PSI - Issue 42

A. Kostina et al. / Procedia Structural Integrity 42 (2022) 425–432 A. Kostina / Structural Integrity Procedia 00 (2019) 000–000

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energy of a cold work. To take into account this effect a constant Taylor-Quinney coefficient has been introduced as a ratio of the released heat to plastic work (integral form) or ratio of rates of these values (differential form). The magnitude of this parameter varies from 0.6 to 1 depending on the material and loading conditions (Clifton et al. (1984), Belytschko et al. (1991), Zhou et al. (1996), Dolinski et al. (2010)). Such a large spread indicates that the value often used in engineering practice, equal to 0.9, is not always correct. Lee et al. (2011) have been studied temperature variation in the process of dynamic compression of 316L stainless steel. It has been demonstrated that Taylor-Quinney coefficient depends on strain, strain rate and initial temperature. The authors have been illustrated that, at a strain of 0.4 and a strain rate of 5000 s –1 , the temperature rises by 140°C. Oliferuk et al. (1996) compared fine-grained and coarse-grained austenitic steel and found that the value of the stored energy in fine-grained material is higher than in coarse-grained. This conclusion agrees with the data given in (Titchener and Bever (1958), Zhang et al. (2018)). Zhang et al. (2018) used infrared thermography to determine the Taylor-Quinney coefficient in aluminum samples deformed at rates of 1100–4200 s –1 . As a result, a strong dependence of this value on the strain rate has been obtained. The above-mentioned works show that accurate description of the energy balance during inelastic deformation requires taking into account not only energy dissipation, but also its accumulation. To develop constitutive relations describing these processes, approaches based on irreversible thermodynamics are widely applied (Callen H. (1960)). The main idea is to introduce internal state variable to describe the energy accumulation induced by the structural changes in the material (Ranc and Chrysochoos (2013)). Historically, the irreversible thermodynamics was originally a linear theory, but then it was combined with theory of the generalized standard material to overcome the limitations associated with its linearity (Halphen and Nguyen (1975)). The formalism of the generalized standard material theory is widely applied (Lemaitre (1985), Lemaitre and Chaboche (2000), Lubliner (2000)) because it ensures the fulfillment of the dissipation inequality by deriving constitutive equations from both thermodynamic and dissipative potentials. In this paper, we apply quasi-standard thermodynamic formalism to derive constitutive relations that describe inelastic deformation of metals in order to accurately calculate the amount of the dissipated energy during fatigue loading of two titanium alloys (Ti-5Al-2V and Grade-2 ). It is assumed that in the isothermal case, the free energy is a function of several parameters: elastic strain, structural parameter, isotropic and kinematic hardening parameters. To derive constitutive relations, a dissipative function is introduced, which consists of two terms, one of which is responsible for the process of plastic deformation, and the other is responsible for the structural evolution of the material. Plastic deformation is associated with dissipative processes generated by the motion and annihilation of dislocations, structural evolution is defined by the initiation and growth of defects. Dissipated energy is obtained as the difference between plastic work and stored energy. To calculate dissipated energy per loading cycle during fatigue crack propagation a stationary crack approach was used. The obtained dependence of energy dissipation per loading cycle on crack length was compared with experimental data obtained by original heat flux sensor (Iziumova et al. (2021), Vshivkov et al. (2016)). 2. Materials and experimental conditions Cyclic experiments on titanium alloys were carried out on Instron 8802 servo-hydraulic machine under constant maximum loading. Each material was tested with two different maximum loadings which were equal to 8.5 kN and 9.5 kN in case of Ti-5Al-2V alloy, 7.5 kN and 8 kN in case of Grade-2 alloy. The stress ratio was 0.1 and loading frequency was 10 Hz. The specimens were made of sheets with the thickness of 3 mm and had V-shape notches. The geometry of the samples is presented in Fig.1. The chemical composition of the studied materials is given in Tables 1-2.

Fig. 1. Geometry of the specimen. All sizes are in mm.