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Transactions, SMiRT-24 BEXCO, Busan, Korea - August 20-25, 2017 Division III

A physically motivated element deletion criterion for the concrete damage plasticity model Alexis Fedoroff1, Juha Kuutti2, and Arja Saarenheimo3 1

Research Scientist, Structural integrity, Technical and Research Centre of Finland, Ltd, Finland Senior Scientist, Structural integrity, Technical and Research Centre of Finland, Ltd, Finland 3 Research Team Leader, Structural integrity, Technical and Research Centre of Finland, Ltd, Finland 2

ABSTRACT An important part of a nuclear power plant (NPP) structural safety assessment deals with aircraft impact on concrete structures. Since full scale tests are virtually impossible to realize, one has to resort to numerical simulations that need to be validated with benchmark tests. In such simulations, it is necessary to push the analysis all the way up to local destruction of the reinforced concrete structure in order to investigate its resistance against punching and penetration. In terms of material modelling one needs to manage the transition from continuum microscale damage at a material point to macroscopic fracture mechanics. In the context of finite element computations, it seems that element deletion is the most effective way to represent macroscopic cracks and disaggregation in concrete. However, the bottleneck of this approach has been, so far, the deletion criterion. This article proposes a physically based element deletion criterion to be used in impact simulations using damage-plastic material models. INTRODUCTION The assessment of the safety and the structural integrity of NPP containment and other critical buildings involves a thorough analysis of accidental dynamic events: seismic, impact and blast. Scientific research on those topics is highly challenging, both from the experimental and computational point of views. Nevertheless, it is important to investigate these phenomena because such research provides essential information for the authorities to perform their regulatory duties associated with nuclear power plant commissioning. This article concentrates on the computational aspects of impact loaded reinforced concrete structures, where impact speed magnitude is around 100m/s. More precisely, this article treats some of the modelling details relative to the concrete material model, in order to simulate both hard missile impacts, where the projectile penetrates the target structure, and soft missile impacts, where the missile deforms but does not cut through the target. The concrete material model used in the computations presented hereafter is the Concrete Damage Plasticity (CDP) model provided by the finite element software Abaqus, (Simulia, 2016). The CDP model is founded on the yield surface proposed in the “Barcelona” concrete model, (Lubliner, Oliver, Oller, & Oñate, 1989), and a hardening-softening behaviour proposed in (Lee, 1996) and (Lee & Fenves, 1998). The hardening-softening behaviour is calibrated to match the uniaxial concrete tests in tension and compression performed at a quasi-static speed. Since in this publication the applications of interest comport high loading velocities, the original CDP model is unable to describe correctly the physical phenomena. It is, therefore, necessary to customize the Abaqus CDP model by introducing a loading speed dependency in the material model as well as element deletion to materialize macroscopic concrete fracture and pulverization during missile impact. This customized Abaqus CDP model shall then be validated against experimental impact benchmark results.

24th Conference on Structural Mechanics in Reactor Technology BEXCO, Busan, Korea - August 20-25, 2017 Division III

Simulation of missile impacts on reinforced concrete structures Experimental study of missile impacts on reinforced concrete structures has recently been undertaken in MEPPEN, (Rüdiger & Reich, 1983) and IRIS, (Vepsä;Saarenheimo;Tarallo;Rambach;& Nebojsa, 2012), projects. The main purpose of those projects was to test the behaviour of reinforced concrete structures under missile impact. Both soft and hard missiles have been tested, and the concrete structure was investigated both for its resistance against punching, bending and vibration damping properties. In a PhD thesis, (Vu, 2013), the PRM concrete model, (Rouquand;Pontiroli;& Mazars, 2007), was implemented in Abaqus to simulate the hard and soft missile tests of the IRIS benchmark using finite elements. The PRM model is a scalar damage-elasticity model coupled with an elasto-plastic model, where the stiffness degradation damage variable is computed as a weighted average of compression and tension damages, respecively. In another publication, (Rodriguez;Martinez;& Marti., 2013), the IRIS benchmark hard and soft missile impacts were simulated using the Abaqus CDP model with confinement stress dependency of the hardening/softening behavior in compression. This is a purely elasto-plastic model without stiffness degradation. The present publication extends the Abaqus CDP model to include strain rate dependence both in compression and in tension, coupled with physically meaningful element deletion criteria both in tension and in compression. It must also be noted that in addition to the simulation of benchmark impact tests, there is a strong need to produce credible simulations of full scale aircraft impacts on nuclear power plant containment buildings. For example, (Arros & Doumbalsky, 2007) simulated with LS-DYNA the impact of a Boeing 747 carrier aircraft on a fictitious nuclear power plant building and compared the results of the coupled simulation with a simulation using the Riera force-time curve, (Riera, 1968). In the publication (Fedoroff, 2016), a Boeing 777 impact on a fictitious nuclear power plant control building was simulated using Abaqus and the CDP material model. One of the conclusions that are recurrent in the cited publications is the necessity to include the dynamic increase of concrete strength in the concrete material model to simulate faithfully hard missile impacts. Dynamic increase of concrete strength

Figure 1 Strain rates for different loading velocities, (fib, 2010)

If a hardening-softening law taken from quasi-static concrete compression and tension tests is used, the CDP model does not provide reliable results in impact simulations. The reason is that the Dynamic Increase Factor (DIF) plays a non-negligible role in high strain rate events. As shown in Figure 1, strain rates in the concrete varying from 1 to 10 s -1 are typically observed in impact tests and even higher strain rates can be observed in blast tests. Studies using the Split Hopkinson Pressure Bar (SHPB) both in compression, (Bischoff & Perry, 1991), (Ross;Tedesco;& Kuennen, 1995) and in tension, (Klepaczko & Brara, 2001), show a noticeable increase in the DIF for strain rates higher than 1 s -1. The increase is even more pronounced in tension tests. Likewise, a dynamic increase in the fracture energy can be observed in SHPB tests, (Weerheijm & Vegt, 2010). Aside from Hopkinson tests, dynamic increase has also been observed in direct tensile tests, (Min;Yao;& Jiang, 2014), with strain rates varying from 10 -6 to 10-1 s-1.


Figure 2 shows the increase of compressive strength as the strain rate increases and Figure 3 the increase of tensile strength. The data has been collected from numerous experiments. Although there seems to be enough experimental data for the scientific community to reach a consensus about the fact that concrete is strain rate dependent both in uniaxial compression and tension, there is not yet clear unanimity about the physical phenomena at microscale that could possibly explain the dynamic increase. Confinement pressure due to inertia effects, rate dependent capillary forces due to pore water content and considerations from fracture mechanics have been proposed as possible causes for dynamic increase of concrete properties.

Figure 2 Compressive dynamic strength increase, (Bischoff & Perry, 1991)

Figure 3 Tensile dynamic strength increase, (Eibl & Bachmann, 1993)

In uniaxial compression tests at moderate strain rates (10 -6 to 10-1 s-1), it has been observed, (Rossi, 1991), that the dynamic increase factor depends on the moisture content of the concrete specimen. Hence, it has been proposed that the pore water generates capillary forces, which are proportional to the deformation speed of the pores themselves and explain the dynamic increase of peak stress. Likewise, the same phenomenon has been put forth for uniaxial traction at moderate strain rates, (Rossi, 1994). However, at high strain rates, from 1 to 100 s-1, a notable dynamic increase of the peak stress is observed independently of the moisture content. In uniaxial compression, the triaxial stress state resulting from a confinement due to the inertia effects has been acknowledged as the main reason behind the dynamic increase of the peak stress, (Li & Meng, 2003). Those conclusions have been made by simulating SHPB tests. Likewise, the Hopkinson tests and simulations reported in (Grote;Park;& Zhou, 2001) and (Park;Xia;& Zhou, 2001) lead to identical conclusions. The tension behaviour of concrete at high strain rates is subject to multiple discussions about the possible physical origin of the dynamic strength increase. The examination of concrete samples in a SHPB test in tension shows, (Klepaczko & Brara, 2001), that the spalling surface of the concrete cuts through the aggregates that lie embedded in the cement mortar. This contrasts with the observation made from quasi-static tensile tests, where the aggregates stay intact through the split surface. Hence, it has been proposed, (Ross;Jerome;Tedesco;& Hughes, 1996) and (Min;Yao;& Jiang, 2014), that at slow strain rates the cracks in the concrete matrix grow around the aggregates, whereas at high strain rates the cracks propagate directly through the aggregates. Hence, depending on the type and size of aggregates, the peak stress and fracture energy increase as the strain rate increases. Element deletion in impact simulations An important topic of discussion that frequently comes forth in publications on impact simulations is element deletion. In high speed event finite element simulations, “high end” continuum enhancement techniques, such as XFEM or adaptive re-meshing to follow crack propagation are too costly to be implemented. Therefore the only way to materialize macroscopic crack initiation and propagation in scabbing, spalling and punch cone formation in a cost-effective way is element deletion. In order to


implement element deletion, first, an element deletion criterion (or multiple criteria) must be chosen. In (Rodriguez;Martinez;& Marti., 2013), a fixed element deletion criterion corresponding to a cutoff value of the equivalent plastic strain in compression of 0.2 is chosen. The justification of this choice is primarily numerical, since the explicit central difference time integration, which is used in the analysis, fails if an element undergoes too large deformations in a time increment. Nevertheless, the choice of a particular cutoff value remains arbitrary. In contrast, (Ågårdh & Laine, 1999) proposes an element deletion criterion where the cutoff value of the equivalent strain depends on the confining pressure. This approach brings more physicality in the deletion criterion, because at low confining pressures the element is removed sooner that at high confining pressures. Our position on the element deletion criterion is that it should be first of all physically based, in other words a cracked element should be removed if its load bearing capacity is lost. In addition to this physical criterion, one should add a criterion that serves computational purposes so that the hard missile impact simulation can be run without numerical errors. The element deletion criteria should, therefore, be low enough to allow the formation of spalling, scabbing and punch cone in concrete, but high enough to prevent elements from disappearing in front of the missile head. DESCRIPTION OF THE CONCRETE MATERIAL MODEL The concrete damage plasticity model shall be briefly explained here after in order to provide a better insight in the phenomena involved in the simulations. A detailed description of the model can be found in (Lubliner, Oliver, Oller, & Oñate, 1989) and (Lee & Fenves, 1998). Although the original CDP model couples the elastic-damage and the elastic-plastic behaviours, for the sake of clarity, the model described here below involves only elasticity and plasticity. The yield surface and flow potential The concrete damage plasticity model assumes a non-associative flow rule. The yield function, originally given in (Lee & Fenves, 1998), is shown in Equation (1) in terms of the current stress tensor s and the cohesion stresses in compression and tension, f c and f t respectively. The yield surface is the zero set in the stress space of the yield function. ö æ ö f 1 æç F (s , f , f ) = 3J 2 (s ) +a I1 (s ) + çç (1-a ) c - (1+a ) ÷÷ s max -g -s max ÷ - f . (1) ÷ c c t 1-a çè ft è ø ø In Equation (1), a and g are constant material parameters and I1 denotes the first stress invariant, 3J 2 denotes the equivalent Mises stress and s max denotes the maximum principal stress. Recall also the Macaulay bracket notation x = 12 ( x + x ) . The flow potential, as suggested by (Lee & Fenves, 1998), is the Drucker-Prager hyperbolic function given by Equation (2) in terms of the current stress tensor and the dilation angle y , the eccentricity e and the initial cohesion stress in tension, f t0 . G (s ) =

(e f t0 tany )2 + 3J 2 (s ) + 13 tany I1(s ) .

(2)

In a general triaxial stress state it is assumed that the cohesive stresses evolve with the equivalent plastic p p strains: in compression f c ( e c ) and in tension f t ( e t ) , where the equivalent strains are defined incrementally in terms of minimum and maximum principal strain increments as follows: p p . (3) e& = -(1 - r(s )) e& and e& = r(s ) e& c min t max The weight coefficient r (s ) has a value of 1 in uniaxial tension, 0.5 in pure shear and 0 in uniaxial compression. In contrast to the hardening/softening assumptions in (Lee & Fenves, 1998), application of the CDP model to impact simulations require field variable dependency, as well. This field variable


dependency, on confinement stress in compression and on total strain rate in tension, is detailed in the following two paragraphs. Hardening/softening behaviour in compression Consider a “confined uniaxial” stress state defined by s = diag( - s cnf , - s cnf , - s axi ) , where the positive quantities s cnf and s axi ³ s cnf denote the confining pressure and the axial stress, respectively. Then the yield condition simplifies to 1 + 2a + g s axi £ s cnf + f c , (4) 1-a From the additive decomposition of small strains into elastic and plastic parts and from the elastic constitutive relations one gets the expression of the equivalent plastic strain in compression, e cp , in term of the axial total strain,

tot e axi

, and the stresses:

s axi - 2n s cnf , (5) E E denotes the elastic modulus and ν Poisson’s ratio. Equations (4) and (5) hold in particular for the peak tot value of the axial stress, s axi, peak (s cnf ) and the corresponding total strain, eaxi, peak ( s cnf ) . Both those values are assumed to be functions of the confining stress. The exact relation can be taken from triaxial experimental data, for example (Gabet, 2006). Alternatively, one can use the relation of the Eurocode, (Bamforth;Chisholm;Gibbs;& Harrison, 2008), which assumes a bi-linear relation for the peak stress and a bi-quadratic relation for the total strain. From Equation (4), one gets the peak cohesion stress in compression and from Equation (5) the plastic strain in compression at peak: 1 + 2a + g f c,peak (s cnf ) = s axi, peak (s cnf ) s cnf , 1-a (6) s axi, peak (s cnf ) - 2n s cnf p tot e c, peak (s cnf ) = e axi, peak (s cnf ) , E Assuming, as suggested in (Lee & Fenves, 1998), an exponential evolution of the cohesion stress: tot e cp = e axi -

f c (e cp , s cnf ) = f c0 ((1 + a c (s cnf )) exp( -bc (s cnf )e cp ) - ac (s cnf ) exp( -2bc (s cnf )e cp ) ) ,

where

f c0 denotes

the initial yield stress and the coefficients

ac (s cnf )

f c,peak (s cnf )

and

b c ( s cnf )

(7)

are defined as follows:

f (s ) æ f c,peak (s cnf ) ö ÷÷ - c,peak cnf - 1 a c (s cnf ) = 2 + 2 çç f c0 f c0 f c0 è ø , æ 1 + a c (s cnf ) ö 1 ÷ bc (s cnf ) = - p ln ç e c,peak (s cnf ) çè 2ac (s cnf ) ÷ø 2

(8)

one gets the following shapes for the cohesive stress curves in compression. The example plot shown in Figure 4 is for a C60/50 concrete grade with material constant values α=0.12 and γ=1.5. In Figure 5 the plot shows the corresponding evolution of the axial stress in terms of the total strain and confinement stress. The confinement dependency is the one suggested in the Eurocode 2: ì 1.0 + 5.0 s cnf s axi, peak (0) , s cnf s axi, peak ( 0) £ 0.05 s axi, peak (s cnf ) s axi, peak (0) = í (9) î1.125 + 2.5 s cnf s axi, peak ( 0) , s cnf s axi, peak ( 0) ³ 0.05 tot tot e axi, peak (s cnf ) e axi, peak ( 0 ) = (s axi, peak (s cnf ) s axi, peak ( 0) )

2


In a general triaxial stress state the confinement stress is evaluated using a linear combination of the current maximum principal stress and largest ever maximum principal stress: s cnf (t ) = 0.2 - s max ( t ) + 0.8 max - s max (t ) (10) t

Figure 4 Cohesion stress evolution

Figure 5 Axial stress evolution

Softening behaviour in tension Consider a uniaxial stress state defined by s = diag( s axi , 0 ,0 ) , where the positive quantity s axi denotes the axial stress in tension. Then the yield equation simplifies to s axi £ f t . From the additive decomposition of small strains into elastic and plastic parts and from the elastic constitutive relations one gets the tot expression of the equivalent plastic strain in compression, e cp , in term of the axial total strain, e axi , and the p tot stresses: e t =e axi + s axi E . Assuming an exponential softening behaviour of the cohesive stress in tension, as suggested in (Lee & Fenves, 1998), one gets the following expression for the cohesive stress in tension in terms of the equivalent plastic strain in tension and the maximum principal total strain rate: æ f (e& tot ) l ö tot tot f t (e tp , e&max ) = f t0 (e&max ) exp çç - t0 maxtot ch e tp ÷÷ . (11) è Gf (e&max ) ø In the material model implementation the strain rate is evaluated as a linear combination of the current strain rate and the highest ever strain rate. The dynamic increase factor for the peak stress can be chosen as a bi-linear relation on a logarithmic scale, as suggested in (fib, 2010). Likewise, a bi-linear relation on the logarithmic scale can be chosen for the dynamic increase factor of the fracture energy, (Weerheijm & Vegt, 2010). Figure 6 shows the values that have been used when computing the cohesive stress softening behaviour, shown in Figure 7, and the axial stress evolution, shown in Figure 8. The fracture energy is defined as the integral under the curves in Figure 7.

Figure 6 Dynamic increase factor in tension Element deletion criteria The proper choice of element deletion criteria is a decisive constituent of the material model, especially when the target is to simulate hard missile impacts. The proposal is a criterion based on some cutoff


values of the equivalent plastic strain in tension and compression, respectively. Looking at figures 4 and 7, one can deduce that such a cutoff value has to depend on the confinement stress for compression and the strain rate for tension. Since the cohesion stress tends in all cases asymptotically to zero, irrespective of the value of the confinement stress or the strain rate, it is tempting to consider a small threshold value for the cohesive stress as a pre-image for the corresponding plastic strain cutoff values.

Figure 7 Cohesion stress evolution in tension

Figure 8 Axial stress evolution in tension

Figures 9 and 10 show the confinement stress dependent plastic strain cutoff value in compression and the strain rate dependent cracking displacement cutoff value in tension, respectively. Both curves are constructed based on a choice of cohesive stress threshold value of 1Pa. This choice of threshold value is small enough for the element under consideration to be able to dissipate more than 99.99% of its energy before being removed. Notice, however, that in case of element deletion in a compression state, the cutoff strain has been given an upper bound of 40%. This upper bound is necessary to avoid deformations that are so large that the explicit time integration solver fails at some point during the impact simulation.

Figure 9 Compressive inelastic strain cutoff

Figure 10 Tensile cracking displacement cutoff

DESCRIPTION OF THE MODEL FOR IRIS P1 BENCHMARK TEST SIMULATION The finite element model used in the IRIS P1 benchmark test simulations consists of a quarter model of the reinforcement concrete slab and a quarter model of the missile. The slab dimensions is 2m by 2m and thickness is 250mm. Bending reinforcement of 10mm diameter bars at 90mm spacing is used in both directions, top and bottom surfaces. Concrete strength corresponds to a C60 concrete grade. The 47kg 168mm diameter hard missile is shot at 136m/s initial speed. Detailed information about the P1 test can be found in (Vepsä;Saarenheimo;Tarallo;Rambach;& Orbovic, 2012).

Figure 7 quarter model

Figure 8 reinforcement


Figures 7 and 8 show the constitutive parts of the quarter FE model. Field variable dependent CDP hardening/softening properties are as per Figures 4 and 7 with element deletion criteria as per Figures 9 and 10. Reinforcing steel is modelled using isotropically hardening Mises yield surface (initial yield stress 540MPa, maximum yield stress 605MPa at 6% plastic strain) coupled with ductile damage initiation at 12% equivalent plastic strain. After damage initiation, the stress decreases exponentially to a plastic strain of 16% at failure. The missile uses linear elastic material models both for the stainless steel cover and the lightweight concrete fill. SIMULATION RESULTS Simulations of the IRIS P1 benchmark test were run for concrete mesh size 0.01m. The formation of the initial confinement zone, where the red colour indicates high confinement pressure (Figure 9), spalling at top surface and formation of tension cracks at bottom surface (Figure 10), shear failure (Figure 11) and punch through (Figure 12) can be identified. Figure 14 shows the cut view of the IRIS P1 benchmark test to be compared with the simulations.

Figure 9 model at 0.1ms

Figure 10 model at 1ms



The speed of the missile tail was monitored both in the experimental test and the FE simulation. Figure 13 shows the evolution of the speed. Note the presence of a local minimum speed shortly after the confinement zone is fully formed, a local maximum speed after shear failure, and asymptotic decrease of the velocity towards the residual value found in the IRIS P1 experiments.

Figure 14 Cut view of IRIS P1 benchmark test concrete slab (quarter of a slab with its mirror image) Figure 13 Missile tail speed


CONCLUSIONS A customized Abaqus CDP model is presented. The object of customization is the confinement stress dependent hardening/softening behaviour in compression as well as the strain rate dependent softening behaviour value in tension. Likewise, a custom element deletion criterion is proposed, based on a confinement stress dependent plastic strain cutoff value in compression and a strain rate dependent cracking displacement cutoff value in tension. The proposed custom CDP model is validated against the IRIS P1 benchmark test, where a hard missile impacts a reinforced concrete wall. The following four qualitative aspects of the simulation were in accordance with the observations made in hard missile impacts: · The formation of a confinement zone in front of the missile head. · Element removal due to tensile cracking at the boundary of the confinement zone. This element deletion materializes in the simulation the spalling and scabbing observed in the experiments. · Element removal due to compressive failure at the boundary of the confinement zone. This element deletion materializes the shear failure observed in the experiments. · Punch through of the missile with the fragmented confinement zone still in front of the missile head. It can be claimed that the proposed material model is successful, at least qualitatively, in simulating hard missile impacts because it avoids unnecessary element deletion inside the confinement zone while enabling element deletion in the most critical tension and shear failure zones. Furthermore, the element deletion criterion is naturally deduced from the softening behaviour, which means that there is no need from the user side to consider this aspect in the simulations. REFERENCES Arros, J.;& Doumbalsky, N. (2007). Analysis of aircraft impact to concrete structures. Nuclear Engineering and Design, 1241-1249. Bamforth, P.;Chisholm, D.;Gibbs, J.;& Harrison, T. (2008). Properties of concrete for use in Eurocode 2. Surrey, UK: Cement and Concrete Industry Publications. Bischoff, P.;& Perry, S. (1991). Compressive behaviour of concrete under high strain rates. Mater. Struct., 425-450. Eibl, J.;& Bachmann, H. (1993). Festigkeisverhalten von Beton unter dem Einfluenss der Dehngeschwindigkeit. Massivbau Baustofftechnologie Karlsruhe, Universität Karlsruhe. Fedoroff, A. (2016). Feasibility study of aircraft impact simulation on nuclear powerplant. Espoo: VTT. Fedoroff, A. (2017). Continuum damage plasticity for concrete modeling, VTT-R-00331-17. VTT. fib. (2010). fib Model Code 2010. Lausanne: International Federation for Structural Concrete. Gabet, T. (2006). Comportement triaxial du béton sous fortes contraintes: Influence du trajet de chargement. Grenoble: Univérsité Joseph Fourier. Grote, D.;Park, S.;& Zhou, M. (2001). Dynamic behavior of concrete at high strain rates and pressures: I. experimental characterization. International Journal of Impact Engineering, 869–886. Hillerborg, A. (1985). The theoretical basis of a method to determine the fracture energy Gf of concrete. Materials and structures, 291-296. Hillerborg, A.;Modéer, M.;& Petersson, P. E. (1979). Analysis of crack formation and crack growth by means of fracture mechanics and finite elements. Cement and Concrete research, 773-782. Klepaczko, J.;& Brara, A. (2001). An experimental method for dynamic tensile testing of concrete by spalling. International Journal of Impact Engineering, 387}409. Lee, J. (1996). Theory and implementation of plastic-damage model for concrete structures under cyclic and dynamic loading. Berkley, California: PhD dissertation, University of Berkley.


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