Smoothed Particle Hydrodynamics for the Linear and

2nd Reading April 10, 2015 9:9 WSPC-255-IJAM S1758-8251 1550032

International Journal of Applied Mechanics Vol. 7, No. 2 (2015) 1550032 (40 pages) c Imperial College Press DOI: 10.1142/S1758825115500325

Smoothed Particle Hydrodynamics for the Linear and Nonlinear Analyses of Elastoplastic Damage and Fracture of Shell

F. R. Ming∗ , A. M. Zhang† and S. P. Wang‡ College of Shipbuilding Engineering Harbin Engineering University Harbin 150001, P. R. China ∗[email protected] †[email protected] ‡[email protected] Received 20 August 2014 Revised 10 November 2014 Accepted 27 November 2014 Published 14 April 2015

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It is a troublesome and focused problem of solid mechanics to solve shell structures with Smoothed particle hydrodynamics (SPH), which is a fully meshfree method. In this paper, an integral model of SPH shell is proposed to more accurately capture the nonlinear strain along the thickness direction. Though the idea is similar to the “Gaussian integral point” in Finite element method (FEM), it is absent and just the first presentation in SPH. Furthermore, focusing on the metal materials, a high-efficiency iteration algorithm for plasticity is derived and the plastic damage theory of Lemaitre–Chaboche is also introduced based on the studies of Caleyron et al. (2011). As for the dynamic fracture of SPH shell, the multiple line segments algorithm is proposed to treat crack adaptively, which overcomes the mesh dependency occurring in mesh method. These algorithms and theories are successfully applied in the integral model of SPH shell of elasticity, plastic damage and dynamic fracture. Finally, the linear and nonlinear analyses of geometry and material are carried out with FEM, the global model and the integral model of SPH shell to prove the feasibility and the accuracy of the integral model. Keywords: SPH shell; global and integral model; linearity and nonlinearity; elastoplasticity; damage; fracture.

1. Introduction The deformation, crack and fracture of shell structures always come out in the production, transportation, assembly and other processes. The studies on these classic subjects are very common [Belytschko and Tabbara, 1996; Bordas et al., 2008; Dai et al., 2013; Li et al., 2000; Liu and Swaddiwudhipong, 1997; Miao et al., 2012; Moës et al., 1999; Rabczuk and Zi, 2007; Unosson et al., 2006; Zenkour, 2013]. When the structure is of small deformation, the strain and the displacement are ‡ Corresponding

author. 1550032-1


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F. R. Ming, A. M. Zhang & S. P. Wang

always regarded as in a linear relation, while with the increase of deformation, the nonlinearity between them becomes more and more obvious, which is the so-called geometric linearity and nonlinearity; in similar, the strain and the stress are in a linear relation in a certain range but it will become nonlinear when the material yields, which is the so-called material linearity and nonlinearity. The linear analysis is relatively simple but it is only an approximation; while the nonlinear analysis is always complicated, including the geometric nonlinearity, the material nonlinearity and the boundary nonlinearity, where the relevant parameters vary nonlinearly but it is closer to the real structural responses. The boundary nonlinearity is not in the scope of the present paper and will be delivered in the subsequent study. In the existing numerical algorithms of solving shell structures, the mesh algorithm is well developed [Belytschko et al., 2000b; Chapelle and Bathe, 2003], but there are also many challenges of dealing with nonlinear problems due to the dependence of mesh [Belytschko et al., 1984; Chapelle and Bathe, 1998; Cho et al., 1998; Wang and Chen, 2004]. Though the meshfree methods spring up in recent years, many scholars agree they have great advantages of coping with nonlinear problems because of the meshfree property [Bordas et al., 2008; Bui and Nguyen, 2011; Caleyron et al., 2012; Cui et al., 2010; Krysl and Belytschko, 1996; Maurel and Combescure, 2008; Miao et al., 2012; Noguchi et al., 2000; Rabczuk and Zi, 2007]. As the oldest meshfree method [Gingold and Monaghan, 1977], SPH is always not competent to deal with solid mechanics problems especially for thin shell structures due to the drawbacks of stability and accuracy. In recent year, the SPH shell model was firstly proposed by the scholar Combescure et al. [Maurel, 2008], who initialized the simulations of thin shell structures with SPH method. Thereafter, the numerical model is applied by Ming et al. [2013a, 2013b]. It has been proved that both the static and dynamic analyses of SPH shell have a high accuracy and a good stability. The SPH shell has overcome two problems: the accuracy and the stability. The main treatments include the approximation function of high-order consistency [Dilts, 1999; Randles and Libersky, 2000], the Lagrangian framework [Belytschko et al., 2000a; Rabczuk, et al., 2004], the stress points [Dyka et al., 1997; Randles and Libersky, 2000], and the dissipative stabilization [Gray et al., 2001; Maurel et al., 2006; Monaghan and Gingold, 1983; Randles and Libersky, 1996], which can be detailed as: • The approximation function of high-order consistency: it contains the moving least square function (MLS) in Moving least square SPH (MLSPH) [Dilts, 1999] and the normalized kernel function in Normalized SPH (NSPH) [Randles and Libersky, 2000], which can guarantee the consistency of the approximation function strictly and thus overcome the boundary defects of the traditional kernel function. They improve the numerical accuracy greatly and benefit the stability. • The Lagrangian framework: it is proposed by Belytschko et al. [2000a], Rabczuk et al. [2004] containing the Lagrangian approximation function [Belytschko et al., 1550032-2


Smoothed Particle Hydrodynamics for the Linear and Nonlinear Analyses

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2000b] and the total Lagrangian equation [Rabczuk et al., 2004]. The core idea of them is to solve the equilibrium equations of SPH shell in the material coordinates and therefore overcome the instability from the Euler kernel function and the updated Lagrangian equation. • The stress points: they are a series of particles distributed among the SPH particles like the Gauss points in FEM. The method was firstly put forward by Dyka et al. [1997] with the purpose of improving the tensile instability while Belytschko et al. [2000a] insisted the employ of stress points improves the instability from the deficient matrices but not the tensile instability. • The dissipative stabilization: it contains the conservative smoothing [Randles and Libersky, 1996], the artificial stress [Gray et al., 2001], the artificial viscosity [Maurel et al., 2006; Monaghan and Gingold, 1983] and so on. They stabilize the program at the cost of energy dissipations and consequently some errors will bring about, so they should be applied appropriately. Though many literatures [Liu and Yang, 2012] have clarified that the stress instability in SPH cannot be eliminated completely, the combinations of the above treatments are able to suppress and weaken the instability significantly and improve the numerical accuracy greatly. However, the SPH shell raised by Maurel [2008] called “global model” here has removed the nonlinear strain due to the expensive computational costs during the simplification and the strain is only simplified to be constant strain and linear strain in the thickness direction, which is barely enough to cope with the linear problems especially for the thick shells. When dealing with nonlinear problems like plastic large-deformation, this treatment adopts the resultant stress algorithm for plasticity which has a poor accuracy especially for the thick shells, thus it cannot obtain the progressive development of plasticity through the thickness. Though the computation costs of the resultant stress algorithm are attractive, when dealing with the thickness shells or severe nonlinear problems, the accuracy is not acceptable with our tries. Based on the researches of Combescure et al. [Caleyron, 2011; Maurel, 2008] and the previous results [Ming et al., 2013a, 2013b], firstly, an integral model will be put forward in the incremental form. In the integral model, some integral points are arranged along the shell thickness direction at the stress point to capture the variations of strain and stress and then the generalized force is obtained by the integration at these integral points. Though the arrangement of integral points in the thickness direction is similar to the “Gauss integral points” in FEM, it is unique in SPH shell. Afterwards, as for ductile metal, an iteratively plasticity algorithm featured by fast-convergence will be derived and applied in the SPH integral model; moreover, the multiple line segments algorithm is proposed to deal with the formation and propagation of crack adaptively, which is firstly presented in SPH shell but shows convenience and robustness. Finally, the analyses of linear and nonlinear problems involving the elasticity, plasticity, plastic damage and dynamic fracture are carried out. 1550032-3



2. Theoretical Background 2.1. Integral model of SPH shell

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The simplifying process of SPH shell includes the definition of configurations, the description of shell motion, the evaluations of strain and stress and the discretization in space and in time [Caleyron, 2011; Maurel, 2008; Ming et al., 2013a, 2013b]. The SPH shell is established in the undeformed configuration. The Green strain is adopted and then transformed to Almansi strain to apply the constitutive relation, and finally the total Lagrangian equation, where the first Piola–Kirchhoff stress is used, is solved to describe the shell responses. In the analysis of nonlinear problems, the incremental theory is always adopted. The model established by Combescure et al. [Caleyron, 2011] has removed the variation of nonlinear strain along the thickness and the stress in the cross-section of shell is obtained by the separation of constant strain and linear strain; the stress point is global and viewed as an independent integral point, so the simplification is called “global model” in our senses. Compared with the global model, the present study will take the nonlinear strain along the shell thickness into account: at a stress point, some integral points are distributed in the thickness direction to obtain the stress in the cross-section of shell, while the SPH particle is still regarded as an independent integral point; afterwards, the integration is performed to gain the resultant stress, and thus the “integral model” is named. In this way, a fresh phenomenon takes place, namely, there are three series of points in the model of SPH shell, as shown in Fig. 1. Only the improvements of the integral model compared with the global model will be discussed in this section. If noting the displacement of a point in the cross-section of shell as u = (u, v, w)T , the displacement in an incremental form in the local configuration of shell can be

SPH SPH SPH SPH SPH SPH SPH SP SP SP SP SP SP SPH SPH SPH SPH SPH SPH SPH SP SP SP SP SP SP SPH SPH SPH SPH SPH SPH SPH SP SP SP SP SP SP SPH SPH SPH SPH SPH SPH SPH SP SP SP SP SP SP SPH SPH SPH SPH SPH SPH SPH SP SP SP SP SP SP

n SP 5 4 3 2 1

Fig. 1. The distribution of SPH particles, stress points and integral points; the symbols: “SPH” indicates SPH particles, “SP” denote stress points and “×” represents integral points.

1550032-4



expressed as:

 δuip = δum − ξip δϕx ,     δvip = δvm − ξip δϕy ,     δwip = δwm ,

d d ∈ − , , 2 2

ξip

(2.1)

where the subscript “ip” and “m” represent the number of integral points and the quantity of shell’s mid-plane, respectively; d is the shell thickness; ϕx and ϕy indicate the rotational angles of the shell’s pseudo-normal with the positive directions of axis x and axis y rotating 90◦ to axis z. The geometric deformation of SPH shell model is depicted in Green stain and then transformed to be Almansi stain to apply the constitutive relation. Therefore, the increment of Green strain E at an integral point is: δEip = 0.5[∇0 δu + (∇0 δu)T + (∇0 u)T ∇0 δu + (∇0 δu)T ∇0 u + (∇0 δu)T (∇0 δu)]. (2.2) The first two terms on the right of the above equation are of small strain corresponding to the Cauchy strain; the subscript “0” indicates the quantities versus the undeformed configuration; moreover,     ∂um ∂uip ∂uip ∂uip ∂ϕx ∂um ∂ϕx − ξ − ξ −ϕ ip ip x   ∂x ∂y0 ∂z0  ∂x0 ∂y0 ∂y0    ∂x0  0        ∂vm  ∂vip ∂v ∂v ∂ϕ ∂v ∂ϕ ip ip y m y    = ∇0 uip =  − ξip − ξip −ϕy  ,  ∂y0 ∂z0   ∂x0 ∂x0 ∂y0 ∂y0   ∂x0       ∂wip ∂wip ∂wip   ∂wm ∂wm 0 ∂x0 ∂y0 ∂z0 ∂x0 ∂y0

by



∇0 δuip

∂δuip  ∂x 0    ∂δvip =  ∂x 0    ∂δwip ∂x0 

∂δuip ∂y0 ∂δvip ∂y0 ∂δwip ∂y0

∂δum ∂δϕx  ∂x − ξip ∂x 0 0    ∂δvm ∂δϕy =  ∂x − ξip ∂x 0 0    ∂δwm ∂x0

(2.3)



∂δuip ∂z0    ∂δvip   ∂z0    ∂δwip  ∂z0 ∂δum ∂δϕx − ξip ∂y0 ∂y0 ∂δvm ∂δϕy − ξip ∂y0 ∂y0 ∂δwm ∂y0 1550032-5

 −δϕx     −δϕy  .    0

(2.4)



The transformation matrices between the undeformed local-configuration, the deformed local-configuration and the global configuration are assumed to be κ0 and κ, and thus the Almansi strain is obtained by: δεip = κ · F−T · κT0 · δEip · κ0 · F−1 · κT

(2.5)

where F is the gradient matrix signifying the extent of deformation; the details of κ and F may refer to the literature of Ming et al. [2013a]. Applying the constitutive equation, the increment of the Cauchy stress is expressed as: δσ ip = f (δεip ).

(2.6)

In the incremental theory, the stress at the time step n is known as σ nip , and at the time step n + 1 is updated as: hence the stress σ n+1 ip = σ nip + δσ ip . σ n+1 ip

(2.7)

can be modified appropriately by judging the stress Therefore, the stress σ n+1 ip state of the integral point, including the elasticity, the plasticity, plastic damage or dynamic fracture (i.e., Secs. 2.2–2.4). Now that the generalized in-plane force T and the moment M per length unit will be drawn through the integration algorithm:

d/2 np σ ip (ξip )dξ = σ ip (ξip )Aip , (2.8) T= −d/2

d/2

M= −d/2

by

ip=1

σ ip (ξip )ξip dξ =

np

σ ip (ξip )ξip Aip ,

(2.9)

ip=1

where Aip is the weight of the integral point; np is the total number of integral points. The Gauss integration and Simpson integration are commonly available for the integration algorithm. Compared with the Simpson integration, the Gauss integration has a higher accuracy with the same integral points and thus the default Gauss integration of 5 points is employed. According to the equilibrium of force and moment for an infinitesimal in the local configuration, the differential equations are obtained as:   ∂Txx Txy ∂2u   + = ρd 2 ,    ∂x ∂y ∂t     ∂Tyx Tyy ∂2v (2.10) + = ρd 2 ,  ∂x ∂y ∂t       ∂Txz Tyz ∂2w   + = ρd 2 ,  ∂x ∂y ∂t   ∂Mxx ∂Mxy ∂ 2 θx     ∂x + ∂y − Txz = I ∂t2 , (2.11)   ∂Myx ∂Myy ∂ 2 θy    ∂x + ∂y − Tyz = I ∂t2 , 1550032-6



where I represents inertia  Txx Txy  N= Tyx Tyy Tzx Tzy

effects. If one notes:   Mxx Mxy 0    0 , M = Myx Myy 0 0 0

0



 0 , 0

  −Txz    S= −Tyz  0

(2.12)

thus the total Lagrangian equation expressed by nominal stress in global configuration can be drawn as: ¨ ∇ · Ng = ρdU,

(2.13)

¨ ∇ · Mg + Sg = Ig Θ,

(2.14)

where the subscript “g” indicates expressions in global configuration; besides, Ng = JF−1 κT Nκ, Mg = JF−1 κT Mκ, Sg = JκT S

(2.15)

moreover, the Jacobi J = |F|. Conducting the particle approximation of the total Lagrangian equations, one gets the discretized form as follows: ¨ i, [Nj − Ni ]g ∇0 Φij = ρdU (2.16) j

¨ i. [Mj − Mi ]g ∇0 Φij + (Si )g = Ig Θ

(2.17)

by

j

The subscript “0” denotes quantities in initial configuration; the moving least square function is selected as the approximation function Φ, which satisfies the consistency of high-order and thus the accuracy and stability are guaranteed. The approximation function is calculated once and for all due to the employ of total Lagrangian framework. Consequently, the translational and angular acceleration can be solved with the differential equilibrium equation and then the position and the pseudo-normal of the shell can be updated; the details may refer to the literatures [Ming et al., 2013a, 2013b]. The integral model could capture the variation of strain and stress more accurately along the thickness direction, thus the accuracy is improved. Combined with the total Lagrangian equation in undeformed configuration, MLS function of high-order completeness, employment of stress points and other stabilized treatments, the stability is also guaranteed. 2.2. Plastic algorithm of SPH shell When the stress at a point of SPH shell excesses the elastic limit, as shown in Fig. 2, the material reaches the inelastic phase, where the variation of stress is no longer linear with the strain. The employ of plasticity algorithms depends on the simplifying process of SPH shell. In the global model, the nonlinear strain along the shell thickness is removed, and thus the whole section of shell will go to plasticity 1550032-7



σ D=0

σs

arctan Ep

Dσ

σ s0

D = Dc

o

εe

ε

s p

ε pc

ε

by

Fig. 2. The relationship of stain and stress for a material experiencing the process of elasticity, plasticity and plastic damages.

at the same time. In general, the resultant stress algorithm for plasticity is used to evaluate the stress in the cross-section, as described in the literatures [Crisfield, 1974; Zeng et al., 2001], which is collectively called the “global model” with the simplifying process in this paper. When solving the nonlinear problems with the global algorithm, especially when the iteration is employed in the modification of stress back to the yield surface, the efficiency is significantly improved. However, the accuracy looks bleak when the shell is thick. The integral model featured by a high accuracy will be raised in this section. The integral model can truly reflect the development of plasticity in the shell thickness direction, though the computation costs increase due to the integral points positioned to perform the integration. It is necessary to seek a reasonable and efficient iteration algorithm with a high accuracy to deal with nonlinear problems. Some plasticity theories have been put forward so far, e.g., Von Mises, Tresca and others. In this paper, the material is of ductile metals established in Von Mises theory and assumed to be isotropic hardening. The transverse shear stress does not take effects on the plasticity. Based on the researches on the global model of Maurel [2008]; Caleyron et al. [2012], the integral model proposed in this paper will be detailed. The yield function of an integral point at the stress point will be presented firstly to judge whether the point yields or not. At the time step n + 1, the yield function is expressed as: f =σ ¯ n+1 − σsn+1 = 0,

(2.18)

where σ ¯ is the equivalent stress and σs is the yield stress. In order to solve conveniently, the yield function is employed as follows: σ n+1 + σsn+1 ) = (¯ σ n+1 )2 − (σsn+1 )2 = 0. f = (¯ σ n+1 − σsn+1 )(¯

(2.19)

According to the Von Mises theory, if the independent components of plane stress are noted as: σ = (σx , σy , σxy )T 1550032-8

(2.20)



the equivalent stress σ ¯ n+1 is [Zeng et al., 2001]: (¯ σ n+1 )2 = (σ n+1 )T Ξ (σ n+1 ),   1 −0.5 0   Ξ= 1 0 . −0.5 0 0 3

(2.21)

(2.22)

If f ≥ 0, the material yields. At the moment, as for the hardening material, the yield surface will change and the stress has to be modified back to the yield surface. The modification of stress of an integral point at the stress point will be detailed as follows. In similar to the global model [Caleyron, 2011; Crisfield, 1974; Maurel and Combescure, 2008; Zeng et al., 2001], there are several steps including the prediction of elasticity, the judgment of stress state (yield or not), the stress modification back to the yield surface and the iteration of Newton–Raphson. (a) The prediction of elasticity: At the beginning of the time step n+1, the stress is predicated as an elastic material and the estimated stress is: n+1 = σ n + Γ∆ε, (2.23) σ where indicates an estimated value; moreover,   1 υ 0  E  υ 1 0  Γ=  . 1 − υ2  1 − υ 0 0 2

(2.24)

(b) The judgment of stress state:

by

To judge whether the material yields or not with Eq. (2.19), if f < 0, the material has not yielded and thus σ n+1 = σ n+1 ; otherwise, the material yields and the stress should be modified as follows. (c) The stress modification back to the yield surface: Owing to only the elastic increment ∆εe contributing to the stress increment ∆σ, the increment of plastic strain ∆εp will act as the permanent deformation, so one obtains: ∆σ = Γ∆εe = Γ(∆ε − ∆εp ).

(2.25)

According to the flow rule of plasticity, the increment of plastic strain∆εp is:  ∆εp = ∂f ∆λ, f ≥ 0, ∂σ (2.26)  ∆εp = 0, f < 0, ∂f where ∂σ is the flow direction of plasticity; ∆λ is a nonnegative coefficient signifying the size of the increment of plastic strain; regarding the definition of the equivalent

1550032-9



plastic strain and the equivalent stress in Von Mises theory, the increment of the equivalent plastic strain ∆¯ εp is: 2 2 2 ∂f 2 ∆¯ εp = (∆εp )ij = (∆λ)2 3 3 ∂σij 2 2 2 2 = Sij (∆λ)2 = σs2 (∆λ)2 . (2.27) 3 3 Therefore, ∆λ =

3∆¯ εp . 2σs

(2.28)

According to Eq. (2.19), regarding the symmetry of Ξ, one gets: ∂f = 2Ξ σ ∂σ substituting Eqs. (2.28), (2.29) into Eq. (2.26), so we get: ∆εp =

3∆¯ εp Ξσ σs

(2.29)

(2.30)

considering the Eqs. (2.23) and (2.25), the total stress is obtained: σ n+1 = σ n + ∆σ = σ n+1 −

3∆¯ εp ΓΞσ n+1 . σsn+1

(2.31)

Therefore the modified stress is:

σ n+1 = L−1 σ n+1

(2.32)

herein,

by

L=I+

3∆¯ εp n+1 ΓΞ. σs

(2.33)

With regard to the hardening of the material, the yield stress should be duly updated. Assuming the strain–stress curve from the uniaxial tensile experiments is known and the material is isotropic hardening, the yield stress can be noted as: εp , σsn+1 = σsn + Ep ∆¯

(2.34)

where Ep is the plastic modulus drawn from the following equation: Ep =

EEt E − Et

(2.35)

herein, Et is the tangent modulus, i.e., the slope of the curve of strain–stress. Therefore, substituting Eq. (2.32) and Eq. (2.34) into Eq. (2.19), we get: fn+1 = σ n+1 T L−T ΞL−1 σ n+1 − (σsn+1 )2 = 0.

(2.36)

There is only one unknown parameter ∆¯ εp in the above equation. After solving it, the new stress will be obtained from Eqs. (2.32) and (2.34). 1550032-10



(d) The iteration of Newton–Raphson: The Newton–Raphson iteration is adopted to solve the above equation in present paper. The iteration form is: = ∆¯ εm ∆¯ εm+1 p p −

fn+1 , fn+1

(2.37)

where “m” is the number of iterations; the derivative of Eq. (2.36) is obtained by: fn+1 = σ n+1 T (L−T ΞL−1 ) σ n+1 − 2σsn+1 Ep = 0.

(2.38)

The detailed form is given in Appendix A. When the iteration is done with Newton–Raphson algorithm, an initial value is requested. It is very necessary to the convergence and the iteration efficiency [Caleyron, 2011; Maurel and Combescure, 2008]. The closer the initial value is to the true value, the easier and the faster the convergence is. The radial return method is always used to get the initial value, but in this paper a more effective solution is presented as: ∆¯ εp =

¯ σ n+1 − σsn , + Ep

(2.39)

by

σ n+1 T Ξσ n+1 , = σ n+1 T KT Ξσ n+1 /¯ σ n+1 , K = where ¯ σ n+1 = n+1 . 3ΓΞ/¯ σ Doing the Newton–Raphson iteration with the above initial value, 1 ∼ 3 iterations are enough to get a precision solution. The occurring times of different iterations among 500 steps selected randomly are listed in Table 1. There are more than 95% of time steps doing one iteration, while less than 1% of time steps doing more than three iterations. The algorithm is featured by fast convergence. The details of the initial value are given in Appendix B. 2.3. Damage algorithm of SPH shell The plasticity of the metal is caused by deviatoric stress while the damage results from the hydrostatic stress or the stress triaxiality. A damaged metal experiences the nucleation, the growth and the connection of cavities and cracks from the microscopic defects to the macroscopic fracture. The paper mainly targets at the plastic damage of ductile metal and the combined Lemaitre–Chaboche damage theory [Lemaitre and Chaboche, 1990] and the Von Mises plastic theory are applied in the study. The procedures are similar to those carried out by Caleyron. As for a metal Table 1.

Occurring times of different iterations among 100 time steps.

Iterations

One

Two

Three

More than three

Occurring times

491

6

2

1

1550032-11



in the condition of stress triaxiality, the damage criterion is given as 2 2 (1 + υ) + 3(1 − 2υ)σtri ε¯p − ε¯sp ≤ 0, f= 3

(2.40)

where ε¯sp is the threshold equivalent plastic strain; σtri = σH /¯ σ is the stress triaxiality; σH is the hydrostatic stress. In the continuum mechanics, the damage variables are used to represent the continuous variation of the damage. The variable D is selected to evaluate the state of damage, and 0 ≤ D ≤ 1. Therefore, the evolution of damage is given by [Caleyron, 2011]: 2 Dc 2 (1 + υ) + 3(1 − 2υ)σtri ∆¯ εp , (2.41) ∆D = c ε¯p − ε¯sp 3 where Dc indicates the critical damage and ε¯cp denotes the critical equivalent plastic strain. The plastic damage theory is always combined with the Von Mises theory or Johnson–Cook plasticity model [Johnson and Cook, 1983]. In this paper, the Von Mises theory is chose. The yield function taking the coupling of plasticity and damage into account is given by: σ ¯ − σs = 0. 1−D

(2.42)

σ ¯2 − σs2 = 0. (1 − D)2

(2.43)

f= A more general form is: f=

According to the flow rule of plasticity, considering the plastic damage, the effective stress is:

by

σ = (1 − D)Γεe and thus the increment of stress is expressed as: σ σ = (1 − D)Γ(∆ε − ∆εp ) − ∆D . ∆σ = (1 − D)Γ∆εe − ∆D 1−D 1−D

(2.44)

(2.45)

The total stress in a new step n + 1 is updated as: σ n+1 = σ n + ∆σ = σ n+1 − (1 − D)

3∆¯ εp σ n+1 n+1 . ΓΞσ − ∆D 1−D σyn+1

(2.46)

That is: σ n+1 = L−1 σ n+1 , where: L = (1 − D)

∆D 3∆¯ εp I. ΓΞ + 1 + 1−D σsn+1 1550032-12

(2.47)

(2.48)



Substituting the above equation into Eq. (2.43), the yield function of the coupling of plasticity and damage is obtained: fn+1 =

σ n+1 T L−T ΞL−1 σ n+1 − (σsn+1 )2 = 0. (1 − D)2

(2.49)

At the moment, ∆¯ εp is the only unknown parameter. The Newton–Raphson iteration is also employed to solve the above equation. The derivative of Eq. (2.49) can be easily drawn as: fn+1 =

σ n+1 T (L−T ΞL−1 ) σ n+1 − 2σsn+1 Ep = 0. (1 − D)2

(2.50)

2.4. Dynamic fracture algorithm of SPH shell

by

In the model of SPH shell, the macroscopic crack can be represented by microscopic damage particles. The crack of SPH shell is not explicitly defined but forms and propagates dynamically and adaptively, especially advantage of treating multiple cracks due to the meshfree property. In the following, the adaptive treatments of cracks with multiple line segments will be presented, which is performed in the initial configuration. As shown in Fig. 3, the crack is constructed by means of the topology consisting of SPH particles and stress points, whereas it must follow the principles: (1) If all the integral points at a stress point has achieved the critical damage, the stress point will be marked as a damaged point and removed from the topology, which means keeping it from interacting with other particles, but it will be added to the crack as an endpoint of line segments; the SPH particles around the stress point will be regarded as free edge and the conditions of free edge will be applied. (2) If all the stress points around an SPH particle have been damaged, the SPH particle will be marked as a fragment moving with the inertia at break; the treatment will retain the opportunity of interacting with other particles and make the mass and energy conserved. (3) The interaction of particles on the opposite side of the crack joined by damaged stress points will be suppressed, which can be carried out by the visibility method [Belytschko and Tabbara, 1996]. In detail, the point to be approximated will be regarded as a light source and the crack is opaque, therefore, only the lighted particles in the support domain will be used to make the approximation, namely, the effective particles for approximation. The above principles should be checked and satisfied at every step once the crack occurs. When there are cracks in the support domain, the visibility method will be applied. The procedures will be arranged as: (1) At the initial time, the stress points in the support domain of each particle (including SPH particle and stress point) will be searched and stored. 1550032-13



SPH

SP

DSP

R-SPH

by

FB

Fig. 3. The formation and propagation of cracks, where SPH indicates SPH particles, R-SPH denotes relaxed particles which will form fragments; SP represents stress points; DSP signifies damaged stress points; FB indicates free edge; the shaded part of the support domain is used to make approximation.

(2) Once the crack occurs, the damaged stress points at every time step will be recorded and added to the topology of cracks according to the nearest distance criterion, then the new line segment for the crack will be numbered, as the right column of Fig. 3 plotted, namely, the macroscopic crack expands; this is critical to the subsequent search of neighboring particles. (3) When determining the neighboring particles, firstly check the damaged stress points in the support domain and then identify the line segments of crack including these stress points. 1550032-14



(4) As the line segments have been determined, one has to find the effective pair of particles to approximate by judging the relative position of every pair of particles and every line segment of crack in the support domain; if the connection of the pairing particle does not intersect with any line segments, the pairing particle is effective for approximation. To judge whether two line segments intersect, as presented in Fig. 4, where the line segment L1 indicates a crack; if the two endpoints of L1 locate at the opposite

4 4

L2

L2

3

3 3

L1 1

L1 2

L2

L1

1

4 1

2

2

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) > 0

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) > 0

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) = 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) > 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) = 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) > 0

(a) Effective pair

(b) Effective pair

(c) Effective pair

L2

L1

by

1

L1

4

2

3

3 L2

3

1

2

L2

L1

2

1

4

4

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) < 0

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) < 0

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) < 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) > 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) = 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) < 0

(e) Invalid pair

(f) Invalid pair

(d) Effective pair

4

L2 L2

L1

L1 1

2

3

1

2

3

3

L1 4

1 4

2 L2

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) = 0

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) = 0

L1 ( x3 , y3 ) ⋅ L1 ( x4 , y4 ) = 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) > 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) = 0

L2 ( x1 , y1 ) ⋅ L2 ( x2 , y2 ) > 0

(g) Effective pair

(h) Effective pair

Fig. 4.

(i) Effective pair

The determination of pairing particles. 1550032-15


F. R. Ming, A. M. Zhang & S. P. Wang Table 2. (1) (2) (3) (4)

Initializing the SPH shell model and relevant parameters; Defining the configurations, constructing the MLS function and its derivatives; Solving the Green strain and Almansi strain of each integral points at the stress points; Solving the increment of stress and the total stress; (a) (b) (c) (d)

(5) (6) (7)

The program flow of SPH shell.

predicting the elastic increment of stress and the corresponding total stress; judging the stress state, if f ≤ 0, it is unyielded, goto (5); otherwise: doing the iterations with an initial value for the increment of equivalent plastic strain; updating the total stress and the yield stress, goto (b);

Transforming the nominal stress of stress points to the stress of SPH particles and solving the equilibrium equation; Updating motion quantities of SPH particles, such as displacement, velocity and normal, etc.; Judging the time, if starting the next timestep, goto (2); otherwise, goto the end.

side of L2 , and conversely the two endpoints of L2 also locate at the opposite side of L1 , the two line segments intersect; regarding more complicate cases, an endpoint of a line segment is on the other line segment, then one can refer to the cases in Fig. 4. It is worth mentioning that since the approximation function of a higher accuracy is employed in SPH shell, e.g., MLS function, with the crack occurring, one has to guarantee there are enough particles to approximate so that the matrix is reversible. Three effective particles are essential to make the approximation because the MLS function of first-order is adopted throughout this paper. Otherwise, the approximation function will be degraded as constant function. In summary, the crack formation and propagation can be adaptively processed, and the mass and energy is conserved.

by

2.5. Program flow of SPH shell The program will become complicated and its efficiency will be reduced when taking the geometric nonlinearity and material nonlinearity into account. Especially when the iterations are done to modify the stress back to the yield surface, the computation costs increase greatly. The flow of the present program is shown in Table 2.

3. Linear and Nonlinear Analyses of SPH Shell The linear and nonlinear analyses of SPH shell are similar to those of traditional methods. The linear analysis is relatively easy while the nonlinear analysis is very complicated. The geometric nonlinearity and material nonlinearity are included in this paper but the boundary nonlinearity is excluded for the future paper. The paper mainly focuses on the analytical method and its efficiency and accuracy, and therefore a very simple model is established, as plotted in Fig. 5. A square shell fixed around starts to vibrate under the uniform step load of q = 1 MPa. The model is 1550032-16



B

A

SPH particles Stress points Built in boundary

L Fig. 5.

Model of a square shell fixed around; A and B represent the test points.

by

Table 3.

The relevant geometric and material parameters.

L/m

d/m

ρ/kg · m−3

E/GPa

υ

σs0 /MPa

Ep /GPa

εsp

εcp

Dc

0.5

0.01

7850

210

0.3

235

0

0.02

0.37

0.24

discretized into 2601 SPH particles and 2500 stress points. All the degrees of freedom of SPH particles at the outermost boundary are confined. The relevant geometric and material parameters are listed in Table 3 where L, d, ρ, E and υ indicate the side length, the thickness, the density, the elastic modulus and the Poisson ratio, respectively. The test points A and B are respectively distributed at the center of the square shell and the midpoint of the boundary but half a particle-spacing away from the fixed boundary, see in Fig. 5. The default number of the integral points is 5 in the integral model. The results of the integral model will be compared with those of the global model and the FEM model. The FEM model is discretized into 2500 4-node shell elements with the positions of all the nodes corresponding with those of SPH particles and the explicit dynamic analysis is conducted with the commercial software ABAQUS. The above computation model seems to be very simple, but in the nonlinear simulation, the structure vibrates with a high frequency under the specific load and the boundary condition, which is a harsh trial for the convergence, the stability and the accuracy. According to our numerical experiments, with the increase of the load, the nonlinear problem of the above model is not easy to simulate even with some large commercial software. The accuracy and reliability of FEM results can be verified by the example in our previous studies referred to Ming et al. [2013b], as shown in Fig. 6 ,where a spherical cap fixed around is forced to vibrate under the inward uniform pressure of 4.14 MPa. 1550032-17



R ϕ

Fig. 6.

0.25

Model of a spherical cap.

SPH Shell S.S & Z.S.Liu. Owen & Hinton FEM

u/d

0.2 0.15 0.1 0.05 0 0

by

Fig. 7.

0.2

0.4 0.6 t/ms

0.8

1

Displacement of the pole point.

The cap has a radius of R = 0.577 m, a thickness of 0.01041 m, a span angle of 26.67◦. The material parameters are as follows: elastic modulus E = 7.24 × 1010 Pa, Poisson ratio υ = 0.3 and the density ρ = 2618 kg· m−3 . The gauging point locates at the pole of the cap. The displacement of the gauging point from SPH shell, Finite element carried out by Swaddiwudhipong and Liu [1996], EFG presented by Owen and Hinton [1980], FEM by ABAQUS [Hibbitt, 2004] are plotted in Fig. 7. Obviously, the results show good agreements, though there are some deviations between them, which may be from the difference of integral schemes. In general, the FEM result is reliable. 3.1. Linear analysis of SPH shell The so-called “linear analysis” means the analysis based on the assumption of small deformation, small strain of linear elasticity material, and the nonlinear term is ignored. The geometric linearity indicates a small displacement and a small strain, 1550032-18



which are in linear relationship, so only the first two terms of Green strain in Eq. (2.2) are kept. The material linearity denotes the linear relationship between the strain and stress; the material is always in elasticity and thus the Hook law is applied to link the strain and the stress. Therefore, the elastic material only with the linear strain considered is used in the linear analysis. The linear analysis is put forward mainly for analyzing quickly in approximately linear case. Under the impact of the step load, the vertical displacement of point A varying with time is plotted in Fig. 8, and the corresponding equivalent stress of integral point 5 is shown in Fig. 9. The vertical displacement and the equivalent stress (the plane of integral point 5) at the typical time of a vibration period obtained from the integral model are shown in Fig. 10. In the linear analysis, the stress at the points with a same distance apart from the mid-plane is same and therefore the equivalent stress of integral points 1 and 5 is not distinguished. It is obvious that the vertical

The vertical displacement varying with time of test point A.

by

Fig. 8.

Fig. 9.

The equivalent stress varying with time of test point A (integral points 1 and 5). 1550032-19



σ / MPa

w / mm

(a)

(d)

(c)

σ / MPa

w / mm

σ / MPa

w / mm

(b)

σ / MPa

w / mm

σ / MPa

w / mm

σ / MPa

w / mm

(e)

(f)

by

Fig. 10. The vertical displacement and the equivalent stress at the typical time of a vibration period obtained from the intergral model; the left part is the displacement and the right one is the equivqlent stress (the plane of integral point 5); (a) t = 0.01 ms, (b) t = 0.28 ms, (c) t = 0.60 ms, (e) t = 2.80 ms and (f) t = 3.44 ms.

displacement obtained from different methods agree well with each other. However, the equivalent stress calculated from the global model has some slight deviators from the results of the integral model and FEM, which may be from the different integration ways as described above. Though the poor accuracy the global model has, the efficiency is improved greatly. Taking the CPU time-consuming per time step of the global model as a reference, with the same computer configuration, the ratio of CPU time-consuming between the global model and the integral model is listed in Table 4. It is obvious that with the increment of integral points in global model, the CPU time-consuming increases. When 7 integral points are applied, the ratio is more than 2.5. However, the timeconsuming is not increase linearly with the number of integral points. When the number of integral points is not beyond 5, the ratio is less than 2. In this way, the number of integral points may be selected by the requirement of accuracy. In a word, when the model is of small displacement or small strain and the precision requirement is not strict, the global model is preferable. 3.2. Nonlinear analyses of SPH shell The nonlinear analyses of SPH shell here are limited to the geometric nonlinearity and material nonlinearity of metal. The geometric nonlinearity is embodied in the Table 4.

The ratio of CPU time-consuming of different Gauss points.

Global model

2 points

3 points

4 points

5 points

6 points

7 points

1

1.75280

1.76852

1.89957

2.06415

2.35873

2.50655

1550032-20



nonlinear term of strain while the material nonlinearity is reflected at the nonlinear elasticity of material, including the elasticity, the plasticity and the coupling of plasticity and damage. With regard to the fracture and cracks, they will be studied in the subsequent papers. 3.2.1. Geometric nonlinearity of elastic shell

by

Compared with the linear analysis, the analysis of geometric nonlinearity of elastic shell has considered the high-order nonlinear term, i.e., the last four terms on the right of Eq. (2.2). If the nonlinear terms are considered, the strain and stress in the nonlinear analysis will be distributed differently even for two points with a same distance apart from the mid-plane of shell and thus the integral points 1 and 5 will be distinguished in the following. It is worth mentioning that in the nonlinear analysis of large-deformation, it is necessary to update the MLS function per several time steps which is similar to update the stiffness matrix in FEM, and thus it is not the total Lagrangian equation in the strict sense at the moment. Considering the geometric nonlinearity, the vertical displacement varying with time of test point A is plotted in Fig. 11. The results from different methods are in good accordance with each other, which has verified the accuracy of the integral model again. The equivalent stress of integral points 1 and 5 at the test point A varying with time is shown in Fig. 12. It is obvious that the stress at the integral points 1 and 5 is no longer same, which reveals the effects of nonlinear bending. By comparison, the results at different integral points of the integral model and FEM agree well with each other, but there are some differences for the global model. The results of global model are very close to those of integral point 5 but far from the integral point 1, which may be mainly from the no consideration of the nonlinear strain in the shell thickness direction. Compared with the linear analysis of the above section, the maximum value of the equivalent stress in the nonlinear

Fig. 11.

The vertical displacement varying with time of test point A. 1550032-21



(a)

(b)

Fig. 12. The equivalent stress varying with time of different integral points at test point A, (a) integral point 1 and (b) integral point 5.

analysis is reduced slightly, but the vibration frequency increases. The influences of nonlinearity are obvious.

by

3.2.2. Geometric and material nonlinearity of elastoplastic shell After the discussion of geometric linearity and nonlinearity separately, the combined geometric nonlinearity and material nonlinearity of elastoplastic shell will be presented. The so-called material nonlinearity indicates the nonlinear relationship between the strain and stress. Based on the Von Mises theory of metal, the nonlinear analyses will be carried out. This section mainly focuses on the method of nonlinear analysis, so a relative simple material, the perfect elastoplastic material (Ep = 0), is used, but all the theories are also suitable for other materials. The same geometric model as the above sections is applied to study the nonlinear responses of the elastoplastic material. The relevant parameters are listed in Table 3. The equivalent stress varying with time at different integral points of test point A is shown in Fig. 13. It is found that the material goes into plasticity soon and the time interval of integral point 1 and integral point 5 going into plasticity is very small, but there are some obvious differences between them. The stress attenuates fast after arriving at the yield stress and then vibrates with a high frequency at integral point 1 while the stress stays at the yield stress for a long time and then vibrates with large amplitude at integral point 5. The equivalent plastic strain varying with time at different integral points is also plotted in Fig. 14. It shows that the plastic strain comes out within a very short time and the elastic strain is so small as to be ignored in many literatures. Furthermore, the results of the integral model and FEM show good agreements, especially when the vibration is of high frequency, the accuracy is also considerable. Actually, there is no distinction of integral points in global model and thus no difference for the point in the thickness section. The equivalent stress and the equivalent plastic strain of the global model are very 1550032-22



(a)

(b)

Fig. 13. The equivalent stress varying with time of different integral points at the test point A, (a) integral point 1 and (b) integral point 5.

by

Fig. 14. The equivalent plastic strain varying with time of different integral points at test point A.

different from the integral model. The reason is mainly from the resultant stress algorithm of plasticity for no consideration of the variation of plasticity along the shell thickness direction, which is the very disadvantage of the global model. The vertical displacement varying with time of test point A is shown in Fig. 15. It is found that the result of integral point accords with that of FEM, but the result of the global model shows a difference. As for the global model of elastoplastic shell, the stress is evaluated from the whole shell section and thus different yield functions and modifications will be employed when the plasticity occurs, which will affect the stress state and even the motion of shell. The vertical velocity and the equivalent plastic strain of the integral model (the plane of integral point 5) at the typical moment in a vibration period are plotted in Fig. 16. 1550032-23



Fig. 15.

The vertical displacement varying with time of test point A.

ε / ×10−2

V / m ⋅ s −1

by

(a)

(d)

(c)

ε / ×10−2

V / m ⋅ s −1

ε / ×10−2

V / m ⋅ s −1

(b)

ε / ×10−2

V / m ⋅ s −1

ε / ×10−2

V / m ⋅ s −1

(e)

ε / ×10−2

V / m ⋅ s −1

(f)

Fig. 16. The vertical velocity and the equivalent plastic strain of the integral model at the typical moment in a vibration period; the left part is the vertical velocity and the right one is the equivalent plastic strain (the plane of integral point 5); (a) t = 0.02 ms, (b) t = 1.20 ms, (c) t = 2.00 ms, (d) t = 2.50 ms, (e) t = 3.00 ms and (f) t = 3.50 ms.

3.2.3. Geometric and material nonlinearity of elastoplastic and damaged shell When the plasticity occurs and the equivalent plastic strain reaches a certain value, the growth of cavities and cracks takes place due to the material’s defects. With the accumulation of microscopic damage, the macroscopic damage will come out. Because of lack of the damage parameters, the damage parameters of high-strength steel will be applied in this model but it does not affect the analytical method. 1550032-24



Based on the continuum damage mechanics and the damage theory of Lemaitre– Chaboche, the damage variable D is selected and the integral model is applied to carry out the study. This section mainly aims to show the feasibility of studying the problems of this sort and lay a foundation for the further study of fracture, crack, etc. The same geometric model as the above sections is established, but the damage σ )p and ∆¯ ε = (¯ εp )d − theory is applied in the material. If we define: ∆¯ σ = (¯ σ )d − (¯ (¯ εp )p , where the superscripts “p” and “d” respectively indicate the values of only considering the plasticity and considering the coupling of plasticity and damage. The differences of the equivalent stress and the equivalent plastic strain at test point B are shown in Figs. 17 and 18, respectively. They show that after damage, the equivalent stress will decrease while the equivalent plastic strain will increase. With the damage variable gradually tending to be stable, the differences will be also stable. It conforms to the definition of the equivalent stress exactly. Considering the damage variable 0 ≤ D ≤ 1 and its irretrievability, when the damage material yields, the equivalent stress is modified as (1 − D)σs , so the equivalent stress will be smaller than the fully plasticity model. Furthermore, the decrease of the stress will cause the decline of resisting the external load and thus will lead to the increase of equivalent plastic strain. When the damage variable is stable, the distributions of the damage variable at the plane of integral point 1 and integral point 5 are shown in Fig. 19. It is found that the damage occurs near the built-in boundary and is different for the planes of different integral points, which reveals that the damage in the shell thickness direction cannot be ignored. 3.2.4. Geometric and material nonlinearity of damaged and fractured shell

by

To assess the fracture of SPH shell further, the uniformly distributed pressure is increased to be q = 5 MPa, in which case the fracture will occur. The above dynamic

Fig. 17. The differences of the equivalent stress and the damage variable of different integral points at test point B between the consideration and no consideration of damage. 1550032-25



Fig. 18. The differences of the equivalent plastic strain and the damage variable of different integral points at test point B between the consideration and no consideration of damage.

by

(a)

(b)

Fig. 19. The damage variable of different planes, (a) the plane of integral point 1, (b) the plane of integral point 5.

fracture algorithm of SPH shell will be employed without regard to the effects of strain rate. In order to verify the results of SPH shell, the same model is also established by FEM with same node distributions and same material parameters. The ductile damage criterion is applied in FEM model with the initial damage strain 0.13. The damage evolution is evaluated with plastic displacement, which is assumed to be linear with strain relaxation, and the failure plastic displacement is set as 2.5 mm. The formation and propagation of cracks are plotted in Fig. 20. Under the distributed pressure, the deflection of the shell increases gradually. The critical plastic displacement firstly occurs at the midpoints of the surrounding boundary and thus the fracture occurs. Under the sustained tensile force, the crack propagates 1550032-26



U/m

t ≈ 1.45 ms U/m

t ≈ 1.55 ms U/m

t ≈ 1.80 ms U/m

by

t ≈ 2.10 ms (a)

(b)

Fig. 20. The formation and propagation of cracks in the shell; the stress points in the last figure of SPH shell are shaded to display the crack more obviously, (a) FEM and (b) SPH shell.

along the two ends. Because of the formation of cracks, the free edge takes place and inward tension. Eventually, the cracks expand to the vicinity of corners and tear with an angle of 45◦ approximately to the fixed boundary. The duration of the whole process is very short. Comparing the crack obtained from FEM and SPH shell, there are some differences at the free edge, which may be associated with the treatments of free edge. Besides, the FEM result shows serrated cracks at the corner but almost straight for SPH shell, which may attribute to the difference of mesh and meshfree method. 1550032-27



0.35 0.3

The present FEM

Lcrack / m

0.25 0.2 0.15 0.1 0.05 0 1.4

Fig. 21.

1.5

1.6 1.7 t / ms

1.8

1.9

The length of the crack varying with time.

0.1 0.08

The present FEM

U/ m

0.06 0.04 0.02 0 0

by

Fig. 22.

0.5

1 t / ms

1.5

2

The deflection at the shell center varying with time.

The crack length varying with time is shown in Fig. 21, where only the stage of straight crack propagation is covered. It is obvious that the cracks propagate quickly at the initiation of crack formation, but slow down near the right-angle corner. Correspondingly, the deflection of the shell center varying with time is presented in Fig. 22, which shows an opposite law with the variation of crack length. The deflection increases slowly at the rapid expansion stage, but accelerates when the crack expansion slows down. The results of SPH shell and FEM show good agreements except some deviations of the amplitude, but the errors are within 1%.

4. Dynamic Fracture of SPH Shell In order to test the ability of SPH shell to solve the nonlinear problems of dynamic fracture further, two benchmarks of flat and spatial curved shell problems are selected to assess the reliability of SPH shell. 1550032-28



P

γ

P

A’

y α

β

R

A (a)

B

B´

O

(b)

Fig. 23. Experimental mechanism and assembly carried out by Simonsen BC [Simonsen and T¨ ornqvist, 2004; Ren and Li, 2012]. (a) Experimental model and assembly and (b) the kinematic mechanism.

4.1. Dynamic fracture of planar SPH shell

by

The experimental model of a flat plate with a pre-crack carried out by Simonsen and T¨ ornqvist [2004] is used to test the planar fracture problem. The experimental mechanism and assembly are shown in Fig. 23. The flat plate gradually forms a ductile crack under the action of tensile mechanism. The crosshead is forced to be uniform motion and thus acts as a displacement boundary condition on the flat plate. Taking the upper arm as an example, the coordinate axes are established as Fig. 23(b). When the triangle OAB moves to OA B , an angle of β will produces, so the displacement of an arbitrary point R on the displacement boundary will be [Ren and Li, 2012]: δx = lOR cos(γ − β) − lOR cos(γ),

(4.1)

δy = lOR sin(γ − β) − lOR sin(γ),

(4.2)

where α and γ can be drawn from the coordinates of point A and R; if the vertical displacement of the crosshead is noted as ∆, according to geometric relation, the rotation angle of the boundary will be: β = α − [π − arcsin(∆/lOA + sin α)].

(4.3)

In this paper, the SPH shell is also modeled and the detailed parameters can refer to Fig. 23. The whole model is discretized as 3599 SPH particles and 3480 stress points. The material parameters are listed in Table 5. Table 5. d/m 0.01

ρ/kg ·

m−3

7800

The relevant parameters.

E/GPa

υ

σs0 MPa

Ep /MPa

εsp

εcp

Dc

200

0.3

270

833

0.13

0.37

0.22

1550032-29



(a)

(b)

Fig. 24. The cracks from experimental and numerical results. (a) Experimental results by Simonsen et al. [Simonsen and T¨ ornqvist, 2004] and (b) numerical results.

t ≈ 2.00 ms

t ≈ 5.50 ms

t ≈ 7.00 ms

t ≈ 9.00 ms

t ≈ 12.00 ms

by

t ≈ 0.65 ms

Fig. 25.

The typical instants of the shell fracture (PEEQ: equivalent plastic strain).

With the motion of the displacement boundary, the crack propagates gradually along the pre-determined direction, and obviously there are some stress concentrations at the tip of the crack. The free edge generated from the crack shows some necking, as plotted in Fig. 24. Finally, a straight crack along the pre-crack is produced and the typical instants of the shell fracture are presented in Fig. 25. It is obvious that there exists a triangular area of plastic strain approximately. The edge of the crack is uneven, which agrees well with the experimental result. 1550032-30



800 Experiment The present

P/kN

600

400

200

0 0

20

40

60 80 ∆ / mm

100

120

Fig. 26. The relationship between the vertical displacement and the tensile force of the crosshead; the experiment carried out by Simonsen and T¨ ornqvist [2004], Ren and Li [2012].

With the motion of the two arms of crosshead, the tensile force measured by the machine varying with the vertical displacement is shown in Fig. 26. When the vertical displacement increases, the tensile force also increases gradually. However, with the propagation of cracks, the stress concentrations slightly relieve, and meanwhile the arm between the forced displacement boundary and the shaft increases, which will cause a decrease of tensile force. As the figure reveals, the tensile obtained from experiment is stable and smooth, while the curve drawn from SPH shell shows some fluctuations, which may be from the stress changes in the process of dynamic fracture, as explained in Ren and Li [2012]. However, it is not hard to see the results from experiment and SPH shell still show good agreements on the whole. The crack length varying with the vertical displacement of the crosshead is presented in Fig. 27. The crack begins to expand until the crosshead moves a short

300

by

Experiment The present

250

Lcrack /mm

200 150 100 50 0 0

50

∆ / mm

100

150

Fig. 27. The relationship between the vertical displacement and the crack length of the crosshead; the experiment carried out by Simonsen and T¨ ornqvist [2004], Ren and Li [2012]. 1550032-31



distance. The crack length has an approximately linear relationship with the displacement of the crosshead. The numerical result has a more smooth transition from zero crack length. In summary, the numerical results accord well with the experiment, and thus the numerical accuracy and stability of SPH shell are verified in dealing with planar fracture problems. 4.2. Dynamic fracture of a cylindrical shell The detonation device in Fig. 28 was proposed by Chao and Shepherd [2005] to carry out the experiment of crack study. The numerical model had been modeled by Song and Belytschko [2009] with XFEM. The device consists of two parts: a detonation tube on the left used for the generation of initial detonation wave and a fixed target specimen on the right for loading. There is a dent at the middle of the specimen with a length of 50.8 mm. The specimen’s material is Al 6061-T6 and the detailed parameters are listed in Table 6. Its size can refer to Fig. 28. In the present case, the crack expansion is driven from the implosion pressure, which can be obtained from the following formula [Song and Belytschko, 2009]: 0, t ≤ x/vcj , (4.4) p(x, t) = pcj exp[−(t − x/vcj )/(3x/vcj )], t > x/vcj ,

by

0.05m

0.038m

where pcj and vcj indicate the Chapmand–Jouguet pressure and the velocity of detonation wave, pcj = 6.2 MPa and vcj = 2390 m· s−1 ; the coordinate x denotes the distance between the source point and the observed point. The propagation of stress wave along the axial direction of the specimen is shown in Fig. 29. As the stress wave propagates right, the amplitude decays gradually. When the stress wave arrives at the pre-crack, the stress concentration takes place

0.0508m 0.05m

Fig. 28.

The experimental model of a detonation tube established by Chao and Shepherd [2005].

Table 6. d/m 0.00089

ρ/kg ·

m−3

2780

Parameters of the detonation tube.

E/GPa

υ

σs0 /MPa

Ep /MPa

εsp

εcp

Dc

69

0.3

275

640

0.13

0.343

0.3

1550032-32



σ eq / MPa

(a)

σ eq / MPa

(b)

σ eq / MPa

(c)

σ eq / MPa

(d)

by

Fig. 29. The propagation of stress wave before the obvious expansion of cracks, (a) t ≈ 22 µs, (b) t ≈ 165 µs, (c) t ≈ 190 µs and (d) t ≈ 210 µs.

quickly; as the time goes on, the sustained stress wave passes through the crack followed by a rapid expansion of cracks. The process of crack propagation is plotted in Fig. 30, where the results of Galerkin carried out by Becker and Noels [2013] are arranged on the left, while the results of SPH shell are displayed on the right column. It is obvious that the two results are in good accordance with each other. As the crack expands some distance along the axial direction, the propagation along the circumferential direction emerges at the front of the crack (the right end). With the crack expansion, the wall of the thin tube begins to overturn due to the inertia effects. The experimental result from Chao is presented in Fig. 31. It is obvious that there are some deviations between the experimental result and the numerical one. The rear of the crack (the left end) in experiment does not show obvious circumferential expansion, but propagates at a certain angle, which may attribute to the random of the experiment or the discretization of numerical simulation. However, the final results from the two methods show good similarities on the whole. The 1550032-33



(1)

t ≈ 230µs

(2)

t ≈ 270µs

(3)

t ≈ 300µs

(4)

t ≈ 330µs

(5)

t ≈ 400µs

by

(a)

(b)

Fig. 30. The expansion of cracks, the right one from the present result; the free edge generated from cracks are colored, (a) result of Galerkin by Becker [Becker and Noels, 2013] and (b) the present result.

Detonation wave

Fig. 31.

The experimental result from Chao [Becker and Noels, 2013].

expansion velocity before the crack bifurcation is presented in Fig. 32. The law of the expansion velocity shows some similarities for the rear crack and the front crack, and the amplitude is both up to 300 m/s under the impact of implosion pressure. In the experiment, the wall thickness of the detonation tube is very thin (about 0.89 mm), 1550032-34



500

500

-1

400

300

Vcrack / m·s

Vcrack / m·s

-1

400

200 The present G. Becker et al. Song et al.

100 0 150

200

t / µs

250

300

(a)

The present G. Becker et al. Song et al.

300 200 100 0 150

200

t / µs

250

300

(b)

Fig. 32. The expansion velocity of the crack before bifurcation [Song and Belytschko, 2009; Becker and Noels, 2013], (a) the rear crack and (b) the front crack.

while the peak pressure is of order 106 Pa, so the fracture shows strong nonlinearity and randomness. There are great difficulties for the numerical simulation and experimental study. Compared with the existing literature and experimental results, the present numerical study reveals the robust of the present SPH shell for dealing with dynamic fracture problems.

by

5. Conclusions The applications of smoothed particle hydrodynamics method in solid mechanics have always been troubled by the drawbacks of accuracy and stability. The SPH shell firstly proposed by Maurel et al. has overcome those thorny problems successfully, which is called “global model” in the paper. However, in the simplifying process of the global model, the nonlinear strain in the thickness direction is removed considering the expensive computational costs and thus there are some limitations in the nonlinear analysis of thick shells; especially when the material is of nonlinearity, such as plastic damage and fracture, the resultant stress algorithm used in the global model has an inadequate accuracy due to no consideration of the plasticity variation along the thickness direction. The integral model is firstly put forward in this paper, i.e., a number of integral points are distributed along the thickness direction at the stress point to capture the nonlinear characteristics of bending. Afterwards, based on the Caleyron’s derivations of the plasticity algorithm for global model, the stress modification algorithms featured by high efficiency for the plasticity material and for the plastic damage and fracture material are derived and then successfully applied to the integral model. Furthermore, the algorithm of multiple line segments is proposed to treat cracks adaptively, which is simple, operable and applicable. Finally, considering the 1550032-35



material of elasticity, plasticity, plastic damage and dynamic fracture, the linear and nonlinear analyses of SPH shell are presented. All the results of the integral model agree well with existing results while only the linear material shows good agreements for the global model. Though the global model has a poor accuracy in the nonlinear analysis, a considerable efficiency is shown. To sum up, the integral model proposed in this paper has a high accuracy and a good stability in the linear and nonlinear analyses, which may provide a reference for the further study.

Acknowledgments The authors appreciate the help of Prof. Alain Combescure from INSA-Lyon in France for his guidance and instructions. Also, this work is supported by the Excellent Youth Foundation of Heilongjiang Province (JC201207), the Excellent Young Scientists Fund of China (51222904) and the Lloyd’s Register Foundation, which helps to protect life and property by supporting engineering-related education, public engagement and the application of research.

Appendix A Owing to −T

L

−1

ΞL

−T −1 3∆¯ εp 3∆¯ εp = I + n+1 ΓΞ Ξ I + n+1 ΓΞ σs σs n+1 σs σsn+1 −1 −1 −T −T . = IΞ + Γ Ξ Ξ I+ Ξ Γ 3∆¯ εp 3∆¯ εp

(A.1)

by

Considering the symmetry of Γ and Ξ, the expression of (A.1) is simplified as: n+1 2 2σ n+1 σs Γ−1 Ξ−1 Γ−1 . (A.2) L−T ΞL−1 = Ξ + s Γ−1 + 3∆¯ εp 3∆¯ εp The derivative of the above equation versus ∆¯ εp is: −T εp − σsn+1 −1 2 σsn+1 Ep ∆¯ εp − σsn+1 −1 −1 −1 2 Ep ∆¯ Γ + Γ Ξ Γ L ΞL−1 = 2 3 ∆¯ εp 9 ∆¯ εp ∆¯ ε2p =−

2 σsn −1 2 σsn+1 σsn −1 −1 −1 Γ − Γ Ξ Γ . 3 ∆¯ ε2p 9 ∆¯ ε3p

(A.3)

Appendix B If the following approximation is made, Eq. (2.30) becomes as [Zeng et al., 2001], ∆εp =

3∆¯ εp 3∆¯ εp Ξσ. Ξσ ≈ σs ¯ σ 1550032-36

(B.1)



Substituting it into Eq. (2.25), we get: εp K)σ n+1 , σ n+1 = (I − ∆¯

(B.2)

K = 3ΓΞ/¯ σ n+1 .

(B.3)

where

Therefore, substituting Eq. (B.3) into Eq. (2.19): εp K)T Ξ(I − ∆¯ εp K)σ n+1 − (σsn+1 )2 fn+1 = σ n+1 T (I − ∆¯ = σ n+1 T Ξσ n+1 − 2∆¯ εp σ n+1 T KT Ξσ n+1 + ∆¯ ε2p σ n+1 T × KT ΞKσ n+1 − (σsn+1 )2 = 0.

(B.4)

Note: σ n+1 = σ n+1 T KT Ξσ n+1 /¯

(B.5)

Equation (B.4) is reduced as: εp ¯ σ n+1 + (∆¯ εp )2 = (σsn+1 )2 . ¯ σ n+1 2 − 2∆¯

(B.6)

According to the physical meaning, the solution of the above equation is unique. We have: εp = σsn+1 . ¯ σ n+1 − ∆¯

(B.7)

Hence, substituting Eq. (2.34) into the above equation, the approximate solution of the increment of equivalent plastic strain ∆¯ εp is: ∆¯ εp =

¯ σ n+1 − σsn . + Ep

(B.8)

by

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Liu, Z. S. and Swaddiwudhipong, S. [1997] “Response of plate and shell structures due to low velocity impact,” Journal of Engineering Mechanics 123(12), 1230–1237. Liu, M. B. and Yang, X. F. [2012] “Improvement on stress instability in smoothed particle hydrodynamics,” Acta Physica Sinica 61(22), 261–268. Maurel, B. [2008] Modélisation par la méthode SPH de l’impact d’un réservoir rempli de fluide, Institut National des Sciences Appliquées de Lyon (INSA de Lyon, Villeurbanne, France), pp. 1–212. Maurel, B. and Combescure, A. [2008] “An SPH shell formulation for plasticity and fracture analysis in explicit dynamics,” International Journal for Numerical Methods in Engineering 76(7), 949–971. Maurel, B., Combescure, A. and Potapov, S. [2006] “A robust SPH formulation for solids,” European Journal of Computational Mechanics/Revue Europ´ eenne de Mécanique Numérique 15(5), 495–512. Miao, Y., He, T., Luo, H. and Zhu, H. [2012] “Dual hybrid boundary node method for solving transient dynamic fracture problems,” Computer Modeling in Engineering & Sciences (CMES) 85(6), 481–498. Ming, F. R., Zhang, A. M. and Cao, X. Y. [2013a] “A robust shell element in meshfree SPH method,” Acta Mechanica Sinica 29(2), 241–255. Ming, F. R., Zhang, A. M. and Yao, X. L. [2013b] “Static and dynamic analysis of elastic shell structures with smoothed particle method,” Acta Physica Sinica 62(11), 27–37. Moës, N., Dolbow, J. and Belytschko, T. [1999] “A finite element method for crack growth without remeshing,” International Journal for Numerical Methods in Engineering 46(1), 131–150. Monaghan, J. J. and Gingold, R. A. [1983] “Shock simulation by the particle method SPH,” Journal of Computational Physics 52(2), 374–389. Noguchi, H., Kawashima, T. and Miyamura, T. [2000] “Element free analyses of shell and spatial structures,” International Journal for Numerical Methods in Engineering 47(6), 1215–1240. Owen, D. R. J. and Hinton, E. [1980] Finite Elements in Plasticity: Theory and Practice (Pineridge Press, Swansea, UK). Rabczuk, T., Belytschko, T. and Xiao, S. P. [2004] “Stable particle methods based on Lagrangian kernels,” Computer Methods in Applied Mechanics and Engineering 193(12), 1035–1063. Rabczuk, T. and Zi, G. [2007] “A meshfree method based on the local partition of unity for cohesive cracks,” Computational Mechanics 39(6), 743–760. Randles, P. and Libersky, L. [1996] “Smoothed particle hydrodynamics: Some recent improvements and applications,” Computer Methods in Applied Mechanics and Engineering 139(1), 375–408. Randles, P. and Libersky, L. [2000] “Normalized SPH with stress points,” International Journal for Numerical Methods in Engineering 48(10), 1445–1462. Ren, B. and Li, S. [2012] “Modeling and simulation of large-scale ductile fracture in plates and shells,” International Journal of Solids and Structures 49(18), 2373–2393. Simonsen, B. C. and T¨ ornqvist, R. [2004] “Experimental and numerical modelling of ductile crack propagation in large-scale shell structures,” Marine Structures 17(1), 1–27. Song, J. H. and Belytschko, T. [2009] “Dynamic fracture of shells subjected to impulsive loads,” Journal of Applied Mechanics 76(5), 051301–051302. Swaddiwudhipong, S. and Liu, Z. S. [1996] “Dynamic response of large strain elastoplastic plate and shell structures ” Thin-Walled Structures 26, 223–239. Unosson, M., Olovsson, L. and Simonsson, K. [2006] “Failure modelling in finite element analyses: Element erosion with crack-tip enhancement,” Finite Elements in Analysis and Design 42(4), 283–297. 1550032-39



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Smoothed Particle Hydrodynamics for the Linear and

Smoothed Particle Hydrodynamics for the Linear and

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