A one-dimensional model to describe flow localization in viscoplastic slender bars subjected to super critical impact velocities

Abstract

In this paper we investigate flow localization in viscoplastic slender bars subjected to dynamic tension. We explore loading rates above the critical impact velocity: the wave initiated in the impacted end by the applied velocity is the trigger for the localization of plastic deformation. The problem has been addressed using two kinds of numerical simulations: (1) one-dimensional finite difference calculations and (2) axisymmetric finite element computations. The latter calculations have been used to validate the capacity of the finite difference model to describe plastic flow localization at high impact velocities. The finite difference model, which highlights due to its simplicity, allows to obtain insights into the role played by the strain rate and temperature sensitivities of the material in the process of dynamic flow localization. Specifically, we have shown that viscosity can stabilize the material behavior to the point of preventing the appearance of the critical impact velocity. This is a key outcome of our investigation, which, to the best of the authors’ knowledge, has not been previously reported in the literature.

Appendix A: Mesh sensitivity analysis

In this section we analyze the mesh sensitivity of finite difference and finite element computations. In the calculations of this section we consider the material properties given in Table 1. The impact velocity is $\hat{V}^{\mathit{imp}}=0.01$.

1.1 A.1 Finite difference model

Figure 9 shows the equivalent plastic strain $\hat{\bar{\varepsilon}}^{p}$ versus the normalized dimensionless coordinate $\hat{Z}/\hat{L}$ for three different mesh densities: 500 nodes, 750 nodes and 1000 nodes. For $\hat{t}=10$ (Fig. 9(a)) the curves obtained for the three different meshes virtually overlap each other. On the contrary, for $\hat{t}=40$ (Fig. 9(b)), we observe a (slight) disagreement in the predictions obtained for 500, 750 and 1000 nodes. Such a mesh dependence of the results is always detected close to the impacted side where large gradients of plastic strain are developed. However, we have checked that it does not have implications in the analysis of the results presented in this paper. A mesh with 1000 nodes has been used to carry out all the finite difference computations reported in Sects. 5 and 6.

Fig. 9

Finite difference results. Dimensionless impact velocity $\hat{V}^{\mathit{imp}}=0.01$. Equivalent plastic strain $\hat{\bar{\varepsilon}}^{p}$ versus normalized dimensionless coordinate $\hat{Z}/\hat{L}$ for three different mesh densities: 500 nodes, 750 nodes and 1000 nodes (reference mesh). Dimensionless time: (a) $\hat{t}=10$ and (b) $\hat{t}=40$. The vertical yellow line delimits the zoomed area (For interpretation of the references to colour in the text, the reader is referred to the web version of this article)

1.2 A.2 Finite element model

Figure 10 depicts the equivalent plastic strain $\hat{\bar{\varepsilon}}^{p}$ versus the normalized dimensionless coordinate $\hat{Z}/\hat{L}$ for three different mesh densities: 1000, 1600 and 2000 elements along the axial direction, respectively. In any case the element aspect ratio is 1:1. The results presented correspond to two different loading times: $\hat{t}=10$ in Fig. 10(a) and $\hat{t}=40$ in Fig. 10(b). No differences are observed in the results presented for the three meshes. A mesh with 2000 elements along the axial direction has been used to carry out the finite element computations reported in Sect. 5.

Fig. 10

Finite element results. Dimensionless impact velocity $\hat{V}^{\mathit{imp}}=0.01$. Equivalent plastic strain $\hat{\bar{\varepsilon}}^{p}$ versus normalized dimensionless coordinate $\hat{Z}/\hat{L}$ for three different mesh densities: 1000 elements along the axial direction, 1600 elements along the axial direction and 2000 along the axial direction (reference mesh). Dimensionless time: (a) $\hat{t}=10$ and (b) $\hat{t}=40$. The vertical yellow line delimits the zoomed area (For interpretation of the references to colour in the text, the reader is referred to the web version of this article)