13th International Conference on Fracture June 16–21, 2013, Beijing, China -5- The value of λе can be determined if its dependence from the external force Р is found. To determine Р, let us present a specimen with a crack as a set of double-cantilever beams: with straight-through notch in width of х and with chevron notch in width of а - х (Fig. 4). We shall find forces Р1 and Р2 for each part, determining equal displacement λе of these force application points. Using equation (5), it is easy to derive expression for the force Р1 affecting the specimen in width of х = 2Δl⋅tg(α/2), which provides displacement of load application points Р1 to the specified value of λе: 3 e 1 λ tg . 4 2 E l b P l α Δ ⎛ ⎞⎛ ⎞ = ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (8) Fig. 4. Presentation of the double-cantilever beam specimen. When applying equation (3), with regard for the width of the chevron-notched specimen equal to а-х, we shall derive an expression for the force Р2, which provides displacement of load application points to the same value of λе: 2 3 e 2 0 0 0 0 λ 2 2 1 tg 4 ctg 2 ctg , 8 2 2 2 E a l b l a l a P a l l l l l α α α − ⎡ ⎤⎡ ⎤ ⎡ Δ ⎤ Δ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ = − + + + ⎢ ⎥⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎣ ⎦ ⎣ ⎦⎣ ⎦ (9) where l = l0 + Δl. From equations (8) and (9), an expression for λе is determined: 2 1 3 e 3 0 0 0 0 8 2 2 2 λ tg 1 tg 4 ctg 2 ctg , 2 2 2 2 Pl l l l a l a Eab a a l l l l α α α α − − ⎡ ⎤ ⎡ ⎤⎡ ⎤ Δ ⎛ Δ ⎞ Δ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎢ ⎥ = + − + + + ⎢ ⎥⎢ ⎥ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦⎣ ⎦ ⎣ ⎦ (10) where Р = Р1+Р2. Equations (7) and (10) were used to calculate fracture energy determining the necessary condition for spontaneous crack propagation in studied materials. Figure 5 presents typical loading diagrams for titanium alloy VТ6 and commercial titanium VТ1-0 with UFG structure obtained under testing of small-size chevron-notched specimens. Both diagrams correspond to loading rate v = 2,0 µm/s. Specimens in length of 18 mm were made of bars in section of 6х6 mm2. Calculations have shown that the value of Gs is maximal at the peak of loading and therefore, can serve as a crack resistance criterion of studied materials at specified geometrical parameters and loading conditions of the specimen. Stress intensity factor is generally used as a crack resistance criterion in engineering fracture mechanics for a cleavage crack:
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