Singular integro-differential equations for a new model of fracture w. curvature-dependent surface tension

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1 Singular integro-differential equations for a new model of fracture with a curvature-dependent surface tension Department of Mathematics Kansas State University January 15, 2015 Work supported by Simons Foundation ( , )

2 Motivation Linear elastic fracture mechanics (LEFM) r Stress singularities at the crack tips: σ + iτ Cr 1/2 for a non-interface crack; σ + iτ Cr 1/2+iγ for an interface crack.

3 Alternative approaches Cohesive and processing zones (Barenblatt, 1959; Dugdale, 1960; Hillerborg 1976). Atomistic and lattice based approaches (Marder and Gross, 1995; Abraham et al, 1997; Holland and Marder, 1998). Atom-to-continuum approaches (Tadmor et al, 1996; Xiao and Belytschko, 2004). Peridynamics (Silling 2000). Surface elasticity/surface tension (Gurtin and Murdoch, 1975; Steigmann and Ogden, 1997, 1999; Slattery et al, 2007; Schiavone at al, ; Sendova and Walton, 2010; Zemlyanova and Walton, 2012; Zemlyanova, 2013).

4 Curvature-dependent surface tension model The linear elasticity model is assumed for the behavior of material in the bulk. A curvature-dependent surface tension acts on the boundary of the crack. First introduced: E.-S. Oh, J. R. Walton, J.C. Slattery, T. Sendova, J. Walton, 2010 for a straight crack. Advantages: Based on physically valid assumptions No ad hoc choices of parameters or material properties Linear elasticity techniques are valid Can be incorporated into industrial FEM codes

5 New results The curvature-dependent surface tension model has been applied to: Straight non-interface crack (Sendova and Walton, 2010). Curvilinear smooth cracks of arbitrary shape (Zemlyanova and Walton, 2012). Straight interface crack (Sendova and Walton, 2010; Zemlyanova, 2013). Curvilinear interface cracks (submitted). Contact problems for a rigid stamp (forthcoming). Main conclusion: No power singularities of the order 1/2 or oscillating singularities. Some components of stresses and strains may have logarithmic singularities.

6 Boundary condition Surface of the crack is subjected to the surface stress: T (ζ) = γp, where P = I n n is a projection tensor. Jump momentum balance condition: grad (ζ) γ + 2 γhn + [[ T ]]n = 0, where n is the unit normal pointing in the bulk of material; div (ζ) and grad (ζ) are the surface divergence and gradient; H = 1 2 div (ζ)n is the mean curvature; γ is the surface tension; T is the Cauchy stress tensor.

7 A straight non-interface crack (Sendova&Walton, 2010) Linear dependence of the surface tension on the mean curvature: γ(x) = γ 0 + γ 1 div (ζ) n. Linearizing the boundary condition under assumption that u i,j, u i,jk are small: σ 12 (x, 0) = γ 1 u 2,111 (x, 0), σ 22 (x, 0) = γ 0 u 2,11 (x, 0).

8 Mathematical techniques Dirichlet-to-Neumann mappings: σ 12 (x, 0) = α 1 u 2,1 (x, 0) + β 1 σ 22 (x, 0) = α 1 u 1,1 (x, 0) + β 1 u 1,1 (r, 0)dr ; r x u 2,1 (r, 0)dr. r x Substituting into the boundary conditions leads to the integro-diffrential equation: γ 0 u 2,11 (x, 0)+ 1 π l l δ 1 u 2,1 (r, 0) + δ 2 γ 1 u 2,111 (r, 0) r x = σ, x ( l, l). Main conclusion: For a curvature-dependent surface tension (γ 1 0) stresses and strains remain bounded at the crack tips.

9 A non-interface crack of arbitrary shape Linear dependence of surface tension on the change of curvature γ = γ 1 (div (ζ) n div (ζ0 )n 0 ). Linearized boundary condition: (σ n + iτ n ) ± (s) = γ { } 1 d 2µ κ 0(s) ds Im(u 1 + iu 2 ) ± t κ 0 (s) Re(u 1 + iu 2 ) ± t +i γ { } 1 d d 2µ ds ds Im(u 1 + iu 2 ) ± t κ 0 (s) Re(u 1 + iu 2 ) ± t where ±" denotes an upper or a lower side of the crack, s is an arc length, κ 0 (s) is the initial curvature of the crack.

10 Muskhelishvili s formulas Stresses and strains in a cracked plane can be expressed through two analytic functions Φ(z) and Ψ(z) (Muskhelishvili complex potentials): (σ n + iτ n )(t) = Φ(t) + Φ(t) + d t dt (tφ (t) + Ψ(t)), 2µ d dt (u 1 + iu 2 ) = κφ(t) Φ(t) d t dt (tφ (t) + Ψ(t)), where µ and κ are elastic constants of the cracked plate.

11 Savruk s Integral Representations Complex potentials Φ(z) and Ψ(z) can be written using Savruk s integral representations: where Φ(z) = Γ + 1 g (t)dt 2π L t z Ψ(z) = Γ + 1 2π L (κ + 1) 1 + πi L + (κ + 1) 1 πi ( g (t)dt t z tg (t)dt (t z) 2 ) ( κq(t)dt t z tq(t)dt (t z) 2 L q(t)dt t z, g (t(s)) = 2µ d ( (u + iv) + (s) (u + iv) (s) ), i(κ + 1) dt 2q(t(s)) = (σ n + iτ n ) + (s) (σ n + iτ n ) (s), s [0, l], are the jumps of stresses and derivatives of displacements. ),

12 Stresses and strains Boundary conditions: (σ n + iτ n ) ± (s 0 ) = γ [ 1 2µ κ 0(s 0 ) κ 0 (s 0 ) Re Ω ± (s 0 ) Im d ] Ω ± (s 0 ) ds 0 where i γ 1 d 2µ ds 0 (σ n +iτ n ) ± (κ + 1) 1 (s 0 ) = ±q(s)+ π [ κ 0 (s 0 ) Re Ω ± (s 0 ) Im d ] Ω ± (s 0 ) + f (s 0 ), ds 0 l 0 g (s)ds (κ 1) s s 0 πi(κ + 1) +(regular terms), Ω ± i(κ + 1) (s 0 ) = ± g (s 0 )+ κ 1 l g (s)ds 2κ + 2 2π(κ + 1) 0 s s 0 πi(κ + 1) +(regular terms). l 0 q(s)ds s s 0 l 0 q(s)ds s s 0

13 System of Cauchy singular integro-differential equations The problem is reduced to the system of two equations: { γ 1 d κ 0 (s 0 ) 4µ ds 0 [ d ds 0 1 l Im g (s)ds π 0 + κ 1 s s 0 π(κ + 1) [ κ 1 l Re g (s)ds + π 0 s s 0 4κ π(κ + 1) l 0 l 0 Re q(s)ds s s 0 + 4κ l π(κ + 1) 0 Re q(s)ds κ 1 l s s 0 π 0 = Im M 1 (g, q)(s 0 ), s 0 [0, l], Im g (s)ds s s 0 where M 1 (g, q)(s 0 ) is a regular linear integral operator. ] Im q(s)ds + s s 0 ]}

14 Unknown functions 4 real unknown functions: Re g (s), Im g (s), Re q(s), Im q(s). 2 additional real conditions: q(s 0 ) + q(s 0 ) = iγ [ ] 1 4µ κ 0(s 0 ) κ 0 (s 0 )(g (s 0 ) g (s 0 )) + i(g (s 0 ) + g (s 0 )) ; q(s 0 ) q(s 0 ) = γ 1 d [ ] κ 0 (s 0 )(g (s 0 ) g 4µ ds (s 0 )) + i(g (s 0 ) + g (s 0 )) 0 [ d = i ds 0 1 κ 0 (s 0 ) (q(s 0) + q 0 (s 0 )) ].

15 Two words about regularization Step 1: Both equations of the system have the form 1 l ϕ(s)ds πi 0 s s 0 = g(s 0 ), s 0 [0, l]. Invert the singular integral, obtain weakly singular integro-differential equations. Step 2: Recast the highest order derivatives as the new unknown functions: ϕ(s 0 ) = Im g (s 0 ); ψ(s 0 ) = Re q (s 0 ). Obtain two weakly-singular integral equations for the new unknowns.

16 Boundedness of the stresses and the derivatives of the displacements Rewrite the boundary condition in the form: (σ n + iτ n )(s 0 ) = γ { } 1 d 2µ κ 0(s 0 ) u sn κ 0 (s 0 )u sτ ds 0 It follows that are bounded, while +i γ 1 d 2µ ds 0 { } d u sn κ 0 (s 0 )u sτ. ds 0 u sn and σ n u sτ and τ n may have logarithmic singularities.

17 Numerical methods and accuracy Unknown functions g (s) and q(s) were approximated by Fourier series; Spline collocation methods; Taylor series. Figure : Convergence for Taylor and Fourier series approximations

18 Semicircular crack under horizontal stretching at infinity Normal and shear stresses. Normal and tangential components of the derivative of displacements.

19 A straight interface crack (Zemlyanova, 2013) Material 1 -l l Material 2 Oscillating singularity: σ + iτ Cr 1/2+iγ. Alternative models: Contact zones (Comninou, 1977). No-slip zones (Mak et al, 1980). Intermediate layer (Atkinson, 1977).

20 Curvilinear interface fracture Figure : A partially debonded inclusion of arbitrary shape. Jump momentum condition: grad (ζ) γ + 2 γhn + [[ T ]]n = 0. Surface tension γ linearly depends on change in the mean curvature of the surface: γ = γ (div (ζ) n div (ζ0 )n 0 ).

21 Boundary conditions (σ n + iτ n ) ± (s) = γ +i γ d 2µ ds { d 2µ κ 0(s) (σ n + iτ n ) + 0 (s) (σ n + iτ n ) (s) = γi d 2µ ds +i γi } ± ds u sn κ 0 (s)u ± sτ { } d ± ds u sn κ 0 (s)u ± sτ, s [0, l 0 ], { } d 2µ κ ± 0(s) ds u sn κ 0 (s)u ± sτ { } d ± ds u sn κ 0 (s)u ± sτ,, s [l 0, L], d dt (u 1 + iu 2 ) + 0 (s) = d dt (u 1 + iu 2 ) (s), s [l 0, L], where γ = γ + on L + 0 and γ = γ on L 0.

22 Mathematical techniques: Integral Representations The complex potentials Φ 0 (z) and Ψ 0 (z) for the inclusion can be written as: Φ 0 (z) = 1 g 0(t)dt 2π L L0 t z + (κ 0 + 1) 1 q 0 (t)dt πi L L 0 t z, Ψ 0 (z) = 1 ( g 0 (t)dt 2π L L 0 t z tg ) 0(t)dt + (t z) 2 (κ 0 + 1) 1 ( κ 0 q 0 (t)dt tq ) 0 (t)dt, πi t z (t z) 2 L L 0 The complex potentials Φ(z) and Ψ(z) for the matrix can be written as: Φ(z) = Γ + 1 g (t)dt (κ + 1) 1 q(t)dt + 2π L L 0 t z πi L L 0 t z, Ψ(z) = Γ + 1 ( g (t)dt 2π L L 0 t z tg ) (t)dt + (t z) 2 (κ + 1) 1 ( κq(t)dt πi t z tq(t)dt ). (t z) 2 L L 0

23 Mathematical techniques: Integral Representations Extend the inclusion/outside matrix to the full complex plane assuming that the stresses and strains are equal to zero outside of the inclusion/matrix: 2q 0 (t) = (σ n + iτ n ) + 0 (t) (σ n + iτ n ) 0 (t), t L L 0, i(κ 0 + 1)g 0 (t) = 2µ d 0 dt (u 1 + u 2 ) + 0 (t), t L L 0, 2q(t) = (σ n + iτ n ) + (t) (σ n + iτ n ) (t), t L L 0, i(κ + 1)g (t) = 2µ d dt (u 1 + u 2 ) (t), t L L 0. The problem can be reduced to the system of eight singular integro-differential equations which can be further regularized.

24 Numerical results Method: Representations of unknown functions by Taylor polynomials. Figure : Graphs of the normal and shear stresses σ + n0, σ n, τ + n0, τ n on the curves L 0 and L

25 Comparison with known results Figure : Graphs of the derivatives of the stresses d(u 1,2 ) + 0 /dt, du 1,2 /dt on the fracture surface L 0 Mechanical parameters: µ 0 = µ = 40, ν 0 = ν = 0.25, γ + = γ = 0.01, γ i = 0.05, σ 1 = 1, σ 2 = 0, α = 0.

26 Advantages of the method Works for the fractures of arbitrary shape. Can be generalized for multiple fractures and multiple inclusions.

27 How to find the parameters? Experimental results for tension loaded nanowires with rough surface: Atomistic elucidation of the effect of surface roughness on curvature-dependent surface energy, surface stress, and elasticity by P. Mohammadi and P. Sharma. Parameters in Steigmann-Ogden model have been computed.