Modified mixed-mode caustics interpretation to study a running crack subjected to obliquely incident blast stress waves

Highlights

• Incident P and S waves, as well as reflected waves, were visualized.

• A modified mixed-mode caustics interpretation was proposed and verified.

• More precise mixed-mode SIFs and crack-tip positions were determined.

• Transient stress superposition and delayed redistribution occur in the crack tip.

Abstract

Crack-wave interaction is common in blast, producing mixed-mode cracks when stress waves are oblique to cracks. How to measure mixed-mode crack-tip stress under blast stress waves is difficult. To this end, optical caustics method was employed to study a running crack in a PMMA plate subjected to obliquely incident blast stress waves, by which stress problem was transformed into optical problem. Incident P and S waves, as well as reflected waves, were visualized as fringe patterns, and mixed-mode crack-tip stress was represented by a caustics pattern (shadow spot). However, mixed-mode caustics patterns under P waves were not consistent with the classical interpretation, hence a modified mixed-mode caustics interpretation was proposed and then verified by P-wave compression and tension loading cases. In modified interpretation, an iteration procedure was proposed to obtain modified stress intensity factors, i.e. $K_I$ and $K_{II}$, and crack-tip positions which were more precise than those by the classical interpretation. P-wave compression phase decreased $K_I$ and crack velocity sharply but produced the largest $K_{II}$, while the following P-wave tension phase had an opposite effect. S waves and reflected waves produced higher $K_I$ and crack velocity with lower K_II. K_II and crack velocity histories explained curved crack path and microscopic fracture surface. Finally, the reason for the success of the modified interpretation was discussed, and it is indicated that transient stress superposition and delayed redistribution in the crack tip are the mechanism underlying the modified interpretation.

Introduction

The interaction between dynamic cracks and stress waves is common in blast applications, such as mining and tunnelling engineering [1,2]. Due to arbitrary crack propagation in multi-hole blasting, blast-induced cracks are always subjected to obliquely incident blast stress waves, thus mixed-mode cracks appear which are more general than mode I cracks in practice. For a mixed-mode crack, the local crack-tip stress is dominated by mixed mode I and mode II stress intensity factors (SIFs) which determine crack propagation. For a better understanding of mixed-mode crack propagation in blast, it is of fundamental importance to evaluate mixed-mode crack-tip stress under blast stress waves.

Mixed-mode crack-tip stress caused by stress waves was studied by dynamic fracture theory [3]. Based on stress superposition, Freund [4], Achenbach [5] and Ma [6] obtained analytical solutions for mode I and mode II SIFs of a crack subjected to obliquely incident stress waves. Specifically, Freund [4] restricted crack remaining straight, while Achenbach [5] and Ma [6] considered a kinked crack deflecting off the straight line. Due to complex blast stress waves, these analytical solutions were not applied to real blast cases, but they provided a critical information, i.e. stress superposition in the crack tip upon arrival of stress waves.

Due to complexity in analytical approaches, laboratory blast experiments are indispensable which were used to measure shock wave pressure [7], stress wave form [8] and dynamic fracturing behavior [9], [10], [11]. In blast experiments, visual observations of blast stress waves and cracks provide useful information for understanding the interaction between stress waves and cracks. Hence, optical methods combining with transparent testing materials are good choices, such as photoelasticity and caustics methods [12]. Despite a wide range of applications, both photoelasticity and caustics methods face challenges in obtaining SIFs from crack-tip optical patterns in blast. Historically, photoelasticity was introduced into blast experiments much earlier than caustics method, due to its advantages in rich full-field data for stress waves and crack-tip stress [13,14]. However, it is difficult to obtain SIFs from highly dense and distorted isochromatic fringes in the crack tip upon arrival of blast stress waves [15]. This challenge is also presented in Ref. [16]. About 30 years later than photoelasticity, caustics method was introduced into blast experiments by Yang [17], to obtain crack-tip SIFs by measuring the diameter of the caustics pattern (shadow spot). Compared with photoelasticity, caustics method provides simple caustics patterns, avoiding dense fringes in the crack tip [18]. The shape of the caustics pattern reflects stress distribution in the crack tip. Caustics patterns are always in an approximately circle shape under general dynamic loading, such as tensile machine [19,20], gas gun projectile [21], drop weight [22] and out-of-plane impact [23], indicating K dominated stress field in the crack tip. However, recent blast experiments [24,25] show that caustics patterns are distorted in an ellipse shape under blast-induced P wave loading, indicating that crack-tip stress is non-K dominated and SIFs obtained by the classical caustics interpretation have large errors. In order to better interpret distorted caustics patterns, a modified caustics interpretation for a mode I crack subjected to blast-induced P waves was built, by which more precise crack-tip SIFs were obtained [24,25].

However, in blast experiments using caustics method, mode I and mode II SIFs of a mixed-mode crack are still obtained by the classical mixed-mode caustics interpretation, which may cause errors if mixed-mode caustics patterns are distorted under blast stress waves. Therefore, a modified mixed-mode caustics interpretation is required. The present work is the extension of the previous work by authors [26] where a mode I crack was subjected to normally incident blast stress waves.

In this paper, a running crack in a PMMA plate subjected to obliquely incident blast stress waves was investigated by caustics method and high-speed photography. Incident P and S waves, as well as reflected waves, were visualized as fringes and distinguished based on different fringe patterns. Upon arrival of blast-induced P waves, crack tip was surrounded by a mixed-mode caustics pattern which was not consistent with the classical caustics interpretation. To solve this problem, a modified mixed-mode caustics interpretation was proposed based on the superposition of geometrical optics and verified by P-wave compression and tension loading cases. An iteration procedure was proposed to obtain $K_I$ and $K_{II}$ in the modified caustics interpretation. SIFs and crack velocity histories were obtained in the whole crack propagation, which explained variations of crack path and microscopic fracture surface well. Finally, the mechanism underlying the modified mixed-mode caustics interpretation was discussed.