From the true stress-strain curves of Ti-6Al-4V alloy, it can be found that Ti-6Al-4V alloy is a typical strain-softening alloy. It is well accepted that for strain-hardening alloys the fracture criteria can be determined by directly comparing the simulations with the destructive experiments in terms of the critical deformation level. However, for strain-softening alloy the DFC cannot be determined directly due to the fact that softening mechanisms such as dynamic recrystallization (DRX) and dynamic recovery (DRV) occur during the deformation processes, and it is difficult to find visible cracks on the surface of deformed sample and the stress-strain curves. Therefore, it is significant to find an indirect way to evaluate the ductile damage criteria for strain-softening alloy. In this research, an indirect research approach has been brought forward, which utilizes physical experiments, numerical simulations and mathematic computations to provide mutual support for determining the critical fracture time.

As a series of samples were compressed on a heat physical simulation machine under different seven deformation temperatures and four strain rates, the true stress-strain data were collected and then inputted in the FE software of DEFORM-2D. Through an integration method using Eq. (2) in the FE software of DEFORM-2D, the damage values at all strains during the simulation processes were obtained. According to cumulative damage theory, the damage value increases with increasing level of deformation, when it reaches the critical damage value, ductile fracture occurs. However, during the simulation procedure of the compression test in DEFORM-2D, the damage value will keep increasing until the last compression step, even if the damage value reaches the critical damage value. Therefore, it is necessary to analyze the damage accumulation process and identify the initial fracture time.

Kachanov [21, 22, 23] explained the “one-dimensional surface damage variable” by considering a damaged body and a representative volume element (RVE) at a point $M$ oriented by a plane defined by its normal $\stackrel{\u20d7}{n}$ and its abscissa $x$ along the direction $\stackrel{\u20d7}{n}$ (as shown in Figure 2) [21]. The value of the damage $D(M,\stackrel{\u20d7}{n},x)$ attached to the point $M$ in the direction $\stackrel{\u20d7}{n}$ and the abscissa $x$ is:

$D(M,\stackrel{\u20d7}{n},x)=\frac{\delta {S}_{Dx}}{\delta S}$(3)Figure 2: Damaged RVE in a damaged body.

where ${S}_{Dx}$ is the area of intersection of all the flaws with the plane defined by the normal $\stackrel{\u20d7}{n}$ and abscissa $x$; $S$ is the total area at the intersection plane. Damage $D$ is bounded as $0\le D\le {D}_{c}$, where ${D}_{c}$ is a critical damage value corresponding to the decohesion of atoms. $D=0$ represents the undamaged RVE material, and $D={D}_{c}$ represents the rupture failure in the remaining resisting area. When the maximum damage value keeps increasing in a very small growth rate near to zero or equal to zero with increasing deformation strain, it means that fractures happens [24].

Based on Kachanov^{’}s one-dimensional surface damage theory, it can be known that the damage value increases with the increasing of time during a compressing simulation. Thus the maximum value seems to appear at the last simulation step which is not the initial fracture step. As the cumulation damage value contributes to identify the critical damage value, it is considerable to analyze the damage cumulating process. Thus, an innovative concept of the sensitive rate of Cockcroft-Latham damage ($\left({R}_{\text{step}}\right)$) in plastic deformation processes is developed as eq. (4) [12]:

${R}_{\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}}=\frac{\mathrm{\Delta}D}{{D}_{\mathrm{a}\mathrm{c}\mathrm{c}}}$(4)In which ${R}_{\text{step}}$ is equals to the ratio of the damage increment at one step ($\mathrm{\Delta}D$) to the accumulated value ($D\mathrm{a}\mathrm{c}\mathrm{c}$). And it is supposed that if the maximum damage value keeps increasing in a very small growth rate near to zero, which means $R\mathrm{s}\mathrm{t}\mathrm{e}\mathrm{p}$ decreases gradually to zero or near zero, ductile fractures appears. The conception of such assumptions is of good consistency to Kachanov’s explanation of damage mentioned above. And then the fracture time, that is, fracture strain or fracture height reduction can be identified and determined.

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