Repositorio Institucional

9th International Conference on Fracture Mechanics of Concrete and Concrete Structures FraMCoS-9

V. Saouma, J. Bolander and E. Landis(Eds)

DOI 10.21012/FC9.301



DISCONTINUOS APPROACH FOR MESO-SCALE MODELING OF POROUS MATERIALS LIKE CONCRETE UNDER HIGH TEMPERATURE



*



†



*



GUILLERMO ETSE , DIEGO SAID SCHICCHI , ANTONIO CAGGIANO AND



*

MARIANELA RIPANI



*

CONICET and University of Buenos Aires, Argentina e-mails: getse@herrera.unt.edu.ar, acaggiano@fi.uba.ar, mripani@fi.uba.ar



†

Stiftung Institut für Werkstofftechnik, Bremen, Germany and Instituto Nacional de Tecnología Industrial (INTI), Buenos Aires, Argentina

e-mail: schicchi@iwt.uni-bremen.de



Keywords: High Temperature, Cohesive fracture, Spalling, Meso-Scale, Porosity, Pore pressure.

Abstract: In this work, the failure behavior of concrete exposed to elevated temperatures is analyzed. A thermo-mechanical and poropressure-based interface model for failure analysis of concrete subjected to high temperature is presented. The model represents an extension of a fracture energy-based interface formulation to account for the damage induced by high temperatures and for the temperature dependent pore pressure and humidity in concrete. Thereby, the non-linear response of the proposed coupled thermo-mechanical interface model for porous materials like concrete is activated under kinematic and/or temperature and/or hydraulic increments (with or without jumps). A simplified procedure is proposed to account for the temperature dependent pore pressure in concrete. This contribution focuses on both the formulation of the novel interface constitutive theory and on the strategy proposed for mesoscopic finite element analysis of concrete failure behavior under different hydro-thermo-mechanical conditions. Finally, some numerical analysis are presented which demonstrated the predictive capabilities of the proposed interface model.



1 INTRODUCTION

The action of high temperature in concrete is a field of much interest and attention due to its crucial influence in strength, durability and serviceability conditions of structural components. As a matter of fact, long-term exposures to high temperature fields and fire strongly affect the most relevant thermomechanical properties of concrete materials such as cohesion, friction, stiffness and strength [1-2].

The exposure to high temperature and/or fire is one of the most destructive events that concrete buildings and structures may suffer [3-4]. The chemical composition, the physical



micro- and meso-structure as well as the moisture content and porosity of concrete drastically change under elevated temperatures [5-6]. The cause of such modifications is represented by the dehydration of the hardened cement paste and the conversion of calcium hydroxide into calcium oxide [7]. As a consequence, during and after long term exposure to high temperature, the most important mechanical features of concrete can adversely be affected. Experimental evidences demonstrate the dramatic and substantial changes of cohesion, tensile and compressive strengths as well as the Young modulus and Poisson's ratio of concrete due to the long term exposure to fire and/or high temperature [8-9].



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Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



When temperature rises, concrete loses weight with a progressive increase of its porosity. Particularly, from 20ºC to 200ºC cement-based materials quickly lose their specific weight as a result of evaporable water loss. Then, when temperature reaches the range between 200ºC and 600°C the weight loss starts to be slower. This process is due to the dehydration of cement paste which deals with the loss of water, chemically combined to the calcium silicate hydrates. After 600 °C the decomposition of magnesium and calcium carbonates, which form part of the concrete matrix, produces further loss of weight which can reach the 10% of the total one [10-12].

Plenty of studies addressed the change of concrete mechanical properties when subjected to high temperature, with special focus on its durability. The study of the pore size distribution in concrete exposed to thermal action (up to 800°C) has been addressed by Janotka and Bagel [13]. This research confirmed that under increasing temperatures the pore size mainly grows and its distribution becomes more and more homogeneous. Porosity of concrete subjected to high temperatures may increase up to 40% of its initial value. It is worth mentioning that porosity rise is not only due to the evaporable water loss, but also to the dehydration of the gel structure formed by the calcium-silicate hydration products when concrete is exposed to elevated temperature [14].

One of the more interesting and crucial phenomena in concrete components when subjected to fire or high temperature is the socalled “spalling”. Such a phenomenon has been studied in literature from both the experimental [15-16] and theoretical [17-18] point of view. The process of spalling is quite complex, however the literature on this matter states that spalling of concrete mainly depends on the porosity of the cement matrix, the amount of water content and the stress state either due to thermal gradients and/or applied mechanical loads. During heating, the water within the concrete is transformed into steam and tends to migrate to colder areas of the concrete matrix. Once the vapor flux reaches coldest zones, it condenses again, forming a



fully saturated water layer. This process typically occurs in the neighborhood of the heated surface inside the concrete piece. This area is commonly called “moisture clog” and is characterized by a low permeability, generating an impermeable barrier to gases passage. Then, the continuous rise of temperature, with the subsequent generation of further vapor gases which cannot escape to colder areas due to the presence of the water barrier, generates pore pressure. Such increment in the pore pressure, added to the stresses induced by thermal strains, mainly activates the process of spalling. It is important to remark that pore pressure acts as a triggering of spalling phenomenon. Once the cracking process starts the pore pressure is quickly dissipated and the further development of spalling does only depends on the thermal stresses increments [5,19,20].

This paper presents a discrete-crack modeling approach aimed at reproducing and predicting the failure behavior of porous materials like concrete when subjected to high temperature fields. In this framework, a zerothickness interface constitutive theory for porous media like concrete under high temperatures is presented, which is an extension of the temperature independent interface model for non-porous materials by Carol et al. [21]. This interface model deals with a novel pressure-dependent dehydration rule accounting for the porosity features of concrete and thermal conditions. The proposed interface model, as shown in this work, allows accurate mesoscopic simulations of concrete cracking and spalling when exposed to fire.

2 COUPLED THERMO–POROELASTIC CONTINUOUS MODEL

The mechanics of porous media provides a useful theoretical framework for the study of transport problems in porous materials as concrete, soils and rocks. In this framework, porous media can be interpreted as multiphase materials composed by a deformable skeleton and pores with one or more saturating fluids.



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Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



2.1 Constitutive equations of thermo-poroelastic materials



The constitutive equations of thermo-poroelastic materials, in terms of total stress

porosity evolution  e and vapor pore

pressure pvp, are described by means of the following equations [22]:



  C :  e  Bpvp  A



e



 B:e  1 M



pvp 



(1)



being C the fourth order elastic tensor,  e the elastic strain tensor, B = bI the Biot's tensor (being b the Biot's coefficient) and I the second order identity tensor. Moreover,   T T0 is the temperature rise (being T and

T0 the actual and reference temperatures,

respectively), A =  I the thermal expansion tensor with  the thermal coefficient, M is the Biot's modulus and, finally,  is the latent

heat of porosity variation.



2.2 Temperature-dependent properties



The elastic properties of concrete play a key role in its overall temperature dependent response. Based on several experimental results [23-25] the dependency of the concrete elasticity modulus E and of the Poisson’s ratio on the temperature rise can be approximated by means of the following functions:



E  E0 1E 



(2)



 0 1 



(3)



where E0 and 0 are the elastic modulus and

Poisson’s ratio at T0, respectively, E and

are degradation parameters to be calibrated (suggested values by the authors for normal

strength concrete are E  0.0014 and

  0.0010 ).

Then, the Biot’s coefficient and modulus depend on the temperature level and can be defined as [26]:



 b, min



T T , b1



  1

b 2



1 bmin  sinb   1 bmin 



,



 1,



T T . b2



(4)



with



T T











b1  



b T T



2



b2



b1







M, 0



T T , tr



M







 



1



M



2



sin  M



  M,







M, res



T T . res











with  







(5)



M T T



2



tr



0



M  M  M



0



res



being Tb1, Tb2, Ttr and Tres (temperature) parameters to be calibrated. Suggested values are Tb1  100º C , Tb2  500º C and Ttr  200º C (this latter represents the trigger temperature when spalling starts).

The Biot’s coefficient b ranges between its minimum value bmin at Tb1 and its maximum

one bmax  1when T≥Tb2 . On the other hand, the Biot’s modulus M decreases from its maximum value M0 at Ttr up to its minimum

one, M res , at Tres  2Ttr T0 .



3 COUPLED THERMO–POROPLASTIC DISCONTINUOUS MODEL

This section reports the temperature dependent interface formulation aimed at analyzing cracking behavior of quasi-brittle porous materials such as cementitious mortar and concrete.

Stresses and relative displacements are defined at the discontinuity mid-plane of each interface. The principle of “effective stresses” applied to the particular case of discontinuities considers that the fluid pressure influences the normal stress component (i.e., similar to the volumetric component in the continuum formulation, see Eq. 1).

Thus, following the formulation by Segura and Carol [27]:



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Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



t  t  jbpvp



(6)



where t   N ,T t and t   N ,T t are the

mid-plane effective and total stress vectors, respectively,  N and  N are the effective and total normal stresses, T is the shear stress component, j the discontinuity Biot’s

coefficient [28] and b  1, 0t is a vector that

introduces the influence of the fluid porepressure acting in the normal direction to the discontinuity axis.

The basic constitutive equations of the proposed elasto-thermo-poroplastic interface model are the following [29]:



u  uel  ucr  uth



uel  Cd1·t



(7)



  t  Cd· u  ucr  uth



where u  [u, v]t is the vector of the relative



displacement rates across the interface, which



is additively decomposed into the elastic,



.



.



plastic and thermal components, uel , ucr and



.



.



uth , respectively; t  [ N ,T ]t



is the



effective stress rate vector defined in the



normal and tangential interface directions.



Moreover, Cd defines the elastic stiffness



matrix, thermally degraded through the



temperature-based scalar variable, d :



Cd  d C



with



C







  



kN 0



0



kT



 



(8)



where kN and kT are the interface normal and tangential elastic stiffness, respectively.

The rate of normal and tangential interface effective stresses can be derived through the rates of relative displacement vector and temperature rise as:



t  Cdep·u  f [T ]



(9)



being Cdep the tangential interface stiffness for elastic degradation while f [T ] the thermal interface effective stresses due to the rate of interface temperature field.



The complete formulation of the elastothermo-plastic rate equations which can be derived from the description of the thermal and plastic interface displacements, the formulation of a temperature dependent yield criterion, the adoption of temperature and fracture-energy dependent evolution laws are omitted in this work for the sake of brevity. A detailed formulation was published by the authors in [29].



4 VAPOR PRESSURE

As previously described in the introduction, due to the moisture evaporation phenomena, the relative humidity of concrete in the heated surface zones tends to decrease to very low values. Consequently a sharp front, separating the moist and dry material, moves inwards and at this “impermeable” front, an intensive evaporation process tries to take place, increasing considerably the so-called vapor pressure. The variation of the pore pressure in concrete with the temperature was experimentally measured by Pereira et al. [30].



1.20



"" 1.00

""



0.80



Experimental measurements



pvp/pvp,0



0.60



0.40



0.20



0.00



0



100 200 300 400 500 600 700



T [ C]



Figure 1: pvp – temperature rule: comparison of the proposed law against poropressure measurements at 20



mm from the heating surface by Pereira et al. [30].



In this work, a simplified procedure to account for the vapor pressure in both continuous and discontinuous models detailed in Sections 2 and 3 is proposed. Particularly, the following expressions have been employed for the pore pressure evolution:



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Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani







0



if T  0



pvp







 



  vp1 T Tvp p e vp,0



if 0  T  Tvp



 



pvp,0e



vp



2



T



Tvp







if



T  Tvp



(10)



where pvp,0, Tvp, vp1 and vp1 are model parameters to be calibrated. Figure 1 shows the plot of the pvp-temperature rule and, as an example, a comparison of the considered law against pore pressure measurements at a depth of 20 mm from the heating surface of a concrete specimen by Pereira et al. [30] is reported.

It is worth mentioning that the classical way for modeling drying processes and vapor diffusion phenomena in porous media like concrete deals with calculating the moisture diffusion which takes place in its porous structure. This can be approached by means of modeling the migration of the “evaporable water” throughout the pore structure of the cementitious composite. Then, the vapor pore pressure should be computed as result of this complex phenomenon dealing with moisture and temperature diffusion phenomena coupled with mechanical effects induced by thermal events. This further development is out of the scopes of this work and it will be certainly considered by the authors in future work of this research line.



4 NUMERICAL ANALYSES

This section proposes some calibrations and numerical applications of the proposed model. For the calibration purpose, experimental results on concrete specimens tested under pure tension are considered. After this, failure processes in concrete under coupled temperature and fracture loading are taken into account for the evaluation of model predictive capabilities.



4.1 Interface model calibration

Tensile tests (Figure 2) performed on concrete specimens after being exposed to room, medium and high temperatures (at residual stages) are considered as a reference.



tensile stress MPa



direct tension - DT

Figure 2: Test set-up of tensile tests.



5



4



3 2



1



0



0



200



400



600



800



Tavg °C Figure 3: Strength decay in direct tensile tests: experimental results [31] and model predictions.



Bamonte & Felicetti (2007) tests



5.00



Residual Test 20 C



4.00



Residual Test 250 C



Residual Test 600 C



Numerical simulations

3.00



tensile stress [MPa]



2.00



1.00



0.00 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

u [mm] Figure 4: Residual stress-crack opening curves: experimental results [31] against numerical predictions.



For this purpose, experimental results presented by Bamonte and Felicetti [31] were considered. As a result of the calibration analysis [29], a decay trend occurred for both the concrete tensile strength and elastic stiffness. Experimental data exhibited a similar behavior against increasing temperatures. Particularly, Figure 3 deals with the variation



5



Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



of concrete peak strength with temperature, obtained from the direct tensile tests by Bamonte and Felicetti [31] and those obtained with the numerical prediction based on the interface model parameters and its internal decay functions as indicated above.

Moreover, the comparison between the same experimental results and numerical data in terms of stresses vs. crack opening displacements, for three different temperature levels, is highlighted in Figure 4. Beyond the general soundness of the interface model predictions for failure behavior of concrete affected by temperature, the numerical results demonstrate the capabilities of the proposed formulation to reproduce the strong sensitivity of the concrete mechanical behavior on the acting temperature.



4.2 Pore-pressure coupled analysis



In this section, the predictions of the temperature-based interface model are evaluated under tensile loads and subjected to thermo-mechanical couplings with or without consideration of the pore pressure effects.



Step-1



Step-2



CF



A prefixed mode I displacement is imposed at ambient temperature



ISO-834 curve of temperature rise is applied (holding the already applied displacement)



Figure 5: Specimen geometry, boundary conditions and considered steps.



The discontinuous model was previously calibrated with the experimental data of residual mechanical properties as proposed in Section 4.1, while the thermal parameters given in Table 1 were considered. Furthermore, the adopted poropressures are those highlighted in Figure 1.

Figure 6 shows the temperature-dependent behavior of a tensile test as predicted by the



proposed model. Particularly, Figure 6(a) shows the imposed displacement-temperature input in the tensile test. In the first step of the test, only mode I displacements are imposed, at room temperature (i.e., 20 °C). Then, in a second step of the test, the ISO-834 temperature curve is applied, while keeping constant the previously applied tensile displacement (similarly to the well-known relaxation test, see Figure 5).



0.010 0.008



Tensile tests

Displacement/Temperature input



Normal separation [mm]



0.006



0.004



0.002



0.000 0



100 200 300 400 500 600 700

Temp [ C]

(a)



Tensile tests



5.0



Stress-Temperature response without pore-pressure



Stress-Temperature response with pore-pressure 4.0



Stress [MPa]



3.0



2.0



1.0



0.0 0.000



0.002 0.004 0.006 0.008 Normal separation [mm]

(b)



0.010



Tensile tests



5.0



Stress-Temperature response without pore-pressure



Stress-Temperature response with pore-pressure 4.0



stress increment due to



3.0



vapor pore pressure



Stress [MPa]



2.0



1.0



0.0 0



100 200 300 400 500 600 700

Temperature [ C]



(c) Figure 6: Stress-temperature behavior with and without



poropressure: (a) displacement-temperature history, (b)



normal stress vs. opening displacements and (c) stress-



temperature response.



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Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



Table 1: Thermal parameters of concrete according to EN 1992-1-2:2004 [32].



T [°C]

20 100 200 400 500 600 700 734 800 900 1000 1100 1200



cp [J kg-1 °C-1] λ [W m-1 °C-1]



900 1470 1000 1100 1100 1100 1100 1100 1100 1100 1100 1100 1100



1.642218 1.49765 1.3317 1.049 0.93225 0.8319 0.74795 0.72314292 0.6804 0.62925 0.5945 0.57615 0.5742



εth [-]

1.12E-07 4.94E-04 1.19E-03 3.18E-03 4.63E-03 6.50E-03 8.88E-03 9.82E-03 1.18E-02 1.20E-02 1.20E-02 1.20E-02 1.20E-02



T [°C]

20 100 200 400 500 600 700 734 800 900 1000 1100 1200



ρ[kg/m-3]

2300 2300 2254 2185 2.16E+03 2.14E+03 2.12E+03 2.12E+03 2.10E+03 2.08E+03 2.06E+03 2.04E+03 2.02E+03



E [GPa]

3.82E+01 33.9216 28.5736 17.8776 12.5296 7.1816 1.8336 0.01528 0.01528 0.01528 0.01528 0.01528 0.01528



ν [-]

0.2 0.184 0.164 0.124 0.104 0.084 0.064 0.0572 0.0572 0.0572 0.0572 0.0572 0.0572



Figure 6(b) and (c) illustrate the results in terms of normal stress-opening displacements and stress-temperature response, respectively, in case of considering (or not) the vapor pore pressure effects. Poropressure produces an increment of the effective normal stress, approaching the failure surface of the interface at a temperature of 209 °C (without poropressure it is reached at 310 °C). Then, the internal parameters, damaged through the temperature-based dehydration rule, softens the response of the specimen up to the complete failure (i.e., the tensile stress becomes null). When considering vapor poropressure, the numerical test shown in Figure 5, indicates that the failure state is reached at about 343 °C while without considering poropressure effects the process is slower and the total failure occurs at 446 °C.



4.3 Wall exposed on one side to the standard ISO fire curve

The description of physical phenomena in heated concrete walls presented in this section follows similar numerical tests available in scientific literature regarding high temperature tests of concrete elements [33-36]. Here, the results of simulations concerning a 12-cm wall (the same thermo-poro-mechanical parameters of Section 4.2 are considered), exposed on one side to the standard ISO-834 (Figure 7) are outlined. The heated surface generates a gradual increase of the element temperature, starting from the surface as illustrated in Figure 8. Due to that an empirical rule is adopted, the corresponding vapor pressure curves at different times are plotted in Figure 10. The maximum values of the vapor poropressure curves (0.7-0.8 MPa) coincide with the position where the temperature reaches approximately 200 °C.



Iso-834 fire



12 cm

Figure 7: Boundary examples for the simulation of a 12-cm wall during exposed to ISO-FIRE 834 heating.



iso - 834 for buildings



1200



T=20+345 Log(8 t+1)



1000



Temp [ C]



800



600



400



200



0 0 2 000 4 000 6 000 8 000 10 000

time [sec]

Figure 8: The temperature development of the buildings fire curve, ISO-834.



7



Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



800



700



657 sec



507 sec



600



369 sec



500



265 sec



187 sec



400



131 sec



300



89.8 sec



Temp [ C]



200



100



0 0 20 40 60 80 100 120

x [mm] Figure 9: Temperature vs. abscissa X at several times.



pvp [MPa]



0.9



0.8



657 sec



0.7



507 sec



0.6



369 sec



0.5



265 sec



0.4



187 sec



0.3



131 sec



0.2



89.8 sec



0.1



0 0 20 40 60 80 100 120



x [mm] Figure 10: Pore-pressure vs. abscissa X at several

times.



Increasing temperature causes concrete material dilatations. Figure 11 highlights the Ux displacement response due to such thermal dilatation. However, this phenomenon can be in part caused to the classical thermal dilatation of the material, and, also can be caused by the progressive crack opening. Concrete cracking and spalling, during heating processes, is mainly caused by the thermal dilatation of the coarse aggregate which leads to high traction stresses in the Interface Transition Zones with the surrounding mortar and, consequently, to the development of local micro-cracks.

When cracks parallel to the heated surface starts to be formed, e.g., due to an external load or as result of constrained thermal dilatations of the surface layer and/or mesoscale components, the vapor poropressure can contribute to the material spalling, and particularly to further increase of the interface jumps as demonstrated in Figure 12.



-60



Ux [mm]



-50



657 sec



507 sec



-40



369 sec



-30



265 sec 187 sec



-20



131 sec



89.8 sec



-10



0 0 20 40 60 80 100 120

x [mm]

Figure 11: Ux displacement vs. abscissa X at several times.



-1.8E-3



u - joints [mm]



-1.6E-3 -1.4E-3 -1.2E-3 -1.0E-3 -8.0E-4 -6.0E-4



657 sec 507 sec 369 sec 265 sec 187 sec 131 sec 89.8 sec



-4.0E-4



-2.0E-4



0.0E+0 0



20 40 60 80 100 120

x [mm]



Figure 12: Interface opening displacement vs. abscissa



X at several times.



5 CONCLUDING REMARKS

This paper addressed the formulation of a novel coupled pore pressure-based and coupled thermo-mechanical interface model for porous materials like concrete when subjected to high temperatures. The principle of “effective stresses”, applied to the particular case of interface discontinuities was considered in conjunction with the influence of the fluid pressure on the normal total stress components. Particularly, this proposal took into account a quadratic hyperbola as a maximum strength criterion for the interface, defining the interaction between maximum shear and normal interface effective stresses. Consequently, the constitutive formulation explicitly considered the thermal effects and the vapor pore pressure into the failure criterion and the resulting post-cracking response through a temperature-based scaling function affecting the strength parameters and



8



Guillermo Etse, Diego S. Schicchi, Antonio Caggiano and Marianela Ripani



the softening law. Numerical analyses, performed with the constitutive model presented in this paper, demonstrated its predictive capabilities in terms of the most relevant aspects of the mechanical behavior of concrete under high temperatures. The proposed interface model can also be employed in mesoscopic analyses using mortar-mortar and mortar-aggregate interfaces aimed at simulating failure processes and spalling phenomena of concrete specimens under high temperatures.

ACKNOWLEDGMENTS

The authors acknowledge the financial support for this work by the SUPERCONCRETE Project (H2020-MSCARISE-2014–n. 645704) funded by the European Union as part of the H2020 Programme.

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