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A simplified implementation of the conventional extended finite element method (XFEM) for dynamic fracture in thin shells is presented. Though this implementation uses the same linear combination of the conventional XFEM, it allows for considerable simplifications of the discontinuous displacement and velocity fields in shell finite elements. The proposed method is implemented for the discrete Kirchhoff triangular (DKT) shell element, which is one of the most popular shell elements in engineering analysis. Numerical examples for dynamic failure of shells under impulsive loads including implosion and explosion are presented to demonstrate the effectiveness and robustness of the method.

In this work, we describe a method for modeling fractured discrete Kirchhoff triangular (DKT) shell elements [

Even though numerous references are available for continuum shell elements, the literature on dynamic crack propagation in shells is quite limited. Cirak et al. [

Mehra and Chaturvedi [

The described implementation scheme is mainly based on the XFEM, but its actual implementation follows the phantom node method [

An elementwise progression of the crack is also employed; that is, the crack tip is always on an element edge. The elementwise crack propagation scheme may cause some noise during the crack propagation with coarse meshes. However, in Song et al. [

The main advantage of the DKT shell element is that a mesh can easily be generated from any kind of surfaces. The geometry of the element is described by three linear shape functions in the reference coordinates. The kinematic of the DKT shell elements is described by superimposing the membrane, the bending, and the rotational (drilling) behavior of shells with different corresponding degrees of freedoms (DOFs) as shown in Figure

Kinematic data of the DKT triangular element: (a) the corotational coordinates, (b) the bending degrees of freedom, (c) the in-plane membrane degrees of freedom, and (d) the drilling degrees of freedom.

However, for further explanation on the salient features of the DKT element, henceforth, we will use

Positive directions of

The discrete Kirchhoff assumption [

To verify the discrete Kirchhoff assumption, one has to add additional shape functions which do not change the nodal values of any field but only are allowed to modify the values on the midpoint. Thus, the rotational DOFs are discretized by

Geometry and local tangential-normal coordinate system of the DKT element.

The different functions

Representation of the crack by the XFEM and the phantom node method with DKT elements.

Based on the phantom node approach [

As with the standard approach to phantom nodes [

The discontinuous velocity fields in the midsurface of the fractured shell elements can be described by

In this work, Newmark scheme for the explicit time integration is used. The time integration procedure is written as

A diagonal mass matrix is frequently used in this explicit time integration scheme because it allows us to avoid a matrix inversion for solving (

However, this explicit integration scheme is conditionally stable, and the stability condition is defined in terms of a maximum time step

In the explicit dynamic analysis method, constructions of lumped mass are essential to ensure the computation of nodal accelerations without implicit solution procedures. However, the mass lumping scheme for cracked elements which employ the XFEM approach is not obvious. To circumvent such difficulties, several methods have been proposed: implicit (in cracked elements)-explicit (in continuum elements) time integration scheme [

In this study, the lumped mass for regular DOFs is diagonalized by the conventional row sum mass lumping technique, but, for the cracked elements, we used the mass lumping scheme that was proposed by Menouillard et al. [

A damage plasticity model that can account for the effects of stress triaxiality and Lode angle was proposed by Xue [

A critical strain based fracture criterion is used to determine the onset point of a poststrain localization behavior of a material, that is, fracture. When the strain at a crack tip material point reaches a fracture threshold, we inject a strong discontinuity ahead of the previous crack tip according to maximum principle tensile strain direction of an averaged strain

Schematic of averaging domain for the evaluation of the fracture criterion: (a) the size of averaging domain and (b) possible principal stress states through the shell depth.

In this study, a cohesive crack model is prescribed along the newly injected strong discontinuity surfaces until the crack opening is fully developed, that is, until cohesive traction has vanished. The roles of a prescribed cohesive model can be summarized as follows.

It can be a remedy to spurious mesh-dependent pathological behaviors by providing a bounded solution at the crack tip. For linear elastic fracture simulations, if the crack tip is not smoothly closed with cohesive forces, finite element solutions are unbounded at the crack tip due to the crack tip stress singularity and a crack path is determined by the surrounding mesh resolution. Also, for fracture in plastic bulk materials, the crack tip stress singularity can be slightly alleviated by plasticity. However, the finite element solutions still depend on the mesh resolution.

If the crack opening displacement is not governed by a cohesive model, the normal stress component to the crack surface suddenly drops to zero due to lack of fracture energy dissipations; note that injecting a strong discontinuity without prescribing cohesive force is the same as creating two free surfaces without dissipating new surface initiation energies. In this case, the total system suffers from an excessive accumulation of elastic energy and this excessively accumulated energy accelerates the crack propagation speed; more discussions on the relationship between crack propagation speed and dissipated fracture energy can be found in Rabczuk et al. [

The cylinder is of length

Setup for implosion induced failure of cylinder.

In this example, we focused on predicting final fracture pattern of two experiments. The first specimen is denoted by IMP26 experiment, where

For numerical analysis, we modeled IMP26 and IMP25 specimens with 49200 and 24000 shell elements, respectively. The average element size is about 1 mm. The material behavior of AL6061-T6 is modeled with the damage plasticity [^{3}, Poisson ratio

The following coordinate system was used: the

A bilinear load curve for the pressure was used in both simulations. The pressure started at zero and was increased to

Loading curve used in the simulation of Texas experiments.

Geometry imperfection is introduced into the radius to evoke circumferential buckling easily. The actual radius with imperfection has the form

Figures

Snapshots of numerical results of IMP26 experiment: (a) initial configuration and the mesh, (b) deformed configuration at time

In IMP25 simulation, we allowed an injection of the discontinuity near the interface of the main part and the extension part of the specimen. We also observed large plastic strain, large damaged or unstable material points along the central buckling lines. However, this may be due to the repulsive forces generated during the contact, so no crack was allowed to initiate in these regions.

Several imperfection magnitudes

Figure

Snapshots of numerical results of IMP25 experiment: (a) deformed configuration at time

The final configurations of different imperfection magnitudes are compared in Figure

Comparison of final configurations of cylinders with different imperfection magnitudes: (a) 0.05% imperfection, (b) 0.1% imperfection, (c) 0.5% imperfection, and (d) 1.0% imperfection.

Comparisons of time history of nodal velocity at the center node with different amplitudes of imperfection.

Chao [

Preflawed cylinder with gaseous detonation loading.

The pressure wave was initiated at the source point and then passed the specimen and the extension tube, causing the original surface notch to form a crack cutting through the cylinder wall and propagate.

Chao [

Experimental results of Shepherd experiment [

The specimen is modeled with 40680 shell elements. The left and the right ends of the numerical models were fully clamped in the simulation. The following fitted exponential-decay curve [^{3}, Poisson ratio

The configurations at different times of numerical results are shown in Figures

Numerical results of Shepherd experiment at different times: deformed configurations with effective stress contour plots at (a) time

Final configurations of Shepherd experiment. (a) Top view. (b) Side view.

The final fracture patterns are shown in Figure

We described a new finite element method for prediction of dynamic fractures in thin shells. The method is incorporated within an explicit time integration scheme and able to represent the crack paths free from initial mesh topologies. For the representation of discontinuities due to cracks, the described method employs a simplified version of the conventional XFEM based on the phantom node method. In this approach, the cracked shell element is treated by two superimposed elements with newly added phantom nodes on the cracked portions.

The method is implemented for the DKT shell element. This facilitates the implementation of the method into standard finite element programs. Another attractive feature of the method is that it provides an easy mesh generation and a relatively low computational cost and this allows large scale nonlinear dynamic fracture problems to be solved efficiently.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The support of the Office of Naval Research under Grants nos. N00014-13-1-0386 and N00014-11-1-0925 is gratefully acknowledged.