M. Polanco-Loria
Department of Fracture Mechanics and Materials Testing SINTEF, Materials Technology, Trondheim, Norway
e−mail: mario.polanco@matek.sintef.no O. S. Hopperstad and T. Børvik Department of Structural Engineering
Norwegian University of Science and Technology, Trondheim, Norway
ABSTRACT
Summary This paper describes a new concrete material model specially developed to handle impact loading situations. The main feature of this model is the simplicity to include pressure sensitivity, strain rate dependency and damage behaviour effects characteristics of concrete materials. Based on the numerical predictions so far, this model represents a good compromise between simplicity and accuracy for large-scale numerical analysis of structures under impact loading.
INTRODUCTION
It is well known that concrete exhibits a rate-dependent behaviour when subjected to high rate straining, with significant increase of dynamic strength. This particular behaviour is rather important under impulsive loading, as it occurs when structures are subjected to impacts or explosions (e.g. strain rates ε ). For these strain rates dynamic strength can be enhanced, with respect to the static one, up to at least 80 % in tension and 25 % in compression.
1
1 −
> s
&
On the other hand, under a multiaxial state of stress the damage mechanisms activated are highly dependent on the loading path. Mazars [1] proposes an interesting classification to illustrate which are the main damage modes involved depending on the loading state as shown in Table 1. The complexity of a numerical model will depend on the objectives to simulate the basic mechanism involved in a certain type of problems or in the generality to consider all types of loading. In particular, for impact and penetration problems these three mechanisms: cracking, shearing and compaction are present at the same time, see Fig. 1. In addition, the modelling complexity is increased because strain rate effects (different for each mechanism) should be included.
te n s io n c r a c k in g c o m p a c tio n c o m p r e s s io n
w ith c o n fin in g p r e s s u r e Fig. 1 Mechanisms activated during impact
projectile
target
Table 1 Main damage mechanisms
Type of loading Particularity Local damage mode loading examples A: tension and
compression (low lateral pressure)
• uniaxial
cracking mode I and mode I+II
cracking mainly in mode II and III (branching) σ
ε
σ1 σ2
C: hydrostatic pressure in compression
• triaxial
no extension hardening and
stiffening
consolidation of the
microporous structure
p
µ
p
Based on these arguments this paper introduces a new concrete material model representing a nice compromise between simplicity and accuracy for concrete structures exposed, in particular, to impact loading conditions.
THE CONCRETE MODEL
The concrete model proposed [2] is an enhanced version of the original model of Holmquist et al. [3] and it uses a factorised formulation as follows:
3
In particular, for the pressure sensitivity the proposal consists of working with one simple continuous function defined as:
{ }
Npressure B P T D
F = ∗+ *(1− ) 2
The material parameters are calibrated from classical results from the literature. The rate sensitivity adopts the formulation proposed by Camacho and Ortiz [4] and largely used by Borvik for metals [5]:
N T B, *and
[ ]
CFrate = 1+ε&* 3
Finally, a reduction of the shear strength on the compressive meridian can be considered by introducing an elliptic function depending on the deviatoric polar angle and the normalized out-of-roundness parameter, , as proposed by Willam and Warnke [6] defined as:
3
FJ θ
e
[ ]
The damage behaviour is treated separately according to each mechanism. For this purpose, three damage internal variables: , representing the brittle, shear and compaction damage, respectively, are introduced.
C
NUMERICAL SIMULATION OF A PENETRATION PROBLEM
The penetration computations are based on a test performed by Hanchak et al. [7] where a square reinforced concrete plate of 610x610x178 mm was impacted. Three layers of square-pattern reinforcement steel rods with a diameter of 5.6 mm were used. The uniaxial compressive and tensile strength of the concrete was fc =48 MPa and ft =4 MPa, respectively. In addition to the pressure-compaction curves, triaxial tests were performed under various confining pressure levels so that shear strength-pressure curves could be established.
A 30-mm, smooth-bore powder gun was used to launch a 0.50 kg ogival-nose steel projectiles with a length of 143.7 mm and a diameter of 25.4 mm. The striking velocity was 434 m/s. 2D axisymmetric elements were used in the computations. In total 5000 elements were used. The erosion element option of LS-DYNA was adopted in the numerical calculations. The numerical model predicts a residual velocity of 205 m/s, which compares fairly well with the experimental residual velocity of 214 m/s reported by Hanchack, see Fig. 2. Consequently, the present model represents a good compromise between simplicity and accuracy for large-scale numerical analysis of structures under impact loading. However, further numerical validation tests are still needed.
0 50 100 150 200 250 300 350 400 450 500
0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.40
Time (ms) Velocity (m/s)
Impact
Residual
Fig. 2 Residual velocity of the projectile after perforation ACKNOWLEDGMENTS
The authors gratefully acknowledge Dr. T. Berstad at Livermore Software Technology Corporation for the numerical implementation of the present model in LS-DYNA.
REFERENCES
[1] Mazars J. (1984) "Application de la mecanique de l'endommagement au comportement nonlineaire et la rupture du beton de structure" These de Doctorate d'Etat, LMT, Universite de Paris, France.
[2] Polanco-Loria M. (2001) "Improvements to the HJC concrete model in LS_DYNA"
SINTEF report STF24 F01286, Trondheim, Norway.
[3] Holmquist T.J., Johnson G.R. and Cook W.H. (1993) "A computational constitutive model for concrete subjected to large strains, high strain rates and high pressures" Proceedings of 14th International Symposium on Ballistics, Quebec, Canada.
[4] Camacho G.T. and Ortiz M. (1997) "Adaptive Lagrangian modelling of ballistic penetration of metallic targets" Computer Methods in Applied Mechanics and Engineering, Vol. 142, pp. 269-301.
[5] Børvik T. (2000) "Ballistic penetration and perforation of steel plates" Ph.D. Dissertation, The Norwegian University of Science and Technology, Norway.
[6] Willam K.J. and Warnke E.P. (1975) "Constitutive model for the triaxial behaviour of concrete" International Association of Bridges and Structural Engineers, seminar on concrete structures subjected to triaxial stresses, IABSE Proc. 19, Italy.
[7] Hanchak S.J., Forrestal M.J., Young E.R. and Ehrgott J.Q. (1992) "Perforation of concrete slabs with 48 MPa and 140 MPa unconfined compressive strengths" International Journal of Impact Engineering, vol. 12, no. 1, pp. 1-7.