Modeling failure of Ti-6Al-4V using damage mechanics incorporating effects of anisotropy, rate and temperature on strength

Abstract

In the context of lessons learned from the second Sandia Fracture Challenge summit and observed shortcomings in our blind predictions, we revisit finite-element-based failure modeling of the Ti-6Al-4V sample. Anisotropy, rate effects, and thermal softening are used to model strength; damage mechanics is used to model failure; a non-local approach is used to mitigate mesh-dependence of results. We obtain upper and lower bound force versus crack-opening displacement curves and the means of the quantities of interest are found to meet the requirement of being within 10 % of the experimental results, but the predictions of more intensive quantities such as principal strain and temperature in the ligaments of the challenge sample are >10 %.

Appendices

Appendix 1: Adiabatic localization strain

Following the procedure presented in chapter 15 of the text by Meyers (1994), the onset of localization can be estimated as follows. Using Eq. (3) in the plastic work equation, Eq. (4), and carrying out the integration, the increase in the homologous temperature can be shown to be

$$\begin{aligned} T^{*}= & {} 1- \exp \left\{ -\frac{{0.9\sigma _{\circ }}}{\rho C_{p}(T_{M}-T_{R})}\right. \\&\times \left. \left[ \frac{{1+C_{2}\log \left( \dot{{\epsilon }}_{p}/\dot{{\epsilon }}_{\circ }\right) }}{C_{1}(n+1)}\right] \left( 1+C_{1}\epsilon _{p}\right) ^{n}\right\} , \end{aligned}$$

where the homologous temperature is defined as $T^{*}=(T-T_{R})/(T_{M}-T_{R})$. Substituting for the homologus temperture in the strength model and setting $d\bar{{\sigma }}/d\epsilon _{p}=0$ gives a non-linear equation which can be solved for the strain assuming a strain rate and using strength model constants from Table 1; solving the non-linear equation gives a strain of approximately 30 %.

Appendix 2: Cutting plane algorithm

We provide a brief description of the cutting-plane algorithm and derive the plastic strain increment, further details may be found in the article by Ortiz and Simo (1986). The computation of the damage is uncoupled from the plastic strain increment; this is tantamount to assuming that damage does not change during the timestep. The task of the subroutine at each time step is to update the stress tensor. At the same time, if the material is in the plastic state, the yield function (here, coupled with damage) has to be satisfied and the equivalent plastic strain has to be updated. In the following, we develop the plastic strain increment to be used in the cutting-plane algorithm. For simplicity, the transformation tensor, $\varvec{{C}}$is not shown here, and our computational implementation uses the diagonalization of the transformed stress tensor using the matrix of eigenvectors, see Eq. (14) of Karafillis and Boyce (1993). The yield function, $\phi$, is linearized around the current values of the state variables (see Fig. 14) as

$$\begin{aligned} \phi\approx & {} \left. \phi (\varvec{\sigma },F)\right| _{n+1}^{(i)}\,+\left. \frac{{\partial \phi }}{\partial \varvec{{\sigma }}}\right| _{n+1}^{(i)}:\left( \varvec{\sigma }-\varvec{{\sigma }}_{n+1}^{(i)}\right) \,+\,\nonumber \\&\left. \frac{{\partial \phi }}{\partial F}\right| _{n+1}^{(i)}\left( F-F_{n+1}^{(i)}\right) response+ \end{aligned}$$

(17)

Fig. 14

Cutting-plane algorithm

Invoking normality of plastic flow gives decrease in stress due to plastic flow as

$$\begin{aligned} \varvec{\sigma }_{n+1}^{(i+1)}-\varvec{\sigma }_{n+1}^{(i)}=-\varDelta \lambda \,\varvec{L}:\varvec{r}_{n+1}^{(i)}, \end{aligned}$$

(18)

where the elasticity tensor is $\varvec{L}=2G\varvec{I} \,{+}\, (\mathcal {K} \,{-}\, \frac{{2}}{3}G)\varvec{1}\otimes \varvec{1}$, where G and $\mathcal {{K}}$ are the shear and bulk modulus, respectively, and $\varvec{r}=\partial \phi /\partial \varvec{\sigma }$. Expression for the strength differential $d\bar{{\sigma }}$ reads

$$\begin{aligned} F_{n+1}^{(i+1)}-F_{n+1}^{(i)}=\left. \frac{{dF}}{d\epsilon _{p}}\right| _{n+1}^{(i)}\varDelta \epsilon _{p} \end{aligned}$$

(19)

Plastic strain increment tensor is given by $d\varvec{\epsilon }^{p}=d\lambda \partial \phi /\partial \varvec{{\sigma }}$, and using

$\bar{{\sigma }}d\epsilon _{p}=\varvec{\sigma }d\varvec{\epsilon }^{p}$ gives

$$\begin{aligned} d\epsilon _{p}=d\lambda \frac{{\varvec{{\sigma }}:\frac{{\partial \phi }}{\partial \varvec{\sigma }}}}{\bar{\sigma }}\,, \end{aligned}$$

where $\bar{{\sigma }}$ can be written in terms of F. Substituting into Eq. (17), and setting $\phi =0$ gives an expression for $\varDelta \lambda$

$$\begin{aligned} \varDelta \lambda =\left. \frac{{\phi }}{\frac{{\partial \phi }}{\partial \varvec{{\sigma }}}:\varvec{L}:\frac{{\partial \phi }}{\partial \varvec{{\sigma }}}-\frac{{\partial \phi }}{\partial F}\frac{{\partial F}}{\partial \epsilon _{p}}\left( \frac{{\varvec{{\sigma }}:\frac{{\partial \phi }}{\partial \varvec{\sigma }}}}{\bar{\sigma }}\right) }\right| _{n+1}^{(i)}. \end{aligned}$$

Once the $\varDelta \lambda$ is known, damage is computed and the stress state is advanced. The cutting-plane algorithm is well-known and no further elaboration is given here.