87

DYNAMIC FRACTURE PROPAGATION IN FIBRE-REINFORCED CONCRETE

Ignacio Rivero, Rena C. Yu∗, and Gonzalo Ruiz

E.T.S. de Ingenieros de Caminos, Canales y Puertos,Universidad de Castilla la Mancha

Avenida Camilo Jose Cela s/n, 13071 Ciudad Real∗E-mail: rena@uclm.es

RESUMEN

El objetivo de este trabajo es modelar la propagacion dinamica de fisuras en vigas de hormigon con fibras solicitadasa flexion en tres-puntos. Se asume una unica fisura cohesiva que cruza la seccion central de la viga. La apertura de lafisura esta gobernada por una ley cohesiva que es sensible a la velocidad de apertura de dicha fisura en cada punto.Por otra parte, las fibras que cosen la fisura se modelan explicitamente. Se tienen en cuenta tanto su comportamiento atraccion como el comportamiento de la intercara de contacto de las fibras con la matriz de hormigon, que se implementausando curvas analıticas tension rasante–deslizamiento. Finalmente, las curvas carga-desplazamiento obtenidas simulandoensayos dinamicos de vigas de hormigon con fibras se ajustan bien a las curvas experimentales.

ABSTRACT

The object of this work is to simulate the dynamic fracture propagation in FRC (fibre reinforced concrete) beams loaded ina three-point bend configuration. We assume that a single cohesive crack is to propagate at the middle section; the openingof this crack is governed by a rate-dependent cohesive law; the fibres around the fracture plane are explicitly represen-ted through truss elements. The fibre behaviour is depicted by an equivalent constitutive law, which is obtained from ananalytical load-slip curve. The proposed model is implemented in the commercial software ANSYS using contact/targetelements for the cohesive fracture. The obtained load-displacement curves are compared with their experimental counter-parts. The good agreement with experimental data testifies to the model’s feasibility.

AREAS TEMATICAS PROPUESTAS: Metodos y Modelos Analıticos y Numericos

PALABRAS CLAVE: Cohesive Fracture, Fibre Reinforced Concrete

NOMENCLATURE

Pull-out tests and steel fibresF Pull-out loads Slip lengthEs Elastic modulusεu Ultimate strainfu Ultimate strengthL Fibre lengthd Fibre diameterLe Embedment lengthLhc Critical embedment lengthµ Friction coefficient

Concrete MatrixEc Elastic modulusft Tensile strengthfc Compressive strengthGF Specific fracture energydmax Maximum aggregate sizewc Critical crack opening displacementw Crack opening velocityw01 Reference opening velocity at low loading ratesn Viscosity indexw02 Reference opening velocity at high loading ratesm Coefficient

1. INTRODUCTION

The industrial use of FRC in structural applications, inparticular SFRC (steel fibre reinforced concrete), has in-creased significantly during the last decades. However, inorder to optimise the design with fibre reinforced mate-rial and spread its applications, a complete understandingand modelling of the material behaviour under both staticand dynamic loading conditions are necessary.

Depending on the fibres are explicitly represented or not,numerical models on FRC can be roughly classified asexplicit or homogeneous ones. Examples of the formerand the later can be found in [1, 2, 3, 4] and [5, 6, 7] res-pectively. But most of them are dealing with static loa-ding conditions. In spite of abundant of experimental re-sults available for FRC materials under impact loading,publications on numerical modelling for dynamic fractu-re propagation are few. In the current work, we model thefibres explicitly based on the approach of Laranjeira [4],the semi-analytical model of Soetens [8] on pullout beha-viour of fibres, and couple them with the cohesive frac-ture endorsed with rate-dependent laws [9, 10] for lowloading rates and impact loading through a drop-weightimpact machine [11].

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88

Nest we explain in detail the finite methodology. Thenumerical model applied to the experimental results ofZhang et al. [12] are given in Section 3. Relevant conclu-sions are drawn in Section 4.

2. NUMERICAL METHODOLOGY

We envision the numerical model to simulate the fracturepropagation of a three-point bend SFRC beam under dy-namic loading conditions in 3D. The main characteristicsof the model are as follows

For simplification, a single cohesive crack is assu-med to propagate at the middle section.

The fibres across the middle section are explicitlymodelled as truss elements which connect the no-des of the meshed solid elements.

The SFRC material farther away from the middlesection is considered to be linear elastic, except a0.1 % damping coefficient is applied when the loadspectra cover high frequencies.

The rate-dependent phenomenon is modelled th-rough, on the one hand, cohesive laws contingenton the current local crack opening velocity; on theother hand, the Pierce model of visco-plasticity[13] available in ANSYS for the steel fibres.

The constitute law for fibres should cover both theuniaxial material behaviour as steel and the bond-slip law for the interface between steel and concre-te.

The above hypotheses, also implicitly embrace those as-sumed by Voo and Foster [2] for their Variable Engage-ment Model. In other words, effects of individual fibrescan be summed over the failure surface to yield the ove-rall behaviour of the SFRC; the geometric centres of thefibres are uniformly distributed in space and equally pro-bable to be oriented in any direction; the bending stiffnessof a fibre is small and energy expended by bending of fi-bres can be neglected.

Next we first explain the employed cohesive laws endor-sed with rate-dependent behaviour, then move on with themodelling of fibre pullouts behaviour.

2.1. Cohesive laws endorsed with rate-dependentbehaviour

The feasibility of the cohesive theory of fracture has beenwell validated since it came to be in the 1960s. The cha-racteristic length define by Hillerborg [14] and the intrin-sic time defined by Camacho and Ortiz [15] for cohesive

fracture are given as follows

lch =EGF

f2t

(1)

tch =ρcwc

2ft(2)

where E is the elastic modulus, GF , the specific fractureenergy, ft the tensile strength; ρ strands for the mass den-sity, c the longitudinal wave speed, wc the critical ope-ning displacement. The length lch is related with the si-ze of the fracture process zone, whereas tch enables usto capture the dynamic effects through static propertieswhen the inertia effect is dominant, see [16]. But whenthe Stephen effect [17], or micro-cracks start to play a sig-nificant role [10], rate-dependent cohesive laws are inevi-table in order to correctly reproduce the observed experi-mental results.

In this work, we employ the cohesive laws contingent onthe current crack opening velocities illustrated in Eq. (3)and Eq. (4) for low and high loading rates respectively.Such a distinction is necessary due to the different failuremechanisms involved, see [18] for details.

σ(w, w) = ft

[1 − w

wc

]g(w), (3)

g(w) = 1 +(

w

w01

)n

σ(w, w) = ft

[1 − 1

h(w)w

wc

], (4)

h(w) = 1 +(

w

w02

)m

In Eqs. (3-4), w01 and w01 are the respective referencecrack opening velocities for low and high loading rates.The viscosity index n is related to the movement of thewater content in concrete [19], whereas m is a coefficientwhich reflects the extent of micro cracks, now modelledas a single cohesive crack. All four parameters are deter-mined by fitting experimental results.

2.2. Constitutive law for steel fibres and bond-slipinterface between the fibre and concrete bulk

As mentioned before, the behaviour of the fibre is cho-sen to cover both the axial deformation and the bond-slipof the interface between concrete and steel fibres. Conse-quently, an equivalent stress-strain relationship is formedfor the pull-out tests, which is not really material pro-perty, we follow this notation Laranjeira [4] who denotedit as constitutive for simplification,

Though it is always preferred to obtain the bond-slip cur-ves from pull-out tests, it is not always posible to performsuch tests for a large number of fibres in different orien-tations. We choose to implement the semi-analytical met-hod by Soetens et al. [8] to construct the necessary cons-titutive law for each fibre orientation on the fly.

Anales de Mecánica de la Fractura, 31 (2014)

89

2.2.1. Each fibre as a new material

It needs to be pointed out that, under the above conside-rations, each fibre behaves as a kind of new material, itsconstitutive law is that given by a pull-out curve. Whenmodelled, each fibre deforms in a way that the displace-ment at its free end is equal to that registered experimen-tally in the pull-out test under the same load. For this newmaterial, the stress is calculated as the pull-out load divi-ded by the fibre cross-section area, the strain as the sliplength divided by the fibre length in the model, see below

σ =F

πd2/4(5)

ε =s

L(6)

where F and s are the load and the slip length in a pull-out test, d is the fibre diameter, L the fibre length.

Since the pull-out curve is the relation between the F ands, L in Eq. (6) is not necessarily the real fibre length,rather the modelled length which would give the correct sunder the the same load F . The same argument is true forthe fibre diameter. From a numerical standpoint, this wayof defining the fibre behaviour allows us to optimise themesh to obtain best results with any limitation imposedby the fibre geometry.

We compare the fibre pull-out in reality with that of theconceptual model in Fig. 1, where the fibre is fixed inone end and it is elongated as the load at the other endincreases. It bears the emphasis that, the slip length s inthe Fig. 1a (pull-out tests), emerges as the displacement sFig. 1b (modelled fibre), which should be measured onlyfor the un-bonded part only. When contained in a concre-te bulk, this s is actually the crack opening displacement,see Fig. 2.

Figura 1: The fibre pull-out behaviour as in reality and inthe conceptual model, as in [4].

2.2.2. Fibre inclination angle

As mentioned before, the fibres can be oriented in anydirection with equal probability. Since the pull-out beha-viour is for a fibre with a specific inclination angle. Adifferent fibre orientation leads to a different constituti-ve law. On the one hand, this may result in analytical

pull-out curves as the only possible resource; on the ot-her hand, this enables us to model all the fibres parallelas in Fig. 2, and let the constitutive laws to take care ofall possible fibre orientations.

11th World Congress on Computational Mechanics (WCCM XI) 5th European Conference on Computational Mechanics (ECCM V)

6th European Conference on Computational Fluid Dynamics (ECFD VI) July 20 - 25, 2014, Barcelona, Spain !!

Numerical Study on the Dynamic fracture propagation in

Fibre-Reinforced Concrete

Ignacio Rivero¹, Rena C. Yu1 and Gonzalo Ruiz1 1 University of Castilla-La Mancha, Avda. Camilo José Cela, 13071 Ciudad Real, Spain

Email: rena@uclm.es, URL: www.uclm.es !!Key Words: Fibre-Reinforced Concrete (FRC), Dynamic Fracture, Rate-dependent Cohesive Law. !!The industrial use of FRC in structural applications, particularly the use of steel fibres, has increased considerably during the last decades. However, in order to optimise the design with fibre reinforced material and spread its applications, a complete understanding and modelling of the material behaviour is still lacking.

The object of this work is to simulate the dynamic fracture propagation in FRC (fibre reinforced concrete) beams loaded in a three-point bend configuration. The proposed model has the following main characteristics: a) a single cohesive crack is allowed to propagate in the middle section. The opening of this crack is governed by a rate-dependent cohesive law proposed in [1]; b) the fibres around the fracture plane are explicitly represented through truss elements, see Fig. 1. The fibre behaviour is depicted by an equivalent constitutive law, which is obtained from an analytical load-slip curve.

!

Fig. 1 Fibres shown to sew the main crack in a three-point-bend beam (left) and the numerical representation through truss elements sharing nodes with the solid elements near the middle section. Figura 2: Observed fibre distribution at the middle sectionand the illustration in the proposed numerical model.

In Fig. 3a), we compare the analytical pull-out curve for azero-fibre orientation contrasted with several experimen-tal reference points. The pull-out curves for four incli-nation angles 15o, 30o, 45o and 60o are given in Fig. 3b)for demonstration purpose. Through Eqs. (5-6), these cur-ves are transformed to the uniaxial elasto-plastic materialmodels to govern the deformation of the fibres.

!

Figura 73 Curva analítica para simular el comportamiento a arrancamiento de las fibras.

Los puntos que definen la curva analítica son los puntos base que sirven para generar las curvas en diferentes orientaciones siguiendo el método de Laranjeira [47]. A partir de la curva de referencia de inclinación 0º se pueden obtener las curvas para las inclinaciones elegidas de 15º, 30º, 45º y 60º. A continuación se muestran los valores necesarios para obtener las distintas curvas:

¥ Le (longitud embebida)=12.5mm

¥ (tension última del acero)=1900MPa

¥ (Resistencia a tracción del hormigón)=5MPa

¥ N (factor que recoge el número de lados –en un lado o en los dos de la fisura-en que se produce el desconchamiento)=1

¥ (factor que tiene en cuenta la configuración del ensayo de arrancamiento)=18

¥ (coeficiente de fricción entre acero y hormigón)=0.6

¥ Lh,crit (longitud embebida crítica)= 5mm

¥ Ef (modulo elástico del acero de las fibras)= 210GPa

¥ (deformación última del acero de las fibras)=3.5%

Los resultados obtenidos se resumen en la Figura 74.

Pull-

out l

oad

(N)

0

125

250

375

500

Slip length (mm)

0 1.25 2.5 3.75 5

Analy-c  Curve Experimental  reference  points

!! 59

(a)Parte III - Modelo numérico

!

Figura 74 Curvas de arrancamiento de las fibras generadas para las 4 orientaciones de referencia.

La longitud que se emplea para simular las fibras es de 50mm, que es igual en este caso a la longitud real de la fibra. De este modo, esta será la longitud para transformar los valores de desplazamiento en la curva de arrancamiento a deformación en el modelo plástico de las fibras en Ansys.

Cuando se empleen las fibras en análisis estáticos o transitorios de baja velocidad se emplearán las curvas completas. En los casos de alta velocidad de carga el modelo de material empleado no permite la definición de la curva completa, pues no permite ramas descendentes, y solamente se tomarán los primeros puntos que constituyen la rama ascendente de las curvas de cada orientación.

7.4. Condiciones  de  contorno  En el modelo, las condiciones de contorno están definidas por los apoyos, la carga del ensayo y la gravedad en los análisis dinámicos. Se considera que tanto los apoyos como la carga, actúan en una sola línea de nodos. En los nodos de apoyo, asociados a los contactos, se limitan todos los desplazamientos. La acción de la carga se simula de diferente forma, dependiendo del tipo de análisis y la velocidad de carga. Para el caso de análisis estático se impone un desplazamiento para simular el proceso de carga. En los análisis dinámicos transitorios de baja y alta velocidad de carga se empleará un archivo con la evolución de desplazamientos en el tiempo, de forma que para cada velocidad se generará un archivo de este tipo.

!7.5. Código  

En la parte “Anexos” se adjuntan los códigos completos con los que se trabaja en Ansys. El código está basado en el código de [48], en el que también se simulaba un ensayo de impacto 3PB.

!

Pull-

out l

oad

(N)

0

125

250

375

500

Slip length (mm)

0 1.25 2.5 3.75 5

15º 30º 45º 60º

! 60

(b)

Figura 3: Analytic pullout curves (a) compared with theexperimental reference points in [8] for inclination angleof zero degrees. (b) for four different inclination angle of15, 30, 45 and 60 degrees.

The different values for the parameters needed to definethese curves in Fig. 3b) are listed in Tab. 1, where Le

Anales de Mecánica de la Fractura, 31 (2014)

90

is the embedment length, fu, the steel ultimate strength,Lhc

the critical embedment length, µ is the friction coef-ficient between steel fibre and the concrete bulk, Es thesteel elastic modulus, εu the ultimate strain of steel.

Tabla 1: Material and model parameters to generate theanalytical pull-out curves.

Le fu µ LhcEs εu

mm [MPa] [mm] [GPa] %12.5 1900 0.6 5 210 3.5

3. NUMERICAL SIMULATIONS

We apply the aforementioned methodology to simulatethe experimental results of Zhang et al. [12] on SFRCprisms conducted on notched beams under three-pointbend configuration. All the beams were of 700 mm inlength, 500 mm in span, with a square cross-section of150 mm in edge length. A pre-notch of 25 mm was cut atthe middle section. The tested loading-line displacementrates range from 10−3 to 103 mm/s. The three lower loa-ding rates 3.3×10−3 mm/s (quasi-static), 0.1 mm/s and3.3 mm/s were carried out under position control usinga servo-hydraulic testing machine. The three higher loa-ding rates 881 mm/s, 1770 and 2660 mm/s are perfor-med employing a drop weight testing device designed andbuilt by Zhang et al. [11].

3.1. Material properties and model parameters

In the work of Zhang et al. [12], a single type of SFRCwas used for all the tests. The siliceous aggregate com-ponent was of 12 mm in maximum size. Steel fibres ofhooked ends, 50 mm in length, 0.75 mm in diameter wereadded as reinforcements with a proportion of 64.5 kg/m3.Independent tests were carried out to characterise boththe reinforced and plain concrete, the obtained propertiesat the age of 65 days are listed in Tab. 2.

Tabla 2: Properties of SFRC and the concrete matrix [12].

ft fc GF E ρ dmax

[MPa] [MPa] [N·m] [GPa] [kg/m3] [mm]Steel Fibre Reinforced Concrete

- 92 - 35.1 2438 12Concrete Matrix

5.1 76 145 32 2297 12

Adjustable parameters for the cohesive laws

The initial values for the rate-dependent parameters forthe cohesive laws described in Eqs. (3-4) are taken fromthe work of Tarifa [18] for high strength concrete wit-hout fibres. The sensibility of these parameters are alsostudied.

Tabla 3: Parameters for the rate-dependent cohesive laws

w01 n w02 m[mm/s] [mm/s]

30 0.16 9E-6 0.24

3.2. Boundary conditions

The two unilateral supports of the beam are modelled th-rough two lines of contact elements, see Fig. 4. Each ele-ment consists two nodes, one is constrained with no de-gree of freedom, the other belongs to the meshed solid.The axial stiffness between the two nodes reflect that ofthe steel cylinders upon which the beam was rested befo-re testing.

For the modelling of quasi-static tests (loading line dis-placement rate, 3.3 ×10−3 mm), the external load is ap-plied as loading displacements, whereas for the rest, aninput file which contains the experimental evolution ofdisplacement in time is prepared.

3.3. Numerical discretization

In Fig. 4, we show one example of the modelled beam,where 8-node solid elements are employed for the bulkmaterial. Near the central section, where the crack is ex-pected to grow, a finer mesh is adopted, fibres are repre-sented by two node truss elements. Even though it is notnecessary, the actual length 50 mm is set for all the fibresin this case. Not shown in Fig. 4 are the contact elementsembedded at the middle section to model the cohesivefracture.

!

Figura 61: Vista de las fibras localizadas en el plano de fractura del modelo propuesto.

El empleo de curvas de arrancamiento experimentales sería más conveniente que el de curvas analíticas, pero es costoso realizar ensayos de este tipo para cada tipo de fibra. En cualquier caso, en el trabajo de Soetens [43] ya se han aplicado curvas analíticas de este tipo con buenos resultados. Además, el empleo de estas curvas permite liberarse de la necesidad de disponer de ensayos de arrancamiento para distintos ángulos, haciendo más independiente el modelo.

El modelo, por tanto, simula por un lado la contribución del hormigón, mediante la ley cohesiva, y la contribución de las fibras, mediante su simulación explícita y comportamiento regido por la comentada ley cohesiva.

Justi&icación  del  modelo  A continuación se justifica brevemente la forma en la que se modelan las fibras, que es el punto más importante a la hora de modelar el HRF en este modelo.

Las fibras se distribuyen de manera aleatoria en la masa de hormigón, teniendo cada una orientación diferente, condicionada en mayor o menor medida por la cantidad de fibras, el tipo de hormigón, la forma de la sección, etc. Cuando el material comienza a ser solicitado a tracción, comienza a producirse el fallo del hormigón y las fibras empiezan a colaborar soportando parte de la carga.

!

Figura 62: Fibras cosiendo la fisura en probeta real [22] (izq.) y en el modelo propuesto (dcha.)

!! 43

Figura 4: Initial mesh and fibre distribution for the calcu-lations.

3.4. Validation against experimental results

Typical experimental load-displacement curves for lowand high loading rates are given in Fig. 5, where the num-ber of visible fibres at the middle section is also given forsome cases. In Fig. 5b), the label before the loading ra-te is the identification number of the tested beam. Noticethat, the structural behaviour is ductile at low loading raterange, whereas it is brittle under impact loading.

In Fig. 6, we compare the load-displacement curve ob-tained from the numerical model where 355 fibres were

Anales de Mecánica de la Fractura, 31 (2014)

91

represented, with that of a beam tested at quasi-static sta-te. 354 fibres were counted at the middle section. Thefitting is surprisingly good for the entire loading process.It needs to be pointed out, however, the comparison is fora specific test, the fibre distribution might have occiden-tally coincided with that of the randomly generated in themodel. But it nevertheless serves well as a validation forour proposed model.

Parte IV - Resultados y discusión

!

Figura 83 Resultados experimentales para velocidad de carga de 3.33mm/s.

Existen tres probetas, 12A13, 12A14 y 12A16, en las que se contaron unas 230 fibras atravesando la sección. Sería de esperar que el comportamiento fuese similar y sin embargo se tienen tres tendencias diferentes. En estos casos la diferencia se puede deber a las posiciones de las fibras y a diferente sensibilidad a la velocidad de carga.

Comparando algunos de los casos anteriores:

!

Figura 84 Comparación de los resultados experimentales a diferentes velocidades bajas de carga.

En general en estos casos de baja velocidad la influencia de la velocidad de carga es muy baja y resulta difícil determinar si las variaciones registradas en los diferentes ensayos se deben a este aspecto o más bien a la propia variabilidad que tiene el material. Además, tomando como

Car

ga (k

N)

0

17.5

35

52.5

70

Desplazamiento (mm)

0 1.25 2.5 3.75 5

12A13-­‐233 12A14-­‐23212A15-­‐333 12A16-­‐231

Load

(kN

)

0

20

40

60

80

Displacement (mm)

0 1.25 2.5 3.75 5

315  fibres  (sta6c) 221  fibres  at  0.1  mm/s306  fibres  at  0.1  mm/s 232  fibres  at  3.3  mm/s233  fibres  at  3.3  mm/s

! 64

(a)

TFM: Simulación numérica del comportamiento dinámico del HRF

!

Figura 87 Resultados experimentales para la velocidad de carga 2650mm/s.

De nuevo los valores de la carga pico son similares, siendo estos valores de mayor magnitud que en los casos precedentes. La forma de la curva es prácticamente la misma, pero en este caso no se observa el efecto de cierre observado para menores velocidades. En este caso el martillo logra romper por completo la sección y no permite a las fibras más que una contribución residual antes del fallo completo. En esta tanda de ensayos también se encontró una probeta con un número excesivamente bajo de fibras, cuyo resultado nos permite evaluar cuánto colaboran las fibras en la parte inicial del impacto.

!

Figura 88 Comparación de los resultados para los casos de alta velocidad de carga.

Comparando los resultados de las tres velocidades se observa una clara dependencia de la velocidad de carga, como se observa en el hormigón en masa [48]. La contribución de las fibras es notable en la rama postfisuración, pero también se detecta su contribución en la parte inicial

Car

ga (k

N)

0

47.5

95

142.5

190

Desplazamiento (mm)

0 0.5 1 1.5 2

12B13-­‐277 12B17-­‐26412B18-­‐61

Load

(kN

)

0

47.5

95

142.5

190

Displacement (mm)

0 1 2 3 4

12B01-­‐881  mm/s 12B11-­‐1750  mm/s 12B13-­‐2650  mm/s

!! 67

(b)

Figura 5: Typical experimental load-displacement curvesrecorded at (a) low and (b) high loading rates.

Parte IV - Resultados y discusión

!

Figura 96 Simulación con 355 fibras para comparar con la curva experimental de 12A04 (354 fibras).

Para este ajuste se ha empleado el mismo valor de predeformación en las fibras (=0.007) y la resistencia a tracción para la ley cohesiva se ha empleado la mitad del valor original, es decir 2.45MPa, si bien se podría haber considerado un valor menor de forma que no perturbase el comportamiento inicial, como se puede ver en el pico que aparece en las figuras Figura 94 y Figura 95. El factor de orientación se ha ajustado en función del número de fibras, empleando la expresión 610. Para los siguientes análisis estáticos se emplearán estos valores de predeformación y reducción de la resistencia a tracción. Para los casos a distintas velocidades estos valores deberán ajustarse de nuevo, pues la velocidad de carga influirá en la respuesta de las fibras.

Sensibilidad  al  número  de  /ibras  Para analizar la influencia del número de fibras en los resultados del modelo se simulan tres casos con distinto número de fibras. El factor de orientación empleado es de 0.8 en todos los casos.

Load

(kN

)

0

17.5

35

52.5

70

Displacement (mm)

0 2.5 5 7.5 10 12.5 15

Exp.  (12A04-­‐354) Numerical  (355)

! 76

Figura 6: Load-displacement curves compared with expe-rimental results at quasi-static loading (3.3×10−3 mm/s).

In Fig. 7, for the case of a beam impacted at the loadingrate of 2650 mm/s, the number of fibres at the middle sec-

tion is 275, two types of visco-plastic law (Perzyna andPierce) for the fibres are employed to capture the rate-dependent behaviour. Note that, neither of them reprodu-ce the post peak trend.

TFM: Simulación numérica del comportamiento dinámico del HRF

!

Figura 927 Comparación de resultados para la velocidad 2650mm/se empleando el modelo de dependencia de la velocidad de deformación de Pierce y el de Perzyna. Simulación con 275fibras y rigidez en contacto de

apoyos de 0.65 108N/m.

!Análisis  para  las  velocidades  de  carga  882mm/s,  1750mm/s  y  2650mm/s  En estos análisis las propiedades del hormigón se mantienen como las obtenidas experimentalmente: Gf=160N/m, ft = 4.9MPa y E = 35.1GPa. El modelo cuenta con 275 fibras, 265 fibras para la velocidad de 882mm/s, con modelo de comportamiento viscoplástico tipo Pierce. Se han hecho simulaciones con dos casos distintos: un primer caso con c1=0.15 y c2=0.25 y otro caso con c1=0.1 y c2=1.

Load

(kN

)

0

50

100

150

200

Displacement (mm)

0 0.5 1 1.5 2

Exp.-­‐2650  mm/s Numerico-­‐PerzynaNumerico-­‐Peirce

!! 91

Figura 7: Load-displacement curves compared with ex-perimental results for impact loading at 2650 mm/s).

4. CONCLUSIONS

We have coupled a cohesive model of fracture endorsedwith rate-dependent laws with an equivalent stress-strainrelation to cover the fibre pullout tests. The numericalmethodology is implemented in a commercial softwareANSYS taking advantage of its automated programmingplatform APDL. Taking into consideration the complica-ted physical phenomena involved in the dynamic fracturepropagation of steel fibre reinforced concrete, the valida-tion against experimental results testifies the feasibility ofthe proposed model.

Parametric studies are not presented due to the page limit,but will be shown in the presentations.

ACKNOWLEDGEMENTS

We acknowledge the financial support from the Ministe-rio de Ciencia e Innovacion, MAT2012-35416, Spain.

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