to body-centered cubic tungsten single crystals under a variety of temperatures, loading orientations, and imposed strain rates.

    more » « less
  4. Herein, experimental and modeling tools are employed to understand the joint area for a diffusion‐bonded 316L stainless steel. Detailed microstructure characterizations by means of optical microscopy, electron backscatter diffraction, and energy‐dispersive X‐ray spectroscopy are coupled to in situ micromechanical testing and micro‐ and nano‐indentation to fully reveal the properties at the joint area. Crystal plasticity finite‐element modeling is performed utilizing the exact microstructure to understand the effect of individual slip systems on transgranular strain fields. It is revealed that the diffusion line is only marginally harder. The final failure of the sample occurred away from the joint area, and both the base metal and bond line, shows evidence of considerable twinning and plastic deformation by (dislocation) slip initiating at grain boundaries. Additional slip systems and slip bands form to propagate across all matrices, blocking further dislocations after the ultimate tensile strength point. The presence of flaws within the weldment is found to be negligible.

    more » « less
  5. Schuh, Christopher A (Ed.)
    The {-1012} tensile twins terminating inside the grains of a deformed Mg-Y alloy were investigated by transmission electron microscopy. The crystallographic features of terminating twins and associated slip structures were quantified and correlated. The local stresses developed at a terminating {-1012} twin were computed using crystal plasticity simulations in order to interpret the observed slip patterns. Results indicate that both basal and matrix glide were involved in accommodating the plastic stresses developed in the vicinity of terminating twins. Along the twin boundary, the defect contrast consistent with that of lattice dislocations and twinning partials was observed. Based on these observations, a dislocation reaction is proposed that establishes an interrelationship between the observed matrix glide and {-1012} twinning in Mg-Y alloys. 
    more » « less