Strength of a cement-based dental material: Early age testing and first micromechanical modeling at mature age

The compressive strength evolution of 37 centigrade-cured Biodentine, a cement-based dental material, is quantified experimentally by crushing cylindrical specimens with length-to-diameter ratios amounting to 1.84 and 1.34, respectively, at nine different material ages ranging from 1 h to 28 days. After excluding strength values significantly affected by imperfections, formulae developed for concrete are i) adapted for inter- and extrapolation of measured strength values, and ii) used for quantification of the influence of the slenderness of the specimens on the compressive strength. The microscopic origin of the macroscopic uniaxial compressive strength of mature Biodentine is investigated by means of a micromechanics model accounting for lognormal stiffness and strength distributions of two types of calcite-reinforced hydrates. The following results are obtained: The material behavior of Biodentine is non-linear in the first few hours after production. After that, Biodentine behaves virtually linear elastic all the way up to sudden brittle failure. The strength evolution of Biodentine can be well described as the exponential of a function involving the square root of the inverse of the material age. The genuine uniaxial compressive strength evolution can be quantified using a correction formula taken from a standard for testing of concrete, which accounts for length-to-diameter ratios of cylindrical samples deviating from 2. Multiscale modeling suggests that 63% of the overall material volume, occupied by dense calcite-reinforced hydration products, fail virtually simultaneously. This underlines the highly optimized nature of the studied material.


ANALYSIS OF THE EQUIVALENT SHEAR STRENGTH OF THE LDCR HYDRATES
In the following, it is checked whether or not the LDCR hydrates could be the microstructural origin of macroscopic failure of mature Biodentine subjected to uniaxial compression. To this end, the macroscopic uniaxial compressive strength, see Eq. (30), is downscaled according to Eqs. The correlation between the lognormal distributions of the indentation modulus M and the indentation hardness H of the LDCR hydrates is illustrated in Fig. S2(a), (b). In the same way, the known lognormal distribution of the indentation modulus M is correlated with the sought lognormal distribution of the equivalent shear strength C of the LDCR hydrates, see Eq. (38). Realistic values of µ C and σ C are identified such that the corresponding straight line in Fig. S2(c) becomes a tangent to the graph showing |σ axi /2| over M . The contact point refers to the 50% quantile of M . As for the LDCR hydrates, this value amounts to 45.1 GPa, see Fig. S2(c) and Table 7. The corresponding lognormal parameters of the distribution of the equivalent shear strength read as The corresponding probability density function of the equivalent shear strength of the LDCR hydrates is illustrated in Fig. S2 (d). Related values of the mode, the median, and the mean value are listed in Table S3. The degree of utilization of all LDCR hydrates is larger than or equal to some 78%. This is smaller than the utilization degree of the HDCR hydrates, compare Figs. S3(a) and 13(a). All LDCR hydrates between the 36.1%-quantile and the 64.6%-quantile have degrees of utilization larger than or equal to 99%, see Fig. S3(b). Therefore, if the lognormal distribution of the equivalent shear strength according to Fig. S2(d) is realistic, then 64.6% − 36.1% = 28.5% of the LDCR hydrates have a degree of utilization larger than or equal to 99% and will, therefore, fail virtually simultaneously. Given that LDCR hydrates make up 12.3% of the volume of Biodentine, see Table 8, 28.5% × 12.3% = 3.5% of the volume of Biodentine will fail at the same time. This does not propose the LDCR hydrates as primary reason for the sudden, well-spread brittle failure of the mature Biodentine specimens under destructive compressive mechanical testing.

FUNDAMENTALS OF STIFFNESS HOMOGENIZATION
Methods of continuum micromechanics allow for stiffness homogenization of representative volume elements (RVE) of microheterogeneous materials. The RVEs fulfil the scale separation principle, i.e. the characteristic size of the microheterogeneities is at least 2 to 3 times smaller than the characteristic size of the RVE (Drugan and Willis, 1996), and the characteristic size of the RVE is by a factor of 5 to 10 smaller than the characteristic size of the structure containing the RVE and/or the characteristic length of the external loading imposed on that structure (Kohlhauser and Hellmich, 2013). Inside the RVE, field equations of linear elasticity are considered. At the boundary of the RVE, uniform strain boundary conditions are imposed where u denotes the displacement vector, x the position vector, and E the imposed uniform macrostrain state.
The heterogeneous microstructure of composites is frequently so gracefully built that it cannot be represented in full detail. As a remedy, the volume of the studied RVE, V RV E , is subdivided into (not necessarily connected) subvolumes occupied by the different microstructural constituents called "material phases": where n p denotes the number of material phases and V i stands for the volume occupied by the i th material phase, such that f i = V i /V RV E quantifies its volume fraction. The material phases are selected such that the matter filling every phase volume V i is characterized by a uniform elastic stiffness tensor C i .
Scale transitions are made possible by means of so-called phase strain concentration tensors A i . They allow for (i) the macro-to-micro scale transition regarding strains (Hill, 1963), also referred to as strainconcentration and strain-downscaling: where ε i stands for the volume-averaged strain of material phase i, and : stands for a double-contracting tensor product, and (ii) the micro-to-macro scale transition regarding stiffness Hill (1963), also referred to as stiffness-homogenization and stiffness-upscaling where C hom denotes the homogenized elastic stiffness of the microheterogeneous material. However, because the very details of the microstructure of an RVE are unknown, phase strain concentration tensors cannot be computed up to analytical precision.
In continuum micromechanics, phase strain concentration tensors are estimated with the help of auxiliary three-dimensional matrix-inclusion problems (Eshelby, 1957;Laws, 1977). n p such problems are introduced; one for every material phase. The elastic stiffness, the ellipsoidal shape, and the orientation in space of the i th material phase are assigned to the inclusion of the i th matrix-inclusion problem, see Fig. S4. The stiffness of the infinite matrices of all these problems, C ∞ , is equal to a characteristic stiffness Fig. S4. Auxiliary matrix-inclusion problem: a three-dimensional ellipsoidal inclusion of stiffness C i is embedded in an infinite matrix with stiffness C ∞ ; the matrix is subjected to uniform far-field ("remote") strains E ∞ . of the microheterogeneous material of interest.
• If the material of interest is a matrix-inclusion composite, the stiffness of the infinite matrices of all auxiliary matrix-inclusion problems is equal to the stiffness of the matrix of the composite of interest. This leads to so-called Mori-Tanaka schemes (Mori and Tanaka, 1973;Benveniste, 1987).
• If the material of interest is a composite with a highly disordered (= "polycristalline") arrangement of the material phases, the stiffness of the infinite matrices of all auxiliary matrix-inclusion problems is equal to the homogenized stiffness of the composite of interest. This leads to so-called self-consistent schemes (Zaoui, 2002;Bernard et al., 2003;Dormieux et al., 2006).
The infinite matrices of all auxiliary matrix-inclusion problems are subjected to the same remote strain state E ∞ . The latter is linked to the strain E imposed on the RVE of the composite of interest, as explained next. The strains inside the inclusions of all matrix-inclusion problems are uniform (= spatially constant) and can be computed analytically where ε i is the strain inside the i th inclusion, I the symmetric fourth-order identity tensor, and P i the Hill tensor of the i th inclusion, see also Dormieux et al. (2006). It is assumed that the uniform strain in the i th inclusion is an estimate of the volume-averaged strain of the i th material phase of the composite of interest. In other words, ε i in Eq. (S7) is used as an estimate for ε i in Eq. (S5). The volume-averaged strains of the material phases of the composite of interest, in turn, must fulfill the strain average rule Inserting ε i according to Eq. (S7) into Eq. (S8) and solving the resulting expression for E ∞ yields the following expression for E ∞ as a function of E Insertion of E ∞ according to Eq. (S9) into Eq. (S7), and comparison of the resulting expression with Eq. (S5) yields the following estimate of the phase strain concentration tensors: Four characteristic features of heterogeneous material have to be know prior to applying Eq. (S10). These features account for elastic stiffness of the material phases, their shapes, orientations, volume fractions, and interactions among them. As regards the latter, two distinct types of interactions are distinguished: polycrystalline and matrix-inclusion type. Polycrystalline interaction, refers to perfectly disordered arrangement of material phases. This manifests in Eq. (S10) in a way that the C ∞ is equal to stiffness of the homogenized RVE C hom . Matrix-inclusion type interaction considers stiffness of the infinite matrix C ∞ in Eq. (S10) equal to stiffness of the real RVE-related matrix (Zaoui, 2002).