Meshfree analysis of dynamic fracture in thin-walled structures

ARTICLE IN PRESS Thin-Walled Structures 48 (2010) 215–222

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Meshfree analysis of dynamic fracture in thin-walled structures C. Gato Harbin Engineering University, College of Mechanical Engineering, Harbin, HLJ, PR China

a r t i c l e in fo

abstract

Article history: Received 11 September 2008 Received in revised form 13 October 2009 Accepted 26 October 2009 Available online 26 November 2009

Analysis and reliability assessment of fracturing thin-walled structures is important in engineering science. We focus on numerical analysis of dynamic fracture of thin-walled structures such as pipes and pressure vessels. Instead of using finite element method, we propose meshfree method that has advantages because its higher order continuity and smoothness and its opportunities to model fracture in a simple way. Therefore, connectivity between adjacent nodes are simply removed once fracture criterion is met. The main advantage of our meshfree method is its simplicity and robustness. & 2009 Elsevier Ltd. All rights reserved.

Keywords: Cylindrical shell Fracture Crack Dynamics EFG Thin-walled structure

1. Introduction The numerical analysis of fracture in thin-walled structures is still a challenge in applied mechanics that poses essential difficulties on the numerical method used [1,10–15,18,22, 26,27,31,35–37,40,42,43,46,47]. Thin-walled structures are used in many components such as pipes, vessels or sheets. Fracture of such structure can have different causes, ranging from static loading to internal pressure and gas explosions or impacts. Modeling of thin-walled structures is important in many engineering applications, e.g. the analysis of sheet metal forming, crash-worthiness test, civil structure design, pressure vessel liability, shipbuilding, defense technology, just to name a few. Especially the modeling of fracture in thin-walled structures imposes severe difficulties on existing numerical methods. Numerical analysis of thin-walled cylindrical structures can be classified into three categories: (1) Numerical analysis based on shell theories. (2) Degenerated continuum, or continuum based approach. (3) Direct three-dimensional (3-D) continuum approach. Among these three approaches, 3-D continuum direct approach is the simplest. The first two approaches lead to complicated theories and the development of particular constitutive models. 3-D continuum models cannot easily be adopted. When the 3-D continuum approach is used in the finite element framework, the E-mail address: [email protected] 0263-8231/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.tws.2009.10.011

method is cumbersome since multiple elements need to be used over the thickness of the thin shell to acquire reasonable gradient fields. This makes the method computationally expensive. Therefore, we use meshfree method instead of finite element method. Meshfree approximations have several advantages for modeling thin-walled structures. The two main advantages are:

 Their shape functions are smooth and higher order continuous



and therefore more accurate than finite element methods. Only a layer of 2–3 nodes over the thickness are required that makes the method computationally efficient. More importantly, the critical time step in meshfree methods based on nodal integration is significantly larger than for three-dimensional finite elements. Ref. [29] have shown that the critical time step is more than 100 times larger. We based the time step on nodal distances in circumferential direction and did not observe any instabilities with such a time step. Neither did we observe an influence when we decreased and increased the chosen time step by a factor of 2–3. Fracture can be incorporated in simple manner, i.e. by removing connectivities between meshfree nodes. Fracture in finite elements can be only realized by computational expensive remeshing procedures.

Most meshfree methods proposed so far have focused on a continuum-based approach. A meshfree thin shell formulation based on Kirchhoff–Love theory and element-free Galerkin (EFG) method [3] has been developed by [17] in the context of small strain, linear elastic framework. Ref. [41] extended this work with the consideration of finite strain, non-linear elastic material and

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focused on fracture. In the following work, [30] have simplified the treatment of cracks in thin shell by using an extrinsic basis. Ref. [8] noted the advantage of meshfree approximations in addressing shear locking in Mindlin type of beams and plates and have developed a meshfree formulation based on the reproducing kernel particle method (RKPM) [21]. This methodology is further extended by [16] with the use of EFG. EFG has been employed by [25] for shell and membrane structures in which bi-cubic and quartic basis functions are introduced in order to avoid shear and membrane locking. Ref. [48] showed that the Kirchhoff mode in the Mindlin plate can be reproduced using EFG or RKPM if secondorder polynomial basis is used in the moving least-squares approximation. By implementing this with a nodal integration and stabilization scheme, they have shown that the formulation is stable and free of shear locking. Ref. [49] developed a free mesh method in which the discrete Kirchhoff theory is combined with the mixed approach. In the case of three-dimensional continuum models, [20] have presented a formulation based on RKPM and have studied non-linear large deformation of thin shells. In this paper, we use three-dimensional meshfree continuum approach. Fracture of the shell is modeled by breaking links between particles once a certain fracture criterion is met. The main advantage of this approach is its simplicity and robustness while maintaining high accuracy at relatively low computational cost. Features such as discretization automatization and adaptivity can easily be added in meshfree methods [9,19,33]. Though—except of a simple fracture criterion—the ingredients of the method are not new, the combination of these ingredients are. The combination of these ingredients are choosen such that a robust and efficient method has been established that can handle complicated fracture patterns of arbitrarily shaped thin shells:

 To our best knowledge, it is the first time that the 3-D

  



continuum approach based on Lagrangian kernels is applied to fracture of thin shells. Ref. [39] have shown that the choice of the kernel is essential for fracture problems and that the commonly used Eulerian kernel leads to artificial fracture. The simple fracture criterion within 3-D approach applied to thin structures ensures a wide application range involving complicated fracture pattern. The meshfree formulation ensures optimal pre- and postprocessing since no mesh is required. Thus, complicated geometries can easily be created. Since meshfree methods—though based on Lagrangian description of motion—can handle large deformation, fluid structure interaction can easily be incorporated. We made already a first step in this direction by studying the fluid and the solid, separately. The coupled system will be studied in the future. The implementation of the method is done in Cþ þ. We did not choose a commercial software since changes of the method can be better implemented in our own software. Moreover, due to the long history of commercial software, the data structure is often not object-oriented and flexible enough.

The efficiency of the method is demonstrated through three examples. The paper is structured as follows: we first describe the numerical method and the constitutive model and fracture criterion. Then, we study three examples: quasi-static crack growth in order to validate our method, plate subjected to fast impact and detonation driven fracture of tube with different flaw sizes. These results are compared to experimental data. At the end, we conclude our paper and give future research directions.

2. Formulation and meshfree method The weak form of the linear momentum in the total Lagrangian description is given as

dW ¼ dWint  dWext þ dWkin ¼ 0

ð1Þ

with

dWint ¼

dWext ¼

Z

rX du : P dO0

O0

Z

du  t 0 dG0 þ

Z

G0t

dWkin ¼

Z O0

O0

R0 du  b dO0

R0 du  u€ dO0

ð2Þ

where b denotes the body force, R0 is initial density, u is displacement, P is the first Piola–Kirchhoff stress tensor, t 0 is the applied traction, rX denotes spatial derivatives with respect to material coordinate and superimposed dots denote material time derivatives. Above field equations are supplemented by boundary conditions: u ¼ u;

XA G0u

n0  P ¼ t0 ¼ t 0 ;

ð3Þ X A G0t S

ð4Þ T

with boundaries G0u G0t ¼ G0 and G0u G0t ¼ 0. Hereby, the index t refers to traction boundaries and the index u to displacement boundaries; n is the normal to the traction boundary. The meshfree approximation uh ðXÞ of a given function uðXÞ can be expressed as the product of the shape functions with nodal parameters uJ as in the finite element method: uh ðXÞ ¼

n X

NJ ðXÞuJ ¼ Nu

ð5Þ

J¼1

with n nodes. In the element-free Galerkin (EFG) meshfree method, the shape functions can be derived from moving leastsquare approximations [3] that result in the following shape functions: NT ðXÞ ¼ pT ðXÞA1 ðXÞPWðXÞ

ð6Þ

where is the moment matrix with moment matrix AðXÞ ¼ PðYÞWðXÞPT ðYÞ

ð7Þ T

is called the moment matrix and the matrix P ðYÞ contains the polynomial basis p that contains polynomials up to the order of two (i.e. quadratic polynomials). Note that linear polynomial completeness is required for convergence in Galerkin methods. The matrix WðXÞ ¼ diagfWI ðX  XI ; hÞVI g;

I ¼ 1; . . . ; n

ð8Þ

contains so-called kernel or weighting functions WI ðX  XI ; hÞ. The kernel function have compact support and the support size is determined by the dilation parameter h (e.g. the radius in circular supports). We used the quartic spline function that is commonly used in meshfree methods: ( 1  6s2 þ8s3  3s4 s r 1 ð9Þ WðX  XI ; hÞ ¼ wðsÞ ¼ 0 s41 with s ¼ ðX  XI Þ=2h for circular support size. Note that it is essential to express the kernel function in terms of material coordinates if material fracture is to be modeled appropriately [5]. More details of the formulation can be found in the meshfree literature, e.g. [3,21].

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The test and trial functions have the structure of Eq. (5). Introducing them into the weak formulation yields ( Z Z Z n n X X duI  rX NI ðXÞP dO0 þ NI ðXÞb dO0 þ NI ðXÞt 0 dG0 I¼1

J¼1

O0

O0

G0t



Z þ O0

where a is the back stress, set to zero in our studies. The thermo viscoplastic flow is governed by the following power law  m s e_ ¼ e_ 0 ð19Þ gðe ; TÞ with

R0 NI ðXÞNJ ðXÞu dO0 ¼ 0

ð10Þ

       e n T  T0 1 gðe ; TÞ ¼ s 1 þ 1  d exp

e0

After some algebraic operations, a matrix form of above equation can be derived: ext int MIJ u€ J ¼  f I þf I

Z O0

ext

fI

Z G0t

int

fI ¼

Z O0

NTI ðXÞt 0 dG0 þ

Z O0

NTI ðXÞb dO0

rX NTI ðXÞP dO0

ð13Þ

Softening in material due to temperature is accounted for by varying material parameters EðTÞ ¼ E0  1:6  106 ðT  T0 Þ  105 ðT  T1 ÞðPaÞ

While meshfree method is used for spatial integration, explicit finite difference scheme is used for integration in time. We chose explicit finite difference scheme since we are mainly interested in dynamic applications.

n ¼ v0 þ5  105 ðT  T0 Þ

3. Constitutive models and fracture criterion

aðTÞ ¼ ð2:2þ 0:0016½T  T0 Þ  105 ðK1 Þ

3.1. Visco-plastic model We use a J2 isotropic hardening viscoplastic model of the type !1=n gðep Þ ¼ s0 1þ

ep ep0

ð22Þ

where E and n are Young’s modulus and the Poisson ratio at temperature T. The material parameters will be defined later on. The constitutive update scheme for the thermoelasto-viscoplastic model largely follows the rate tangent modulus approach developed by [28]. 3.3. Fracture criterion

where 1=m is reference plastic strain rate and 1=m is strain rate sensitivity exponent. 3.2. Thermo-viscoplastic model In this section, we outline the constitutive relation of the thermo-elasto-viscoplastic solid adopted from [50] in order to evaluate the stress term in Eq. (1). The rate form of the constitutive equation reads as follows:

tr ¼ C : ðD  Dvp  aT_ IÞ

ð16Þ

where C is the first order elasticity tensor, D is the symmetric part of the velocity gradient L, tr ¼ t_  W  t  t  W is the Jaumann rate of the Kirchhoff stress where W is the antimetric part of the velocity gradient, a is the thermal expansion coefficient and I is the second-order identity matrix. The viscoplastic overstress model here is based on von Mises ! 3e_ Dvp ¼ s~ ð17Þ 2s with with s ¼ t  1=3 trðtÞI;

s0 ðTÞ ¼ s0  1:5  103 ðT  T0 Þ2 ðPaÞ

ð14Þ

where s0 is yield stress, ep and ep0 are the total and reference plastic strains, respectively and 1=n is hardening exponent. Rate dependent behavior is modeled with power law of type !1=m e_ p p seff ¼ gðe Þ 1þ p ð15Þ e_ 0

s~ ¼ s  a

ð20Þ

ð12Þ

ð11Þ

RNI ðXÞ NTJ ðXÞ dO0

¼

k

In Eqs. (19) and (20), e_ 0 is a reference strain rate, m is the rate sensitivity parameter, s0 is the yield stress, e0 ¼ s0 =E is the corresponding reference strain and E is Young’s modulus, n is the strain hardening exponent, T0 is a reference temperature and d and k are thermal softening parameters. The function gðe ; TÞ is the stress–strain relation measured at quasi-static strain rate of e_ at temperature T . The equivalent plastic strain e is defined as Z t Z t rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 vp e¼ e_ dt ¼ D : Dvp dt ð21Þ 3 0 0

with MIJ ¼

217

s ¼ 3=2s~ : s~

ð18Þ

Fracture is incorporated by a stress-based criterion. If the maximum principal tensile stress exceed 3 times the tensile strength s0 of the material, then the links between neighboring particles are broken. In other words, adjacent nodes are excluded from the sum in Eq. (5). This is a similar approach as in the visibility method [2,4], material point method [23,44] or the cracking particles method [32,34] but computationally more effective. More sophisticated models that will also include cohesive zone models will be studied in the future.

4. Results For better illustration of our figures, we plot only outer layer of nodes and use background mesh obtained by triangulation though we emphasize that is only done for graphical purpose. The boundaries can easily be identified due to our simple cracking criterion that removes connectivities. Therefore, we take advantage of algorithm described in [7]. 4.1. Quasi-static fracture In order to quantitatively validate our method, we consider quasi-static tearing problem as illustrated in Fig. 1. A 0.8 mm thick square plate with initial crack is subjected to out-of-plane tearing forces F. The plate is clamped at the edges parallel to the initial crack. The model of Section 3.1 is used with material

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employ visco-plastic model described in previous section. The material parameters for the shell are: Young’s modulus 71,000 MPa, Poisson ratio 0.3, density 2700 kg=m3 and yield strength 150 MPa. The material parameters of the sphere are: Young’s modulus 210,000 MPa, Poisson ratio 0.3, density 7800 kg=m3 and yield strength 1400 MPa. We refined the mesh in the plate and the sphere several times and used unstructured mesh. The coarsest mesh had 12,356 nodes for plate and the finest mesh had 63,234 nodes for plate. The sphere was discretized with 652 nodes for coarsest mesh and 2763 nodes for finest mesh. The displaced plate after perforation for coarse and fine mesh is shown in Fig. 3. The failure pattern and the displaced plate is very similar for all meshes. The diameter of the hole created by the impacting sphere is almost identical for the simulations we tested, i.e. from 12,356 nodes to 63,234 nodes (shell model). Also four petals occur in all simulations. The sizes of the petals are almost identical though their exact position is slightly different. We note that it is difficult to obtain the same number of petals in numerical simulations, see e.g. [45]. Petalling is often observed for thin structures under fast impact loading [30]. This shows that our method is able to model complicated failure pattern independently of mesh size.

F

203mm

40mm

F

203mm

Fig. 1. Quasi-static out-of-plane tearing problem.

1600 4.3. Detonation-driven fracture of cylindrical shells

1400

Total Load [N]

1200

Experiment 5634 nodes 34231 nodes

1000 800 600 400 200 0 0

10

20 30 40 50 Transverse Displacement [mm]

60

Fig. 2. Total load–transverse displacement curve for quasi-static out-of-plane tearing problem.

parameters: Young’s modulus 210 GPa, Poisson ratio 0.3 and yield strength 305 MPa according to the experimental tests by [24]. Different unstructured discretizations are tested starting from 5634 nodes. The finest discretization is 34,231 nodes. The total load versus the transverse displacement is shown in Fig. 2. The simulation predicts the experiment with reasonable accuracy and is independent of the mesh refinement.

We numerically study the experiments of detonation-driven fracture done by [6]. The test-setup consists of a detonation tube of 152 cm length to which a thin-walled aluminum tube is attached. The lengths of aluminium tube range from 45.7 to 89.6 cm. The inner tube radius is 1.975 cm and thickness of shell is 0.89 mm. While the lower end of the device is closed, a thin diaphragm seals up the other end. The aluminum tube contains notches of various lengths at midspan of the aluminium tube. The entire apparatus is filled with a combustible mixture of ethylene and oxygen. Initial pressure varies from 80 to 180 kPa. The mixture is thermally ignited at the closed end and the combustion transitions quickly to a detonation. When it enters the test specimen, the detonation is close to the Chapman–Jouguet (CJ) limit of quasi-stationary self-sustained propagation. Its velocity is between 2300 and 2400 m/s and the pressure values in the fully recreated CJ state range from 2.6 to 6.1 MPa (depending on the initial pressure). The detonation is modeled by applying pressure boundary conditions at the inner wall of the shell. Such pressure boundary condition is shown in Fig. 4. The cylindrical shells are modeled with up to 280,000 nodes. At least 140,000 nodes were needed until convergence of the solution was achieved. We considered experiments with 3 notch lengths: shot 1 with notch length 2.5 cm, shot 2 with notch length 5.1 cm and shot 3 with notch length 7.6 cm. The thermo-viscoplastic constitutive model explained earlier is used with material parameters according to [6]: Young’s modulus 70,000 MPa, Poisson ratio 0.3, density 2700 kg=m3 and yield strength 275 MPa. In the following, we discuss the results of the numerical analysis.

4.2. Impact problem We consider academic example of impact of spherical projectile onto a 10 mm thick square plate. The plate is clamped at all boundary and has dimension 1000  1000 mm. The sphere has a diameter of 40 mm and impact velocity is 300 m/s. Both, the sphere and the plate are modelled with the same constitutive model but the sphere is stiffer than the plate and not allowed to fracture. This is certainly a simplification but we were mainly interested in the fracture pattern of the thin plate. We

4.3.1. Shot 1 At first, the crack propagated in straight line before the crack closer to the detonation curved in an 45 degree angle. Then it continued to propagate in circumferential direction. The deformed shell from the experiment is shown in Fig. 5. This behavior is predicted well in the numerical simulation, Fig. 5. The cracks are arrested after propagating approximately 3/4 of the circumference.

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219

Fig. 3. Displaced configuration of impacted shell.

6

5

Pressure [MPa]

coarse fine 4

3

2

1

0 0

0.1

0.2

0.3

0.4

0.5

0.6

Fig. 5. Displaced configuration and effective stress of the detonation-driven fracture of cylinder; notch length is 1 in.

Time [ms] Fig. 4. Pressure boundary condition for detonation-driven fracture of cylindrical shells.

4.3.2. Shot 2 The results are presented in Fig. 6 that include also the tube after the experimental. The crack pattern was similar except that the farther crack (from the detonation) branched into two circumferential cracks that propagated the entire circumference until fracture. The simulation captures the basic failure pattern, i.e. short crack propagation and crack propagation in circumferential direction. However, the simulation does not predict the crack curving on the side closer to the detonation. Instead, the crack branches and propagates in two direction in circumferential direction as opposed to only one as observed in the experiment. Also the complete failure on the other side was not predicted by the simulation. We attribute these differences to

our simplified fracture model and the fact that fluid–structure interaction (FSI) is neglected. FSI can have significant influence on the pressure in the gas.

4.3.3. Shot 3 The results are presented in Figs. 7 and 8. In the experiment, the shell fractured into three large pieces. A middle segment was separated from its ends and contained several internal cracks. The simulation predicts this failure pattern. Cracks propagated straight from the notch before they branch into circumferential direction and some minor branches. These cracks propagated over the entire circumference and lead to the complete failure (on both sides) of the tube. This was also predicted in our numerical simulation. Displaced configurations of the simulation at different times are shown in Figs. 7 and 8.

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Fig. 6. Displaced configuration and effective stress of the detonation-driven fracture of cylinder at different times; notch length is 2 in.

Fig. 7. Displaced configuration and effective stress of the detonation-driven fracture of cylinder at different times; notch length is 3 in.

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simplified failure model and the pressure boundary conditions. Coupled analysis might improve the results. Therefore, our future research will focus on more advanced fracture models of the shell by e.g. using cohesive zone models and discrete cracks as proposed e.g. by [13,22,38,51]. We will also develop coupled analysis so that the pressure of detonation is transferred to the shell more accurately.

References

Fig. 8. Displaced configuration and effective stress of the detonation-driven fracture of cylinder at different times; notch length is 3 in. (a)–(b) Simulation and (c) experiment.

5. Conclusions We developed meshfree method for fracture of thin-walled shells. The method is characterized by its simplicity and robustness. Nevertheless, it can predict complicated dynamic failure of thin-walled structures. This was demonstrated for three examples:

 Quasi-static fracture.  Impact.  Detonation-driven fracture. The first example was the first test of our method in quasi-static setting. We demonstrated that fracture can be predicted accurately and independent from mesh size. The second example was to test our method and we showed that complicated fracture can be predicted independent of mesh. For all meshes, four petals developed and the size of impact-crater was the same. Experimental data of detonation-driven fracture of thin-walled shells was found in [6]. The detonation was modeled as pressure boundary conditions on the inner walls of the shell. We considered three different experiments with different notch length: shot 1 with short notch, shot 2 with long notch and shot 3 with very long notch. Our method predicted the failure of the shells with shots 1 and 3 accurately. The largest discrepancies occurred for shot 2. We attribute these discrepancies to our

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