Mechanical properties and size effects in lightweight mortars containing expanded perlite aggregate

Abstract

The study presented here investigates the effect of density in cementitious mortar on its mechanical properties under quasi-static loading. The reduction in density was achieved through the addition of expanded perlite as a lightweight aggregate into cement paste by volume replacement of cement in the ratio from 0 to 8. This yielded a range of densities between 1000 and 2000 kg/m³. The compressive and flexural response of these mixes were determined for geometrically scaled specimens to study the size effect. Some mixes were reinforced with polymer microfibres and the Mode I fracture toughness parameters were evaluated through flexural testing of notched beams. When compared with a reference Portland cement paste, the compressive strength and elastic modulus scaled as the cube of the density, while the fracture toughness varied linearly with it. The study shows that the specimen size effect on compressive and flexural strength decreases with a drop in the density of the mix and also with fibre reinforcement. On the other hand, the specimen size effect on the critical crack mouth opening displacement was more pronounced at lower densities.

Appendix: Derivation of the stress intensity factor and fracture toughness

The critical stress intensity factor or fracture toughness is a measure of the material’s resistance to crack growth. It is sometimes represented by the R-curve presented as the stress intensity factor (K _I) plotted against the effective crack length divided by the depth of the section (a _eff/d). The method used to determine the fracture toughness for a specimen was adopted from Kraft et al. [25] by varying the notch length of each specimen to find the corresponding stress intensity factor. Instead of multiple specimens with varying initial notch lengths, the progressively growing crack of one sample is used. The stress intensity factor is dependent on the nominal stress in the section and the crack length. The nominal stress was defined as per Eq. 3, and solved for four point bending where, P was the overall load applied to the system, b, d and S were respectively the width, the depth and the span of the beam.

$$ \sigma = {\frac{PS}{{bd^{2} }}} $$

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While the derivation for stress intensity factors is based on LEFM, the assumption that the fracture process zone ahead of the crack tip is small enough to have negligible plasticizing effect is not valid for cement based materials. An effective crack is used to model the existing crack as a straight traction-free crack of a length equal to the length of the true crack plus the length of the fracture process zone and is shown in Fig. 18. This leaves only an elastic material outside of the crack and therefore allows the use of LEFM. The method to find the effective crack length was adopted from Banthia and Sheng [26] who used a compliance calibration.

Fig. 18

Schematic diagram of an effective crack in cement based materials

In order to determine the length of an effective crack, a _eff, one must first relate the crack length and the crack mouth opening displacement, CMOD, by an equation from LEFM. Such equations may be found in stress analysis handbooks and Eq. 4 is taken from Tada et al. [27] for a beam of finite width, with a single edge notch undergoing pure bending. In Eq. 4 ‘a’ is the crack length and ‘b’ is the characteristic length of the specimen, which in this case is also the height of the beam. The multiplying function V(a/b) adjusts the original expression which was derived for an infinite plate containing a centre crack, to be applicable to the given case. For a single edge notch in a beam subjected to pure bending, V(a/b) is taken from Tada et al. [27] as expressed in Eq. 5. The correction factor so expressed was found to be accurate to within 1% error for any a/b.

$$ {\text{CMOD}} = {\frac{4\sigma a}{E}}V\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) $$

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$$ V\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) = 0.8 - 1.7\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) + 2.4\left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right)^{2} + {\frac{0.66}{{\left( {1 - {\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right)^{2} }}} $$

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Since the crack mouth opening displacement was not measured directly during the test, it was interpreted from the load versus deflection diagram as suggested by Armelin and Banthia [28]. As shown there in, during the course of a bending test, the CMOD bears a linear relationship with the midspan deflection and is about 33% higher in value, Eq. 6.

$$ {\frac{\text{dCMOD}}{{{\text{d}}\delta }}} = {\frac{{c \cdot {\text{d}}\theta \left( {h - c} \right)}}{{{\text{d}}\theta \frac{3}{2}h}}} = \frac{4}{3} $$

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By assuming that the material ahead of the effective crack is linear elastic, Eq. 4 may be rearranged to determine its elastic modulus for each point along a load–CMOD graph as shown by Eq. 7. Further, the elastic modulus is a constant and therefore it must be true that by taking a point at the initial portion of the load–CMOD curve where the crack has not progressed further than the initial notch length, a _o, the elastic modulus so derived may be equated to that at a point further along the load–CMOD curve corresponding to a new crack length, a _eff, which is now the length of an effective crack corresponding to that point on the load–CMOD curve. In this manner, one evaluates the corresponding effective crack length, a _eff.

$$ E = 6\left( {{\frac{P}{\text{CMOD}}}} \right){\frac{{a_{\text{o}} S}}{{bd^{2} }}}V\left( {{\raise0.7ex\hbox{${a_{\text{o}} }$} \!\mathord{\left/ {\vphantom {{a_{\text{o}} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) = 6\left( {{\frac{P}{\text{CMOD}}}} \right){\frac{{a_{\text{eff}} S}}{{bd^{2} }}}V\left( {{\raise0.7ex\hbox{${a_{\text{eff}} }$} \!\mathord{\left/ {\vphantom {{a_{\text{eff}} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) $$

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Thus, at each load level, the corresponding stress intensity factor may be determined by Eq. 8. Again, as this equation is derived from the condition of an infinite plate with a centre crack, a geometric correction factor, F ₁(a/b), is applied to adjust the original result for practical cases. This factor for an edge notch with finite width undergoing pure bending is presented in Eq. 9 and according to Tada et al. [27], it is known to have an accuracy of better than 0.5% for any value of a/b less than 1. The peak value in the K _I–a _eff graph is the critical stress intensity factor (K _IC), or the fracture toughness. The corresponding critical crack length (a _effc) was also noted.

$$ K_{\text{Ieff}} = \sigma_{n} \sqrt {\pi a_{\text{eff}} } F_{1} \left( {{\raise0.7ex\hbox{${a_{\text{eff}} }$} \!\mathord{\left/ {\vphantom {{a_{\text{eff}} } b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) $$

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$$ F_{1} \left( {{\raise0.7ex\hbox{$a$} \!\mathord{\left/ {\vphantom {a b}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$b$}}} \right) = \sqrt {{\frac{2b}{\pi a}}\,{\tan }\left( {{\frac{\pi a}{2b}}} \right)} {\frac{{0.923 + 0.199\left( {1 - { \sin }\left[ {{\frac{\pi a}{2b}}} \right]} \right)^{4} }}{{{ \cos }\left( {{\frac{\pi a}{2b}}} \right)}}} $$

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