PSI - Issue 33

Victor Rizov et al. / Procedia Structural Integrity 33 (2021) 428–442 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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3. Numerical results The results reported in this section of the paper are obtained by applying the time-dependent solutions to the strain energy release rate derived in the previous section. The strain energy release rate is expressed in no-dimensional form by using the formula ) /( G G E b UP N  . First, the evolution of the strain energy release rate with the time is investigated. For this purpose, calculations of the strain energy release rate are carried-out at various values of the time. It is assumed that 0.010  b m, 0.015  h m, 0.7  n , 0.7  m and 0.001  F u m. The influence of the crack location along the thickness of the beam on the time-dependent strain energy release rate is investigated too. The crack location along the thickness of the beam is characterized by h h / 1 ratio. In order to evaluate the influence of the crack location, calculations are carried-out at three h h / 1 ratios. The evolution of the strain energy release rate and the influence of the crack location are illustrated in Fig. 4 where the strain energy release rate in non-dimensional form is plotted against the non-dimensional time at three h h / 1 ratios. The time is expressed in non-dimensional form by using the formula UP UP N t tE  /  . It is evident from Fig. 4 that the strain energy release rate decreases with the time. This finding is attributed to the stress relaxation. It can be observed also in Fig. 4 that the strain energy release rate decreases with increasing of h h / 1 ratio. The influence of the material gradient in the thickness of the beam on the strain energy release rate is also studied. The material gradient is characterized by UP LW E E / and UP LW   / ratios. Calculations are performed at various UP LW E E / and UP LW   / ratios. The results obtained are shown in Fig. 5 where the strain energy release rate in non-dimensional form is plotted against UP LW E E / ratio at three UP LW   / ratios. The curves in Fig. 5 indicate that the strain energy release rate increases with increasing of UP LW E E / ratio. One can observe also in Fig. 5 that the increase of UP LW   / ratio leads to increase of the strain energy release rate. The evolution of the strain energy release rate with the time for the case of material with different viscoelastic behaviour in tension and compression is also analyzed. For this purpose, the strain energy release rate in non dimensional form is plotted against the non-dimensional time in Fig. 6 at three cUP cLW   / ratios. It can be observed in Fig. 6 that the strain energy release rate decreases with the time. The curves in Fig. 6 show that the strain energy release rate increases with increase of cUP cLW   / ratio. The effects of cUP cLW E E / and cUP tUP E E / ratios on the strain energy release rate are evaluated in Fig. 7. It can be observed in Fig. 7 that the strain energy release rate increases with increasing of cUP cLW E E / and ratios. The strain energy release rate in non-dimensional form is plotted against F u in Fig. 8 for both cases (material with identical viscoelastic behaviour in tension and compression and material with different viscoelastic behaviour in tension and compression). The curves in Fig. 8 indicate that the strain energy release rate increases with increasing of F u . One can observe also in Fig. 8 that the strain energy release rate for material with different behaviour in tension and compression is higher than that for material with identical behaviour in tension and compression. 4. Conclusions The influence of the stress relaxation on the strain energy release rate for a lengthwise crack in a functionally graded cantilever beam configuration is analyzed. The stress relaxation is treated by using a linear viscoelastic model consisting of a spring and a dashpot. The modulus of elasticity and the coefficient of viscosity are distributed cUP tUP E E /