FEASIBILITY STUDIES OF A PLASTICITY-BASED CONSTITUTIVE MODEL FOR ULTRA-HIGH PERFORMANCE FIBER-REINFORCED CONCRETE

Transcription

1 FEASIBILITY STUDIES OF A PLASTICITY-BASED CONSTITUTIVE MODEL FOR ULTRA-HIGH PERFORMANCE FIBER-REINFORCED CONCRETE J.M. Magallanes, Y. Wu, K.B. Morrill, J.E. Crawford Karagozian & Case, 2550 N. Hollywood Way, Burbank, CA 91505, USA ABSTRACT A plasticity-based constitutive model for concrete is calibrated to existing and newly acquired material characterization data pertaining to ultra-high performance fiber-reinforced concrete (UHPFRC). Nonlinear least square regression and the response surface methodology are applied to estimate the constitutive model parameters based on the available data. The model is then employed in Lagrangian finite element calculations, the results of which are compared with data from laboratory and field blast tests. Although the UHPFRC material response database is incomplete and several key phenomenological aspects of UHPFRC remain unquantified, the constitutive model yields good qualitative comparisons with observed behaviors in the UHPFRC panel tests. Quantitative comparisons show that the computed responses are sensitive to model parameters related to the tension/extension behaviors of the material, of which little is currently known. These results demonstrate the feasibility of developing a plasticity-based constitutive model for UHPFRC but indicate that additional material characterization and structures data is needed to improve and generalize the model for other ultra-high performance concrete materials. INTRODUCTION Ultra high performance concretes (UHPC), especially those of the reactive powder concrete type (Richard and Cheyrezy, 1995; Roux et al., 1996), are cementitious materials consisting of cement, sand, silica fume, silica flour, superplastizer, water, and a variety of different types of fibers. Various mixtures are used in practice, each designed to improve the mechanical and/or durability properties of concrete. UHPC materials employing fibers may be termed fiber reinforced concrete (FRC) or UHPFRC, to add to the list of industry vernacular, and can have unconfined compressive strengths upwards of 200 MPa and compressive-to-tensile strength ratios similar to normal-strength concrete (NSC). Past concrete testing and constitutive modeling has focused on NSC that typically has unconfined compressive strengths less than 40 MPa. A number of plasticity constitutive models have been developed for NSC (see Chen (1982)) and some have been shown to compare well against experiments. Examples include those described by Malvar et al. (1997), Mould and Levine (1994), and Schwer and Murray (1994). Tensoral-based constitutive models have also been developed, including the Microplane suite of models by Bažant et al. and variants of it (Bažant et al., 2000; Cusatis et al., 2006), but these are clearly more computationally expensive and are not currently available to the engineering community at large for use in commercial finite element (FE) codes. The plasticity-based models remain a key tool for analysts studying the behavior of reinforced concrete (RC) structures subjected to extreme dynamic loads. Because of its unique properties, there is at present no general constitutive model that captures UHPFRC behaviors in a manner suitable for use in performing first-principle FE calculations. Thus, there is a strong need to improve analytic modeling capabilities for these materials. This is especially the case in engineering studies and designs where an efficient model capable of capturing the most concerning material behaviors is highly desirable.

2 In this paper the feasibility of developing a plasticity-based constitutive model for UHPFRC is investigated. The present study focuses on a particular UHPFRC material marketed commercially as Ductal (Cavill, 2005; Cavill and Rebentrost, 2006). A recent release of the K&C concrete (KCC) model (Malvar et al., 1994; Magallanes et al., 2010a), which is of the internal damage variable class of plasticity models, is employed in a parameter estimation study using a limited set of published and proprietary materials data gathered by the authors. The KCC model is capable of modeling confinement effects, ductile or brittle deformations in both compression and tension/extension, and dilatancy due to shear, among a number of other key concrete behaviors (Magallanes, 2008). As described in the next section, these are behaviors that are present in UHPFRC data. The calibrated model is then deployed in FE models using the explicit Lagrangian FE code LS-DYNA (LSTC, 2007), the results of which are then compared with experimental data to assess its performance. MATERIAL BEHAVIORS Beyond a few triaxial compression test results (Akers et al., 1998; Williams et al., 2009), data from a few studies reporting basic elastic properties (Graybeal, 2006; Graybeal and Davis, 2008), and sparse data from several proprietary material characterization efforts (e.g., ARA, 2009), there is a distinct lacking of comprehensive data sets showing the full range of behaviors needed to develop a constitutive model suitable for performing physics-based FE calculations. Moreover, it is a challenge for state-of-the-art laboratory testing to measure the unique behaviors presented by UHPFRC for a number of reasons, which are only mildly related to its enhanced compressive strength. These include: Post-failure regime. Post-peak response of geomaterials fall into one of two classes undergoing uniaxial compression (Fig. 1a). Class I behavior in the post-peak regime is stable in the sense that work must be supplied to the sample to effect any further shortening. In contrast, failure in Class II materials is self-sustaining (Wawersik and Fairhust, 1970). The Class I/II classifications have been substantiated in experiments in which boundary conditions and sample lengths were varied in order to separate material properties from structural effects (Hudson et al., 1972; Van Mier and Vonk, 1991) and by a number of bifurcation models (Rudnicki and Rice, 1975; Costin, 1983). The post-failure behavior of UHPFRC, which exhibits behaviors common to both concrete and rock (Fig. 1b), has not been well characterized, especially for low confining pressures. Successful material experiments on Class II and difficult Class I materials require stiff testing machines with fast response times and additional means for managing the release of energy stored in the test system, which are the primary reason for the lack of data in the post-failure regime shown in Fig. 1b. Volume strains Axial strains (a) Post-failure classes [30]. (b) Data for concrete, rock, and UHPC. Fig. 1. Behavior of concrete and rock under uniaxial compression.

3 Tensile fracture. Tensile fractures in concrete are highly localized and are also manifested by softening in the post-failure regime. Tensile softening has been shown to affect the accuracy of numerical simulations when compared against the results of field tests (Crawford and Malvar, 1999). Based on the limited data that is available for UHPC in unconfined tension (Graybeal, 2006; Isaacs et al., 2009), current and developmental versions of UHPFRC exhibit much greater fracture energy absorption capacities than concrete or rock. Extension behaviors. One behavior of concrete common to all geomaterials is an increase in compressive strength with increasing confinement. Fig. 2 illustrates two typical types of quasi-static laboratory tests conducted to characterize this behavior: triaxial compression (TXC) and triaxial extension (TXE). The stress paths are plotted as stress difference (SD) versus mean normal stress (σ m ). The TXC and TXE tests are first loaded hydrostatically to a specified hydrostatic pressure. While maintaining the confining pressure in the radial direction (σ r ) fixed, the axial stress (σ a ) is next increased for the TXC tests, while σ a is decreased for the TXE stress path tests. Loading NSC samples to failure in TXE tests is made relatively straightforward using conventional triaxial testing equipment. UHPFRC specimens, on the other hand, are difficult to load to failure because of their enhanced tensile properties. Where NSC TXE specimens generally fail while σ a is compressive, UHPFRC can be unloaded to a zero axial stress while still resisting the radial stresses without failure. Consequently, little is known regarding the materials failure surface on the tensile meridian, including its post-peak behavior (Magallanes et al., 2010b). Dilatancy. Geomaterials including concrete subjected to deviatoric loading exhibit dilatancy, which is another important behavior to capture in both analysis and design of RC structures (Rudnicki and Rick, 1975; Crawford and Malvar, 1999). Two mechanisms engender this behavior in concrete: local failures due to distributed microcracking and the formation of shear bands as microcracks grow and coalesce along discrete planes. Because most UHPC formulations omit coarse aggregate from the concrete mix design, and because various UHPC designs have fibers embedded throughout its cementitious matrix, the significance of dilation from shear to the mechanical response of the material is not well understood. Stress Difference, SD ( a - r ) Failure Surface for Compressive Meridian Hydrostatic compression, 3 HC 1 Failure Surface for Extension Meridian Mean normal stress, m ( a + 2 r ) / 3 Triaxial extension, TXE Triaxial compression, TXC Fig. 2. Common stress paths. METHODOLOGY FOR CALIBRATING THE CONSTITUTIVE MODEL Model Description The KCC model is a three-invariant plasticity model, where the failure surface is derived by interpolating between two of three independent surfaces ( σ y, σ m, and σ r ) using an internal damage variable, λ. The yield function is defined as: where f I J, J, 3 J I,, 1, J 3 (Eq. 1)

4 J r 3 m y y m (Eq. 2) rj 3 m r r m where I 1 is the first invariant of the stress tensor, J 2 and J 3 are the second and third invariants of the deviatoric stress tensor, σ y, σ m, and σ r are three independent surfaces describing the strength limits for the compressive meridian, η(λ) is a nonlinear function that ranges from zero to unity for λ λ m and from unity to zero when λ > λ m, and λ is accumulated as a function of the effective plastic strain using three damage accumulation parameters, b 1, b 2, and b 3 (Malvar et al., 1994). The three independent surfaces, denoted here as σ i, use a simple function to account for the effects of pressure (p=i 1 /3). Three parameters a 0i, a 1i, and a 2i (9 parameters total for the three surfaces) define each of the failure surfaces: p i a0i (Eq. 3) a a p 1i As seen in Eq. 2, each of the surfaces are functions of a third invariant, J 3, via the function, r', which is based on using a formulation proposed by William and Warnke (see Chen (1982)). Strain hardening and softening behaviors are efficiently modeled by formulating the plasticity model in this way. The η(λ) damage function provides the mechanism by which the KCC model may reproduce hardening until the damage is sufficient to reach surface σ m and then softening until σ r is reached. Such a feature allows modeling brittle, ductile, and brittle-toductile transition behaviors. Fig. 3a shows an illustration of a typical axial stress (σ a or σ 11 ) versus axial strain (ε a or ε 11 ) response for concrete subjected to TXC loading. Subsequent to loading the concrete specimen hydrostatically (Point 0), sample behavior is approximately linear up to roughly 35-65% of its peak strength (Point 0 to Point 1) and then exhibits strain hardening up to the peak strength (Point 2). Strain-softening is then observed to a residual strength who s value also depends on the level of confinement (Point 3 and on). Fig. 3b illustrates the functionality of Eq. 2, where γ (the current yield surface) is interpolated between the yield and maximum surfaces when λ λ m and the maximum and residual surfaces for λ > λ m. For the trial stress state in the plasticity algorithm, the plastic potential function is defined as: 2i * * * g 3 J I,, (Eq. 4) 2 1 J 3 where ω is a dilatancy parameter. The dilatancy parameter is unity for a fully-associative flow rule or less than unity for a partially-associative flow rule. Using Eq. 4, one can derive a consistency condition for the plasticity algorithm (Malvar et al., 1994). A classical radial return algorithm is then used to correct the trial stress. The KCC model was developed with the intent of being used in conjunction with any equation of state (EOS) model that could capture volumetric hardening, and for this reason, is a very simple and flexible model capable of modeling the known UHPFRC behaviors. Additionally, the KCC model incorporates rate effects via a radial rate enhancement to the failure surfaces based on the effective deviatoric strain rate. This provides a means to enhance strength as a function of loading rate that is easily calibrated to test data, in the manner described by Magallanes et al. (2010a).

5 2 Stress ( 11 ) Strain ( 11 ) (a) Illustration of a typical axial stress versus axial strain response under TXC loading. Stress Difference, Maximum, m p< 0 p> 0 2 Residual, r Yield, y f c 1 3 Compression meridian -f t f c /3 0 Pressure, p (mean normal stress) p r Extension meridian p m p y (b) Illustration showing the failure surfaces in the p-sd plane. Fig. 3. Failure surfaces for the KCC model. Parameter Estimation Methodology Parameters of material models can be difficult to evaluate when they cannot be measured directly. This is especially challenging with complex models like the KCC model, which uses parameters having both physical significance and several others that do not have physical meanings (e.g., the damage parameters). In general, two calibration methods are employed: (1) where each of the parameters is evaluated independently and (2) where parameters are grouped and then taken as a group to evaluate simultaneously. Parameters estimation may then be performed using nonlinear least square regression (NLSR) (Draper and Smith, 1981) or variants of response surface methods (RSM) (Myers and Montgomery, 1995). Investigators have noted deficiencies with the first approach rooted in errors propagating through the data analysis (Senseny et al., 2003), while the second approach may suffer from lack of sufficient data or even the quality and precision of the dataset. Highly correlated and nonphysical parameter estimates can be estimated using either of these methods. Also at issue is the adequacy of the functionality embedded in the material model, which may need to be enhanced to better correlate the data.

6 Given the sparcity of the UHPFRC dataset, a combination of the two methods (i.e., NLSR and RSM) was used to identify the model parameters for the UHPFRC, without any modification to the constitutive model equations. Quasi-static laboratory material data for the material was assembled and reduced in a format suitable for calibration of the KCC model parameters. Strain-rate effects were neglected for this present study, primarily out of necessity given the lack of high load rate data (Issacs et al., 2009). The first step in estimating parameters was performed using NLSR for the parameters that are directly measurable from experiments. These variables included the elastic material properties, the equation of state (in the form of tabulated compaction pairs), and the three groups of failure surface coefficients (a 0i, a 1i, and a 2i ). Results from the NLSR regression on the maximum failure surface ( σ m ) parameters are shown in Fig. 4a, which was performed using an iterative Levenberg-Marquardt optimization algorithm to minimize the root mean squared (RMS) error between the data and the model defined by Eq. 3. The correlation between each of the three parameters shown in Fig. 4a is quite high but is similar to results noted by other investigators (Senseny et al., 2003). One can expect the correlations to be reduced with more data, since experience and intuition suggests these variables to be fairly independent. Data and Model Model Residual Error Iteration (a) NLSR to the maximum failure surface, σ m. (b) RSM optimization for damage. Fig. 4. Parameter identification for the KCC model using the calibration dataset. The next set of parameter estimates involved simultaneous parameter updates for the KCC damage function and damage accumulation parameters using RSM. The LS-DYNA FE code was employed to iteratively minimize the RMS error between several Ductal TXC tests and the results from simulations using a single finite element with appropriate boundary conditions. Results from a three stage update for the TXC simulations are illustrated in Fig. 4b. An initial set of model parameters was estimated based on our previous experience with the model. The first stage consisted of an optimization to fit the dilatancy parameter (ω) and the damage parameter b 1. The second stage consisted of a global minimization of the RMS error by treating the damage function η(λ) as a set of dependent variables. This second stage was successful in minimizing the RMS error across the data set, but yielded an awkward shape. (This is not uncommon with parameter identification techniques, which in this case, occurred because of discrepancies between several of the low pressure TXC data curves for the UHPFRC material.) A third and final stage was subsequently employed to achieve a local optimization to better capture the hardening and softening transitions present in a selection of the TXC data. Results using the final values for the set of parameters estimated using these procedures are shown in Fig. 5, which are designated the mean model parameters. The TXC stress paths

7 (Fig. 5a) indicate that the parameter update scheme provided a reasonable set of parameter estimates. Although the accuracy of the data in the post-peak regime is not known (recall the difficulties with experimental measurements at low confining pressures), these features were present in the calibration data and correctly preserved in the parameter update scheme. Fig. 5b show results of unconfined tension load paths from single element simulations. The peak strength is close to 10 MPa and the fracture energy is 15 MPa-mm, which are values consistent with the quantities reported in the literature. For example, in the few available direct tension tests that attempt to measure UHPFRC softening, Graybeal (2006) reported static peak strengths between 6 to 10 MPa with and 9 to 20 MPa-mm of fracture energy per unit area (over 200 times that for NSC). Our results shown in Fig. 5b are consistent with the peak strengths and energy absorption properties, but because the current KCC model uses one damage function to control the material rheology for both compressive and tensile loads, the resulting extension curves appear awkward for this stress path. Given more data for UHPFRC in extension, the model can be improved by adding a second independent damage function for these types of loading paths that can more properly represent the data. G f = 15 MPa mm (a) Triaxial compression, including unconfined. (b) Unconfined tension. Fig. 5. Verification of the calibrated model for typical stress paths. RESULTS OF FINITE ELEMENT CALCULATIONS Several Lagrangian FE models were developed for Ductal panel tests to investigate the accuracy of predictions obtained using the calibrated model. These experiments were originally conducted to explore the structural behavior of the panels and unfortunately are not of code validation quality; hence there are uncertainties in the measurements of the response of the test articles. The tests can, however, function to establish the feasibility of the described material model. Both laboratory and field tests were conducted by the University of Adelaide using 1 m wide by 2 m long by 0.1 m thick Ductal panels reinforced with mild steel reinforcing bars (Ciccarelli et al., 2008). For each panel, FE models were constructed using 8-node solid elements to model the concrete (employing single-point integration) and for the steel reinforcement, beam elements were employed. Nodes for the beam elements were coincident with the solid nodes and merged; hence, no slip between the concrete and the reinforcement was allowed. Boundary conditions were modeled to represent conditions similar the test specimen; panels were simply supported with some unquantified level of rigidity provided by flexible steel angles. Wood shims used in some of the tests were not included in the FE model, as their effect on the response is unclear. For the field tests, the explosive loads were estimated using the

8 BlastX code (Britt and Lumsden, 2004) and the spherical tabular Composition B model, using the total explosive masses. Our load estimates assumed an infinite reflecting surface and thus did not account for airblast clearing effects. A result for a quasi-static four-point bending laboratory test is shown in Fig. 6 along with the numerical results. The test loading conditions were simulated in the model by imposing a constant velocity rate of 10 mm/s at the top of two columns composed of linear compressible continuum elements having no shear strength. In FE codes using explicit time integration, this method provides a means of calculating the resistance of the structures subjected to displacement-controlled loading, including their hardening and softening responses. The results indicate that the failure pattern in the simulation was similar to that observed in the test; tension cracks initiated on the tensile side of the panel near the center, between the two load points, and then propagated towards the compression face (Fig. 6a/6b). The load versus displacement behavior as computed using the constitutive model (designated the mean model ) exhibits a similar pre-peak loading branch (Fig. 6c), but reaches a maximum strength of 240 kn, compared to the test peak of over 300 kn. The postpeak softening behavior for the mean model is similar to the trend present in the test data, but as shown in Fig. 6c, is sensitive to the hourglass stabilization control used in calculation. In general, the hourglass method yielding the lowest total hourglass energy resulted in the lower postpeak strengths. Additional parametric calculations indicate that the response of the slabs were highly sensitive to constitutive model parameters related to the tensile strength and tensile fracture energy characteristics. Response sensitivities for the tensile strength, f t, and the tensile damage parameter, b 2, are presented in Fig. 7a and 7b, respectively. In each of those figures, the test results are plotted along with results from the mean model and two additional curves representing results obtained by perturbing either b 2 or f t to plausible low and high values. For UHPFRC tensile strength, these were determined to be 3 MPa and 30 MPa. For b 2, -1.0 and were selected, the latter of which induces more fracture energy absorption in each fractured finite element (Fig. 5b). Peak strengths range from 150 kn to 450 kn for f t values of 3 MPa and 30 MPa, respectively. The parameter b 2 shows less sensitivity to the peak strength but does affect the post peak behavior of the panel. Similar results were obtained with FE models created to simulate the field experiments, which are slightly more difficult to interpret because of the known uncertainties in test boundary conditions and blast loads. Nevertheless, sensitivity studies also indicated high sensitivities to both b 2 and f t. Fig. 8 shows displacement histories as functions of time for three tests fielded using similarly designed UHPFRC panels. Test 11 resulted in an elastic response with no damage (Fig. 8a). Test 08 showed moderate damage with a peak deformation of 72 mm (Fig. 8b). Test 12 resulted in a complete slab failure (Fig. 8c, note that the gauge data from Test 12 broke during the test and the data is suspect). For each of these figures, the displacement-histories recorded using LVDT gages are compared with results using our mean model and responses obtained by perturbing b 2 and f t in the bounds described in the preceding example. For each of these tests, the mean model compares well with the test data after considering the uncertainties in the test fielding and instrumentation. By inspecting each of the figures, it is apparent that the results are relatively insensitive to f t and b 2 at low levels of displacement, while the calculations become highly sensitive to these same parameters at high levels of deformation. This satisfies intuition, as one can expect the tensile strength and post peak softening behavior of the material to dominate when the structure is loaded in the inelastic regime.

9 Bottom of slab (a) Posttest photograph (b) Computed damage. (c) Comparison of force versus displacement. Fig. 6. Comparison of predicted results for a quasi-static four-point bending test. (a) Sensitivity to tensile strength, f t. (b) Variation of tensile fracture energy, b 2. Fig. 7. Sensitivity study for the model parameters governing the materials tensile behavior.

10 Sensitivity to f t Sensitivity to b 2 (a) Test 11 Sensitivity to f t Sensitivity to b 2 (b) Test 08 Sensitivity to f t Sensitivity to b 2 (c) Test 12 Fig. 8. Comparison of displacement histories for three blast tests.

11 CONCLUSIONS The results of the studies presented in this paper demonstrate the feasibility of developing a plasticity-based constitutive model for UHPFRC materials. In this study, the parameters of the K&C concrete model were calibrated directly to a limited set of materials data for Ductal using parameter identification techniques. These procedures yielded a reasonable set of model parameters for this particular set of data. When employed in FE calculations for structural panel tests, the model produced good qualitative and quantitative comparisons, despite unavoidable uncertainties in boundary conditions, loading, and instrumentation. The studies indicated a high sensitivity to model parameters related to the tensile strength and rheology of the material, of which little is currently known. The sensitivity was minor when the structure remained elastic and became more severe when the structure was loaded to an inelastic response. Although these results show promise with regards to modeling UHPFRC materials with plasticity models, more basic research is needed pertaining to the behavior of UHPFRC materials under quasi-static and dynamic loads. Such research would allow improvement and generalization of constitutive models, plasticity-based or not, which are needed for a variety of analytic and design efforts. ACKNOWLEDGEMENTS This work was supported in part by a Small Business Innovative Research project and internal K&C funding. A special thanks to Dr. Mark Rebentrost and Dr. Gavin Wight of VSL- Australia for their insight regarding the structural panel tests using the Ductal material. REFERENCES [1] Akers, S.A., Green, M.L., Reed, P.A. (1998), Laboratory Characterization of Very High-Strength Fiber-Reinforced Concrete, US Army Corps of Engineers, SL-98-10, Vicksburg, MS, November, [2] Applied Research Associates (2009), personal communication. [3] Bažant, Z.P., Ferhun, C.C., Carol, I., Adley, M.D., and S.A. Akers (2000), Microplane model M4 for concrete, I: formulation with work-conjugate deviatoric stress, Journal of Engineering Mechanics, Vol. 126, No. 9, pp [4] Britt, J.R., and M. G. Lumsden (1994), Internal Blast and Thermal Environment from Internal and External Explosions: A User s Guide for the BLASTX Code, Version 3.0, Science Applications International Corporation, St. Joseph, Louisiana, SAIC (Limited Distribution) [5] Cavill, B. (2005), An ultra-high performance material for resistance to blasts and impacts, 6th Asia- Pacific Conference on Shock and Impact Loads on Structures, Perth, Australia, Dec. 7-9, [6] Cavill, B., and M. Rebentrost (2006), Ductal An ultra-high performance material for resistance to blast and impacts, fib 2nd Int. Congress, Naples, Italy, pp. 11, June 19-21, [7] Chen, W.F. (1982), Plasticity in Reinforced Concrete, McGraw Hill, New York. [8] Costin, L.S. (1985), Damage mechanics in the post-failure regime, Mech. Mat., vol. 4, No. 2, pp , July, [9] Ciccarelli, J., Henderson, A., Jordans, K., and B. Noack (2008), Resistance against explosive loading of metal foam retrofitted and ultra high strength concrete structural members, University of Adelaide, Research Report, October, [10] Crawford, J.E., and L.J. Malvar (1999), User s and theoretical manual for the K&C concrete model, Karagozian & Case, Burbank, TR , (Limited Distribution) [11] Cusatis, G., Bažant, Z.P., and L. Cedolin (2006), Confinement-shear lattice CSL model for fracture propagation in concrete, Comput. Methods Appl. Mech. Engrg., Vol. 195, pp [12] Draper, N.R. and H. Smith (1981), Applied Regression Analysis, John Wiley and Sons, Inc., 2nd ed.

12 [13] Graybeal, B.A. (2006), Material Property Characterization of Ultra-High Performance Concrete, Federal Highway Administration, FHWA-HRT , August, [14] Graybeal, B.A., and M. Davis (2008), Cylinder or Cube: Strength Testing of 80 to 200 MPa Ultra- High Performance Fiber-Reinforced Concrete, ACI Materials Journal, Vol. 105, No. 6, pp [15] Hudson, J.A., S.L. Crouch, and C. Fairhurst (1972), Soft, stiff, and servo-controlled testing machines ; a review with reference to rock failure, Eng. Geol., Vol. 6, No [16] Issacs, J.B., Magallanes, J.M., Rebentrost, M., and G. Wight (2009), Exploratory Dynamic Material Characterization Testing on Ultra-High Performance Fibre Reinforced Concrete, Proceedings of the 8th International Conference on Shock and Impact Loads on Structures, Adelaide, Australia, December 02-04, [17] Livermore Software Technology Corporation (2007), LS-DYNA user s manual Version 971, Livermore, CA, May, [18] Malvar, L.J., J.E. Crawford, J.E. Wesevich, and D. Simons (1997), A plasticity concrete model for DYNA3D, Int. J. of Imp. Eng., vol. 19, No. 9-10, pp [19] Magallanes, J.M. (2008), Importance of concrete material characterization and modeling to predicting the response of structures to shock and impact loading, Structures Under Shock and Impact X, Eds. N. Jones, WIT Press, Southampton. [20] Magallanes, J.M., Wu, Y., Malvar, L.J., and J.E. Crawford (2010a), Recent Improvements to Release III of the K&C Concrete Model, Proceedings of the 11th International LS-DYNA Users Conference, Dearborn, MI, June 6-8, [21] Magallanes, J.M., Martinez, R.M., Wawersik, W.R., Del Frate, R., Morrill, K.B. (2010b), A test apparatus for measuring the mechanical behaviors of ultra-high performance concretes under quasistatic tension/extension load paths-phase I: Feasibility study and concept design, Karagozian & Case, Burbank, CA, TR , May (Limited Distribution) [22] Mould, J.C. and H.S. Levine (1994), A Rate-Dependent Three Invariant Softening Model for Concrete, Mechanics of Materials and Structures, Voyiadjis, G.Z., Bank, L.C., Jacobs, L.J., Eds., Elsevier Science. [23] Myers, R.H., and D.C. Montgomery (1995), Response Surface Methodology. Process and Product Optimization using Designed Experiments, Wiley, [24] Richard P., and Cheyrezy M. (1995), Composition of Reactive Powder Concrete, Cement and Concrete Research, vol. 25, No. 7, 1995, pp [25] Roux, N., Andrade, C., and Sanjuan, M.A. (1996), Experimental Study of Durability of Reactive Powder Concretes, Journal of Materials in Civil Engineering, Vol. 8, No. 1, 1996, pp [26] Rudnicki, J.W., and J.R. Rice (1975), Conditions for localization of deformation in pressuresensitive dilatant materials, J. Mech. Phys. Solids, Vol. 23, No [27] Schwer, L.E., and Y.D. Murray (1994), A three-invariant smooth cap model with mixed hardening, International Journal for Numerical and Analytic Methods in Geomechanics, Vol. 18, pp [28] Senseny, P.E., Brodsky, N.S., DeVries, K.L. (2003), Parameter evaluation for a Unified Constitutive Model, ASME Journal of Engineering Materials and Technology, Vol. 115, pp , April, [29] Van Mier, J.G.M., and R.A. Vonk (1991), Fracture of concrete under multiaxial stress recent developments, Mat. Structures, Vol. 24, No. 61. [30] Wawersik, W. R., and C. Fairhurst (1970), A study of brittle rock fracture in laboratory compression tests, Int. J. Rock Mech. Min. Sci., 7. [31] Williams, E.M., Akers, S.A., and P.A. Reed (2006), Laboratory characterization of SAM-35 concrete, U.S. Army Corp of Engineers Engineering Research and Development Center, Technical Report TR [32] Williams, E.M., Graham, S.S., P.A., Reed, and T.S. Rushing, (2009), Laboratory characterization of Cor-Tuf Concrete With and Without Steel Fibers, U.S. Army Corp of Engineers Engineering Research and Development Center, Draft Technical Report, May, 2009.