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Meshfree simulation of concrete structures and impact loading
Accepted Manuscript Title: Meshfree simulation of concrete structures und impact loading Author: R. Drathi, A.J.M. Das, A. Rangarajan PII: DOI: Reference:
S0734-743X(15)00219-5 http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.013 IE 2597
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
17-1-2015 29-7-2015 28-10-2015
Please cite this article as: R. Drathi, A.J.M. Das, A. Rangarajan, Meshfree simulation of concrete structures und impact loading, International Journal of Impact Engineering (2016), http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Meshfree Simulation of Concrete Structures und Impact Loading R. Drathi, A.J.M. Das, A. Rangarajan IIT Kanpur, Kanpur, India, e-mail: [email protected] Highlights • Uncertainty analysis determining effects of material parameters on impact resistance of concrete • Determining the importance of physical effects on impact resistance of concrete • Computational analysis of impact resistance with meshfree method Abstract The impact resistance of concrete structures is of major importance in engineering application. Computational methods are increasingly used for such types of applications but face difficulties due to the complex physical behaviour involving large deformations and large strains. Meshfree methods seem ideally suited to deal with these type of problems. In this manuscript, we present stochastic simulations based on the element-free Galerkin method to predict upper and lower bounds of the impact resistance of concrete structures. We account for stochastic distribution of material parameters and validate our results with benchmark experiments conducted by the group of Hanchak. Keywords: Meshfree Method, Impact Resistance, Strain-rate, Fracture Introduction 1. Computational modeling of concrete under impact loading remains one of the key challenges in Civil Engineering. Besides computational methods, constitutive models are an important ingredient of any mechanical model. For high dynamic loading it is important to accurately capture the material response under extreme pressure loading and the so-called strain rate effect. Popular constitutive models based on damage mechanics include the work by [1, 2, 3, 4]. In [5], the authors extended the famous Johnson-Cook (JC) [6] model to concrete materials; the JC model accounts for strain rate and temperature effects and also plastic deformations. A quasi-continuum plasticity approach capturing the dynamic buckling strength of sandwich structures was proposed by [7]. Coupled damage-plasticity models were proposed by for instance by [8, 9, 10, 11, 12, 13]. In this work, we employ a constitutive model proposed in [14]. It employs a dynamic damage variable that delays the damage evolution in order to take the strain-rate effect into account. This dynamic damage variable depends on previous damage increments and the associated damage rates. In [15], the authors extended their scalar damage model for isotropic damage to anisotropic damage by introducing a vectorial damage. Another important ingredient to model the impact resistance of concrete is the computational method. Many studies are baed on finite element analysis [16, 17, 18, 19, 20, 21, 22, 23, 24]. Often, element-deletion methods were exploited in order to allow for large deformations and complete perforations [25, 26]. Meshfree or meshless methods are good alternatives to FEM [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] as they can model large deformation and perforations without additional techniques and much loss of accuracy. An overview and computer implementation aspects of meshfree methods (MM) is discussed in [38]. MM have also been used to model impact events. For example, the authors in [39] accurately predicted the penetration depth and residual impact velocities compared to the experiments done by the authors in [40]. These authors reported finite element simulations underpredict the impact resistance of concrete. While dynamic fracture in MM was initially captured quite naturally by separation of nodes [41, 42, 43, 44, 45, 46, 47], the authors in [48, 32] for example found that such an approach might lead to numerical fracture. More sophisticated approaches such as XFEM (extended finite element method) [49, 50] or smoothed extended finite element method [51, 52, 53], extended MM [54, 55, 56, 57, 58, 59, 60, 61, 62, 36, 63], extended isogeometric analysis formulations [64, 65], the phantom node method [66, 67, 68, 69] or smoothed phantom node approaches [70], recent multiscale methods [71, 72, 73, 74, 75] or efficient remeshing techniques [76, 77, 78, 79, 80, 81, 82] might also be applied to dynamic fracture [83, 84]. However, while they seem well suited to capture a moderate number of propagating cracks, their performance to capture a large number of cracks in a large deformation setting still needs to be shown. A compromise to above mentioned approaches is the cracking particles method (CPM) [85, 86]. In the CPM, fracture is modelled by set of cracked particles. Several improvements have been incorporated into the CPM [87, 88, 89, 90, 91, 92]. Since MM are computational costly, they have been coupled to finite element methods [30, 92]. The two mentioned formulations have also been applied to predict the impact resistance of concrete structures. The majority of the publications (see the list above) are focused on deterministic approaches but it is well known that this can lead to unrealistic crack patterns as the ones predicted in [39]. In these simulations, cracks are too close to each other. Stochastic approaches such as introducing some randomness in the tensile strength [93] can alleviate this unrealistic behaviour [94, 95, 96]. However, none of these simulations consider stochastic material parameters though it is barely possible to calibrate the material parameters uniquely and exactly. On the other hand, every computational method needs to be validated. Classical benchmark problems for impact resistance of concrete include the experiments by [97, 98, 99, 100, 101, 102, 103]. The experiment exploited in this manuscript was carried out by the authors in [104]. In these experiments, concrete specimen were subjected to impactors with various velocities. In summary: We present stochastic simulations to predict the impact resistance of concrete. The element-free Galerkin (EFG) method is exploited in combination with a viscous damage-plasticity model [14]. A simple node splitting algorithm described in [89] has been exploited in order to avoid artificial fracture. It can be considered as a special case of the CPM. Finally, our simulations are validated by comparison to experimental data from our own laboratory and from Hanchak et al. [104]. 2. Governing Equations and Discretization We solve the equation of motion that can be stated in weak form by W int W ext W kin = 0
W in t =
W ext =
W k in =
ij
: ij d
u iti d t
u i bi d
u iu i d
(1)
, W ext and W int indicating the kinetic energy, external work and internal energy, respectively; Ω is the domain and t u = , with t u = 0 is the external boundary consisting of traction and displacement boundary conditions indicated by the subscript t and d, respectively. The components of the linear strain tensor is denoted by ij and ij are the components of the Cauchy stress tensor; the components of the traction and body force vector are given by ti and bi, respectively; ui are the components of the displacement field, ρ is the density and the superimposed ‘dot’ stands for material time derivatives. As already mentioned in the introduction, we employ the EFG method [105] to discretize the displacement field. It can be shown that the EFG approximation is given by u ( x , t ) = N I ( x ) u I (t ) (2) W kin
IS
uI being the nodal parameters of the displacement field, which are unequal to the physical displacement values at that point, or in other words u(xI) ≠ uI. The shape functions are denoted by NI(x). They are obtained from miminization of a discrete L2 norm which finally leads to:
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T
N (x) = p (x) A
1
(3)
(x) B (x)
n
A (x) =
w(X X
T
I
is the moment-matrix and B(x) is computed as
) p(X I ) p (X I )
I =1
B ( x ) = w ( x x 1 )p ( x 1 ) w ( x x 2 )p ( x 2 )
w ( x x n )p ( x n )
(4) w(X − XI) denoting the weighting function and p (X) the polynomial basis. More details are given in [105]. Exponential kernel function and linear basis functions are chosen, i.e. p T ( x ) = 1 x y . Substituting the discretization, eq. (2) into the weak form of the equation of motion, eq. (1), leads to the well-known system of equations: (5) M D = F e x t F in t where the nodal parameters uI(t) of the displacement field are stored in the global vector D. It can be shown that the mass matrix is obtained by T
M =
N N d T
(6)
The external force vector and internal force vector is given by Fext =
F in t =
N b d T
N t d T
t
B d T
(7)
N and B being matrices containing the EFG shape functions and their spatial derivatives, respectively. We use a stabilized [48] nodally regularised [32] element-free Galerkin method [105]. We take advantage of the updated Lagrangian kernel formulation presented in [39] to ensure the stability of the method while simultaneously maintaining the applicability to extremely large deformations needed for dynamic fracture and fragmentation. 3. Constitutive Model The employed constitutive model is based on the approach presented by Rabczuk et al. [14]. While the original approach is a coupled damage-plasticity model, we removed the plasticity part from the formulation which reduces the number of material parameters. Subsequently, we summarize the basic equations of this constitutive model. The strain rate ij is decomposed into an elastic part ije and a damage part ijd : ij
e
=
ij
d
(8)
ij
The stress-strain relation can be written as ij = 1 D C ijk l
(9) D = DS + DD being a damage variable which is decomposed into a static part DS and a dynamic part DD, γ is a function accounting for high hydrostatic pressure response and C ijkl denotes the components of the elasticity tensor. The formulation has been implemented in rate form as suggested in [14]. We use the same damage surfaces in compression and tension F d = c1 J 2 e
ci,
i = 1,...,4
d
c
J 2 c3 e
2
(a )
kl
c4 I1
e
e ,m ax
2 d
(10)
= 0
being material parameters, κd denotes the effective damage strain, (a )
second invariant of the elastic strain tensor and is adopted to model the damage evolution: DS = 1 e
e d 0 e d
e ,m ax
indicates the
a
th
e
I1
is the first invariant of the elastic strain tensor,
e
J2
is the
eigenvalue of the elastic strain tensor. A classical exponential function
g
d e0
D S = 0 d < e0
(11) where ed, e0 and g are material constants. A dynamic damage evolution is introduced decaying the static damage evolution. It is defined by DS
t
DD =
=0
t
H (t ) d
(12)
H(t − τ) being a monotonically decreasing history function which decays from 1 to 0 with a specific time: H (t ) = e
t * L ( ) d
L ( d ) = l *
with 4.
0 = 1s
h
* w ln d
* r d
,
* d
=
d ( ) 0
(13)
1
. Results As stated in the introduction, we used our own experiments and the experiments by Hanchak et al. [104] to validate the proposed computational model. In the experiments by Hanchak and co-workers as illustrated in Fig. 1, concrete structures were subjected to impact loading with different velocities of the impactor. They tested two types of concrete, a high performance concrete (HPC) and standard concrete; the latter one had a compressive strength of 50 MPa. Since our own concrete had a similar compressive strength, we will use the latter one in order to validate our model. In the Hanchak experiments, all concrete specimen were perforated completely. Initial studies showed discrepancies between the results of 2D models and full three-dimensional models; we therefore show only the results of our 3D simulations. We needed roughly 2 million nodes to obtain a convergent residual velocity of the impactor and the convergent fracture pattern. We assumed stochastic material parameters and the mean values of all material parameters are shown in Table 1. They have been classified into five different categories: 1. Initial parameters of the intact material, i.e. elasticity modulus, poisson’s ration and initial density. 2. Parameters associated to the damage evolution. These parameters can be further classified into parameters associated to the static damage evolution e0, ed and gd (the key parameters are ed and gd controlling the ductility and maximum strength of the concrete) and the parameters associated to the dynamic damage evolution c t 1 and c t 2 which determine how the dynamic damage decays in time. 3. The parameters associated to the damage surface c1 to c4. 4. The reduction factors rc and rt controlling the amount of tensile damage done by compressive loading and vice versa. 5. The parameters associated to the pressure-dependence of concrete. The parameters associated to the damage surface are assumed to be deterministic while all other parameters are subjected to a uniform PDF
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(probability distribution function) with a standard deviation of 5% and 10%, respectively. We used sampling method (LHS) modifying only one class of material parameters while fixing all other material parameters. A sensitivity analysis (SA) as presented in [106, 107, 108, 109, 110] would lead to more quantitative results but such studies will be done in the near future. The minimum and maximum residual velocities of the impactor for simulations based on a standard deviation of 5% and 10% are summarized in Table 2 for the different initial impact velocities. The values of the deterministic simulation based on the mean values according to Table 1 are shown in Table 2, too. Obviously, uncertainties in the material parameters substantially affect the residual velocity of the impactor. The first parameter set (associated to the initial parameters of the intact material) as well as the fourth parameter set (associated to the reduction factors) barely influence the residual velocity. This is indicated by the fact that the scatter in the output is much smaller than the scatter in the input. On the other hand, the second parameter set (associated to the damage evolution) and the fifth parameter set (associated to the pressure dependence of concrete) significantly influence the residual velocity of the impactor. This observation holds for both standard deviation of 5% and 10%, respectively. The most pronounced influence is obtained by the parameter set related to the dynamic strength increase. We also carried out simulations and sampling methods to determine upper and lower bounds of the residual impact velocity varying two parameter sets while keeping all other parameter sets constant. As the third and fifth parameter sets seems to be the most influential ones, those parameters are varied. However, the minimum and maximum residual velocities barely changed. As can be seen, there is a good agreement between experimental data and computational predictions. The last benchmark problem is a concrete specimen subjected to contact detonation. The set-up is illustrated in Figure 2. The explosive is modelled with the JWL-EOS (John-Wilkonson-Lee equation of state) using the EFG discretization. Due to the small amount of the charge, the concrete specimen was not perforated. The penetration depth and the effective diameter at the upper surface was measured. The same mean values of the material parameters as listed in Table 1 are adopted. Also the PDF and standard deviations of the previous example are employed. Table 3 depicts the computational results. Similar observations as made for the Hanchak experiments hold also for this experiment. However, the uncertainties in the input parameters are smaller. The largest effect is attributed to the parameter set 3 related to the dynamic damage evolution followed by parameter sets 2 and 5. The smallest influence are related to parameter sets 1 and 4. 5. Conclusion We presented stochastic simulations predicting the impact resistance of concrete. Stabilized nodal integrated EFG method is exploited to discretize the displacement field. This method is particularly suitable for problems involving large deformations as they commonly occur under impact and explosive loading. A scalar dynamic damage model has been used accounting for the pressure dependence of concrete as well as the strain-rate effect. The computational method has been is validated through two benchmark problems: 1. The impact experiments by Hanchak et al, who measured the residual velocity of the impactor and 2. Our own experiment where concrete specimen are subjected to a contact detonation. In our own experiments, we measured the penetration depth as well as the effective damage diameter at the upper surface. Our computational results are in good agreement to the experimental results. We employed sampling method to account for uncertainties in material parameters. 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Rabczuk, Molecular dynamics/xfem coupling by a three-dimensional extended bridging domain with applications to dynamic brittle fracture, International Journal for Multiscale Computational Engineering 11 (6) (2013) 527–541. [75] H. Talebi, M. Silani, T. Rabczuk, Concurrent multiscale modeling of three dimensional crack and dislocation propagation, Advances in Engineering Software 80 (2015) 82–92. [76] P. Areias, T. Rabczuk, Finite strain fracture of plates and shells with configurational forces and edge rotation, International Journal for Numerical Methods in Engineering 94 (2013) 1099–1122. [77] P. Areias, T. Rabczuk, D. D. da Costa, Assumed-metric spherically-interpolated quadrilateral shell element, Finite Elements in Analysis and Design 66 (2013) 53–67. [78] P. Areias, D. D. da Costa, J. Sargado, T. Rabczuk, Element-wise algorithm for modeling ductile fracture with the rousselier yield function, Computational Mechanics 52(6) (2013) 1429–1443. [79] P. Areias, T. Rabczuk, D. D. da Costa, Element-wise fracture algorithm based on rotation of edges, Engineering Fracture Mechanics 110 (2013) 113 – 137. [80] H. Nguyen-Xuan, G. Liu, S. Bordas, S. Natarajan, T. Rabczuk, An adaptive singular es-fem for mechanics problems with singular field of arbitrary order, Computer Methods in Applied Mechanics and Engineering 253 (2013) 252–273. [81] P. Areias, T. Rabczuk, P. Camanho, Initially rigid cohesive laws and fracture based on edge rotations, Computational Mechanics 52(4) (2013) 931 – 947. [82] P. Areias, T. Rabczuk, P. Camanho, Finite strain fracture of 2d problems with injected anisotropic softening elements, Theoretical and Applied Fracture Mechanics 72 (2014) 50–63. [83] G. Liu, Y. Hu, Q. Li, Z. Zuo, Xfem for thermal crack of massive concrete, Mathematical Problems in Engineering 2013 (2013) Article ID 343842. [84] T. Rabczuk, S. Bordas, G. Zi, A three-dimensional meshfree method for continuous multiple crack initiation, nucleation and propagation in statics and dynamics, Computational Mechanics 40 (3) (2007) 473–495. [85] T. Rabczuk, T. Belytschko, Cracking particles: A simplified meshfree method for arbitrary evolving cracks, International Journal for Numerical Methods in Engineering 61 (13) (2004) 2316–2343. [86] T. Rabczuk, J.-H. Song, T. Belytschko, Simulations of instability in dynamic fracture by the cracking particles method, Engineering Fracture Mechanics 76 (6) (2009) 730–741. [87] H. Wang, S. Wang, Analysis of dynamic fracture with cohesive crack segment method, CMES-Computer Modeling in Engineering & Sciences 35 (3) (2008) 253–274. [88] N. Sageresan, R. Drathi, Crack propagation in concrete using meshless method, CMES-Computer Modeling in Engineering & Sciences 32 (2) (2008) 103–112. [89] T. Rabczuk, G. Zi, S. Bordas, H. Nguyen-Xuan, A simple and robust three-dimensional cracking-particle method without enrichment, Computer Methods in Applied Mechanics and Engineering 199 (37-40) (2010) 2437–2455. [90] T. Rabczuk, R. Gracie, J.-H. Song, T. Belytschko, Immersed particle method for fluid-structure interaction, International Journal for Numerical Methods in Engineering 81 (1) (2010) 48–71. [91] F. Caleyron, A. Combescure, V. Faucher, S. Potapov, Dynamic simulation of damage-fracture transition in smoothed particles hydrodynamics shells, International Journal for Numerical Methods in Engineering 90 (6) (2012) 707–738. [92] Y. Chuzel-Marmot, R. Ortiz, A. Combescure, Three dimensional sph-fem gluing for simulation of fast impacts on concrete slabs, Computers and Structures 89 (23-24) (2011) 2484–2494. [93] T. Rabczuk, G. Zi, S. Bordas, H. Nguyen-Xuan, A geometrically non-linear three dimensional cohesive crack method for reinforced concrete structures, Engineering Fracture Mechanics 75 (16) (2008) 4740–4758. [94] T. Rabczuk, J. Eibl, Numerical analysis of prestressed concrete beams using a coupled element free galerkin/finite element method, International Journal of Solids and Structures 41 (3-4) (2004) 1061–1080. [95] T. Rabczuk, J. Akkermann, J. Eibl, A numerical model for reinforced concrete structures, International Journal of Solids and Structures 42 (5-6) (2005) 1327–1354. [96] T. Rabczuk, T. Belytschko, Application of particle methods to static fracture of reinforced concrete structures, International Journal of Fracture 137 (1-4) (2006) 19–49.
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[97] R. John, S. P. Shah, Mixed mode fracture of concrete subjected to impact loading, ASCE Journal of Structural Engineering 116 (1990) 585–602. [98] P. Rossi, A physical phenomenon which can explain the mechanical behavior of concrete under high strain rates, Materials and Structures 24 (1991) 422–424. [99] D. Frew, S. Hanchak, M. Green, M. Forrestal, Penetration of grout and concrete targets with ogive-nose steel projectiles, International Journal of Impact Engineering 18 (1996) 465–476. [100] D. Frew, S. Hanchak, M. Green, M. Forrestal, Penetration of concrete targets with ogive-nose steel rods, International Journal of Impact Engineering 21 (1998) 489–497. [101] T. Borvik, A. Clausen, O. Hopperstad, M. Langseth, Perforation of aa5083-h116 aluminium plates with conical-nose steel projectilesexperimental study, International Journal of Impact Engineering 30 (2004) 367–384. [102] W. Riedel, M. Wicklein, K. Thoma, Shock properties of conventional and high strength concrete: Experimental and mesomechanical analysis, International Journal of Impact Engineering 35 (2008) 155–171. [103] P. Forquin, W. Riedel, J. Weerheijmh, Dynamic test devices for analyzing the tensile properties of concrete, Understanding the Tensile Properties of Concrete 181e (2013) 137–180. [104] S. Hanchak, M. Forrestal, E. Young, J. Ehrgott, Perforation of concrete slabs with 48 mpa (7 ksi) and 140 mpa (20 ksi) unconfined compressive strengths, International Journal of Impact Engineering 12 (1992) 1–7. [105] B. T., L. Y.Y., G. L., Element-free galerkin methods, International Journal for Numerical Methods in Engineering 37 (1994) 229–256. [106] N. Vu-Bac, T. Lahmer, Y. Zhang, X. Zhuang, T. Rabczuk, Stochastic predictions of interfacial characteristic of carbon nanotube polyethylene composites, Composite Part B: Engineering 59 (2014) 80–95. [107] N. Vu-Bac, T. Lahmer, H. Keitel, J. Zhao, X. Zhuang, T. Rabczuk, Stochastic predictions of bulk properties of amorphous polyethylene based on molecular dynamics simulations, Mechanics of Materials 68 (2014) 70–84. [108] H. Ghasemi, R. Rafiee, X. Zhuang, J. Muthu, T. Rabczuk, Uncertainties propagation in metamodel-based probabilistic optimization of cnt/polymer composite structure using stochastic multi-scale modeling, Computational Materials Science 85 (2014) 295–305. [109] N. Vu-Bac, M. Silani, T. Lahmer, X. Zhuang, T. Rabczuk, A unified framework for stochastic predictions of young’s modulus of clay/epoxy nanocomposites (pcns), Computational Materials Science 95 (2015) 520–535. [110] M. Silani, H. Talebi, S. Ziaei-Rad, P. Kerfriden, S. Bordas, T. Rabczuk, Stochastic modelling of clay/epoxy nanocomposites, Composite Structures 118 (2014) 241–249.
Figure 1: Hanchak impact experiment Figure 2: Explosion experiment Table 1: Parameters of the constitutive model Young’s Poisson ratio density modulus E [g/mm3] [GPa] 52 0.22 2700 e0 ed gd
c t1
ct2
0.55
0.0001 c1 0.01234 rt 1.2 av
0.0003 c2 0.0252 rc 18.8 bv
2.3 c3 0.7821
0.032 c4 0.3464
ev
e v , th
0.7
3.4
0.02
0.007
Table 2: Residual velocity ([m / s]) of the impactor in dependence of the impact velocity ([m / s]) and the scatter in the input parameters: The first line of the same impact velocity refers to a 5% scatter while the second line to the 10% Impact Determ Set 1 Set 2 Set 3 Set 4 Set 5 Experiment speed 360 75 74-75 73-76 74-76 75-76 74-76 67 360 74-75 73-77 72-77 75-76 73-76 430 223 222-223 220-229 219-231 221-224 221-230 214 430 222-224 215-230 213-234 221-225 215-232 750 628 625-631 618-634 615-635 623-632 619-631 615 750 624-632 610-639 601-645 621-633 611-641 1060 957 952-961 942-971 938-974 950-962 943-973 947 1060 951-961 934-980 928-989 949-963 932-984 Table 3: Penetration depth and damage diameter for the explosion experiment: The first line refers to a scatter of 5% while the second line refers to the 10% scatter in the input Set 2 Set 3 Deterministic Set 1 Set 4 Set 5 Experiment Diameter [cm] 57 56.7 - 57 56-57.2 55.6 - 57.5 56.5 - 57 56.4-57.3 56 Diameter [cm] 56.7 - 57 55.5-57.3 55 - 57.8 56.1 - 57.3 56.3-57.3
Penetr. depth [cm] Penetr. depth [cm]
28
27.8-28 27.8-28
27.3-28.2 27.-28.5
27-28.4 26.3-28.8
27.5-28.1 27.-28.3
27.7-28 27.6-28
26
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S0734-743X(15)00219-5 http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.013 IE 2597
To appear in:
International Journal of Impact Engineering
Received date: Revised date: Accepted date:
17-1-2015 29-7-2015 28-10-2015
Please cite this article as: R. Drathi, A.J.M. Das, A. Rangarajan, Meshfree simulation of concrete structures und impact loading, International Journal of Impact Engineering (2016), http://dx.doi.org/doi: 10.1016/j.ijimpeng.2015.10.013. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Meshfree Simulation of Concrete Structures und Impact Loading R. Drathi, A.J.M. Das, A. Rangarajan IIT Kanpur, Kanpur, India, e-mail: [email protected] Highlights • Uncertainty analysis determining effects of material parameters on impact resistance of concrete • Determining the importance of physical effects on impact resistance of concrete • Computational analysis of impact resistance with meshfree method Abstract The impact resistance of concrete structures is of major importance in engineering application. Computational methods are increasingly used for such types of applications but face difficulties due to the complex physical behaviour involving large deformations and large strains. Meshfree methods seem ideally suited to deal with these type of problems. In this manuscript, we present stochastic simulations based on the element-free Galerkin method to predict upper and lower bounds of the impact resistance of concrete structures. We account for stochastic distribution of material parameters and validate our results with benchmark experiments conducted by the group of Hanchak. Keywords: Meshfree Method, Impact Resistance, Strain-rate, Fracture Introduction 1. Computational modeling of concrete under impact loading remains one of the key challenges in Civil Engineering. Besides computational methods, constitutive models are an important ingredient of any mechanical model. For high dynamic loading it is important to accurately capture the material response under extreme pressure loading and the so-called strain rate effect. Popular constitutive models based on damage mechanics include the work by [1, 2, 3, 4]. In [5], the authors extended the famous Johnson-Cook (JC) [6] model to concrete materials; the JC model accounts for strain rate and temperature effects and also plastic deformations. A quasi-continuum plasticity approach capturing the dynamic buckling strength of sandwich structures was proposed by [7]. Coupled damage-plasticity models were proposed by for instance by [8, 9, 10, 11, 12, 13]. In this work, we employ a constitutive model proposed in [14]. It employs a dynamic damage variable that delays the damage evolution in order to take the strain-rate effect into account. This dynamic damage variable depends on previous damage increments and the associated damage rates. In [15], the authors extended their scalar damage model for isotropic damage to anisotropic damage by introducing a vectorial damage. Another important ingredient to model the impact resistance of concrete is the computational method. Many studies are baed on finite element analysis [16, 17, 18, 19, 20, 21, 22, 23, 24]. Often, element-deletion methods were exploited in order to allow for large deformations and complete perforations [25, 26]. Meshfree or meshless methods are good alternatives to FEM [27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37] as they can model large deformation and perforations without additional techniques and much loss of accuracy. An overview and computer implementation aspects of meshfree methods (MM) is discussed in [38]. MM have also been used to model impact events. For example, the authors in [39] accurately predicted the penetration depth and residual impact velocities compared to the experiments done by the authors in [40]. These authors reported finite element simulations underpredict the impact resistance of concrete. While dynamic fracture in MM was initially captured quite naturally by separation of nodes [41, 42, 43, 44, 45, 46, 47], the authors in [48, 32] for example found that such an approach might lead to numerical fracture. More sophisticated approaches such as XFEM (extended finite element method) [49, 50] or smoothed extended finite element method [51, 52, 53], extended MM [54, 55, 56, 57, 58, 59, 60, 61, 62, 36, 63], extended isogeometric analysis formulations [64, 65], the phantom node method [66, 67, 68, 69] or smoothed phantom node approaches [70], recent multiscale methods [71, 72, 73, 74, 75] or efficient remeshing techniques [76, 77, 78, 79, 80, 81, 82] might also be applied to dynamic fracture [83, 84]. However, while they seem well suited to capture a moderate number of propagating cracks, their performance to capture a large number of cracks in a large deformation setting still needs to be shown. A compromise to above mentioned approaches is the cracking particles method (CPM) [85, 86]. In the CPM, fracture is modelled by set of cracked particles. Several improvements have been incorporated into the CPM [87, 88, 89, 90, 91, 92]. Since MM are computational costly, they have been coupled to finite element methods [30, 92]. The two mentioned formulations have also been applied to predict the impact resistance of concrete structures. The majority of the publications (see the list above) are focused on deterministic approaches but it is well known that this can lead to unrealistic crack patterns as the ones predicted in [39]. In these simulations, cracks are too close to each other. Stochastic approaches such as introducing some randomness in the tensile strength [93] can alleviate this unrealistic behaviour [94, 95, 96]. However, none of these simulations consider stochastic material parameters though it is barely possible to calibrate the material parameters uniquely and exactly. On the other hand, every computational method needs to be validated. Classical benchmark problems for impact resistance of concrete include the experiments by [97, 98, 99, 100, 101, 102, 103]. The experiment exploited in this manuscript was carried out by the authors in [104]. In these experiments, concrete specimen were subjected to impactors with various velocities. In summary: We present stochastic simulations to predict the impact resistance of concrete. The element-free Galerkin (EFG) method is exploited in combination with a viscous damage-plasticity model [14]. A simple node splitting algorithm described in [89] has been exploited in order to avoid artificial fracture. It can be considered as a special case of the CPM. Finally, our simulations are validated by comparison to experimental data from our own laboratory and from Hanchak et al. [104]. 2. Governing Equations and Discretization We solve the equation of motion that can be stated in weak form by W int W ext W kin = 0
W in t =
W ext =
W k in =
ij
: ij d
u iti d t
u i bi d
u iu i d
(1)
, W ext and W int indicating the kinetic energy, external work and internal energy, respectively; Ω is the domain and t u = , with t u = 0 is the external boundary consisting of traction and displacement boundary conditions indicated by the subscript t and d, respectively. The components of the linear strain tensor is denoted by ij and ij are the components of the Cauchy stress tensor; the components of the traction and body force vector are given by ti and bi, respectively; ui are the components of the displacement field, ρ is the density and the superimposed ‘dot’ stands for material time derivatives. As already mentioned in the introduction, we employ the EFG method [105] to discretize the displacement field. It can be shown that the EFG approximation is given by u ( x , t ) = N I ( x ) u I (t ) (2) W kin
IS
uI being the nodal parameters of the displacement field, which are unequal to the physical displacement values at that point, or in other words u(xI) ≠ uI. The shape functions are denoted by NI(x). They are obtained from miminization of a discrete L2 norm which finally leads to:
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T
N (x) = p (x) A
1
(3)
(x) B (x)
n
A (x) =
w(X X
T
I
is the moment-matrix and B(x) is computed as
) p(X I ) p (X I )
I =1
B ( x ) = w ( x x 1 )p ( x 1 ) w ( x x 2 )p ( x 2 )
w ( x x n )p ( x n )
(4) w(X − XI) denoting the weighting function and p (X) the polynomial basis. More details are given in [105]. Exponential kernel function and linear basis functions are chosen, i.e. p T ( x ) = 1 x y . Substituting the discretization, eq. (2) into the weak form of the equation of motion, eq. (1), leads to the well-known system of equations: (5) M D = F e x t F in t where the nodal parameters uI(t) of the displacement field are stored in the global vector D. It can be shown that the mass matrix is obtained by T
M =
N N d T
(6)
The external force vector and internal force vector is given by Fext =
F in t =
N b d T
N t d T
t
B d T
(7)
N and B being matrices containing the EFG shape functions and their spatial derivatives, respectively. We use a stabilized [48] nodally regularised [32] element-free Galerkin method [105]. We take advantage of the updated Lagrangian kernel formulation presented in [39] to ensure the stability of the method while simultaneously maintaining the applicability to extremely large deformations needed for dynamic fracture and fragmentation. 3. Constitutive Model The employed constitutive model is based on the approach presented by Rabczuk et al. [14]. While the original approach is a coupled damage-plasticity model, we removed the plasticity part from the formulation which reduces the number of material parameters. Subsequently, we summarize the basic equations of this constitutive model. The strain rate ij is decomposed into an elastic part ije and a damage part ijd : ij
e
=
ij
d
(8)
ij
The stress-strain relation can be written as ij = 1 D C ijk l
(9) D = DS + DD being a damage variable which is decomposed into a static part DS and a dynamic part DD, γ is a function accounting for high hydrostatic pressure response and C ijkl denotes the components of the elasticity tensor. The formulation has been implemented in rate form as suggested in [14]. We use the same damage surfaces in compression and tension F d = c1 J 2 e
ci,
i = 1,...,4
d
c
J 2 c3 e
2
(a )
kl
c4 I1
e
e ,m ax
2 d
(10)
= 0
being material parameters, κd denotes the effective damage strain, (a )
second invariant of the elastic strain tensor and is adopted to model the damage evolution: DS = 1 e
e d 0 e d
e ,m ax
indicates the
a
th
e
I1
is the first invariant of the elastic strain tensor,
e
J2
is the
eigenvalue of the elastic strain tensor. A classical exponential function
g
d e0
D S = 0 d < e0
(11) where ed, e0 and g are material constants. A dynamic damage evolution is introduced decaying the static damage evolution. It is defined by DS
t
DD =
=0
t
H (t ) d
(12)
H(t − τ) being a monotonically decreasing history function which decays from 1 to 0 with a specific time: H (t ) = e
t * L ( ) d
L ( d ) = l *
with 4.
0 = 1s
h
* w ln d
* r d
,
* d
=
d ( ) 0
(13)
1
. Results As stated in the introduction, we used our own experiments and the experiments by Hanchak et al. [104] to validate the proposed computational model. In the experiments by Hanchak and co-workers as illustrated in Fig. 1, concrete structures were subjected to impact loading with different velocities of the impactor. They tested two types of concrete, a high performance concrete (HPC) and standard concrete; the latter one had a compressive strength of 50 MPa. Since our own concrete had a similar compressive strength, we will use the latter one in order to validate our model. In the Hanchak experiments, all concrete specimen were perforated completely. Initial studies showed discrepancies between the results of 2D models and full three-dimensional models; we therefore show only the results of our 3D simulations. We needed roughly 2 million nodes to obtain a convergent residual velocity of the impactor and the convergent fracture pattern. We assumed stochastic material parameters and the mean values of all material parameters are shown in Table 1. They have been classified into five different categories: 1. Initial parameters of the intact material, i.e. elasticity modulus, poisson’s ration and initial density. 2. Parameters associated to the damage evolution. These parameters can be further classified into parameters associated to the static damage evolution e0, ed and gd (the key parameters are ed and gd controlling the ductility and maximum strength of the concrete) and the parameters associated to the dynamic damage evolution c t 1 and c t 2 which determine how the dynamic damage decays in time. 3. The parameters associated to the damage surface c1 to c4. 4. The reduction factors rc and rt controlling the amount of tensile damage done by compressive loading and vice versa. 5. The parameters associated to the pressure-dependence of concrete. The parameters associated to the damage surface are assumed to be deterministic while all other parameters are subjected to a uniform PDF
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(probability distribution function) with a standard deviation of 5% and 10%, respectively. We used sampling method (LHS) modifying only one class of material parameters while fixing all other material parameters. A sensitivity analysis (SA) as presented in [106, 107, 108, 109, 110] would lead to more quantitative results but such studies will be done in the near future. The minimum and maximum residual velocities of the impactor for simulations based on a standard deviation of 5% and 10% are summarized in Table 2 for the different initial impact velocities. The values of the deterministic simulation based on the mean values according to Table 1 are shown in Table 2, too. Obviously, uncertainties in the material parameters substantially affect the residual velocity of the impactor. The first parameter set (associated to the initial parameters of the intact material) as well as the fourth parameter set (associated to the reduction factors) barely influence the residual velocity. This is indicated by the fact that the scatter in the output is much smaller than the scatter in the input. On the other hand, the second parameter set (associated to the damage evolution) and the fifth parameter set (associated to the pressure dependence of concrete) significantly influence the residual velocity of the impactor. This observation holds for both standard deviation of 5% and 10%, respectively. The most pronounced influence is obtained by the parameter set related to the dynamic strength increase. We also carried out simulations and sampling methods to determine upper and lower bounds of the residual impact velocity varying two parameter sets while keeping all other parameter sets constant. As the third and fifth parameter sets seems to be the most influential ones, those parameters are varied. However, the minimum and maximum residual velocities barely changed. As can be seen, there is a good agreement between experimental data and computational predictions. The last benchmark problem is a concrete specimen subjected to contact detonation. The set-up is illustrated in Figure 2. The explosive is modelled with the JWL-EOS (John-Wilkonson-Lee equation of state) using the EFG discretization. Due to the small amount of the charge, the concrete specimen was not perforated. The penetration depth and the effective diameter at the upper surface was measured. The same mean values of the material parameters as listed in Table 1 are adopted. Also the PDF and standard deviations of the previous example are employed. Table 3 depicts the computational results. Similar observations as made for the Hanchak experiments hold also for this experiment. However, the uncertainties in the input parameters are smaller. The largest effect is attributed to the parameter set 3 related to the dynamic damage evolution followed by parameter sets 2 and 5. The smallest influence are related to parameter sets 1 and 4. 5. Conclusion We presented stochastic simulations predicting the impact resistance of concrete. Stabilized nodal integrated EFG method is exploited to discretize the displacement field. This method is particularly suitable for problems involving large deformations as they commonly occur under impact and explosive loading. A scalar dynamic damage model has been used accounting for the pressure dependence of concrete as well as the strain-rate effect. The computational method has been is validated through two benchmark problems: 1. The impact experiments by Hanchak et al, who measured the residual velocity of the impactor and 2. Our own experiment where concrete specimen are subjected to a contact detonation. In our own experiments, we measured the penetration depth as well as the effective damage diameter at the upper surface. Our computational results are in good agreement to the experimental results. We employed sampling method to account for uncertainties in material parameters. 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Figure 1: Hanchak impact experiment Figure 2: Explosion experiment Table 1: Parameters of the constitutive model Young’s Poisson ratio density modulus E [g/mm3] [GPa] 52 0.22 2700 e0 ed gd
c t1
ct2
0.55
0.0001 c1 0.01234 rt 1.2 av
0.0003 c2 0.0252 rc 18.8 bv
2.3 c3 0.7821
0.032 c4 0.3464
ev
e v , th
0.7
3.4
0.02
0.007
Table 2: Residual velocity ([m / s]) of the impactor in dependence of the impact velocity ([m / s]) and the scatter in the input parameters: The first line of the same impact velocity refers to a 5% scatter while the second line to the 10% Impact Determ Set 1 Set 2 Set 3 Set 4 Set 5 Experiment speed 360 75 74-75 73-76 74-76 75-76 74-76 67 360 74-75 73-77 72-77 75-76 73-76 430 223 222-223 220-229 219-231 221-224 221-230 214 430 222-224 215-230 213-234 221-225 215-232 750 628 625-631 618-634 615-635 623-632 619-631 615 750 624-632 610-639 601-645 621-633 611-641 1060 957 952-961 942-971 938-974 950-962 943-973 947 1060 951-961 934-980 928-989 949-963 932-984 Table 3: Penetration depth and damage diameter for the explosion experiment: The first line refers to a scatter of 5% while the second line refers to the 10% scatter in the input Set 2 Set 3 Deterministic Set 1 Set 4 Set 5 Experiment Diameter [cm] 57 56.7 - 57 56-57.2 55.6 - 57.5 56.5 - 57 56.4-57.3 56 Diameter [cm] 56.7 - 57 55.5-57.3 55 - 57.8 56.1 - 57.3 56.3-57.3
Penetr. depth [cm] Penetr. depth [cm]
28
27.8-28 27.8-28
27.3-28.2 27.-28.5
27-28.4 26.3-28.8
27.5-28.1 27.-28.3
27.7-28 27.6-28
26
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