Fracture prediction of thin plates under localized impulsive loading. Part II: discing and petalling

doi:10.1016/j.ijimpeng.2004.07.011

International Journal of Impact Engineering

Volume 31, Issue 10, November 2005, Pages 1277-1308

https://doi.org/10.1016/j.ijimpeng.2004.07.011 Get rights and content

Abstract

The onset of fracture and subsequent propagation of radial cracks of thin clamped circular plates under localized impulsive loading were predicted analytically and numerically for discing and petalling stages with increasing intensity of applied impulse and various radii of loaded area. The equivalent plastic strain times the average stress triaxiality was introduced as a ductile fracture criterion in the numerical simulation. The strain hardening law and critical damage/fracture function was calibrated from tensile test on round specimen and a parallel numerical simulation. Based on the critical damage value, and calculated distributions and histories of stress and strain, the initiation site and extent of fracture were predicted for a range of loading radii and intensity of applied impulse. It was clearly demonstrated that the crack length and final deformed shapes of plates are strongly influenced by the spatial distribution and intensity of impulsive loading. A comparative study on the propagation of radial cracks was also presented. Finally, the numerically obtained crack length was shown to agree well with the closed form solution derived earlier by one of the present authors.

Introduction

Failure by discing and petalling of thin plates can occur under both impact and localized high intensity loading. There has been considerable research on the perforation of thin metallic plates by cylindrical and conical projectiles over the last few decades [1], [2], [3], [4], [5]. The petalling problem was idealized by simple hole enlargement models in the above studies. The first analytical model taking into account the petalling of the plate and a radial crack propagation process was proposed by Landkof and Goldsmith [6], where the solution was based on an energy balance and plastic hinge theory. They also performed a thorough experimental study. In a much more recent development, a detailed analysis of petal formation in thin plates by conical and spherical projectiles was presented by Atkins et al. [7], where the solution was correlated with tests. A closed form solution for the petalling failure mode of circular plates subjected to localized high intensity of impulsive loading was developed by Wierzbicki [8], where the total energy absorbed by the system and the number of petals as well as the final deformed shape of the plate was determined as a function of plate flow stress, thickness, and parameters of the external loading. The solution proposed by Wierzbicki [8] was constructed using the concept of the Crack Tip Opening Displacement (CTOD) and it captured the process of curling away the petals from the flat plate with the associated expenditure of bending energy.

The immediate objective of the present paper (Part II) is to determine the onset of fracture and subsequent propagation of radial cracks in a class of clamped thin plates with various radii and intensity of applied impulse. This paper is constructed in the following way. The accumulated equivalent plastic strain with stress triaxiality as a weighting function is introduced as ductile fracture criterion in Section 2. Moreover, the above fracture criterion is transformed to the space of principal tensile strains in a sheet under the assumption of plane stress plasticity. In Section 3, the calibration procedure to determine the true stress–strain curve and the critical damage parameters associated with the postulated ductile fracture criterion is presented. In Section 4, the closed form solution for the petalling mode proposed by Wierzbicki [8] is reviewed and the final expression for the normalized crack length presented. In Section 5, extensive numerical studies involving discing and petalling failure modes of thin circular plates are discussed. A special attention is paid to the numerical aspect of the selection of a unique number of radial cracks. The evolution of stress, strain, and resulting damage at the critical locations of fracture during crack propagation is presented. Furthermore, the numerically obtained extent of crack propagation is compared with analytical solution [8].

Section snippets

Representation in the space of the average stress triaxiality and the equivalent plastic strain to fracture $((σ_{m} / \bar{σ})_{av}, {\bar{ε}}_{f})$

It is well known that fracture in ductile metals occur through the nucleation and growth of micro-voids and their coalescence to form and propagate a micro-crack. In the present paper, the condition for fracture was assumed to be identical to the local ductile fracture criterion for uncracked bodies formulated and studied by Wierzbick et al. [9] and Bao and Wierzbicki [10], [11].

Similar fracture criteria were postulated earlier by others, see for example, Refs. [12], [13], [14], [15], [16]. It

Plasticity and strain hardening

In the numerical simulations and analytical solutions of thin plates under localized impulsive loading presented in Parts I [35] and II, high strength steel AH36 was used. Therefore, calibration procedure is presented for that material. Because the form of the weighting function was determined from previous study [10], it suffices to perform only one calibration test for fracture. Smooth round specimen with 6.35 mm diameter, shown in Fig. 4, was machined from a 25.4 mm thick AH36 steel plate. A

Petalling solution of impulsively loaded plate

The theory used in this section was developed by Wierzbicki [8]. Suppose that a system of n radial cracks is formed at a point in an infinite plate dividing it into n symmetric petals, Fig. 10. The central angle of the petal is denoted by $2 θ$ so that $θ = \frac{π}{n}$ and the instantaneous length of crack $l_{c}$ can be found from $l_{c} = a \cos θ,$ where a is the instantaneous radius of the radial crack. Note that $l_{c}$ is considered as the process parameter in the present study. The material is assumed to be rigid–plastic

Finite Element (FE) solution with fracture

Numerical results for the petalling failure mode of impulsively loaded, clamped thin circular plates are presented in this section. The considered clamping radius, thickness, and material of circular plate were R₀=5416 mm, h=16 mm, and AH36 steel, respectively. In the present numerical simulation, ductile fracture is assumed to be controlled by Eq. (4). When the equivalent strain weighted by stress triaxiality reaches the critical value at all of the integration points in an element, this element

Comparison and conclusions

The main points of present studies can be summarized as follows:

–
The equivalent plastic strain to fracture ${\bar{ε}}_{f}$ modified by the stress triaxiality $σ_{m} / \bar{σ}$ was introduced as a ductile fracture criterion. The present fracture criterion was calibrated by performing finite element calculation of uni-axial tensile test for specimen of the high strength structural AH36 steel. The key parameters, i.e. stress triaxiality, equivalent plastic strain, and critical damage value Dc, as well as accurate

Acknowledgements

The present work was sponsored by the MURI project through the Office of Naval Research. Thanks are due to Engineering System International and Altair Computing for their continuous support with the finite element programs of PAM-CRASH and mesh generator program of HyperMesh.

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Citation Excerpt :
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Fracture prediction of thin plates under localized impulsive loading. Part II: discing and petalling

Abstract

Introduction

Section snippets

Representation in the space of the average stress triaxiality and the equivalent plastic strain to fracture((σm/σ¯)av,ε¯f)

Plasticity and strain hardening

Petalling solution of impulsively loaded plate

Finite Element (FE) solution with fracture

Comparison and conclusions

Acknowledgements

Int J Solids Struct

Int J Impact Eng

Int J Mech Sci

J Mech Work Tech

Acta Metall

Int J Mech Sci

Int J Mech Sci

Int J Impact Eng

Eur J Mech A/Solids

Eur J Mech A/Solids

Eur J Mech A/Solids

Eng Fract Mech

Comput Methods Appl Mech Eng

Int J Impact Eng

The formation of enlargement of circular holes in thin plastic plates

Quat J Mech Appl Math

Mechanics of high speed projectile perforation

J Franklin Inst

Normal perforation of a thin plate by truncated projectiles

J Franklin Inst

An approximate theory of armor penetration

J Appl Phys

Hole flanging and punching of circular plates with conically headed cylindrical punches

J Strain Anal

Representation in the space of the average stress triaxiality and the equivalent plastic strain to fracture $((σ_{m} / \bar{σ})_{av}, {\bar{ε}}_{f})$