Evolution of growth of interacting near-by micro-cavities and macroscopic damage amplification

Abstract

The elastic stress in the ligament between two near-by holes (two-dimensional cylindrical cavities) depends singularly on the ligament thickness small parameter ζ = δ/R (where δ is the ligament thickness and R the hole radius), as ζ becomes vanishingly small, with the exponent a of ζ depending on the loading and the geometry. The damage of a solid containing a periodic distribution of pairs of near-by holes is treated within the theory of asymptotic homogenization in a two-scale model. By renormalization of the unit cell with the scale y = x/ε, where ε is the periodicity length between the pairs of near-by holes (a characteristic scale of the microstructure), the leading term in the singular amplification of the stress in the ligament between the holes is carried analytically to the macro-scale. The evolution of the growth of the holes (in the micro-scale) is governed by Eshelby mechanics (in this case the Budiansky and Rice path-independent M integral), which can be considered as a dissipation mechanism as the defects evolve. In the unit cell, the rate of change of energy in the (self-similar) growth of the holes balances the rate of change of the strain energy in the volume as the ligament thickness decreases, and this, by the renormalization, is carried to the macroscopic level and defined as the energy-release rate of “damage” at that point. In the strain hardening regime, the amplification of the damage is determined for two cases of distribution of pairs of interacting near-by small holes near large ones, with the larger (singular as ~ζ^−1/2) damage amplification occurring for a pressurized hole near a large unpressurized one, and being larger than the one when the loading is tension at infinity, for the same geometry.

Keywords

Growth of voids void coalescence damage path-independent integrals elasticity energy dissipation asymptotic homogenization two-scale model

1. Introduction

In brittle solids where the elastic stresses still govern the fracture process in the strain hardening regime, it is important to understand how the stress amplifies in the ligament between holes as the ligament thickness becomes vanishingly small [1]. In Figure 1 are shown two populations of cavities (in two dimensions): a small hole near a large hole, two small holes between large ones, and a small hole that has already coalesced with a large one, under tension T at infinity. The stress amplification in the ligament as a function of the ligament thickness ζ = δ/R has been obtained analytically in several cases: for a small hole near a large one (the Mindlin problem [2]) it has been shown [3] to be σ ~ζ^−1/2T, in the ligament of a small hole between two large ones (the Koiter problem [4]) to be σ ~ζ⁻¹T, between two small holes between two large ones [5] σ ~ζ⁻¹T, and between a small hole that has already coalesced with a large one to be σ ~ζ⁻²T [5]. To underline the importance of this asymptotic behavior, for ligament thicknesses of the order of ζ ~ 10⁻⁶, the stresses in the ligament, correspondingly, would amplify as σ ~ 10³T, σ ~ 10⁶T, σ ~ 10¹²T, and no numerics can accurately capture 10¹² differences in the stress within the solid. It thus becomes obvious that the asymptotic behavior of the stress amplification in the ligament is critical for quantifying the rate of acceleration of the coalescence of near-by holes. If a solid contains a periodic distribution of nearby holes that are growing (by different mechanisms discussed e.g. by Kienzler [6], or by dislocation emission [7]), then, consequently the ligament between the holes decreases, which results in singular stress amplification in the ligament. This effect can be carried analytically to the macroscopic scale as damage amplification by the rigorous and systematic technique of asymptotic homogenization, as in Markenscoff and Dascalu [8] for pairs of near-by interacting micro-cracks, for quasi-brittle solids in the strain hardening regime [9]. The damage of a solid containing a periodic distribution of pairs of near-by holes is treated here within the theory of asymptotic homogenization in a two-scale model. By renormalization of the unit cell with the scale y = x/ε, where ε is the periodicity length between the pairs of near-by holes (a characteristic scale of the microstructure), the leading term in the singular amplification of the stress between the holes is carried analytically to the macro-scale in the asymptotic homogenization development.

In the unit cell, the energy dissipation and evolution of the growth of the holes (in the micro-scale) is governed by Eshelby [10] mechanics, namely by the path-independent M integral [11] when the hole is growing self-similarly. The rate of the work of the driving forces during the incremental growth of the holes balances the rate of change of the strain energy in the volume of the unit cell, and this change of the strain energy is carried, by the renormalization, to the macroscopic scale as the energy-release rate of “damage”. The damage evolution law for the growth of holes, analogous to the one for crack propagation involving the J integral, is obtained based on the criterion that the macroscopic energy-release rate of damage at a given position reaches a critical value of the M integral (at the scale ε) for the hole to grow. It may be noted here that the path-independent J, L, M integrals can be considered as a dissipative mechanism of a purely elastic system due to the extension, rotation and self-similar growth (respectively) of an inhomogeneity, as shown by Gupta and Markenscoff [12]. Two cases of damage amplification of the homogenized solid in the strain hardening regime are treated analytically: periodically distributed small holes singularly interacting with near-by large ones under tensile loading at infinity, and pressurized small holes near large unpressurized ones. In the strain hardening regime, larger damage amplification occurs when pressurized holes are near large ones (unpressurized) rather than if the loading is tension at infinity.

2. Stress amplification in vanishingly small ligaments between holes

2.1. The Mindlin problem: a hole in a half-plane

Mindlin [2] solved in bipolar coordinates the problem of a hole in a half-plane under tension T at infinity (Figure 2), correcting a previous solution by Jeffery [13]. The solution of Jeffery–Mindlin is given in terms of a series, with the hoop stress at the hole given by

2 T [1 - \frac{2 \sinh^{2} α_{1} \sin^{2} β}{{(\cosh α_{1} - \cos β)}^{2}}] + 2 T (\cosh α_{1} - \cos β) \cdot {\frac{1}{2} \csc h α_{1} + 2 e^{- 2 α_{1}} \cos β + \sum_{n = 2}^{\infty} N_{n} \cos n β}

(1)

with

N_{n} = \frac{n^{2} \sinh α_{1} \cosh n α_{1} - n \sinh n α_{1} \cosh n α_{1}}{\sinh^{2} n α_{1} - n^{2} \sinh^{2} α_{1}} - 2 n (n \sinh α_{1} - \cosh α_{1}) e^{- n α_{1}}

(2)

The above series does not converge uniformly, as α₁ in Equations (1) and (2) tends to zero (with the ligament thickness δ tending to zero), and a singular asymptotic treatment was needed and was performed by Callias and Markenscoff [3], who obtained the singular term of the hoop stress at B to be

σ_{B} ~ 2 \sqrt{2} T ζ^{- 1 / 2} + O (1)

(3)

where ζ = δ/R.

This agrees with Duan et al. [14] who obtained the result by a different method, but with making the assumption that the stress distribution varies linearly in the ligament. This assumption is consistent with the approach of beam approximation of the ligament (inner expansion) attached to the rigid solid (outer expansion). The Mindlin [2] solution for a thin ligament also gives σ_A = 0, with σ_C = 4T (Figure 2). Mindlin [2, 15] was interested in the stress amplification singularity and he performed experiments [2] that showed the variation of the hoop stress rate with the ligament thickness. Moreover, by integrating the stress, Mindlin obtained the total force P transmitted through the ligament that is found to behave asymptotically, in the small parameter ζ, as

P ~ \sqrt{2} T R ζ^{1 / 2} + O (1)

(4)

One may remark that, although the total force P tends to zero as the ligament thickness becomes vanishingly small, the stress behaves singularly with the ligament thickness because the force is not vanishing fast enough [16].

As a beam theory approximation, used by Markenscoff and Dundurs [16] and stipulated by Keller [17], the ligament is approximated by a beam of parabolically changing cross-section d(x) [4]:

d (x) = δ + \frac{x^{2}}{2 R}

(5)

The range of validity of the beam theory approximation in x (the direction normal to the ligament thickness) is determined by the value of x for which the correction term equals the leading term, that is:

\frac{x^{2}}{2 R} = δ or x = \sqrt{2 R δ}

(6)

(see, also, Markenscoff [18]).

The significance of the Mindlin problem in the growth of holes is that the free edge of the half-plane in the Mindlin problem can be considered as the surface of a large hole with radius tending to infinity. We will be using the above stresses in the asymptotic homogenization analysis below for the growth of a small hole near a large one under remote tension.

2.2. Pressurized hole near a large hole

Dundurs and Ely [19] gave the closed-form solution of a hole under pressure in a half-plane (Figure 3), which will be considered here as the boundary of a large hole. The asymptotics for small ligament thickness yield [16]

σ_{A} = \frac{4 p}{ζ (2 + ζ)} ~ 2 p ζ^{- 1}

(7)

σ_{B} = σ_{C} = p

(8)

and, actually, the hoop stress is equal to p at all points around the hole. The beam theory approximation also holds, in this case the deformation being concave on opposite sides of the Mindlin problem (a), and the stress being singular at the surface of the large hole (Figure 3).

3. Asymptotic homogenization for a solid with periodic distribution of pairs of near-by interacting holes

3.1. Two-scale model and effective behavior of a solid containing a periodic distribution of near-by pairs of holes

The effective behavior of a solid containing holes (cylindrical cavities) is studied by the method of asymptotic homogenization [20 –22]. The damage parameter of the macroscopic solid will be defined by the energy-release rate for the growth of holes by applying Eshelby mechanics [9, 23] (also see Markenscoff and Gupta [24]), and balance of rate of energies at the unit-cell level, and carrying it by the homogenization to the macro-level. Here, as in Dascalu [25] and Markenscoff and Dascalu [8], a two-scale model will be used. We assume a (locally) periodic distribution of pairs of near-by holes (Figures 4(a) and (b)), so that a micro-structural length-scale ε can be defined. The radius R of the holes, as well as the distance δ between them, may vary smoothly inside the body. It will be assumed everywhere in the sequel that δ << R <<ε. The two distinct scales are represented by x (the macroscopic variable) and $y = \frac{x}{ε}$ (the microscopic variable). Following standard asymptotic homogenization [20, 21]:

u^{ε} (x, t) = u^{(0)} (x, y, t) + ε u^{(1)} (x, y, t) + ε^{2} u^{(2)} (x, y, t) + \dots

(9)

σ^{ε} (x, t) = \frac{1}{ε} σ^{(- 1)} (x, y, t) + σ^{(0)} (x, y, t) + ε σ^{(1)} (x, y, t) + \dots

(10)

where u⁽ⁱ⁾(x,y,t), σ ⁽⁰⁾(x,y,t), x ∈ Y_s, y ∈ Y are smooth functions and Y-periodic in y and t is a loading parameter of the macroscopic problem.

In the problem of near-by holes in the unit-cell problem, the displacement u⁽¹⁾ is the solution of the holes problem in an infinite medium (presented in Section 2), since the hole-pairs are assumed to be far away from the unit-cell boundaries, that is, the hole radii are much smaller than the periodicity length, ε, and similar discussions apply with regard to the periodicity boundary conditions as in Markenscoff and Dascalu [8]. The solution to the problem u⁽¹⁾, which has as loading the u⁽⁰⁾ problem (under periodicity boundary conditions), is formally written (as in Dascalu [25])

u^{(1)} (x, y) = ξ^{p q} (y) e_{x p q} (u^{(0)}) (x)

(11)

where the characteristic functions ξ^pq(y) are elementary solutions of the O(1) problem for the set of linearly independent macroscopic (applied) deformations e_xpq(u⁽⁰⁾) = δ_pq. The stress is

σ_{i j}^{(0)} = c_{i j k ℓ} (e_{x k ℓ} (u^{(0)}) + e_{y k ℓ} (u^{(1)}))

(12)

By introducing the mean value operator $< \cdot > = (1 / | Y |) \int_{Y_{S}} \cdot d y$ , where |Y| is the measure of Y, we can prove [20] that

\sum_{i j}^{(0)} \equiv < σ_{i j}^{(0)} > = C_{i j k ℓ} (δ, R) e_{x k ℓ} (u^{(0)})

(13)

where

C_{i j k ℓ} (δ, R) = \frac{1}{| Y |} \int_{Y_{s}} (c_{i j k ℓ} + c_{i j m n} e_{y m n} (ξ^{k ℓ})) d y

(14)

are the homogenized coefficients. The effective constitutive relation (13) should be used in the macroscopic equilibrium equation, to be solved for given macroscopic boundary conditions. These homogenized coefficients are consistent with the classical theory of homogenization and can be numerically computed by solving the unit-cell problem for different values of radius R and ligament thickness δ.

4. Eshelby mechanics for the growth of holes and damage amplification due to interacting near-by holes

4.1. (a) Damage amplification for the Mindlin problem (a small hole near a large hole under remote tension)

Let us consider a unit cell containing a small hole of radius near a large one, and that the small hole grows by a radius increase $δ \dot{r}$ , where $\cdot \equiv \frac{d}{d t}$ , $\frac{d}{d t}$ being a time derivative of the process of growth of holes (actually t can be any monotonically increasing parameter). We will consider the driving forces on the boundaries of the two holes and will balance the rate of work done by the driving forces during the growth of the holes to the rate of change of the strain energy in the volume of the unit cell, and we will evaluate them to the leading order in the ligament thickness.

We will observe that in the Mindlin problem, discussed in Section 2.1, the stresses in the ligament at the free half-plane boundary (large hole) are

σ_{x x} = σ_{y y} = σ_{x y} = 0

(15)

so that [[W]] = 0 and $\vec{T} = 0$ , on the boundary of the large hole and, consequently, the Eshelby [23] driving force on it is

[[W]] - \vec{T} \cdot [[\frac{\partial \vec{u}}{\partial n}]] = 0

(16)

We consider the unit cell, where, for a purely mechanical system, the rate of work of the driving forces [23] on the boundaries S = S₁ + S₂ of the two holes (as they grow by an increment $δ \dot{r} = r \dot{ℓ}$ of their radius each) is balanced by the rate of change of the total potential energy $\prod = \int_{v} W d v - B [u]$ (with W being the strain energy density and $B [\vec{u}]$ the potential energy of the loading in the notation of Budiansky and Rice, 1973), or E_tot in the notation of Eshelby (1970), so that in the unit cell we write:

\int_{S} ([[W]] - \vec{T} \cdot [[\frac{\partial \vec{u}}{\partial n}]]) δ \dot{r} d S + \frac{d Π}{d t} = 0

(17)

and this is the evolution equation for the rate of the growth of the holes. Because of (16), to the leading order in the ligament thickness, only the surface of the small hole contributes to the driving forces on the boundaries. For a self-similar growth $(δ \dot{r} = r \dot{ℓ})$ of a hole equation (17) yields

\dot{ℓ} \int_{S^{ε}} ([[W]] - \vec{T} \cdot [[\frac{\partial \vec{u}}{\partial r}]]) r d S = \dot{ℓ} [[M]] = \dot{ℓ} \int_{S^{ε}} (W r - r σ_{r r} \frac{\partial u_{r}}{\partial r}) r d θ = - \frac{d Π}{d t}

(18)

\dot{ℓ} M = - \frac{d Π}{d t}

(19)

where in (18) the surface integral is the M integral in 2D:

M = \int_{S} (W r - r σ_{r r} \frac{\partial u_{r}}{\partial r}) r d θ

(20)

for a circular hole (with respect to the center of the hole) [11] (also see Lubarda and Markenscoff [26]). The (total) M integral will be path-independent for any contour in the region of analyticity in the domain [27].

For the Mindlin problem, to the leading order in the small ligament thickness (thus avoiding the computation of the total M integral in bipolar coordinates), there is no contribution to the M integral from the large hole, and all the contribution comes from the small hole.

In order to compute the right-hand side of Equation (18), we observe that the leading order contribution to the volume integral as the hole grows and the ligament thickness decreases will come from the singular terms in the stress in the region of validity of the ligament asymptotic approximation. If the hole grows by δr (Figure 5), the ligament thickness decreases by

δ d (x) = δ r \cos θ ~ δ r

(21)

for small θ (in the range between ( $- \arctan \frac{\sqrt{2 R δ}}{R}$ , $+ \arctan \frac{\sqrt{2 R δ}}{R}$ )), from which follows that for self-similar growth:

\frac{d δ}{d t} = \frac{d r}{d t} = r \dot{ℓ}

(22)

In view of (11), the right-hand side of (18) is written by asymptotic homogenization in terms of the macroscopic zero-th order problem, in the unit cell Y = 1. To the leading order in the ligament thickness δ, (δ<< R <<ε), the derivatives with respect to δ will be of higher order for the $B [\vec{u}]$ terms, so that we have

\begin{array}{l} - \frac{d}{d t} \int_{V^{ε}} W d V & = - \frac{d}{d t} \int_{V^{ε}} \frac{1}{2} c_{i j k ℓ} e_{y_{k ℓ}} (u^{(1)}) e_{y_{i j}} (u^{(1)}) d y = - \frac{d}{d R} (\int_{V^{ε}} \frac{1}{2} c_{i j k ℓ} e_{y_{k ℓ}} (u^{(1)}) e_{y_{i j}} (u^{(1)}) d y) \frac{d R}{d t} \\ = - \frac{1}{2} \frac{d C_{i j k ℓ} (δ)}{d (δ)} e_{x k ℓ} (u^{(0)}) e_{x i j} (u^{(0)}) \frac{d δ}{d t} = - \frac{d \prod}{d δ} \frac{d δ}{d t} = - \frac{d \prod}{d t} \end{array}

(23)

For a hole growing self-similarly $(δ \dot{r} = \dot{ℓ} r)$ , in view of Equations (22) (evaluated at r = R) and (23), Equation (19) for the evolution of the growth of holes written in dimensionalized lengths in the unit cell yields

M^{ε} = - ε^{2} \frac{1}{2} \frac{R}{ε} \frac{d C_{i j k ℓ} (δ / ε)}{d (δ / ε)} e_{x k ℓ} (u^{(0)}) e_{x i j} (u^{(0)})

(24)

where M^ε denotes the M integral in the length-scale ε, where all the lengths are dimensionalized by ε in M^ε, and where ε² on the right-hand side of (24) is due to the scaling of M^ε with M in the unit cell: M^ε = ε²M (so that the units of both sides in Equation (24) are ones of force).

The right-hand side of Equation (24) depends on the macroscopic loading at x, and, from (24), we may define (analogously to Dascalu [25], for cracks) the macroscopic “energy release rate of damage” for the interacting holes of the Mindlin problem:

Y = - \frac{1}{2} \frac{R}{ε} \frac{d C_{i j k ℓ} (δ / ε)}{d (δ / ε)} e_{x k ℓ} (u^{(0)}) e_{x i j} (u^{(0)})

(25)

In Equation (25) the ratio $\frac{R}{ε}$ , the radius of the hole over the characteristic length of the periodicity of the microstructure, is independent of ε in asymptotic homogenization, and is a micro-structural parameter expressing the dilution of the pairs of holes in the matrix.

If the laws for the growth of holes are assumed to be

M^{ε} \leq M^{c r} \dot{ℓ} \geq 0 \dot{ℓ} (M^{ε} - M^{c r}) = 0

(26)

where M^ε is the M integral in the unit cell, M^cr is defined to be a material parameter for the growth of holes at the scale ε, then we have an evolution law for damage due to the growth of holes, analogous to the one of crack propagation, as in Dascalu et al. [22], Dascalu [25] and Markenscoff and Dascalu [8]. Kienzler et al. [6] included in the evolution of the growth of holes a surface energy term (their Equation (8)) extending the Griffith criterion to the rate growth of holes, and Kienzler [28] made a historical account of an argument of Griffith and G.I. Taylor about the energy change due to the growth of holes and cracks, all resolved by the M integral.

The homogenized stiffness coefficient can be computed analytically to the leading order in the ligament small parameter. Thus, we now evaluate the damage amplification (in equations (24) and (25)) for the Mindlin problem in the unit cell by evaluation of the homogenized coefficient

\frac{d C_{i j k ℓ} (δ)}{d (δ)}

by integrating the right-hand side of (14) analytically to the leading order.

From the stress analysis in Section 2.1, for the total force transmitted through the ligament d(x) we have

P (x) ~ 2 \sqrt{2} T R^{1 / 2} d {(x)}^{1 / 2}

(27)

and, for the leading order contributions to (25), integrating in the range of validity ( $- \sqrt{2 R δ} < x < \sqrt{2 R δ}$ ), we have, with dimensionalized lengths in the unit cell

C_{1111} ~ 2 \sqrt{2} T \frac{R}{ε} \frac{δ}{ε}

(28)

which induces an amplification of the damage in Equation (25):

\frac{d C_{1111}}{d (δ / ε)} ~2 \sqrt{2} T \frac{R}{ε}

(29)

which is constant. We may note again that the ratio $\frac{R}{ε}$ is independent of ε in asymptotic homogenization and it expresses the micro-structural parameter of the dilution of the pairs of holes in the matrix.

4.2. (b) Damage amplification for a pressurized hole near a big hole

The stresses for this loading have been given in Equations (7) and (8). The unit cell comprising a small pressurized hole near a large one is shown in Figure 6. The rate work of the driving forces on the total surface S = S₁ + S₂ of both the small pressurized hole S₁ and the large hole S₂ (the axis y = 0), as the radii of the holes are growing by δr each (and the ligament thickness consequently decreases) balances the rate of change of the strain energy in the unit volume.

In this case the M integral around the pressurized hole is computed, and considering that the stresses are σ_rr = −p, σ_θθ = −p at all points around the hole, it is evaluated to be

\int_{2 π} ([[W]] - T \cdot [[\frac{\partial u}{\partial r}]]) R d θ = \int_{0}^{2 π} [\frac{R}{4 μ} [(1 - v) (σ_{r r}^{2} + σ_{θ θ}^{2}) - 2 v σ_{r r} σ_{θ θ}] - R σ_{r r} \frac{\partial u_{r}}{\partial r}] d θ = 0

(30)

Thus, the total contribution to the M integral will come only from the large hole, which is the free surface. The driving force on the free surface of the large hole gives a non-zero contribution, since the stress σ_xx experiences a discontinuity, and hence $[[W]] \neq 0, \vec{T} = 0$ at y = 0, and the work of the driving force in (18) as the ligament thickness decreases, to the leading order is calculated from the range of validity of the asymptotic expansion in the x direction in Figure 6, which is O(δ^1/2). The estimation shows that in view of the order of the stress amplification given by equation (7) the order of the damage amplification will be singular as ~ζ^−1/2 Thus the damage amplifies faster if the small hole interacting with a large one is pressurized versus loading of remote tension for the same geometry.

5. Conclusions

In the strain hardening regime, by using a two-scale model and asymptotic homogenization, we showed that the singular elastic stress amplification in the ligament between holes amplifies the “damage” at the macroscopic scale for a solid containing distributed pairs of near-by holes in two different examples of loading. The case of dislocation slip in the ligament (accounting for plastic deformation) has also been considered by Lubarda and Markenscoff [29] and Markenscoff and Lubarda [30], and it was shown that the stress amplifies in the ligament square-root singularly, which will also result in damage amplification. As with the growth of micro-cracks, the energetics of the evolution of all defects are governed by Eshelby mechanics at the micro-level (unit-cell level), treated here for the growth of holes by means of the path-independent Budiansky and Rice [11] M integral, and the dissipated energy at that level is carried analytically (by asymptotic homogenization) to the macroscopic level, and defined as the “energy-release-rate of damage” for the growth of holes.

Footnotes

Acknowledgements

Discussions with Shailendra Pal Veer Singh are acknowledged as well as the support of the National Science Foundation grant CMS #1129888. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily represent the views of the National Science Foundation.

Conflict of interest

None declared.

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