Analysis of macroscopic crack branching patterns in chemically strengthened glass

Abstract

Residual stress profiles were introduced in sodium aluminosilicate glass disks using an ion-exchange process. They were fractured in two loading conditions: indentation and biaxial flexure. The fractal dimension of the macroscopic crack branching pattern called the crack branching coefficient (CBC), as well as the number of fragments (NOF) were used to quantify the crack patterns. The fracture surfaces were analyzed to determine the stresses responsible for the crack branching patterns. The total strain energy in the body was calculated. The CBC was a good measure of the NOF. They are directly related to the tensile strain energy due to the residual stress profile for fractures due to indentation loading. However, in general for materials with residual stresses, CBC (or NOF) is not related to the strength or the stress at fracture, or even to the total stored tensile strain energy. Instead, the CBC appears to be related, in a complex manner, to the distribution of stresses in the body. Therefore, in general, the characterization of the CBC of fractured materials cannot be used to ascertain the prior stress distribution.

Appendices

Appendix A: Determination of Depth of Ion Exchange

At any point in the thickness of the sample, z, the concentration, C_z, of K⁺ is given by

$$\left( {{C_z} - {C_{\text{o}}}} \right)/\left( {{C_{\text{s}}} - {C_{\text{o}}}} \right) = 1 - {\text{erf}}\left( Z \right)$$

((A1))

where C_o is the bulk concentration of K⁺, C_s is the surface concentration, and erf () is the error function. Z is given by

$$Z = z/[2{\left( {D{\beta }} \right)^{1/2}}]$$

((A2))

where β is the exchange time (in seconds), and D is the diffusion coefficient defined by

$$D = {D_{\text{o}}}\exp \left[ { - Q/\left( {kT} \right)} \right]$$

((A3))

where D_o is the diffusion coefficient prefactor, Q is the activation energy for diffusion, k is the Boltzmann’s constant, and T is the temperature of the molten salt. For this glass, D_o = 7.3 × 10⁻⁹ m²s⁻¹, and Q = 0.75 eV.24 For T = 723.15 K, D = 4.329 × 10⁻¹⁴ m²s⁻¹. The point of maximum ion penetration, δ, was taken as the position where C_z = C_δ, such that C_δ is experimentally indistinguishable from C_o. Electron microprobe analyses were conducted on polished cross sections of several disks from the 3-, 6-, 12-, 24-, and 48-h exchanges to determine the depth such that C_δ = C_o. These exchange depths were used in Eq. (A2) to determine a value for Z. The average of these values, Z = 1.2, was then used to determine δ for each exchange condition via Eq. (A1). These values were 53, 75, 105, 149, 211, and 298 μm, respectively, for the 3-, 6-, 12-, 24-, 48-, and 96-h exchange.

Appendix B: Stress State in Piston-On-Three-Ball Biaxial Flexure

The stress states in ball-on-three-ball and piston-on-three-ball biaxial flexure have been studied extensively.25, 26, 27, 28, 29, 30 For the piston-on-three-ball configuration used, the maximum radial and tangential stresses at the center are equal and are given by

$$ {{{\sigma }_{\max }} = }{\left[ {3P\left( {{\text{l + }}v} \right)/\left( {4{\pi}{{\text{t}}^2}} \right)} \right]*\left\{ {1 + 2\ln \left( {a/b} \right) + \left( {{\text{l}} - v} \right)} \right.} \\ {}{\left. {\left[ {{\text{l}} - {b^2}{\text{/}}\left( {2{a^2}} \right)} \right]{a^2}{\text{/}}\left[ {\left( {1 + v} \right){R^2}} \right]} \right\}} \\ $$

((B1))

where P is the load, ν is Poisson’s ratio (ν = 0.22 for Corning 0317 glass), a is the radius of the circle of support balls, b is the radius of the loading piston flat (b < a), R is the radius, and t is the thickness of the disk. Outside the load radius, the radial stress σ_r and the tangential stress σ_θ are given by

$${{{\sigma }_{\text{r}}}\left( r \right) = \left[ {3P\left( {{\text{l + }}v} \right){\text{/}}\left( {4{\pi }{t^2}} \right)} \right]*\left\{ {2\ln \left( {a{\text{/}}r} \right) + \left( {{\text{l}} - v} \right)} \right.} \\{\left. {\left[ {{\text{l}} - {r^2}{\text{/}}{a^2}} \right]{a^2}{b^2}{\text{/}}\left[ {2\left( {{\text{l + }}v} \right){r^2}{R^2}} \right]} \right\}\quad r \geqslant b} \\ $$

((B2))

$${{{\sigma }_{\theta }}\left( r \right) = \left[ {3P\left( {{\text{l + }}v} \right){\text{/}}\left( {4{\pi }{t^2}} \right)} \right]*\left\{ {2\ln \left( {a{\text{/}}r} \right) + \left( {{\text{l}} - v} \right)} \right.} \\{\left. {\left[ {4 - {b^2}{\text{/}}{r^2}} \right]{a^2}{\text{/}}\left[ {2\left( {{\text{l + }}v} \right){R^2}} \right]} \right\}\quad \quad r \geqslant b,} \\ $$

((B3))

in which r is the radial distance from the center. The stress profile was assumed to decrease linearly over the region 0 < r < b from the value σ_max to the value σ_r(b) and σ_θ(b) in the radial and tangential directions, respectively. At any location r, the stress through-the-thickness of the sample falls off linearly, from its maximum value at the tensile surface to zero at the midpoint, with the other half of the sample under compression.

Appendix C: Total Strain Energy Calculation

The strain energy density u_o in a biaxial state of stress is given by35

$${u_0} = \left[ {1{\text{/}}\left( {2E} \right)} \right]\left( {{\sigma }_x^2 + {\sigma}_y^2 - 2v{{\sigma}_x}{{\sigma }_y}} \right)$$

((C1))

where E is the Young’s modulus (E = 71.7 GPa for Corning 0317 glass). Here, the contributions of the shear stresses to the strain energy are ignored. For the samples fractured in biaxial flexure, the bending stress is one of the sources of energy. This strain energy was calculated by expressing the stresses in Appendix B in terms of cylindrical coordinates, and integrating Eq. (C1) numerically (Maple 9, Maplesoft, Waterloo Maple, Inc., Waterloo, ON, Canada). The range of integration was: 0 < r < 3.5 mm, 0 < θ < 2π, and −t/2 < z < t/2. The value of r = 3.5 mm was chosen because it closely corresponds to the value over which the CBC and NOF for the biaxial samples [Figs. 3(g)–3(l)] were evaluated. The tensile component of this bending strain energy is half of the value calculated. For the equibiaxial state of stress due to the ion exchange, the tensile strain energy was calculated by setting σ_x= σ_y= σ_t, and integrating Eq. (C1) over the volume of the disk in tension. The strain energy due to the residual tensile stress was normalized to the volume over which the numerical integration was performed, and the results are plotted for comparison in Fig. 6. For the biaxial samples, both the bending and residual tensile energies contribute to the fracture process, whereas for indentation-induced fracture, only the residual tensile energy drives the fracture process.