Asymptotic Fracture Modes in Strain-Gradient Elasticity: Size Effects and Characteristic Lengths for Isotropic Materials

Abstract

Size-effects characterize the fracture process of many engineering materials. Their modelling calls for material constitutive relations which are not indifferent, as standard elasticity, to variations of scale: strain-gradient elasticity or plasticity have often served the purpose.

The three classical crack opening problems of fracture mechanics are here solved within the framework of linear strain-gradient elasticity for the most general isotropic material. Apart from the Lamé constants, this is completely identified by five additional moduli, modelling its micro-structural characteristics. This general setting allows for recovering, as particular cases, previous analyses in (Gourgiotis and Geogiadis in J. Mech. Phys. Solids 57(11): 1898–1920, 2009), relative to modes I and II for the so-called Simplified Mindlin materials (17), and in (Radi in Int. J. Solids Struct. 45(10): 3033–3058, 2008), relative to mode III for couple-stress materials. More importantly, having a rather refined material description allows for understanding how the energy release rate is affected by the actual values of the characteristic lengths. Hence we demonstrate the strengthening effect of suitable microstructures and their optimality in order to provide cohesive like actions on the crack lips.

Despite being limited to the linear elastic hypothesis, the solutions found constitute an useful insight to develop more complex, possibly nonlinear, constitutive relationships for the strain-gradient terms as for plastic or viscoelastic materials.

Appendices

Appendix A

As suggested by Fig. 1 the actual crack opening displacement (COD) results from a matching between the asymptotic solution (37), which on the upper lip θ=π, reads:

$$ w_a(r)= C_{I\!\!I\!\!I} \biggl( \dfrac{3}{16 (\ell_{s}/\ell_t)^2-3}+1 \biggr) r^{3/2}, $$

(60)

and the far-field solution suitably shifted by a distance b; this last, again on the lip θ=π, reads $w_{f}(r) \propto K_{I\!\!I\! \!I} \sqrt{r-b}$. We estimate the distance d by asking the C ¹ continuity between the two CODs and the equality of the strain and strain-gradient energies. These requests are tantamount to solve the three equations:

$$ w_a(d)=w_f(d),\qquad \dfrac{\partial w_a}{\partial r} \bigg|_{d} =\dfrac {\partial w_f}{\partial r}\bigg|_{d},\qquad \psi_2(w_a) (d)=\psi_1(w_f) (d), $$

(61)

in the three unknowns d, b and C _III /K _III ; here ψ ₁ and ψ ₂ account respectively for the strain and strain-gradient energy densities. For the antiplane case one easily obtain:

$$ d = \dfrac{\ell_t \sqrt{3} }{16 \sqrt{2}} \sqrt{\dfrac{\ell_t^4}{\ell_s^4}-32 \dfrac{\ell_t^2}{\ell_s^2}+128 }, $$

(62)

which is plotted in Fig. 4. A similar reasoning allows to estimate d for the inplane and shear opening modes, see Fig. 5.

Appendix B

We report the expression for the constants in (44), (45) and (45), together with their specialization to couple-stress materials (15) and to Simplified Mindlin materials (17). In particular, the k constants in Eqs. (44)–(45) are:

(63)

with

(64)

The expressions for the J-integral in (57) are:

(65)

(66)

(67)

where $\varDelta = (5 \ell_{e}^{2}+3 \ell_{s}^{2} )^{2} (9 \ell_{b}^{2}+24 \gamma_{3}+16 (\ell_{e}^{2}-9 \ell_{s}^{2} ) ) (-15 \ell_{b}^{2}+40 \gamma_{3}+16 (\ell_{e}^{2}+15 \ell_{s}^{2} ) )^{2}$.

For the constitutive case (15), Eqs. (63) reduce to:

(68)

while clearly J ₂₂=J ₂₄=J ₄₄=0 as ℓ _e =0 in (15).

For the SM materials (17), Eqs. (63) reduce to:

(69)

while

(70)