PSI - Issue 18

Plekhov O. et al. / Procedia Structural Integrity 18 (2019) 711–718 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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experimental studies of the energy balance during irreversible deformation lets us to assume that an energy stored in the material due to the defects evolution is one of the key parameter failure simulation. The model of energy balance describing stored part of mechanical energy can be useful for the study of various phenomena that accompany the process of plastic deformation and failure. For instance, the shear bands formation, the necking and other types of instabilities in the deformation process. In work Benzerga et al. (2005) the stored energy is calculated as a difference between the internal energy associated with the dislocation structure in the current and in the initial states based on a discrete dislocation plasticity model. As one of the results of the work, the link between stored energy value and the Bauschinger effect was shown. The different ways of stored energy evolution estimation based on the analysis of stress-strain relation for material under investigation were published in works Aravas (1990), Szczepinski (2001), Oliferuk et al. (2009). The works Chaboche (1993), Vshivkov et al. (2016) are present the approaches for calculation of the energy of plastic deformation based on internal variables. Analysis state-of-the-art theoretical studies devoted to the description of the energy balance in the metals under irreversible deformation allows us to conclude that the stored energy is integral, experimentally measured parameter associated with current state of material structures, and for its adequate description of the process we have to propose a phenomenological relations describing the evolution structural material defects. To propose a cheap and effective technique for verification of the model we used the results of coupled experimental approach, which combine the infrared thermography and contact measurement of heat flux Vshivkov et al. (2016). Based on the experimental results we develop a thermodynamic internal variable model of heat dissipation in metals. It allows us to simulate the heat dissipation under plastic deformation of metals, to study the failure process based on the stored energy criterion and to derive a new simulation technique for describing of the process of fatigue crack propagation. The key point of the proposed theoretical model is form of thermodynamic potential of the system. The new internal thermodynamic variable is introduced as defect induced deformation and calculated based on the solution of statistical problem of mesodefect evolution. The developed model is adapted for using in standard final-elements numerical code (Abaqus) to simulate energy dissipation under quasi-static and cyclic loading in 3D geometries. 2. Mathematical model of energy balance under irreversible deformation of metals A thermomechanical process has to obey the first and second laws of thermodynamics. In the case of small deformations, these equations involve the following thermodynamic quantities: volumetric mass  , specific internal energy e , strain and stress tensors  and  , heat supply r , heat flux vector q , specific Helmholtz free energy F and specific entropy S , absolute temperature field T . In particular,  is the small strain tensor,  is the Cauchy stress tensor and p is the structural sensitivity parameter. For the sake of simplicity, we will consider the corresponding scalar process. We assume that the specific heat 2 2 ( / ) c T F T     doesn`t depends on the current structural state or 2 and the stress response function  is also independent of the structural parameter. As a result of this 0 F 

2          p T 

assumptions, the partial derivative F p   can be represented as a linear function of T :

( ) ( ) F p Tf p g p     .

(1)

Then we introduce functions 1 ( ) s p , 1 ( ) E p , such that 1 '( )

( ) s p f p   ,

1 '( ) ( ) E p g p  (symbol « ' » denotes

derivative). Integration of (1) lets us obtain expressions for the free energy F and entropy S :

(2)

,

( )         , ( , ) e T T

( , , ) e S p T F T s p 

( , , ) e F p T Ts p E p    ( )

( )   

( , ) e T 

1

1

1

where function 1 ( ) s p is a structural entropy.