Explicit nonlinear finite element approach to the Lagrangian-based coupled phase field and elasticity equations for nano
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O R I G I NA L A RT I C L E
Mahdi Javanbakht
· Hossein Rahbar · Milad Ashourian
Explicit nonlinear finite element approach to the Lagrangian-based coupled phase field and elasticity equations for nanoscale thermal- and stress-induced martensitic transformations Received: 23 January 2020 / Accepted: 4 August 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract In this paper, a nonlinear finite element procedure is developed to solve the coupled system of phase field and elasticity equations at large strains for martensitic phase transformations at the nanoscale. The transformation is defined based on an order parameter which varies between 0 for austenite to one for martensite. The phase field equation relates the rate of change of the order parameter to the Helmholtz free energy. Both the Helmholtz free energy and the transformational strain tensor depend on the order parameter, and the coupling between the phase field and elasticity equations occurs due to the presence of elastic energy in the Helmholtz free energy and the transformational strain in the total strain. The deformation gradient tensor is introduced as a multiplicative decomposition of elastic and transformational gradient tensors. A staggered strategy is used to solve the coupled system of equations so that first, the principle of virtual work is utilized to obtain the integral form of the Lagrangian equation of motion. It is then discretized and solved using the Newton–Raphson method which gives the displacements and consequently, the deformation gradient tensor. Next, since the order parameter is obtained from the phase field equation from the previous time step, the elastic deformation gradient tensor and the first Piola–Kirchhoff stress can be calculated and substituted in the phase field equation to find the order parameter for the current time step. The weighted residuals method is used to derive the finite element form of the phase field equation and the explicit method is used for its time discretization. The finite element procedure is well verified by comparing the obtained results with those from other simulations. Examples of thermal- and stress-induced phase transformations are presented, including austenite–martensite interface propagation and growth of preexisting martensitic regions. The developed algorithm and code can be advanced to solve kinetic problems coupled with mechanics at large strains for phase transformations and similar phenomena at the nanoscale. Keywords Phase transformations · Phase field · Large strains · Total Lagrangian · Explicit method List of symbols λ β η t Δt ψ ψ e, ψ θ , ψ c, ψ ∇
Kinetic energy Gradient energy Phase order parameter Time Time increment Helmholtz free energy Elastic, thermal, crystalline and gradient energies
Communicated by Andreas Öchsner. M. Javanbakht (B) · H. Rahbar · M. Ashourian Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran E-mail: [email protected]
M. Javanbakht et al.
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