The effect of indentation-induced microcracks on the elastic modulus of hydroxyapatite

Abstract

The presence of microcracks in materials affects a wide range of mechanical properties including elastic modulus, Poisson’s ratio, fracture strength, and fracture toughness. The microcrack-induced reductions of the Young’s modulus, E, and Poisson’s ratio, υ, are functions of the size, geometry, and number density of microcracks. In this study, an array of Vickers indentation-induced microcracks was placed on the surfaces of two hydroxyapatite (HA) specimens with totals of 391 and 513 indentations per specimen. This study tests the validity of theoretical studies of microcrack damage-induced changes in E and υ, where the changes are expressed either by (i) the volumetric crack number density, N and (ii) the crack damage parameter, ε. All elasticity measurements were done via resonant ultrasound spectroscopy. For both the HA specimens included in the study and alumina specimens indented in an earlier study [J Mater Sci 38:1910. doi: 10.1007/BF00595764, 1], E and υ decreased approximately linearly with increasing microcrack damage. The slopes of the E and υ versus N and ε are also computed and compared to the available theoretical models.

Appendices

Appendix 1

Equations to calculate S _N [the product of a crack orientation and a crack geometry parameter in Eq. (9)], for each modulus/microcrack model with a particular crack geometry employed in this study.

In the ROM model [53], the overall modulus was taken as the sum modulus of the undamaged layer and the damaged layer. The strain in the undamaged layer was set equal to the strain in the damaged layer. In the DBV model [53], the Bernoulli–Euler beam equation was applied with the boundary conditions that both the bending moments and the shear force were zero at the beam ends.

In the ROM model [53], S _N [Eq. (9)] was calculated using Eq. (12)

$$ S_{N} = f \cdot \frac{{2\langle A^{2} \rangle }}{\pi \langle P\rangle }. $$

(12)

In the DBV model [53], S _N was expressed as Eq. (13)

$$ S_{N} = \frac{{r^{2} + 3}}{{(r + 1)^{2} }} \cdot f \cdot \frac{{2\langle A^{2} \rangle }}{\pi \langle P\rangle }. $$

(13)

In both the Eqs. (12) and (13), f is the microcrack orientation parameter, 〈A ²〉 is the mean of the square of the crack surface area and 〈P〉 is the mean crack perimeter. In Eq. (13), r is the ratio between the half the crack length and the specimen thickness [53].

Equations to calculate S _N for different crack geometries in the ROM model are summarized in Eqs. (14)–(17).

ROM model with unmodified slit crack geometry

$$ S_{N} = \frac{{\pi^{2} (1 - u_{0}^{2} )}}{2}\frac{{2(2c^{2} )^{2} }}{4\pi c} $$

(14)

ROM model with modified slit crack geometry

$$ S_{N} = \frac{{\pi^{2} (1 - u_{0}^{2} )}}{2}\frac{{2(2c^{2} - \pi a^{2} /2)^{2} }}{\pi (4c + \pi a)} $$

(15)

ROM model with unmodified half ellipse crack geometry

$$ S_{N} = \frac{{16(1 - u_{0}^{2} )}}{3}\frac{{2(\pi c^{2} /2)^{2} }}{{\pi^{2} c}} $$

(16)

ROM model with modified half ellipse crack geometry

$$ S_{N} = \frac{{16(1 - u_{0}^{2} )}}{3}\frac{{2(\pi c^{2} /2 - \pi a^{2} /2)^{2} }}{\pi (\pi c + \pi a)} $$

(17)

The value of S _N for DBV model is calculated via multiplying each corresponding S _N value for ROM model by $\frac{{r^{2} + 3}}{{(r + 1)^{2} }},$ where r is the ratio between the half the crack length and the specimen thickness.

Appendix 2

Equations used to predict the slopes, S _p, of the ΔE/E ₀ versus ε plots for HA specimens in this study and alumina [1].

The relative change in Young’s modulus, ΔE/E ₀ in DBV model [from Eqs. (11) and (28) in Case’s article [53]) is given by

$$ \Updelta E/E_{0} = \frac{{r^{2} + 3}}{{(r + 1)^{2} }}\frac{r}{r + 1} f\,G\,N_{\text{l}} , $$

(18)

where f is the microcrack orientation parameter, for aligned microcracks in penny, half penny, ellipse, and half ellipse crack shapes (from Table II in Kim and Case’s article [54]), f is defined as

$$ f = 1 6( 1- u_{0}^{ 2} )/ 3 $$

(19)

for aligned slit crack shapes (from Table II in Kim and Case’s article [54])

$$ f = \pi^{ 2} ( 1- u_{0}^{ 2} )/ 2. $$

(20)

G is the geometry parameter [from Eq. (4) in Kim and Case’s article [54]]

$$ G = \frac{{ 2\langle {\text{A}}^{ 2} \rangle }}{{\pi \langle {\text{P}}\rangle }}. $$

(21)

For DBV model with modified half ellipse crack geometry (from Table II in Kim and Case’s article [54])

$$ G = \frac{{(a + c)(c - a)^{2} }}{2}. $$

(22)

For the DBV model with a modified slit crack geometry (from Table II in Kim and Case [54])

$$ G = \frac{{2(4c^{4} - 2\pi a^{2} c^{2} + \pi^{2} a^{4} /4)}}{{4\pi c + \pi^{2} a}}. $$

(23)

N _l is the microcrack number density in the microcracked surface layer.

N _l is the number of microcracks/(specimen length × specimen width × depth of the microcrack-damaged layer), r is the depth of the microcracked layer/depth of the unmicrocracked layer.

In the plot of ΔE/E ₀ versus ε for HA specimens in this study and alumina specimens [1], ε is defined as GN _l. The theoretically predicted slope, S _p of ΔE/E ₀ versus ε is expressed as

$$ S_{\text{p}} = \frac{{r^{2} + 3}}{{(r + 1)^{2} }}\frac{r}{r + 1}f. $$

(24)