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Implicit CP FEM scheme for twinning materials

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Abstract

Many computational papers have dealt with twinning in a crystal plasticity framework. Considerable former works have proposed constitutive theories based on continuum mechanics and crystal plasticity to consider the complex response of materials deformed by twinning. However, many of the former computational twinning papers did not provide the detailed implementation procedures that were used to produce their simulation results. Given that the mechanical response of materials deformed by twinning is highly nonlinear and complex, a sophisticated numerical scheme is needed. The complex interaction among various slip and twin modes causes different kinds of interaction hardening in their slip/twin systems and therefore makes the mechanical response of twinning materials change abruptly. Such a change causes difficulty in convergence and typically requires the use of small time steps, making the simulation time lengthy. To deal with this highly nonlinear and rapidly changing mechanical response of twinning materials effectively, an implicit time integration scheme that considers the constitutive theories of twinning materials is proposed in this study. The proposed implicit computational scheme for twinning materials is derived in a way that the scheme is effectively implemented in a large deformation finite element (FE) code. This study uses the experimental data of single and poly crystal magnesium shown in Refs. [1, 2] and successfully reproduces the strain-strain responses. The simulation results indicate that twinning causes more numerical difficulties than the slip process. The proposed implicit time integration crystal plasticity FE method (CP FEM) scheme provides better efficiency, accuracy, stability, and robustness than previously used explicit CP FEM schemes, especially in dealing with the numerical difficulties associated with twinning.

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Abbreviations

a :

Length of a side in the basal plane hexagon

C e :

Elasticity modulus tensor

c :

C-axis length of the HCP atomic structure

D :

Symmetric part of the velocity gradient

F:

Deformation gradient

f β :

Twin volume fraction of the β -th twin system

h :

Self-hardening modulus

K :

Bulk modulus

L :

Velocity gradient

\({\underline {\hat m} ^\alpha }\) :

Slip direction of the α -th slip system

\({\underline {\hat m} ^\beta }\) :

Twin direction of the β -th twin system

N s :

Number of slip systems

N tw :

Number of twin systems

n :

Rate-dependent law power constant

q αβ :

Latent hardening modulus between two modes of α and β

R*:

Deformation gradient related to elastic lattice rotation

\({{\bar r}^\alpha }\) :

Relative activity of the α -th (twin or slip) mode

\({\underline {\hat s} ^\alpha }\) :

Slip plane normal of the α -th slip system

\({\underline {\hat s} ^\beta }\) :

Twin plane normal of the β -th twin system

\({{\bf{\hat T}}^ \ast }\) :

Second Piola-Kirchhoff stress at \({\hat B}\) configuration

\(\underline T \) :

Traction vector

\(\underline U \) :

Displacement vector

V*:

Deformation gradient related to elastic stretching

W :

Antisymmetric part of velocity gradient

Ẏ α :

Plastic strain rate in the α -th slip or twin system

Ẏ 0 sl :

Referential strain rate for slip

Ẏ 0 tw :

Referential strain rate for twinning

Y tw :

Constant related to shear strain

ε*:

Small strain elasticity tensor

λ :

Lamé constant

μ :

Shear modulus

σ :

Cauchy stress

σ m :

Hydrostatic part of the Cauchy stress

σ′:

Deviatoric part of the Cauchy stress

τ :

Kirchhoff stress

τ 0 :

Initial CRSS

τ α :

Stress magnitude in the α -th slip/twin system

τ α0 :

CRSS in the α -th slip/twin system

τ (i)0 :

CRSS in the i-th slip/twin mode

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Acknowledgments

This work was supported by a grant from Seed grants for Transitional and Exploratory Projects at Southern Illinois University Edwardsville, whose title is “investigation on the effect of anisotropy on the ductile fracture process of metals.”

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Authors and Affiliations

  1. Mechanical Engineering, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026, USA

    Soondo Kweon & Daniel S. Raja

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  1. Soondo Kweon
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  2. Daniel S. Raja
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Correspondence to Soondo Kweon.

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Soondo Kweon is an Associate Professor in the Mechanical Engineering Department at Southern Illinois University Edwardsville, USA. He received his Ph.D. in Mechanical Engineering from the University of Illinois at Urbana-Champaign. His research interests include crystal plasticity, finite element methods, solid mechanics, ductile fracture, and J2 plasticity.

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Kweon, S., Raja, D.S. Implicit CP FEM scheme for twinning materials. J Mech Sci Technol 35, 4585–4603 (2021). https://doi.org/10.1007/s12206-021-0928-y

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