A stochastic model for the size dependence of spherical indentation pop-in

Abstract

A simple stochastic model is developed to determine the pop-in load and maximum shear stress at pop-in in nanoindentation experiments conducted with spherical indenters that accounts for recent experimental observations of a dependence of these parameters on the indenter radius. The model incorporates two separate mechanisms: pop-in due to nucleation of dislocations in dislocation-free regions and pop-in by activation of preexisting dislocations. Two different types of randomness are used to model the stochastic behavior, which include randomness in the spatial location of the dislocations beneath the indenter and randomness in the orientation of the dislocations, i.e., randomness in the stress needed to activate them. In addition to correctly predicting the experimentally observed average maximum shear stress at pop-in, the model also correctly describes the scatter in pop-in loads and how it varies with indenter radius. Monte Carlo simulations are used to validate the model and visualize the scatter expected for a limited number of tests.

Appendices

Elastic stress field under a spherical indenter at an applied load F

For an isotropic elastic material with Young’s modulus E and Poisson’s ratio ν, the stress components in a cylindrical coordinate system (r, θ, z) for indentation by a spherical indenter with radius, R, and contact radius, a, are given by32,33

$$\matrix{ {{{{\sigma }}_{\rm{r}}}} \hfill & = \hfill & {{{{\sigma }}_{\max }}\left\{ {{{1 - 2v} \over 3}{{{a^2}} \over {{r^2}}}\left( {1 - {{\left( {{z \over {\sqrt u }}} \right)}^3}} \right) + {{\left( {{z \over {\sqrt u }}} \right)}^3}{{{a^2}u} \over {{u^2} + {a^2}{z^2}}}} \right.} \hfill \cr {} \hfill & {} \hfill & {\left. { + {z \over {\sqrt u }}\left[ {u{{1 - v} \over {{a^2} + u}} + \left( {1 + v} \right){{\sqrt u } \over a}{{\tan }^{ - 1}}{a \over {\sqrt u }} - 2} \right]} \right\},} \hfill \cr } $$

((12))

$${{{\sigma }}_z} = - {{{\sigma }}_{{\rm{max}}}}{\left( {{z \over {\sqrt u }}} \right)^3}{{{a^2}u} \over {{u^2} + {a^2}{z^2}}},$$

((13))

$${{{\tau }}_{rz}} = - {{{\sigma }}_{\max }}\left( {{{r{z^2}} \over {{u^2} + {a^2}{z^2}}}} \right)\left( {{{{a^2}\sqrt u } \over {{a^2} + u}}} \right),$$

((14))

where

$$u = {1 \over 2}\left\{ {\left( {{r^2} + {z^2} - {a^2}} \right) + {{\left[ {{{\left( {{r^2} + {z^2} - {a^2}} \right)}^2} + 4{a^2}{z^2}} \right]}^{1/2}}} \right\}.$$

((15))

Other important relations are

$${{{\sigma }}_{\max }} = {{3F} \over {2{{\pi }}{a^2}}},$$

((16))

$$a = {\left( {{{3FR} \over {4{E_{\rm{r}}}}}} \right)^{{1 \over 3}}},$$

((17))

and

$${1 \over {{E_{\rm{r}}}}} = {{1 - {v^2}} \over E} + {{1 - v_{\rm{i}}^2} \over {{E_{\rm{i}}}}},$$

((18))

where E_i and E are the moduli of the indenter and specimen and ν_i and ν are their Poisson’s ratios, respectively.

The maximum principal shear stress (τ_max) and the corresponding angle (θ) that it makes with the r-axis [see Fig. 7(a) ] can be calculated from these equations using Mohr’s circle concepts. They are given by34 }

$${{{\tau }}_{\max }} = {{{{{\sigma }}_1} - {{{\sigma }}_3}} \over 2},$$

((19))

where

$${{{\sigma }}_{1,3}} = {{{{{\sigma }}_r} + {{{\sigma }}_z}} \over 2} \pm {\left[ {{{\left( {{{{{{\sigma }}_r} - {{{\sigma }}_z}} \over 2}} \right)}^2} + {{\tau }}_{rz}^2} \right]^{1/2}},$$

((20))

and

$${{\theta }} = {{{\pi }} \over 4} + {1 \over 2}{\tan ^{ - 1}}{{{{{\sigma }}_r} - {{{\sigma }}_z}} \over {2{{{\tau }}_{rz}}}},$$

((21))

Area enclosed within a maximum principal shear stress contour

The area enclosed within a maximum principal shear stress contour, τ, as schematically shown by the shaded region in Fig. 5, can be determined by dimensional analysis. Let the spatial coordinates be normalized by the contact radius, a, and the stress components by the elastic modulus, E, such that

$$\matrix{ {\bar r = r/a} & {{\rm{and}}} & {\bar z = z/a,} \cr } $$

((22))

where the bars over the symbols (e.g., $\bar r,\bar z$) represent the nondimensional quantities. The enclosed area, A, within a maximum principal shear stress contour can then be nondimensionalized as

$$\bar A = A/{a^2},$$

((23))

and the stress components shown in Appendix A can be rewritten in nondimensional form as

$${{{\bar \sigma }}_r},{{{\bar \sigma }}_z},{{{\bar \tau }}_{rz}},{{\bar \tau = }}{{{{{\sigma }}_{\max }}} \over E}f\left( {\bar r,\bar z,v} \right).$$

((24))

Hence,

$${{\bar \tau }} \propto {{{{{\sigma }}_{\max }}} \over E} \propto {F \over {{{\pi }}E{a^2}}}.$$

((25))

From Eqs. (B2) and (B4), the maximum principal shear stress can be written as

$$\matrix{ {{{\bar \tau }} \propto {F \over {{{\pi }}EA}}} & {{\rm{or}}} & {{{\bar \tau }} \propto {F \over {{{\pi }}A}}} \cr }.$$

((26))

Thus, for a maximum principal shear stress contour of τ (see Fig. 5), the enclosed area can be written as

$$\matrix{ {{{\tau }} = k{{F - {F_{{\tau }}}} \over A}} & {{\rm{or}}} & {A\left( {F,{{\tau }}} \right) = k{{F - {F_{{\tau }}}} \over {{\tau }}}} \cr },$$

((27))

where k is a proportionality constant that is a function of Poisson’s ratio ν, and F_τ is the minimum load required to generate a maximum shear stress of τ in the specimen. The value of F_τ can be determined from Hertzian elastic contact theory through the following relation:

$${F_{{\tau }}} = {1 \over 6}{\left( {{{{\pi }} \over {0.31}}} \right)^3}{\left( {{R \over {{E_{\rm{r}}}}}} \right)^2}{{{\tau }}^3}.$$

((28))

The value of the constant k can be obtained by fitting the numerically determined areas enclosed in the contours of maximum principal shear stresses for Hertzian elastic contact as a function of Poisson’s ratio. It is well approximated by Eq. (1b)

$$k = 0.022{v^2} - 0.0226v + 0.1245.$$

((29))

Probability of pop-in due to activation of preexisting dislocations

In the proposed weak-link model [see Fig. 5(b)], pop-in occurs by activation of dislocations at a load F if at least one dislocation is activated within the τ_crss contour. Let the specimen have n dislocations randomly oriented and spaced. The probability of at least one dislocation being activated is one minus the probability that none of the dislocations are activated. For dislocations not to be activated within the τ_crss contour, either there should be no dislocations within the contour or the shear stress acting on the dislocations should be lower than their activation stress. This probability p(F) can be written as follows:

$$\matrix{ {p\left( F \right)} \hfill & = \hfill & {1 - \left[ {p\left( {{\rm{no}}\,{\rm{dislocations}}\,{\rm{within}}\,{{{\tau }}_{{\rm{crss}}}}\,{\rm{contour}}} \right)} \right.} \hfill \cr {} \hfill & {} \hfill & { + \forall \,{\rm{dislocations}}\,{\rm{within}}\,{{{\tau }}_{{\rm{crss}}}}} \hfill \cr {} \hfill & {} \hfill & {\left. {{\rm{contour:}}\,p\left( {{{{\tau }}_{\rm{r}}} < {{{\tau }}_{{\rm{crss}}}}} \right)} \right].} \hfill \cr } $$

((30))

Note that for a given dislocation

$$p\left( {{{{\tau }}_{\rm{r}}} < {{{\tau }}_{{\rm{crss}}}}} \right) = 1 - {p_{{\rm{act}}}}\,,$$

((31))

where p_act is the probability of activation.

To calculate the probability that no dislocations are activated within the τ_crss contour, the area inside the τ_crss contour can be discretized into N smaller equal-area elements of area ΔA [see Fig. 5(b) ]. The probabilities of no dislocations being activated within each area element can then be multiplied to obtain the probability of no dislocations being activated within the entire τ_crss contour.

For a specimen having n dislocations (n → ∞) with a dislocation density ρ, the probability of finding i dislocations within an area is given by26 }

$$p\left( i \right) = {{\rm{e}}^{ - {{\rho \Delta }}A}}{{{{\left( {{{\rho \Delta }}A} \right)}^i}} \over {\left( i \right)!}}.$$

((32))

Hence, the probability that there are no dislocations within an area ΔA is

$$p\left( 0 \right) = {{\rm{e}}^{ - {{\rho \Delta }}A}}.$$

((33))

The probability that none of the dislocations within the infinitesimal area elements ΔA are activated can then be written as

$$\matrix{ {\forall {\rm{dislos}}\,{\rm<Subscript></Subscript>}\,{{{\tau }}_{{\rm{crss}}}}\,{\rm{contour}}\,:p\left( {{{{\tau }}_j} < {{{\tau }}_{{\rm{act}}}}} \right)} \cr { = \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {p\left( i \right){{\left( {1 - {p_{{\rm{act}}}}} \right)}^i}} } \cr }.$$

((34))

Substituting Eqs. (C4) and (C5) in Eq. (C1) and multiplying the probabilities of all the area elements (1 to N), we obtain

$$\matrix{ {p\left( F \right)} \hfill & = \hfill & {1 - \mathop {\lim }\limits_{N \to \infty } \prod\limits_{j = 1}^N {} } \hfill \cr {} \hfill & {} \hfill & {\left[ {{{\rm{e}}^{ - {{\rho \Delta }}A}} + \mathop {\lim }\limits_{n \to \infty } \sum\limits_{i = 1}^n {{{\rm{e}}^{ - {{\rho \Delta }}A}}} {{{{\left( {{{\rho \Delta }}A} \right)}^i}} \over {\left( i \right)!}}{{\left( {1 - {p_{{\rm{act}}}}} \right)}^i}} \right].} \hfill \cr } $$

((35))

The above equation can be simplified to

$$p\left( F \right) = 1 - \exp \left( { - {{\rho }}{{A\left( {F,{{{\tau }}_{{\rm{crss}}}}} \right)} \over N}\mathop {\lim }\limits_{N \to \infty } \sum\limits_{j = 1}^N {{p_{{\rm{act}}}}} } \right).$$

((36))

Probability of activation (p_act) of preexisting dislocations randomly oriented relative to the direction of maximum principal shear stress (case 2)

For the dislocation to be activated within the j^th contour of maximum principal shear stress τ_j, the resolved shear stress acting to move it on its glide plane should be greater than τ_crss. The probability of such an event can be written using Eq. (7) as

$$\matrix{ {{p_{{\rm{act}}}}} \hfill & = \hfill & {p\left( {\left| {{{{\tau }}_{\rm{r}}}} \right| \ge {{{\tau }}_{{\rm{crss}}}}} \right) = p\left[ {\left| {{{{\tau }}_j}\cos 2\left( {{{\theta }} - {{\alpha }}} \right)} \right| \ge {{{\tau }}_{{\rm{crss}}}}} \right]} \hfill \cr {} \hfill & = \hfill & {p\left[ {\left| {\cos \left( {2\left( {{{\theta }} - {{\alpha }}} \right)} \right)} \right| \ge {{{{{\tau }}_{{\rm{crss}}}}} \over {{{{\tau }}_j}}}} \right].} \hfill \cr } $$

((37))

The absolute value of the resolved shear stress is used in this analysis as the activation depends only on the magnitude of the shear stress.

The probability of activation p_act can then be calculated as the ratio of the range of angles that satisfy the condition, |cos[2(θ − α)]| ≥ τ_crss/τ_j to the total range of angles for 2(θ − α), which is 5π (−π to 4π). Since the cosine function is cyclical and its absolute value is of interest in the present case, we can calculate the range of angles that satisfy the above inequality for a smaller range such as 0 to π and then extend it to the total range (−π to 4π) by multiplying by 5. The range of angles between 0 and π that satisfy the condition |cos[2(θ − α)]| ≥ τ_crss/τ_j is 2θ*, where θ* = cos⁻¹(τ_crss/τ_j). Hence, the probability of activation for the entire range of angles is

$${p_{{\rm{act}}}} = {{2{{{\theta }}^*} \times 5} \over {4{{\pi }} - \left( { - {{\pi }}} \right)}} = {{2{{{\theta }}^*}} \over {{\pi }}} = {2 \over {{\pi }}}{\cos ^{ - 1}}\left( {{{{{{\tau }}_{{\rm{crss}}}}} \over {{{{\tau }}_j}}}} \right).$$

((38))