structures under shock & impact vi, c.a. brebbia & n. jones … · 2014-05-20 · type i,...

Numerical study on impulsive local damage of

reinforced concrete structures by a sophisticated

constitutive and failure model

M. Itoh\ M. Katayama*, S. Mitake , N. Niwa , M. Beppir* & N.

Ishikawa

1 Science & Engineering Division, CRC Research Institute, Inc., Japan

Nuclear Power Engineering Corporation, Japan

National Defense Academy, Japan

Abstract

This study offers a new constitutive and failure model for the concrete whichcovers a wide range of the impact phenomena. The authors extended the staticnon-uniform hardening plasticity model developed by Han and Chen, mainly tothe high strain rate region. The model, called 'Dynamic Drucker-Prager CAPModel', was implemented in a hydrocode AUTODYN™ through the subroutineswhich are open to the user. Then, a number of numerical analyses were carriedout by using the material model, which simulate the impact tests on reinforcedconcrete structures by the aircraft. In the experiment, the engine was modeled bya rigid or a deformable missile on three scales: full, 1/2.5 and 1/7.5. The pene-tration, perforation and scabbing modes of the reinforced concrete were numeri-cally simulated in good agreement with the experiment. The numerical resultswere discussed over the comparison with the Degen's damage equation.

1. Introduction

The concrete indicates a complicated behavior in the compressive and the tensileregion, especially when subjected to the severe impact loading. Therefore, anumber of material properties are indispensable to describe such highly non-linear and dynamic phenomena. On the other hand, it is general that only thelimited properties are measured in the usual material test of the concrete, i.e.density, elastic moduli and static compressive strength. So it is of great use if the

Structures under Shock & Impact VI, C.A. Brebbia & N. Jones (Editors) © 2000 WIT Press, www.witpress.com, ISBN 1-85312-820-1

570 Structures Under Shock and Impact VI

present scheme provides us the recommended values of the dynamic materialproperties based on the correlation between the static compression strength andthe other properties. We adopted two-parameter Drucker-Prager criterion insteadof the four- or five-parameter failure surface used by Han and Chu in the staticnon-uniform hardening plasticity model [1].

In this paper we show the numerical results only on a relatively high-velocity(100-400 m/s) impact problem as a concrete structure. However, we will de-monstrate and verify in other opportunities that the present material model is alsoapplicable to the lower velocity impact problems of the concrete.

2 Dynamic Drucker-Prager cap model

In this section we present a new constitutive equation for concrete which is basedon the static non-uniform hardening plasticity model developed by Han andChen [1]. They utilize three different types of yield surfaces: an initial yield sur-face, a loading surface and a failure surface. The initial yield surface defines theelastic limits in multi-axial stress states. The failure surface is a bounding surface,i.e. no stress state is allowed to exist outside it. The shapes of the initial yieldsurface and the failure surface remain unchanged in the stress space duringloading. However, the loading surface changes its shape non-uniformly from theinitial yield surface to the failure surface with the development of the effectiveplastic strain.

The proposed constitutive equation utilizes the basic framework of the Han-Chen model except the following three major modifications: 1) the strain ratedependency is considered, 2) The two-parameter Drucker-Prager criterion isadopted instead of the four- or five-parameter failure surface of the Han-Chenmodel in order to minimize the number of material constants, 3) The non-associative radial return method is used to impose the plastic flow condition.

2.1 Failure surface

First we define the static failure surface. The Drucker-Prager criterion is ex-pressed in terms of the stress invariant as

0, (1)

where /, and J^ are the first invariant of the Cauchy stress tensor and the sec-

ond invariant of the deviatoric tensor respectively, a and k are material con-stants.

Multiplying V3 on both sides of eqn (1) yields

(2)

in terms of the equivalent stress a^ = 3J and the pressure p = -Ij3

(positive in compression). The yield function may be written as / = a^ -o^

and the yield criterion requires / = 0 when the plastic flow takes place. The


Structures Under Shock and Impact VI 571

yield stress <jy is then expressed as

c^=3V3a;, + V3&. (3)

This yield stress grows without bounds as the pressure develops in compression.

In order to remedy this we introduce the maximum yield stress a"'°* :

(4)

The material constants a and k may be expressed as the functions of the com-pressive strength /J and the tensile strength // ,

Secondly we incorporate the strain rate effect. A dynamic compressive

strength /J and a dynamic tensile strength // are obtained [2], as

(6)

(7)

where £ is the strain rate £ = j(2/3J£ £ and £•• the strain rate tensor. Sub-

stituting the dynamic strengths (//,//) for the corresponding static ones

(fc>ft) of eqn (5) gives the dynamic slope a and the dynamic cohesion k .

We utilize these to define a dynamic failure surface and a dynamic yield stress:

/ = a, - (^ = 0, d^ = /M^a"\33c^ + 3 A. (8)

2.2 Loading surface

The loading surface is defined to have a strain hardening property with the de-velopment of the effective plastic strain £^ . Its shape matches precisely the

shape of the initial yield surface when e^ = 0 , and expands non-uniformly until

it conforms to the failure surface when £^ = e^ . For this feature, we let

/ = - o/,P =0. (9)

The shape factor s (0 < < s < 1 ) above prescribes the three types of yield

surfaces according to its values: the initial yield surface (s = ky\ the loading

surface (ky < s < 1 ) and the failure surface ( s = 1 ).

2.2.1 Hardening parameter: K^ and K^,

As stated above, the strain hardening feature is incorporated by use of the shape



factor, which is a function of hardening factors: K^ and K^ . We propose the

following functional form:

The factors K^ and K^ are given by replacing £ with ^=5(e^/e^) and

^ =5(£p I el), where B[ and el are the dynamic compressive strain and the

dynamic tensile strain respectively. We assume that these dynamic strains areobtained from the respective static ones by using the same enhancement factorsfor the strengths in eqn (6) and eqn (7).

2.2.2 Shape factor: sWe define the shape factor as a combination of functions s = s(s , s^ , s^ ) in the

same manner as [3]. It is divided into three parts according to the loading pres-sure as,

Tension part ( /?,„„ < p < 0 ): s, (^ ) - , (11)(\ tx* _ %/•Ej,,/?)= KQI +— - - —p , (12)

T»C

Compression part (& < p ): s,(e,,, p)= -p + P-c (13)

\y ~~^>c)

The material constant ^ in eqn (12) and eqn (13) denotes the horizontal coor-

dinate (i.e. pressure) of the apex of the cap surface. The foot ^ of the loading

surface is given by

g=#-,cj, (14)

where A is a material constant. Note that ^ increases to infinity as K^ , de-

fined by eqn (10), approaches unity with the development of e^ .

2.3 Material properties

Dynamic material data for concrete are scarcely obtained in general because ofthe high cost of experiments. This sets an obstacle to a practicing engineer whenmodeling a concrete material. Therefore, it is the purpose here to present repre-sentative values for the material properties of the proposed model. Note thatmost of them are evaluated only from the static compression strength.

1) Uniaxial static compressive strength // : 20-50 (MPa)

2) Uniaxial static tensile strength /': O.I/; (MPa).

3) Uniaxial static compressive strain z[ : 2.7 x 10 {l45/«.'(MPa)}"* (-).

4) Uniaxial static tensile strain £,' : 10~* (-).


Structures Under Shock and Impact VI 573

5) Initial value of the hardening factor ky : 0.3-0.5 (-).

6) Abscissa value of the cap apex ^ : ///3 (MPa).

7) Factor controlling the foot of the yield surface A : 2(l - ky )/,' / 3 (MPa).

8) Maximum yield stress a™* : Maximum shear stress T^ (MPa).

We recommend using T^ for a™* when the latter is not available. The

value of T^ is evaluated using the Mohr-Coulomb law as follows:

(15)

where c and (f) are the cohesion and the friction angle when £ takes its

maximum value ( « 10* ),

cos(j) = 2 n/(n + 1), sinQ =(n- \}/(n + 1), (16)

(17)

(18)

3 Numerical Analysis

In order to verify the present material model of the concrete, we carried out thenumerical analysis to simulate an experimental test program conducted by Mutoet al. [4]. The main purpose of the reference test is to investigate the local dam-age of the reinforced concrete structure caused by the accidental aircraft impact.The test program consists of three scale models for F-4 Phantom fighter: 1/7. 5-,1/2.5- and full-scale models. Two types of projectiles, rigid and deformable,were adopted to model the engine part of the aircraft in the experiment. A two-dimensional coupled hydrocode based on the explicit finite difference method,AUTODYN-2D™ [5], was used throughout in the present study.

3.1 Geometrical model

Figure 1 summarizes the geometrical models used in the axisymmetric numericalanalysis. The targets are square in the experiment, while they were assumed to bethe circular panels with the equivalent sectional areas. As shown in the upperhalf of each model in Figure 1, almost all the parts were modeled by the Lagran-gian frame of the reference, shell elements were applied to the reinforcement andthe thin parts of the 1/7.5-scale deformable missile. Therefore, the reinforcementwas also modeled by the thin circular plate with the equivalent mass. It should benoted that the bending moment was taken into account for the shell elements inthe missile, while was ignored for the shell element modeling the reinforcement,i.e. was assumed to be membrane. Each lower half indicates the numerical meshused in the calculation. The concrete panel was constrained at the radial end tothe axial and radial directions. To the interface between the missile and the con-crete panel, the slide/impact interaction boundary condition without friction was



a) 1/7.5-scale model, b) 1/7.5-scale model,Type I, rigid Type II, defoemablematerial location

c) 1/2.5-scale model,Type I, rigid

applied, and all theelements in the con-crete, missile andreinforcement werealso enabled to inter-act with the elementsthat exist in the samecomponent aftersubjected to a seriousdeformation. The ca-pabilities of the inter-action and the nu-merical erosion trig-gered by the maxi-mum geometricstrain enable us tosimulate the compli-cated deformationprocesses.

3.2 Material model

The material modelof the hydrocodeconsists of the equa-tion of state, consti-tutive relation andfailure criteria. Table1 indicates the sum-mary of the materialdata for the concreteused in the numericalanalysis. The differ-ent concrete and rein-forcement were usedfor each scale modelin the experiment, sothe material data ofthem in the numericalanalysis have some

differences as shown in Table 1. And note that two kinds of concrete with fairlydifferent compressive strengths were used for the 1/2.5-scale model. The linearequation of state and the strain-hardening model was applied to the reinforce-ment, and it was assumed to be fractured physically and eroded numerically atthe strain of 16.6 % (true). The steel material of the missile was assumed to bebasically the same as the reinforcement, but was assumed to be fractured physi-cally at the strain of 16.6 %, thereafter eroded numerically at the geometric strain

d) full-scale model,Type I, deformable

a') 1/7.5-scale model,Type II, rigid

Figure 1: Geometrical models and numerical meshesin the analysis.


Structures Under Shock and Impact VI

Table 1. Material data of the concrete.

575

ScaleEquation of StateConstitutive ModelDensity (kg/nf)Young Modulus (GPa)Poisson Ratio ( — )Bulk Modulus (GPa)Shear Modulus (GPa)Hydro-Tensile Limit (MPa)Compressive Strength (MPa)Tensile Strength (MPa)Erosion Strain (%)

1/7.5Linear

Present Model230023.80.213.29.92-6.025.52.55200

1/2.5Linear

Present Model2300

26.8 30.602

14.9 16.711.2 12.7

-6.032.3 42.13.23 4.21

200

Real SizeLinear

Present Model230022.80.212.79.52-6.023.52.35200

of 200 %. The Mie-Gruneisen type shock Hugoniot equation of state and theelastic-perfectly-plastic model was applied to the aluminum material, and it wasassumed to be fractured physically at the strain of 33.6 %, thereafter eroded nu-merically at the geometric strain of 200 %.

3.3 Numerical cases and results

The cases and results performed in the numerical analysis are summarized in Ta-ble 2-4, respectively, for 1/7.5-, 1/2.5-, full-scale model tests reported in the ref-erences [4]. Additionally, complementary calculations were performed for the

Table 2. Summary of the 1/7.5 model comparing the analysis with the test.

No.

1-1

1-2

i •}

1-4

1-5

1-6

1-7

1-8

,/

analysistest

analysistest

analysistest

analysistest

analysistest

analysistest

analysistest

analysistest

thickness(mm)6060150150150150180180

,18P_180180180210210350350

imp. vel.(m/s)194194198198143143216216126126104104213213198198

res. vel.(m/s)138143

-1.32N/A-1.94N/A-0.546N/A-3.32N/A-2.32N/A-1.68N/A-11.8N/A

projectile

deformabledeform able

Rigid—Rigid—

RigidRigidRigidRigidRigidRigidRigidRigidRigidRigidRigidRigid

front dia. (mm)ver.-38178

N/A

155

N/A

N/A

N/A

N/A

320

hor.#%f171

N/A

185

N/A%jN/A

N/A&*N/A

302

ave.155175232N/A169170236N/A173N/A181N/A199N/A189311

depth(mm)-;W'"~~91.3

32.3N/A"'66.8

27.1N/A24.8N/A47N/A3742

rear dia. (mm)ver.•*.%420

N/A

590

N/A

N/A

N/A~~ -N/A

i-

hor.71%335

N/A

440

N/A

N/A

N/A

N/A

,M

ave.292378592N/A482515294N/A202N/A222N/A268N/A

status

perforatedperforatedscabbed

perforatedscabbedscabbedscabbed

just perforatedscabbedscabbedscabbedscabbedscabbedscabbed

penetratedpenetrated

Table 3. Summary of the 1/2.5 model comparing the analysis with the test

N~

2-12-22-32-42-52-6

High Str.High Str.High Str.NormalNormalNormal

Thickness(mm)350450550450550600

Imp. Vel.(m/s)206223216211211211

Res. Vel(m/s)37.8-6.29-5.64

i_-1.30-7.34-6.36

Front Dia.(mm)764662682660626664

Depth(mm)

:/%7' -"8779.431193.484

Rear Dia.(mm)16421562135421401204 j1302

AnalysisPerforatedScabbedScabbedScabbedScabbedScabbed

StatusTest

PerforatedScabbedScabbed

Just PerforatedScabbedScabbed



Table 4. Summary of the full-scale modelcomparing the analysis with the test.

^Experimental result: 2.3 -2.8 mmNo.

3-1

Thickness(mm)1600

fap. Ve(m/s)213

Res. Vel.(m/s)-2.71

Front Dia(mm)2.12*

Depth(mm)167

StatusAnalysisPenetrated

TestPenetrated

(c')Exp. 1-8

Figure 2: Numerical and experimental resultsfor 1/7.5-scale model cases.

1/7.5-scale and rigidmissile case in order tocompare the presentnumerical results withDegen's damageevaluation equation [6].

Table 5 shows the numericalconditions and results for thecomplemental analysis cases.

In Figure 2, (a), (b) and (c)depict the deformed profiles ofthe concrete panels and mis-siles simulated by the presentnumerical analysis, respec-tively, for the case 1-1, 1-3 and1-8 shown in Table 2. Theseresults should be comparedwith the corresponding ex-perimental results below in thefigure: (a'), (b') and (c'). Thenumerical results can be con-sidered to simulate the overalldeformations of the reinforcedconcrete panels.

Figure 3 shows the com-parison of the calculated de-formation of the reinforcedconcrete panels with differentcompressive strengths for the1/2.5-scale model. We can seethat the panel of (a) has lessdamage than that of (b), becau-se of its higher compressivestrength, although it impactedat a higher velocity.

Figure 4 compares the ve-locity history at the tail of themissile between the calculationand the experiment for the full-scale model. All the experi-mental results are obtained bythe real engine impacts. Un-fortunately, there are no reports

on the history of the simplified missile impact tests, so the numerical result can-not be compared exactly with those four experimental results. However, the factthat the simplified missile in the calculation decelerated earlier than the real


Structures Under Shock and Impact 17 577

Table 5. Summary of the complemental 1/7.5-scale model.

No.

I'-lr-2r-sr-4r-5

Thickness(mm)1502101201201600

Mass Imp. VelJ Res. Vel.(kg)6103.63.63.6

(m/s)198213100200400

Perforation9.81294-0.044417.9

StatusAnalysisPerforatedPerforatedScabbedPerforatedPerforated

Degen Eq.PerforatedPerforatedScabbedPerforatedPerforated

elastic

y=211m/s=32.3 MPa

4[at 20 msec(a) Cal. 2 - 2 (b)"Cal. 2 - 4

Figure 3: Compressive strengtheffect (1/2.5-scale).

engines in the experiment iscoincident with the experi-mental results about the de-formations of the concretepanels: one of them is per-forated, two are scabbedand one is (imperfectly)penetrated, while the nu-merical result shows pene-

trated.In Figure 5, all the 1/7.5-scale numerical

results including the complemental cases areplotted together with the Degen's equationcurve. The scabbed points (grayed) shouldbe considered as non-perforated points like apenetrated point, according to the meaningof the equation. This figure indicates thatthese numerical results are correspondentwith the Degen's equation except the case 1-2.

4. Discussion and conclusions

By using the material data derivedin the section 2, which can bespecified only from the elasticmoduli and the static compressivestrength of the concrete, twentynumerical analyses on the high-velocity impact were carried outin the section 3. Figure 2 indicatesthat the present numerical modelcan nearly predict both the cra-tering in the front side and thescabbing in the rear side. Figure 3demonstrated that the differencesof the elastic moduli and the staticcompressive strength cause a sig-

nificant difference in the deformations of the reinforced concrete panels, alt-hough the numerical result is still underestimated than the experimental one. Wecan understand, from Figure 4, that the behavior of the deformable missile mightbe appropriately simulated. Figure 5 and Table 2-5 also show that the presentnumerical results are almost coincident with the experimental results as well asthe Degen's equation from the viewpoint of the ballistic limit for the rigid mis-sile impact on the reinforced concrete.

In this study axisymmetric model was applied for all the calculations, so the

16 20Time (ms)

Figure 4: Velocity history of the missile(full-scale).



reinforced concrete was modeled as a circular panel. However, there might existthe distribution of the damage in the concrete in the circular direction. Thismeans that the experimental result, as well as Degen's equation, has the greaterpossibility of the perforation than the 2-D calculation. Moreover, the reinforce-ment was modeled as a circular thin plate, so the plate contributed to prevent thepanel from the perforation unless it is eroded numerically.

Work in near future will concentrated on expanding and enlarging the analy-sis area by the present material model for the concrete, from relatively low-velocity through higher velocity impact problems.

Acknowledgement — The authors wish to gratefully acknowledge Dr. S. Tamura, CRC Research In-stitute, for his performing the numerical analyses.

References

[1] Han, D. J. & Chen. W. F., A nonuniform hardening plasticity model for con-crete materials. Mechanics of Materials, 4(4), pp. 283-302, 1985.

[2] Yamaguchi, H. et al., Stress-strain relationship for concrete under high tri-axial compression part2 rapid loading. Trans. Archit. Inst. Japan., 396, pp.50-59, 1989 (m JqpaM&se).

[3] W. F. Chen , Constitutive Equations for Engineering Materials Volume2:Plasticity and Modeling (Chapter 6). Theory of Concrete Plasticity, Elsevier:Amsterdam, pp. 840-849, 1994.

[4] Muto, K. et al., Experimental studies on local damage of reinforced concretestructures by the impact deformable missiles part 1~4. Trans, of 10thStructural Mech. in Reactor Tech., J, Anaheim, pp. 257-284, 1989.

[5] Bimbaum, N. K. et al., AUTODYN—an interactive non-linear dynamicanalysis program for microcomputer through supercomputers. Trans, of 9thStructural Mech. in Reactor Tech., B, Lausanne, pp.401-406, 1987.

[6] Degen, P. P., Perforation of reinforced concrete slabs by rigid missiles. Jour-nal of the Structural Division, ASCE, 106, No. ST7, 1980,

D = 101 mmfc = 25.5 MPaV = 215 m/s

=> 350I 300g 250\ 200- 150§> 100'- 50

n

Ii, O Penti Q Scat

rv3trated © !)bed (Compl.)

sgen's EquationScabbed # Perforation• Perforated (Compl.)

^ %. qqrTrZL — ""', c

//_f ~

u-

^ •#

M •~

i — " —1

EE 300%250

1 | 200e 150

* 50n

[/

/

ko

LM

i

i5—

o

^

i

W — '/c=2

01 mm.6kg5.5 MPa

x-<s

^

100 200 300 400Projectile Mass (kg) impact Velocity (m/s)

Figure 5: Ballistic limit comparing the present numerical results withDegen' equation (1/7.5-scale).


structures under shock & impact vi, c.a. brebbia & n. jones … · 2014-05-20 · type i,...

Documents