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A model of dynamic recrystallization in polycrystalline materials is investigated in this work. Within this model a probability distribution function representing a polycrystalline aggregate is introduced. This function characterizes the state of individual grains by grain size and dislocation density. By specifying free energy and dissipation within the polycrystalline aggregate an evolution equation for the probability density function is derived via a thermodynamic extremum principle. For distribution functions describing a state of dynamic equilibrium we obtain a partial differential equation in parameter space. To facilitate numerical treatment of this equation, the equation is further modified by introducing an appropriately rescaled variable. In this the source term is considered to account for nucleation of grains. Then the differential equation is solved by an implicit time-integration scheme based on a marching algorithm [2]. From the obtained distribution function macroscopic quantities like average strain and stress can be calculated. Numerical results of the theory are subsequently presented. The model is compared to an existing implementation in Abaqus as well.In this report a mathematical model stemming from the work of Hackl and Renner [1] is investigated. In this model the aggregate of polycrystalline materials during dynamic recrystallization was presented by the distribution function f (D, ρ) of grain states which characterize individual grains, with grain size D and dislocation density ρ. f dΩ gives the probability of finding a grain in a state lying in the domain dΩ. Balancing the grain number for the control volume and introducing a new rescaled variable s giveThe meaning of material parameters can be referred to [1,2]. The source function h, the grain's number of nucleation process has been integrated into (1). New Theory of Grain NucleationThe source function h can be determined based on the following minimization problemThe rate of potential energy is given byψ h = π 6 Ω D 3 ψhdΩ, where ψ = µb 2 r 2 + 4γ D is the Helmholtz free energy. Moreover, the dissipation potential for the source function takes in the form∆ h = k 2 π 2 Ω D 2 h 2 dΩ, where k is a material parameter.Furthermore, since h is the nucleation process thenLet us suppose that the grain nucleation happpens along a curve r = p(r 0 , D) = r 0 D α . Moreover, grain size should be smaller after the dynamic recrystallization, then a new variable w, where D 3 h = − dw dx , is introduced. Then our variational

This paper is devoted to developing a mathematical model of dynamic recrystallization phenomena based on the earlier work of K. Hackl and J. Renner [1] for polycrystalline materials. In this model a variational approach employing a distribution function for dynamic recrystallization processes of polycrystalline materials is presented. It is based on a marching algorithm at the microscale as well as a homogenization procedure to arrive at the macroscale. The newly proposed theory is now used to improve the variational approach. Then this theory is applied within the homogenization procedure. A comparison of the present model with existing phenomenological ones is given.A micromechanical model for dynamic recrystallization in polycrystal was first introduced by Hackl and Renner [1]. This model is based on variational principles and leads to a variational problem at the microscale. Nguyen and Hackl continued to develop this model by proposing a numerical implementation [2]. Within our approach, internal variables include grain size D, dislocation density ρ, plastic strain ε p , and dislocation strain ε d . We already proposed specific form of Helmholtz free energy and of dissipation potentials as well as constitutive assumptions on dissipative processes and a volume constraint. Thus evolution equations for the internal variables are directly derived via the principle of the minimum of the dissipation potential (PMDP) [3]. As a result, the variational equation accompanied by the volume constraint is given as follows(1)A detailed description of the above material model as well as a parameter study can be found [1] and [2]. During the development, a new proposal of the nucleation theory is presented. PMDP is also employed in this theory. The rate of Helmholtz free energy and the dissipation for the nucleation process have the following formṡUsing the thermodynamic principle, the source function or the nucleation rate becomes Numerical treatmentTo deal with this problem, a two-scale scheme has been implemented. This scheme comprises of a marching algorithm at the microscale and a simple return mapping at the macroscale. For the full description of the marching algorithm, the readers can refer to [2,4]. This box below will illustrate the link between the problem at the microscale and counterpart at the macroscale.