Toughness of fibre reinforced hydraulic lime mortar. Part-2: Dynamic response

Abstract

The seismic rehabilitation of historical masonry buildings necessitates a quantitative understanding of the repointing mortar under variable strain rates. In Part-1 of this paper, plain and fibre reinforced hydraulic lime mortar specimens were examined under compression, flexure and direct shear to evaluate the post-crack response under quasi-static loading. It was seen that although the fibres enhance the flexural toughness of hydraulic lime mortar, the material is weakest in Mode I fracture. In Part-2 of this paper, the authors describe the strain rate sensitivity of hydraulic lime mortar on the basis of impact testing of notched beams. The mixes were identical to those examined in Part-1, and the dynamic response was evaluated using a drop-weight impact machine for strain rates in the range of 10⁻⁶ to 10 s⁻¹. The authors found that compared to fibre reinforced Portland cement-based mortar and concrete, the flexural response of hydraulic lime mortar is more sensitive to strain rate.

Appendix 1: Evaluation of effective crack length and the stress intensity factor

Let:

P :

= applied load

S :

= span

B :

= beam width

D :

= beam depth

Δ:

= midspan deflection

CMOD :

= crack mouth opening displacement

σ :

= flexural stress

a _o , a _eff :

= initial crack length and effective crack length (Note: a _eff  ≥ a _o )

C :

= compliance = CMOD/P

E:

= modulus of elasticity

We thus have, C(a _o ), as the initial compliance and C(a _eff ), the compliance at a crack length a _eff (>a _o ). The compliance at any instant of loading may be experimentally derived from the load, P, and the crack mouth opening displacement, CMOD, as shown in Fig. 11.

Fig. 11

Compliance found experimentally for any load and corresponding CMOD

The compliance corresponding to Point A′, which represents a general loading condition on Fig. 11 is given by:

$$ C\left( {\alpha_{eff} } \right) = {\frac{CMOD\left( a \right)}{P}} $$

(6)

The initial compliance may be found as:

$$ C\left( {\alpha_{o} } \right) = \frac{CMOD}{P} $$

(7)

where CMOD and P correspond to the bending over point, A, in Fig. 11.

In a flexural test, the theoretical compliance is influenced by the span-to-depth (S/D) ratio as well as the extent of the crack. If α = (a/D) and β = (S/D). Thus, the general expression for the compliance for β ≥ 2.5 is given by Guinea et al. [22] as follows:

$$ \begin{aligned} C\left( \alpha \right) & = \frac{72}{432EB}\alpha \left( { - 7.8224 + 6.3904\beta - 0.6327\beta^{2} } \right) \\ & + \frac{72}{864EB}\alpha^{2} \left( {11 - 0.88\beta } \right)\left( {1.24 - 0.14\beta } \right) \\ & - \frac{72}{432EB}\alpha^{3} \left( { - 1.24 + 0.14\beta } \right)^{2} \\ & + \frac{72}{EB}\left\{ {{\frac{2.65\alpha \beta }{{512\left( {1 - \alpha } \right)}}}\left( { - 4.64 + 0.13\beta } \right)} \right\} \\ & + \frac{72}{EB}\left\{ {{\frac{{\alpha \left( {2.65\beta } \right)^{2} \left( {2 - \alpha } \right)}}{{512\left( {1 - \alpha } \right)^{2} }}}} \right\} \\ & - \frac{72}{EB}\left\{ {{\frac{{\alpha \left( {16.48 + 45.19\beta } \right)^{2} }}{{82944\left( {1 + 3\alpha } \right)}}}} \right\} \\ & + \frac{72}{EB}\left\{ {{\frac{{ - 21.5296 + 63.1104\beta - 10.339\beta^{2} }}{4096}}} \right\}{ \ln }\left( {1 - \alpha } \right) \\ & + \frac{72}{EB}\left\{ {{\frac{{5350.0672 + 14129.7728\beta - 1482.6839\beta^{2} }}{995328}}} \right\}{ \ln }\left( {1 + 3\alpha } \right) \\ \end{aligned} $$

(8)

Substituting for β = 3, Eq. 8 reduces to the following:

$$ C\left( \alpha \right) = \frac{1}{EB}\left\{ {\begin{array}{*{20}c} {0.942\alpha + 0.571\alpha^{2} - 0.112\alpha^{3} - {\frac{1.584\alpha }{1 - \alpha }}} \\ { + {\frac{{8.888\alpha \left( {2 - \alpha } \right)}}{{\left( {1 - \alpha } \right)^{2} }}} - {\frac{20.07\alpha }{1 + 3\alpha }} + } \\ {1.362\,{ \ln }\left( {1 - \alpha } \right) + 2.448\,{ \ln }\left( {1 + 3\alpha } \right)} \\ \end{array} } \right\} $$

(9)

Now, the crack mouth opening displacement corresponding to any applied load, P, is given by CMOD(α,β) and is expressed as a function of both the crack length and the span-to-depth ratio [22]. For the present case of β = 3, the CMOD may be expressed as follows:

$$ \begin{aligned} CMOD(\alpha ) & = {\frac{24P}{EB\sqrt \pi }}\left\{ {0.0052 - 1.12\alpha + 0.2456\alpha^{2} } \right. \\ & - 0.176\alpha^{3} + 0.086\alpha^{4} - 0.01148\alpha^{5} \\ & - 0.227\,{\text{ln(1 + }}\alpha ) { + 0} . 6 1 6\,{\text{ln(1 + 3}}\alpha )\\ & - \left. {{\frac{0.879}{1 - \alpha }} + {\frac{0.8745}{{(1 - \alpha )^{2} }}}} \right\} \\ \end{aligned} $$

(10)

As mentioned earlier, for the impact tests, the CMOD was measured using high speed cameras. On the other hand, for the quasi-static tests, the mid-span deflection, Δ, was taken to be equal to the CMOD. This was justified on the basis of the observation by Armelin and Banthia [16]. Upon relating the theoretical result of CMOD(α) for a given value of the applied load, P, from Eq. 10 with the experimental value, one obtains the corresponding value for the modulus of elasticity, E. Since the elastic modulus remains constant throughout the extension of the crack, it is found from the bending over point, corresponding to the initial crack length, a _o . Substituting this value of E in Eq. 9, results in the theoretical evaluation of the compliance corresponding to P, at any instant. Equating the experimentally found compliance at any point, A′, as given by Eq. 6, with the corresponding theoretical value from Eq. 9, we may determine the corresponding effective crack length, a _eff .

Finally, the compliance is related to the specific energy release rate, G, and in turn to the stress intensity factor, K _I , as given by the well-known relationships in Broek [23]:

$$ G = {\frac{{P^{2} }}{2BD}}{\frac{dC\left( \alpha \right)}{d\alpha }} $$

(11)

$$ K_{1} = \sqrt {GE} $$

(12)

Based on the effective crack length, a _eff , derived in Eq. 10, the stress intensity factor for the three point bend test was determined as follows [21]:

$$ K_{I} (\alpha ) = {\frac{6M}{{D^{5/2} }}}\left[ {{\frac{\sqrt \alpha }{{(1 - \alpha )^{3/2} (1 + 3\alpha )}}}} \right] $$

(13)

$$ \left\{ {p_{\infty } (\alpha ) + \frac{4}{3}\left[ {p_{4} (\alpha ) - p_{\infty } (\alpha )} \right]} \right\} $$

(14)

where

$$ M = \frac{PD}{2}; $$

(15)

$$ p_{4} (\alpha ) = 1.9 + 0.41\alpha + 0.51\alpha^{2} - 0.17\alpha^{3} $$

(16a)

and

$$ p_{\infty } (\alpha ) = 1.99 + 0.83\alpha - 0.31\alpha^{2} + 0.14\alpha^{3} $$

(16b)