A coupled thermo-hygro-chemo-mechanical model for the simulation of spalling of concrete subjected to fire loading

The presented research work contributes to the realistic simulation of the stress state within fireloaded concrete in order to attain insight into the development and occurrence of the critical state right before and during the event of spalling. A coupled thermo-hygro-chemo-mechanical code simulating the stress state as a consequence of both thermo-hygral and thermo-mechanical processes is presented together with an embedded strong-discontinuity model which is capable of capturing and tracking the propagation of a crack evolving in concrete as a quasi-brittle material. Combination of the two mentioned models is currently under way. With the resulting coupled model, it will be possible to take into account all major couplings, allowing to realistically simulate the spalling process.


INTRODUCTION
During spalling of fire-loaded concrete, the cross-sectional area of the concrete member is reduced, seriously affecting the integrity of the structure.Spalling is mainly attributed to two types of processes: thermo-hygral and thermo-mechanical processes.Thermo-hygral processes refer to the build-up of vapor pressure inside the concrete pores.Thermo-mechanical processes refer to the thermally-induced, restrained deformation of concrete.In order to realistically assess and predict the susceptibility of a concrete structure to spalling, both types of processes need to be taken into account, making a thermohygro-chemo-mechanical approach necessary.
In this paper, the ingredients of such a model are presented, considering the main couplings between energy and mass transport as well as the mechanical behavior of fire-loaded concrete.Combining the coupled model (which gives insight into the stress evolution in heated concrete) with a code capable of simulating the crack propagation within concrete will lead to a simulation tool for simulating and predicting the spalling process in fire-loaded concrete.The presented research work aims at development of a model assisting researchers and engineers to re-analyze spalling experiments and to evaluate the efficiency of specific fire-protecting methods.

LIST OF SYMBOLS b hom
Biot's coefficient acquired by multiscale homogenization

Thermo-hygro-chemical model
The coupled thermo-hygro-chemical model is formulated based on the balance equations for energy and mass presented in [1].
The balance equation of the water species reads The balance equation of dry air reads (2)

IWCS 2013
The balance equation of energy reads Detailed information on the material parameters and their evolution with temperature can be found in [1,2].

Effective-stress model
The isothermal macroscopic effective-stress theory [3,4] can be described as where the Biot's coefficient b can be acquired by, e.g., multiscale homogenization (see [5] for details].The Bishop's coefficient can be defined as a function of the saturation degreeS w .In heated concrete, Eq. ( 4) is reformulated, considering the shrinkage strain E SH according to [6], leading to where E T is the thermal strain tensor and E LITS represents the strain appearing in concrete subjected to combined mechanical and thermal loading.Based on the approach presented in [7], a relationship between E LITS and is given in [8,9]], reading Tr(E T ) 3 (6) where k LI T S = 0.6 was considered within the numerical simulation.A comparison between simulation and experimental results is shown in Figure 1 (see [8,9] for details.

Tensile strength under biaxial compression
Considering a concrete block, fire loading of one surface leads to biaxial compressive loading near the surface, causing a decrease of the tensile strength along the direction perpendicular to the heated surface.This can be considered by, e.g., a linear reduction of the temperature-dependent tensile strength as a function of the above mentioned compressive loading (see Figure 2 and [2]).

Model and material parameters
For the numerical investigation of spalling, a concrete member heated from one side was simulated using a 2D axisymmetric model (see Figure 3).The minimum element size was approximately 1 mm.The fire load was considered to increase linearly from 20 • C to 1200 • C within the first 300 s.After that, it was 05001-p.3set constant at 1200 • C. The material parameters at room temperature are shown in Table 1, information on the temperature-dependent functions for the respective material parameters can be found in [10].The intrinsic permeability k was varied in order to perform a parameter study (see Table 1).

Results
Figures 4 and 5 show selected numerical results for t = 200 s and t = 300 s.Hereby, the effective stress perpendicular to the heated surface (y-direction, see Fig. 3), which can be calculated by = (v ⊗ v) : ( + b hom p g ) with v = (0, 1).was compared with the temperature-dependent tensile strength f t under biaxial compression in order to obtain indications for the initiation of spalling.The results of and f t are shown together with the gas pressure p g .The numerical values for gas pressure p g are well in the range of experimental values obtained under comparable test parameters (concrete mix, heating curve etc.), see [11,12] for details.The numerically-obtained distributions of gas pressure p g and stress perpendicular to the heated surface are of similar shape, resulting from the underlying thermo-hygral processes (gas pressure influencing the mechanical stresses).Comparison of the results obtained with different values for the intrinsic permeability k (see Figs. 4 and 5) shows that increases considerably with decreasing k.On the other hand, the tensile strength f t perpendicular to the heated surface is reduced by the combination of increasing temperature and biaxial compression, reflecting the underlying 05001-p.5  thermo-mechanical processes.Hence, both types of processes considered to be responsible for spalling are captured by the presented model.Considering the simulation results, no spalling is predicted in simulation (a) with k = 10 −17 m 2 , but the numerical results indicate that spalling may occur in simulation (b) with k = 10 −18 m 2 at 300 s.

SIMULATION OF CRACK PROPAGATION
For simulation of the fracturing process during spalling, the strong-discontinuity embedded approach (SDA) presented in [13] will be implemented into the simulation tool.For this purpose, the energybased crack-tracking strategy presented in [14] is modified by the authors to be incorporated into the SDA (see [15] for details).For validation purposes, numerical examples of (i) an L-shaped panel and (ii) a pull-out test in isothermal conditions were simulated, showing the applicability of the used model.
The geometric dimensions and the material parameters as well as the loading and boundary conditions of the L-shaped panel, experimentally investigated in [16], are shown in Figure 6.The numerically-obtained force-displacement curve and crack path are shown in Figure 7.
Within simulation of the pull-out test, a 2D axisymmetric analysis was performed (see Fig. 8).As pointed out in [17], the pull-out test has two types of possible failure modes: tensile failure and compressive failure.In the presented benchmark, only tensile failure was considered.The numerically-obtained load-displacement curve is shown in Figure 9. Since other papers (see [18,19]) considered one quarter of the structure, F/4 is shown in the force-displacement curve.Good agreement is observed between the results obtained by the authors and numerical results from other simulations.The observed good performance of the developed model leads to the conclusion that the chosen simulation tool for crack propagation is capable of simulating the fracturing process during spalling.

CONCLUSIONS AND ONGOING WORK
In this paper, the ingredients of a coupled thermo-hygro-chemo-mechanical model for simulating the stress evolution within fire-loaded concrete were presented.Hereby, thermo-hygral as well as thermomechanical processes were taken into account, since both types of processes are considered to contribute to spalling.The envisioned model shall be able to simulate the stress state in heated concrete right before and during the spalling process considering all relevant processes.
Preliminary numerical results obtained with the thermo-hygro-chemo-mechanical model illustrated the contribution of both above-mentioned processes to the concrete's susceptibility to spalling.Thermohygral processes cause an increase in pore pressure in consequence of vaporization of pore water.Subsequently, the (effective) tensile stresses in the concrete matrix are considerably increased.Parameter studies showed that these tensile stresses increase with decreasing permeability of concrete, since less water vapor is able to escape the pore system.Thermo-mechanical processes, on the other hand, cause a build-up of compressive stresses parallel to the heated surface, leading to a decrease of the tensile strength perpendicular to the heated surface.This was considered by introducing a dependency of the (temperature-dependent) out-of-plane tensile strength on the biaxial in-plane compressive stresses.
The numerical examples showed the increased spalling risk of concrete with low permeability.
Currently, the presented thermo-hygro-chemo-mechanical model is extended to the simulation of the cracking process initiating spalling.For this purpose, a strong-discontinuity approach was adopted and validated by re-analysis of experiments as well as numerical studies of other research groups.Once the presented approach is implemented in the model, it will be possible to simulate crack propagation during the spalling event.This will enable for consideration of the effect of cracking on the pressure within the concrete's pore space as well as transport of water vapor towards and within the opening crack.
Finally, the developed code will enable for simulation of the main couplings between energy/mass transport and the mechanical loading situation within fire-loaded concrete, yielding realistic simulation of the stress state right before and during the spalling of near-surface concrete layers.This will enable researchers and engineers to re-analyze spalling experiments and to evaluate the safety of concrete and concrete structures under fire loading and the efficiency of fire-protecting methods.

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IWCS 2013 f b tensile, compressive and biaxial compressive strength of concrete[Mpa] h specific enthalpy of vaporization [J kg −1 ] MATEC Web of Conferences 1 second-order unity tensor [-] k intrinsic-permeability tensor of concrete [m 2 ] k rg , k rw relative permeability of gas and liquid phase [-] l specific enthalpy of dehydration [J kg −1 ] ṁdehydr change of mass dehydration products per uinit volume of concrete [kg m −3 s −1 ] ṁvap change of vapor mass per unit volume of concrete [kg m −3 s −1

Figure 1 .
Figure 1.Comparison of numerical results for E LITS with experimental data (see[8,9] for details).

Figure 2 .
Figure 2. Reduction of the tensile strength f t ( b , T ) in case of thermal loading from the surface.

Figure 6 .
Figure 6.L-shaped panel test: geometric properties and material parameters as shown in [16].

Figure 9 .
Figure 9. Pull-out test: comparison of numerically-obtained force-displacement curve and crack path with numerical results reported in [18, 19].

Table 1 .
Material parameters (values at room temperature).