Available online at www.sciencedirect.com Mathematics and Computers in Simulation 80 (2010) 1694–1712 An approach for modelling concrete spalling in finite strains C.E. Majorana a , V.A. Salomoni a,∗ , G. Mazzucco a , G.A. Khoury b a Department of Structural and Transportation Engineering, Faculty of Engineering, University of Padua, via F. Marzolo 9, 35131 Padua, Italy b Department of Civil and Environmental Engineering, Skempton Building, Imperial College, London SW7 2AZ, UK Received 3 November 2008; received in revised form 22 April 2009; accepted 25 May 2009 Available online 6 June 2009 Abstract A new approach for modelling concrete spalling process is here proposed, taking into account a fully nonlinear-displacement/strain theory able to catch complex interactions between pressure, thermal and mechanical fields. The micro-structural modelling of concrete under fire conditions is derived from a mechanical and thermodynamic consistent theory and it is strictly related to a self-consistent, carefully extracted set of experimental data, in order to make a correct validation and calibration of the numerical F.E. procedures and codes. Even if appearing as a first but successful example, it is shown that a procedure accounting for coupled material and geometric nonlinearities is able to attain valuable and realistic numerical results concerning spalling process in concrete. © 2009 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Concrete spalling; Micro-structural modelling; Finite strains 1. Introduction Explosive spalling is a very violent form of spalling characterised by the forcible separation of pieces of concrete, accompanied by a typically loud explosive noise. It normally occurs within the first 30–40 min of exposure to fire. It is also stochastic. For specimens from the same batch, and under identical conditions, some could spall while others do not. Given suitable environmental conditions, in terms of load and thermal attack (i.e. high heat influx), all concrete can display the capacity for explosive spalling [32]. Explosive spalling can take place as a single explosion or a series of explosions, each removing a thin layer of concrete ranging from 100 mm to 300 mm in length [52] and 15 mm to 20 mm in depth [17], capable of causing physical damage on impact. In many cases, explosive spalling is restricted to the unreinforced part of the section and usually does not proceed beyond a reinforcing layer. Multiple spalling is more likely in high strength concrete than in ordinary strength concrete, as experienced in past years in the Great Belt tunnel fire in Denmark [29]. This paper takes its origin from a stress–strain model of heated concrete [40,49] based on extensive testing carried out by Khoury [31,33–35,39] on the strain behaviour of concrete during two heat cycles to temperatures up to 600 ◦ C. Strain measurements of three concretes during two cycles of heating with and without compressive load over a period of 14 days allow the identification, and separation, of strain components for the transient and steady state temperature conditions as functions of temperature. Contractive strain components during first heating are dominated by the Load Induced Thermal Strain (LITS) with shrinkage being the second strain component. The contractive strains during the ∗ Corresponding author. Tel.: +39 049 8275590; fax: +39 049 8275604. E-mail address: salomoni@dic.unipd.it (V.A. Salomoni). 0378-4754/$36.00 © 2009 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2009.05.011 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 1695 subsequent period at constant temperature comprise both creep and shrinkage. The former would contain delayed transient creep (relevant to the temperature in question) as well as constant temperature creep. While LITS is absent during cooling and subsequent thermal cycle, another important strain component appears which is expansive and which is related to the cracks that develop during cooling and subsequent thermal cycle. These crack strains depend on the type of concrete, the temperature level and the load level. The residual strain is measured directly and modelled indirectly as the sum of the irrecoverable contractive and expansive components. Hence, a proper micro-structural modelling of concrete under fire conditions should be developed accomplishing two main requirements: (1) it should be derived from a mechanical and thermodynamic consistent theory; (2) it should be strictly related to a self-consistent, carefully extracted set of experimental data, in order to make a correct validation and calibration of the numerical F.E. procedures and codes. For heat and mass transport purposes, several approaches can be used at macro or meso-levels (the latter approach is being developed by the first two authors, see e.g. [63]) [18,19]. Aggregates and cement matrix are usually smeared (at macrolevel), but fluids can be separated or not, giving rise to different degrees of insight in the heat and mass behaviour. However, for computational purposes both are used in the scientific community, due also to calculation time needed and available computer power. Also the separation of aggregate from matrix, needed for fully understanding and predicting spalling behaviour, is becoming more affordable, as far as 3D parallel computation will become economically feasible. Full description of phase changes, sorption–desorption behaviour at high temperature/fire and related strains is progressively achieved in the analysis. From the mechanical point of view, the stress–strain constitutive relationship for concrete under high temperature and fire needs to take into account several effects like thermal, shrinkage, creep, LITS, damage, plastic and other chemical contributions that must be carefully checked against experiments. Connection between experimental data and established mechanical theories can give reasonable answers for predicting the behaviour of concrete under such heavy loading and thermal conditions. However the authors think that a crucial step forward to predict and describe the kinetics of spalling will be obtained if the theoretical framework described above will be inserted in a fully nonlinear-displacement/strain theory able to catch the complex interactions between pressure, thermal and mechanical fields when parts of concrete are attaining dynamic unstable disruption. The comprehensive strain model obtained from the tests is then directly integrated in the mechanical part of a 3D fully coupled thermo-hydro-mechanical numerical model of heated concrete developed by the first two authors, described in this paper along with the test results [42,45,47,50,61,62,66]. The generated F.E. code is called NEWCON3D, in which concrete is treated as a multiphase system where the voids of the skeleton are partly filled with liquid and partly with a gas phase [10,11,24]. As regards the mechanical field, NEWCON3D couples creep and shrinkage [4–9,12,13], chemo-thermo-mechanical damage [25,44,53,54,59] and plasticity effects under medium and high temperature levels. An original development is here presented to analyse the onset and process of spalling using the theory of finite strains. Usually the starting of spalling has been studied in the past by means of a small strain approach and no attempts have been carried out to describe the whole spalling process. Hence the scope of this paper is to state that for describing the entire spalling process a more general mechanical framework, as it is the finite strains one, is needed. The first part of the paper deals with the description and historical background of the spalling problem in concrete, to present the main experimental results found in the past and to establish that for stating a rigorous thermo-hygromechanical mathematical approach it is necessary to use a complete and reliable data set able to give information on the key strain experiments carried out on heated concrete. Such experiments have been mainly performed at the Imperial College, UK, and this paper is a result of the collaboration between this Institution and the University of Padua, Italy. On this basis, a mathematical description of the phenomena can be given, as presented in the second part of the paper. Some representative examples clearly show how the approach chosen is the only one capable to entirely describe the spalling process of heated concrete. 2. The phenomenon of (explosive) spalling Many factors have been identified in the literature review as influencing explosive spalling in concrete; their influence is summarised as follows: • Heating rate has a major influence on the occurrence of explosive spalling. The probability, and severity of explosive spalling increase with increase in the heating rate [57,58]. However, when spalling does occur it does so within a consistent temperature range, regardless of heating rate [1,2]. 1696 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 • Heating profile: Heating from two faces is more likely to encourage spalling than heating from one face only [57,58]. Because of their exposure to fire on one side only, slabs behave more favourably than beams with regard to explosive spalling. Beams are usually exposed on three or four sides. It is, therefore, preferable to employ structural members with simple external shapes without pronounced projecting features. • Section size: Explosive spalling is unlikely to occur in very thin sections, because moisture tends to escape more readily, thus preventing the build-up of pore pressures. Experimental evidence also suggests that explosions are less likely in thick members greater than about 200–300 mm [3,20,21,26,56]. Explosive spalling is, therefore, most likely to occur in ‘medium’ size sections. The thicknesses of the nuclear containment walls are greater than these dimensions, so that the concrete used for such structures is less likely to experience explosive spalling. • Sections shape: Spalling is also more likely in sections with ‘rapidly’ changing cross-section. Corners, especially acute-angled ones, have a marked spalling tendency [57,58]. The ideal sections are, therefore, plain surfaces and rounded corners. • Moisture content: Spalling can occur if the moisture content of ordinary strength concrete is more than 2% by weight (5% by volume) [20,57,58]. It appears that at lower values of moisture content, which hardly ever occurs in practice, it is unlikely to bring about explosive spalling of ordinary strength concrete even under unfavourable test conditions. For a given set of conditions, explosive spalling is less likely for concretes with moisture contents less than 3% by weight [16,56]. It is judged that the concrete in the nuclear containments is at a relative humidity of 65%, which is equivalent to 3% moisture content by weight. The concretes used in the nuclear containments also can be classified mostly as “normal strength”. These factors combined are indicators of a low probability of explosive spalling. However, very dense high strength concrete can experience spalling in fire even with low moisture contents of 2.3–3.0% by weight. This is due to the low porosity and permeability of such concrete, whereby even the release of chemically bound water can contribute significantly to pore pressures. These factors more than compensate for the increased tensile strength. • Pore pressures: Vapour pressures play an important role in explosive spalling. Significant pore pressures can build up in low permeability concrete with high moisture contents. The magnitude of pressure depends upon the level of pore saturation. In partially saturated pores, the vapour pressure is the saturation vapour pressure (SVP). In empty pores, the vapour pressure is the pressure of superheated steam, while in fully saturated pores it is hydraulic pressure. Pore pressures of about 0.7–2.1 N/mm2 were measured under conditions similar to fire exposure [23,65,67]. Such pressures, being less than the tensile strength of normal concrete, do not cause failure by themselves but contribute to explosive spalling in combination with other stresses. However, Khoylou [41] predicted hydraulic pore pressures in fully saturated pores that are orders of magnitude greater than the tensile strength of concrete. The hydraulic pressure reduced with decrease in initial pore saturation level, and occurred at higher temperatures. • Permeability is an important factor influencing the critical pressure level, because it affects the rate of vapour release [67]. It has been suggested that spalling is unlikely to occur if the permeability of the concrete is greater than about 5 × 10−11 cm2 [27]. Concrete of higher quality generally possesses higher density and therefore offers higher resistance to flow. • Age of concrete: The influence of age of a concrete on its susceptibility to spalling has been subject of conflicting reports. The majority of reports, however, suggest that the probability of spalling reduces with age, although this may be due to the lower moisture level in older concretes. • Strength of concrete: Ironically, poor quality concrete is superior to good concrete in spalling. Concrete that is classed as “high performance” at room temperature – because of its high strength, low permeability and consequently good durability – is in fact a “low performance” concrete at high temperature because of the increased susceptibility to spalling [28,30,64]. Higher strength is achieved by reducing the water/cement ratio. In recent years, this was augmented by the use of silica-fume that produced a dense concrete of very low permeability. Silica-fume concrete (not used in the reactors) has a high susceptibility to explosive spalling even at low heating rates. However, in general, reducing the w/c ratio would enhance pore pressure spalling (via lower permeability) but reduce thermal stress spalling (via higher strength). • Compressive stress and restraint: Applied loads, and restraint, increase the susceptibility of concrete members to spalling [20,56]. An increase in compressive stress, either by reduction in section size or an increase in loading, encourages explosive spalling. The initial compressive stress in the exposed layer of concrete may not by itself promote spalling. However, high compressive stresses – caused by restraint to thermal expansion – develop when the rate of heating is such that the stresses cannot be relieved by creep quickly enough. Combinations of compressive C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 • • • • • • • • • 1697 stresses (above 2 N/mm2 ) and moisture contents (above 3.3% by weight) make the occurrence of spalling likely in a fire. Type of aggregate: Although the test results from different authors are inconsistent, it can generally be concluded that the likelihood of thermal stress explosive spalling is less for concrete containing a low thermal expansion aggregate. The risk of explosive spalling increased the following order: lightweight, basalt, limestone, siliceous, Thames River gravel. However, this only applies for concrete with relatively dry aggregates, since it has been shown that lightweight aggregate concrete has a high susceptibility to spalling if the aggregate is saturated. Aggregate size: The available evidence suggests that increasing the aggregate size promoted explosive spalling [17,52]. Cracking: Internal cracking has a dual and opposite effect upon explosive spalling. While microcracks facilitate the escape of moisture during heating and thus relieve pore pressures, they also facilitate the process of spalling by providing a source for crack propagation. Explosive spalling is expected to be less likely in the top cap region than in the containment wall because of the potential presence of more micro-cracks in the lightly loaded bottom region of the top cap. Reinforcement: Test results suggest that spalling is less likely with the provision of reinforcement in the central region a slab, and occurred at a higher heating rate [56]. The presence or absence of reinforcement was found to be a more important factor in spalling than the quantity of reinforcement. However, congestion of steel bars or tendons, with only small spaces between them is considered to induce the formation of cracks, and may therefore promote spalling. Cover to reinforcement: If the cover exceeds 40 mm for dense or 50 mm for lightweight aggregates concrete, there is a danger of concrete spalling. Concrete cover thicknesses of 15 mm or less, seem less prone to serious spalling, probably because the mass of unsupported concrete is not large [52]. Supplementary reinforcement: The use of a light mesh cover does not prevent spalling but could limit the extent of spalling and significantly improve the performance of the columns with regards to fire resistance [70]. Supplementary reinforcement protected concrete columns from the effects of spalling, but not from the phenomenon itself. Control of spalling by use of supplementary reinforcement is sometimes used in sections where the cover to the outer bars exceeds 40 mm. However, supplementary reinforcement is difficult to place in thin sections, such as ribbed floors. Supplementary reinforcement provides two benefits: limiting of fire damage and easier repair of the structure. It is recommended that such a mesh be used only in cases where high standards of fire resistance have to be met [20,21]. Steel fibres: The addition of a steel fibre mesh reinforcement did not eliminate the explosions in very dense silicafume concrete cylinders subject to heating at a rate of 1 ◦ C/min [28]. In fact the increase in tensile strength produced a more violent explosion because of the sudden release of a greater amount of energy. Polypropylene fibres: Recently, it was found that the addition of 0.05–0.1% by weight of polypropylene fibres in a concrete mix completely eliminated explosive spalling even in high strength concrete (60–110 N/mm2 ) but not necessarily in ultra high strength concrete (>150 N/mm2 ). Air-entrainment: The use of air-entraining agent could remove the risk of explosive spalling [17]. The addition of the agent had the effect of reducing the moisture content and increasing the absorption value. In effect, it reduced the pore saturation thus alleviating pore pressures. There are two forms of explosive spalling, both influenced by external loading; pore pressure spalling and thermal stress spalling. They act singly or in combination depending upon the section size, the material, and the moisture content. Pore pressure spalling has been predicted using a “moisture clog model” [68], “vapour drag forces model” [56] or an “idealised spherical pore model” [1]. The main factors that influence pore pressure spalling are the permeability of the concrete, the initial water saturation level, and the rate of heating. Pore pressure spalling may apply by itself only for small unloaded specimens. For larger specimens, the pore pressure will have to be considered together with both the thermal and load stresses before the likelihood of explosive spalling can be assessed. Thermal stress spalling: At a sufficiently high heating rates, ceramics and dry concrete can experience explosive spalling. This is attributed to excessive thermal stresses generated by rapid heating and demonstrates that factors other than pore pressures may contribute to explosive spalling. Heating concrete generates temperature gradients that induce compressive stresses close to the heated surface (due to restrained thermal expansion) and tensile stresses in the cooler interior regions. Surface compression may be augmented by load or prestress, which are super-imposed upon the 1698 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 Fig. 1. Combined thermal stress and pore pressure in triggering spalling [37]. thermal stresses. However, very few concrete structures are loaded to levels where the necessary failure stress state is reached. This makes thermal stress spalling – by itself – a relatively rare (but not impossible) occurrence. Combined thermal stress and pore pressures: Explosive spalling generally occurs under the combined action of pore pressure, compression in the exposed surface region, and internal cracking (Fig. 1). Cracks develop parallel to the surface when the sum of the stresses exceeds the tensile strength of the material. This is accompanied by a sudden release of energy and a violent failure of the heated surface region. C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 1699 Fig. 2. Discretization adopted in the numerical analyses of construction of a spalling abaqus. The most effective methods to reduce the risk of explosive spalling include [36]: (a) the use of a thermal barrier, (b) employing polypropylene fibres, (c) employing an air-entraining agent, (d) using thicker sections, and (e) use of low thermal expansion aggregate. In addition, appropriate use of reinforcement could limit the extent of spalling, though not prevent the phenomenon itself. The risk of spalling is also reduced if the moisture content is low and the permeability of the concrete is high. 3. From experiments to modelling Taking into account the works [31,33–35,39] and once separated and quantified the strain components during the various stages of the heat cycles, the total strain can be shown to be the superposition of individual strain components each related to the specific strain inducing mechanism within the concrete, so that [47,62]: σ,T,m σ,T,m 0,T,m 0,T,m σ,T,m σ,T,m εσ,T,m t,tot = ε0,el + ε0,plas + ε0,th + ε0,sh + εt,cr + ε0,crack (1) in which superscripts refer to the effects contributing to strain, i.e. stress σ (zero when an independence on the stress level occurs), temperature T and moisture content m; subscripts allow for differentiating among elastic, plastic, thermal, shrinkage, creep (dependent on time t) and crack-induced strains. Concrete is treated as a multiphase system where the voids of the skeleton are partly filled with liquid and partly with a gas phase. The liquid phase consists of bound water (or adsorbed water), which is present in the whole range of water contents of the medium, and capillary water (or free water), which appears when water content exceeds so-called solid saturation point Sssp [22], i.e. the upper limit of the hygroscopic region of moisture content. The gas phase, i.e. moist air, is a mixture of dry air (non-condensable constituent) and water vapour (condensable gas), and is assumed to behave as an ideal gas. The approach here is to start from a phenomenological model [45,49,50,66], originally developed by Bažant and Thonguthai, e.g. [10,11], in which mass diffusion and heat convection–conduction equations are written in terms of relative humidity, to an upgraded version in which its nonlinear diffusive nature is maintained as well as the substitution of the linear momentum balance equations of the fluids with a constitutive equation for fluxes, but new calculations of thermodynamic properties for humid gases are implemented too to take into account different phases as well as high ranges of both pressure and temperature; to enhance the model’s predictive capabilities, a predictor–corrector procedure is supplemented to check the exactness of the solution. The reader is referred to [61] and to [47,62] for details about the field equations for the coupled heat and mass transfer of concrete materials and the description of the mechanical field in small strains, respectively. Starting from experimental evidences and additional data from literature, first numerical analyses in small strains have been conducted in order to construct appropriate 3D design nomograms defining when spalling is likely to occur or not on walls (Fig. 2) made of three different types of concrete: an O.P. C30, a C60 and a C90 concrete, whose material characteristics (for C60 and C90 only) are reported in Tables 1 and 2. The boundary conditions used in the numerical simulations correspond to a heating characterized by an ISO-Fire profile and mixed convective–radiative heat exchange. The variables considered for the sensitivity analysis are mainly the saturation level and the intrinsic permeability. The criterion adopted to define spalling conditions (and the phenomenon of spalling itself) is linked to the value reached by three main parameters; i.e., it is here assumed that spalling is likely to occur when, within concrete, D (damage) >50%, T (temperature) = 460 ± 50 K and Pv (water vapour pressure) = 0.8–1.0 MPa. Such values represent the critical threshold from a numerical point of view but they are strictly linked with experimental observations of the phenomenon. The numerical results are compared and discussed with reference to the main experimental results obtained for standard concretes and available in literature; these ones are briefly summarised in Fig. 3 (C30), focussing on three main spalling indicators, such as moisture content, concrete thickness (linked to the section size) and applied stress. The results are presented within a tentative nomogram. 1700 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 Table 1 Main properties at 20 ◦ C for concrete C60. Parameter Value Porosity, n Intrinsic permeability, k0 (m2 ) Apparent density, ρ k0 (kg/m3 ) Specific heat, Cp (J/kg K) Thermal conductivity, λ (W/m K) Young’s modulus, E (GPa) Poisson’s ratio, ν Compressive strength, fc (MPa) Tensile strength, ft (MPa) 0.082 10−15 to 10−18 2564 855 1.92 42 0.22 60 4 Table 2 Main properties at 20 ◦ C for concrete C90. Parameter Value Water/binder ratio, w/b Binder content (kg/m3 ) Aggregate type and size MIP porosity, n (%) Water intrinsic permeability, k0 (m2 ) Young modulus, E (GPa) Poisson’s ratio, ν Thermal conductivity, λ (W/m K) Specific heat, Cp (J/kg K) 0.29 OPC: 510.5 + micro-silica: 50.5 gabbro, maximum size of 16 mm 6.0 2 × 10−19 36.7 0.18 2.1 851 Two “uncertainty” areas exist, due to the fact that moisture content and concrete thickness can combine independently on the applied stress; however, it clearly appears that moisture contents comprised within the range 3–5% by weight associated to concrete thicknesses variable between 40 mm and 110 mm can produce, under stress levels higher than 5 N/mm2 , spalling phenomena. Fig. 3. Spalling nomogram for standard concretes (C30). See ref. [71]. C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 1701 Fig. 4. Spalling nomogram related to saturation degrees and permeability (C30). A nomogram more directly applicable to high performance concrete is presented in Fig. 4 (C30): in fact, different combinations of saturation degrees and permeability can describe a wide range of high performance (unstressed) concretes. This statement is in fact supported by the results of the numerical analyses, as reported in the following. The results form the numerical analyses have been collected in terms of saturation degree, depth of concrete from heated surface and permeability. In Fig. 5 the main results are shown, for both C60 and C90 concretes: generally speaking, it can be said that spalling is unlikely to occur when permeabilities are higher than 10−16 m2 ; within 10−17 to 10−16 m2 spalling is uncertain, whereas for lower values spalling is likely to occur, independently on the saturation degree. Fig. 5. Spalling nomograms for C60 and C90 high performance concretes. 1702 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 Regarding concrete thickness and, correspondingly, concrete size, the numerical results must be considered in view of the experimental evidences reported before. It is hence clear that any mechanism suggested to explain the explosive spalling of concrete must account for the following observed behaviour: • Given suitable environmental conditions, in terms of load and high heat transfer into the concrete, all concrete can display the capacity for spalling. • Medium concrete sections are more susceptible to spalling than thick sections and very thin sections. • Applied loads increase the susceptibility of concrete members to spalling. • Water-cured concrete specimens are particularly susceptible to spalling. • High strength concrete has an increased susceptibility to spalling. • If it occurs, explosive spalling take place within the first 40 min of heating in fire. • For a given set of conditions, explosive spalling of normal strength concretes (as in nuclear reactors) is less likely with moisture contents less than 3% by weight. • The expansion of a concrete gives a measure of its likelihood to spalling. Concrete that has its thermal expansion restrained is more likely to spall. By corollary, concrete with a low thermal expansion is less likely to spall. • The likelihood of explosive spalling increases with increase in the heating rate. • In conclusion, the development of (essentially) pore pressure within heated concrete can result in explosive spalling [15,55,60]. The main factors that influence pore pressure spalling are the permeability of the concrete, the initial water saturation level, and the rate of heating. Pore pressure explosive spalling occurs more frequently: ◦ at high heating rates, that result in the rapid build-up of pore pressures; ◦ in concretes with lower water/cement ratios, because of their reduced permeability; ◦ in water saturated concrete because of the high initial water saturation; ◦ in small specimens because of the relatively high rate at which they can be heated. Again, the provisions of preventing or minimising the explosive spalling of HPCs can be suggested as follows: (a) for grades lower than C80/95 with the maximum content of silica fume less than 6% by weight of cement, the rules for normal-weight concrete are applied; (b) for grades lower than C80/95 with the maximum content of silica fume greater than 6% and higher grades concrete, it is assumed that spalling can occur in most situations for concrete exposed directly to the fire so that appropriate measures must be provided. 4. A new approach to spalling in finite strains As largely evidenced by experimentation (see e.g. [38]), explosive spalling is characterized by a sequence of sudden detachments of large or small pieces of material from the concrete element; such a phenomenon has usually been modelled in the past via linear geometry approaches, rarely obtaining satisfactory predictions or an unique characterization of the spalling process. Hence the proposed approach can be considered as a first step towards the above objective and has potentialities for industrial applications. Such an approach substitutes the infinitesimal stress–strain relationship (see (1)): dεel = dε − dεpl − dεd − dεcrack − dεcreep − dεsh − dεT − dεlits − dε0 (2) where at the r.h.s., total, plastic, damage, crack, creep, shrinkage, thermal, lits, and autogeneous (other chemical contributions) strains appear, with the finite strain multiplicative decomposition of the deformation gradient [46,48]: F = Fel · Fpl · Fd · Fcrack · Fcreep · Fsh · FT · Flits · F0 (3) The theoretical developments and the associated numerical procedures followed to reach the form (3), can be found in different works, starting from [66] where it has been assumed that F = Fel · Fpl (4) hence a multiplicative decomposition of the elastic and plastic strains has been adopted (Fig. 6), finding the elastoplastic fully consistent tangent matrix (Hessian) in the hypothesis of a hardening material model. In this context the existence of an intermediate configuration has been stated (Fig. 7). C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 1703 Fig. 6. Multiplicative decomposition of the deformation gradient. Then in [14] it has been posed that F = Fel · Fd (5) therefore an elasto-damage multiplicative decomposition has been taken, finding again the Hessian of the transformation, up to its algorithmic form. As a third step, in [48], the assumed decomposition was F = Fel · Fpl · Fd (6) to take into account an elastic–plastic-damage approach in finite strains/displacements, whereas in [51] a thermomechanical model in finite strain has been used, including plasticity, as F = Fel · Fpl · FT (7) Fig. 7. Intermediate configuration. 1704 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 Fig. 8. Proposed approach for multiplicative decomposition when studying concrete in finite strains. The expression F = Fel · Fd · Fcreep · Fsh · FT (8) has already been established in [43] and it is now upgraded as shown in (3). The classical approach of finite strains, dealing with the multiplicative decomposition of the deformation gradient, takes the final form proposed in (3) with the definition of an intermediate configuration which takes the form depicted in Fig. 8, as proposed here by the authors. Since in this case the initial geometrical domain changes as far as the process evolves, both the equations of heat transfer and mass transfer have to be rewritten, together with the momentum balance (the same expressions apply for y and z directions in Eqs. (9) and (10)). Starting from the heat transfer equation presented in [43, eq. 2], geometric changes can be taken into account applying the chain rule to the Fourier equation as follows (as successfully used in [51] for the case of steel at high temperature): qx = −kx ∂T ∂x̄ ∂x̄ ∂x ∂qx ∂ dqx = dx̄ = ∂x̄ ∂x̄ (9a)  ∂T ∂x̄ −kx ∂x̄ ∂x  dx̄ = −  ∂2 T ∂x̄ ∂T ∂2 x̄ ∂kx ∂T ∂x̄ + kx 2 + kx ∂x̄ ∂x̄ ∂x ∂x̄ ∂x ∂x̄ ∂x̄∂x  dx̄ (9b) in which x̄ is the deformed configuration, qi is the thermal flux and ki the conductivity coefficients along each direction, respectively. Similarly, the mass transfer equation presented in [43, eq. 1], needs to be changed applying again the chain rule as follows: mx = −Cx dmx = ∂h ∂x̄ ∂x̄ ∂x ∂ ∂mx dx̄ = ∂x̄ ∂x̄ (10a)  −Cx ∂h ∂x̄ ∂x̄ ∂x  dx̄ = −  ∂2 h ∂x̄ ∂h ∂2 x̄ ∂Cx ∂h ∂x̄ + Cx 2 + Cx ∂x̄ ∂x̄ ∂x ∂x̄ ∂x ∂x̄ ∂x̄∂x  dx̄ (10b) where mi is the mass and Ci the diffusion coefficients along each direction, respectively. In this version of the generated FE code, the changes in the advection terms of the heat transfer equation are being included, while the strain coupling term in the mass transfer equation relates to the Lagrange total strain E. 1705 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 The associated boundary conditions of the new equations remain the traditional ones: (1) prescribed humidities and temperatures (Dirichlet conditions) or (2) prescribed water vapour flows and heat flows (Neumann conditions); mixed boundary conditions (Robbins conditions) are not applied in this context. As far as the equilibrium equations are concerned, the starting point is again the formulation used in [43], where geometrical nonlinearities were introduced as well as viscoelasticity coupled with damage. The restriction of small strains is now removed. Hence, in the geometrically nonlinear description used the 2nd symmetric Piola–Kirchhoff stress tensor S is adopted together with the associated Lagrangian strain E, both referred to the material configuration. The stress and strain tensors S and E are energetically conjugated, preserving the material frame indifference under changes of reference frame. The initial geometry is referred to a Cartesian orthogonal frame. A total Lagrangian formulation is used to calculate the needed quantities with reference to the initial undeformed configuration. The equation of motion can be expressed in the undeformed domain V as  S0 δE0 dV 0 = R (11) V where R is the external virtual work. The strain variation and stress components are evaluated as 1 0 δ[u + u0j,i + u0k,i u0k,j ] 2 i,j δEij0 = (12) 0 0 Sij0 = |J0 |x̄i,m σmn x̄j,n (13) with ui displacement components, xi spatial coordinates and J the Jacobian, referred to the material configuration, as indicated by the apex 0. Viscoelastic constitutive relationship taking into account damage effects is expressed in finite strain formulation [43] using the second Piola–Kirchhoff stress tensor and the associated Lagrangian strain. Coupling of viscoelasticity and damage follows a procedure of the type introduced in [69], where viscous response was characterized by a linear rate constitutive equation and a convolution representation was used to apply viscoelastic models with linearized kinematics to the nonlinear geometrical case. Continuum damage mechanics was employed there to take into account degradation of stiffness. Since in the latter paper the formulation was stated for rubber-like material, modifications were adopted to accommodate the theory to the case of concrete-like materials, as in [43] and here. We do not repeat the whole details for finding the mechanical equilibrium equations, since they can be found in same reference, but changes are reported here as specified in what follows. After introducing the FEM discretization in space, the final form of the equation of motion (11) is found as  [B0 + BL ]T {1 − D0 }C1 [B0 + BL ] dV 0 V +  V0 −  V ⎡ ⎛ GT ⎣{1 − D0 } ⎝C1 t E0 + C2 EO0 + C3 t E0 − ⎡ ⎛ 1 2 N Q0µ + µ=1 1 [B0 + BL ]T ⎣{1 − D0 } ⎝C1 t E0 + C2 EO0 + C3 t E0 − 2 1 2 N µ=1 N µ=1 ⎞⎤ R0µ ⎠⎦ G dV 0 1 Q0µ + 2 N µ=1 ⎫ ⎬ u=R ⎭ ⎞⎤ R0µ ⎠⎦ dV 0 (14) In Eq. (14), B0 , BL and G are the initial, linear and nonlinear strain matrices, respectively; Ci are visco-elastic matrices [43]; Q0µ and R0µ are load vectors of the Maxwell chain units [49,51]. In the actual version of the theory, plastic and LITS strains are incorporated in the autogeneous finite strain term EO0 , while crack coincides with an upper bound of the damaged finite strain (e.g. when D = 0.99). R vector relates to pure mechanical loads. The solution of the system of heat and mass transfer and of the equilibrium equations is found at each time step for the whole time transient in which spalling process happens. It is remarked that sensibly high strains occur in localized regions when pieces of concrete explodes and in the very final stage of the phenomenon when collapse of the whole structure is determined by the complete loss of stiffness: LITS, plastic and damage (crack) strains become too high to be sustained by the material and structure involved. 1706 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 Fig. 9. Modelling scheme for the spalling analysis of a concrete column. Table 3 Material data adopted in the numerical analyses. Concrete Rck (MPa) Elastic modulus (MPa) Poisson’s ratio fck (MPa) fctk (MPa) Density (t/mm3 ) Conductivity (J/(s mm ◦ C)) Specific heat (J/(t ◦ C)) Expansion coefficient (◦ C−1 ) 30 31220 0.2 24.9 1.82 2.4 × 10−9 1.37 × 10−3 8.80 × 108 0.000012 5. Numerical example A 20 mm × 20 mm column (Fig. 9) has been studied, subjected to a thermal load of maximum 400 ◦ C, reached in 30 s, whose material data are listed in Table 3. The hygro-thermal field (see e.g. Fig. 10 referring to a typical section) has been obtained through the research F.E. code NEWCON3D and (once transformed it in a single, equivalent, thermal field) the data passed to ABAQUS© to catch the detachment of pieces (caused by spalling) through the activation of contact mechanisms. The procedure is updated at each time station. This approach, which maintains the fully coupled features of NEWCON3D, allows for both estimating large displacements/rotations/strains due to spalling and to reproduce the “onion” effect typical in spalling processes, characterized by the detachment of pieces of different dimensions from an unaltered (or less altered) concrete core. Fig. 10. Relative humidity evolution (first, intermediate and last time step). C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 1707 Fig. 11. Step-wise history of spalling events. An example of the spalling evolution process on a beam heated in a furnace can be seen in www.promat.it. It can be observed (Fig. 11) that the process duration is of about 21 min and the different events of spalling (in the upper and lower plates and in the web) occur at various jumping times. To properly describe such a result, each finite element has been glued to the surrounding ones through 3D contact elements (Fig. 12) characterized by a Lagrangian multiplier contact algorithm. Additionally, the contact condition is driven by the Mazars’ damage law: D= ⎧ ⎨ 0 A (1 − A)K0 ⎩1 − − B(ε̃−k ) 0 ε̃ e if ε̃ ≤ ε0 if ε̃ > ε0 (15) where A and B are material coefficients, ε0 is the limit linear elastic strain, ε̃ the equivalent strain and K0 the initial value of the softening parameter. The reader is referred to [49,53,54] for additional details. In this way, when contact exists, the stress–strain law is associated to a damage model: after a maximum stress peak is reached, stresses approach zero following the softening curve produced by applying Mazars’ model and a zero stress generates an open contact. A value D = 1 is in fact directly associated to the situation of no-contact: one element or more elements (or groups of elements) are detached and the thermal field is updated; this effect cannot be attained when performing small strains analyses. Hence, to trigger the detachment, a small perturbation (displacement) is introduced and applied to two randomly chosen elements, but belonging to the zones subjected to highest thermal levels, as it is known experimentally that Fig. 12. Connection among finite elements through 3D contact conditions. 1708 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 Fig. 13. Deformed mesh and thermal curves of the first layer. Fig. 14. Deformed mesh and thermal curves of the second layer. spalling takes place in such regions; the deformed mesh at end analysis, together with the contour maps of the updated thermal field, are depicted in Figs. 13–17 referring to one-quarter section and a single layer of elements: the phenomenon of spalling is clearly reproduced, as also evidenced by Fig. 18, where all contact elements characterised by D = 1 and surrounding ones have been removed. Fig. 15. Deformed mesh and thermal curves of the third layer. C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 1709 Fig. 16. Deformed mesh and thermal curves of the fourth layer. Fig. 17. Deformed mesh and thermal curves of the fifth layer. Fig. 18. Final deformed shape (quarter and half section) of the spalled column. 6. Conclusions To resume the main aim of this paper, it has been here proposed to study spalling processes through a new approach accounting for large displacements/rotations/strains in order to reach a better and more realistic simulation of the processes kinetics. 1710 C.E. Majorana et al. / Mathematics and Computers in Simulation 80 (2010) 1694–1712 To reach such an objective, an appropriate theoretical framework should be used for describing phase changes, sorption–desorption behaviour at high temperature/fire and related strains, plus a stress–strain constitutive relationship for concrete under high temperature accounting for thermal, shrinkage, creep, LITS, damage, plastic and other chemical contributions (that must be carefully checked against experiments). All these contributions are being inserted in a fully nonlinear-displacement/strain theory able to catch the complex interactions between pressure, thermal and mechanical fields when parts of concrete are attaining dynamic unstable disruption, as occurs during the spalling process of highly heated concrete. It has been demonstrated, even at a first stage, the potentialities of the approach whose numerical results are in agreement with others available in literature (referring only to a small strains context for catching the onset of spalling phenomenon), and which additionally offers a more deep, complete and extensive ability to describe the whole process of concrete spalling under high temperature regimes, up to the complete failure of both material and structure. References [1] A.A. Akhtaruzzaman, The effect of transient and steady state temperatures on concrete, PhD Thesis, Imperial College, London, 1973. [2] A.A. Akhtarnrzaman, P.J. 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