Thermal induced stress and associated cracking in cement-based composite at elevated temperatures––Part I: Thermal cracking around single inclusion
Introduction
Thermal cracking induced by thermal mismatch has been one of the problems in a cement-based composite material under elevated temperatures. For a multi-phase material, the eigenstrains deriving from the heterogeneous deformations among phase components inevitably cause cracking in the composite, even though it is under a uniform temperature field. Experimental results [1] have shown that this type of cracking significantly reduces the strength and elastic modulus of a cement-based composite. However, the entire thermal cracking process (initiation, propagation and linkage of cracks) and the associated stress distributions under elevated temperatures are difficult to quantify experimentally, mainly because of the limitation of equipment and the complex structure of a composite material.
In order to understand the failure mechanism of a composite material due to thermal effects, many mathematical models have been proposed [1], [2], [3]. In these models, the driving stresses for the crack initiation and propagation are the heterogeneous eigenstresses, which develop in and around the restraining inclusion. These eigenstresses might be caused by thermal expansion, shrinkage [4], [5], initial strains and misfit strains. Timoshenko and Goodier in 1970 [6] proposed a closed-form solution for the axisymmetric problem of a circular inclusion concentrically embedded in the circular disc of another phase material with different thermal and mechanical properties. Hsueh et al. [7] transformed a composite with a microstructure of square-array, hexagon-array, brick-array grains, as well as the actual microstructure of random-array grains into a simple composite-circle analytical model. The residual thermal stresses were predicted reasonably well using the proposed linear elastic solutions except for the model microstructure of brick-array grains. A modified version of Timoshenko and Goodier’s solution incorporating the longitudinal strain proposed by Gentry and Husain [2] was also used to study the differential pressure developed in the interface between concrete and a composite rod. As for a 40 °C temperature increase, the concrete was modeled with a linear-elastic and nonlinear tension-softening material model using a finite element approach. The calculated results showed that the large spacing of the rods and the thick concrete cover were helpful to reduce the tensile stress in concrete as well as the potential for thermally induced cracking. Based on a fracture mechanics model, Timoshenko and Goodier’s solution was adopted by Dela and Stang [3] to calculate the crack growth with time in a high-shrinkage cement paste with a single aggregate disc. The experimentally measured stresses in the selected circular aggregate were employed to predict the stresses distributed in cement paste and the crack growth at a crack tip close to the aggregate in terms of a given stress intensity factor.
Although the above-mentioned models deepen the understanding on thermal stress and cracking, essentially, none of them can simulate the entire thermal cracking process from crack initiation to propagation. Hsueh’s and Russell’s models can determine the stress distribution around a single inclusion in the composite before crack initiates. Dela’s model was suitable to calculate the critical stress value when an existing crack starts to grow. The stress distribution represented by this model would be invalid as soon as the crack is extended. A fracture mechanics model is able to study the growth of existing single crack, but it is not suitable to explain the initiation and coalescence of cracks. More importantly, the phase materials of a cement-based composite are often heterogeneous so that the effect of change in microstructure (mesostructure) on the macroscopic behavior is difficult to be studied by using an analytical model.
Consequently, a numerical method appears to be an effective tool to model cracking processes. Substantial progress [8], [9] has been achieved in numerical simulation of failure occurring in a cement-based composite at ambient temperatures. However, a satisfactory model to simulate the cracking processes caused by the thermal induced stresses in a heated cement-based composite is still not available.
The aim of this paper is to propose and verify a mesoscopic thermoelastic damage (MTED) model, that can numerically simulate the formation, extension and coalescence of cracks in a cement-based composite material (cement-based matrix + aggregate inclusion), caused by the thermal mismatch of the matrix and the inclusion under uniform temperature variations and free boundary conditions. Numerical studies of the effects of the thermal mismatch between the matrix and a single circular inclusion on the stress distribution and crack development are also presented.
Section snippets
Numerical model
In the MTED model, phase materials of a composite are considered to be heterogeneous following the Weibull distribution. Tensile and shear cracking at meso-scale occur if the stress in the composite subjected to high temperatures satisfied with the failure criteria of Coulomb–Mohr with tension cutoff.
Model validation
Fig. 3b shows the comparison of the thermal stresses around the single inclusion of Specimen no. 1 calculated from the T-MFPA program, and from the analytical solutions (Eqs. (10) and (11)) derived from the classical theory of thermo-elasticity [6], [17].
It is evident that under an elastic and undamaged state, an excellent agreement between the stresses obtained from the two different approaches has been obtained.
Thermal cracking history of square specimens
Fig. 5 shows the effect of thermal mismatch on the thermal induced damages and fracture processes of Specimen no. 2 (Group 1) and Specimen no. 4 (Group 2) due to increasing temperatures. Fig. 6 illustrates the influence of the mean strength of the inclusions on the crack development in each group. Detailed descriptions of crack formation of the specimens are shown below.
Effect of thermal mismatch
Although the four specimens are subjected to uniform temperature changes, local stress concentration occurs around the inclusion due to the thermal mismatch between the matrix and the inclusion.
When the CTE of the inclusion is greater than that of the matrix, the inclusion in Specimen no. 2 is stressed under a state of statistically hydrostatic compression due to the restriction from the matrix, and the matrix is under a general bi-axial state of stresses (tensile/compressive and shear
Thermal crack patterns
Three types of thermal cracks have been identified: radial cracks, tangential cracks and inclusion cracks, in a cement-based composite. All these cracks are schematically shown in Fig. 8. The formation and propagation of these cracks are dependent on the difference in CTE between the inclusion and the matrix. Although all these cracks are located in different places, all of them are produced by the tensile eigenstresses derived from the thermal mismatch. From the consideration of the
Conclusions
In this paper, a MTED model was formulated and then incorporated in the established MFPA program for the simulation of the crack formation, extension and coalescence in a cement-based composite (aggregate inclusion + cement mortar matrix) at elevated temperatures. Based on the studies of four specimens, some fundamental characteristics of thermal cracks, caused by different CTE of the inclusion and the matrix, are summarized as follows:
- (1)
When the CTE of the inclusion is greater than that of the
Acknowledgements
The materials presented in this paper are some of the findings of the G-V848 research project entitled “Thermal Stress and Associated Damage in Concrete at Elevated Temperatures” of The Hong Kong Polytechnic University. The project is also partly supported by the NNSF of China (No. 50174013).
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