PSI - Issue 13

A. Vshivkov et al. / Procedia Structural Integrity 13 (2018) 1189–1194 Author name / Structural Integrity Procedia 00 (2018) 000–000

1190

2

nature of this process. A number of approaches, Yates (2008), Mokhtarishirazabad (2017), Izumi (2014), Short (1989), has been developed to study the processes of nucleation and propagation of fatigue cracks based on the J-integral, the work of plastic deformation, the size of the zone of a plastic deformation, the amount of dissipated energy. It is well known that irreversible deformation and failure of metals deformation is accompanied by the structural evolution at all scale levels and leads to energy accumulation and dissipation. Investigation of thermodynamics of deformation and failure is a key issue in solid mechanics. The analysis of the kinetics of damage accumulation, the process of crack nucleation and kinetics of the crack development allows specialists to predict the time of structure failure and to perform in proper time a partial replacement or repair of deteriorated units of complex structures. The real engineering structure works under complex types of loading. So, it is of considerable interest to study the behavior of materials under mixed loading conditions that combine mode 1 and 2. In sufficiently plastic structural materials, the propagation of the crack begins when the plastic deformation near its tip becomes large (of the order of a 10 percent). The previous study of the authors was mainly focused on an equation describing the evolution of plastic work at the crack tip under uniaxial loading. In this work, following by Raju (1972), the plastic work and, as a consequence, heat dissipation at crack tip were divided into two parts corresponding to reversible (cyclic) and monotonic plastic zones. Analysis of this approximation has shown the independence of heat dissipation in cyclic plastic zone from the crack advance. This dissipation is fully determined by the spatial size of a cyclic plastic zone and characteristic diameter of the yield surface. This approach gives well know correlation between fatigue crack rate and dissipated energy, Izyumova (2014) and Ranganathan (2008). This equation based the hypothesis about link between the elastic solution and the elastic-plastic deformation in the fatigue crack tip using the Young’s modulus and the secant plasticity modulus by Dixon (1965).

1 2

s G        G

ef

el

(1)





,

ij

ij

Equation (1) was originally proposed as an empirical expression for a central crack in a wide sheet loaded in tension normal to the line of the crack. The present work is directed on checking this assumption by numerical simulation and experimental measurement of the elastic-plastic zone near the fatigue crack tip.

Nomenclature ε

strain tensor shear modulus

G

secant shear modulus

G S

γ

maximum shear strain = ε 1 – ε 2 principal strain in plane of sheet intensity of plastic deformation

ε 1 , ε 2

ε ie

σ

stress tensor

σ ys

Mises yield stress

σ ys0

initial Mises yield stress

σ m

Mises stress

u

displacement vector

C

fourth-order stiffness tensor

λ

Lamé constant

2. Experimental setup A series of samples made from titanium alloy Graid-2 were tested in the servo-hydraulic biaxial test system Biss BI-00-502, located in Kazan Scientific Center of Russian Academy of Sciences, figure 2. The geometry of the samples is shown in Figure 1. During tests the samples were subjected to cyclic loading of 10 Hz with constant stress amplitude and different biaxial coefficient η=Px/Py (1, 0.7, 0.5, 0). The crack length in the course of the experiment was measured