Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case

Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case Gianni Dal Maso, Gianluca Orlando (SISSA) Rodica Toader (Univ. Udine) R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 0 / 10

Gradient damage models coupled with plasticity The starting point of our analysis is a gradient damage model coupled with plasticity, recently proposed by Alessi, Marigo, and Vidoli. The relevant variables are the displacement u, the elastic and plastic parts of the strain e and p, and the damage variable α. We consider the case of antiplane shear for an isotropic and homogeneous material: the reference configuration is a bounded open set in R n (n = 2 for three-dimensional bodies), R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 1 / 10

Gradient damage models coupled with plasticity The starting point of our analysis is a gradient damage model coupled with plasticity, recently proposed by Alessi, Marigo, and Vidoli. The relevant variables are the displacement u, the elastic and plastic parts of the strain e and p, and the damage variable α. We consider the case of antiplane shear for an isotropic and homogeneous material: the reference configuration is a bounded open set in R n (n = 2 for three-dimensional bodies), u and α are scalar functions defined on, R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 1 / 10

Gradient damage models coupled with plasticity The starting point of our analysis is a gradient damage model coupled with plasticity, recently proposed by Alessi, Marigo, and Vidoli. The relevant variables are the displacement u, the elastic and plastic parts of the strain e and p, and the damage variable α. We consider the case of antiplane shear for an isotropic and homogeneous material: the reference configuration is a bounded open set in R n (n = 2 for three-dimensional bodies), u and α are scalar functions defined on, e and p are vector functions from into R n. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 1 / 10

Gradient damage models coupled with plasticity small strain elasto-plasticity: additive decomposition u = e + p. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 2 / 10

Gradient damage models coupled with plasticity small strain elasto-plasticity: additive decomposition u = e + p. α min α 1, with α min ]0, 1[. α = 1 represents the sound material; α = α min is the maximum possible damage; constitutive relation: σ = αe elastic energy: Q(e, α) := 1 σ e dx = 1 α e 2 dx 2 2 stress constraint: σ κ(α), κ: [0, 1] R is a continuous nondecreasing function, 0 κ(0) κ(1) = 1 and κ(α) > 0 for α > 0 plastic potential: H(p, α) := κ(α) p dx R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 2 / 10

Gradient damage models coupled with plasticity small strain elasto-plasticity: additive decomposition u = e + p. α min α 1, with α min ]0, 1[. α = 1 represents the sound material; α = α min is the maximum possible damage; constitutive relation: σ = αe elastic energy: Q(e, α) := 1 σ e dx = 1 α e 2 dx 2 2 stress constraint: σ κ(α), κ: [0, 1] R is a continuous nondecreasing function, 0 κ(0) κ(1) = 1 and κ(α) > 0 for α > 0 plastic potential: H(p, α) := κ(α) p dx energy dissipated by the damage process: W(α) := b W(α) dx + l α 2 dx b, l > 0, W : [0, 1] R is a continuous decreasing function with W(1) = 0. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 2 / 10

The minimum problem The model proposed by Alessi, Marigo, and Vidoli is based on the minimization of the total energy E(e, p, α) := Q(e, α) + H(p, α) + W(α), defined for e L 2 (; R n ), p L 1 (; R n ), and α H 1 (; [α min, 1]), with the constraint e + p = u for some u W 1,1 () satisfying prescribed boundary conditions. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 3 / 10

Dependence on ε We study the asymptotic behavior of the functional E(e, p, α) when the minimum problem forces the damage to be concentrated along sets of codimension one. This means that α must be close to 1 on great part of the domain and can be close to 0 just on sets of codimension one. To force this behavior we assume that the three constants α min, b, l depend on a small parameter ε > 0 in a precise way: α min = δ ε, b = 1/ε, and l = ε, with δ ε /ε 0 as ε 0. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 4 / 10

Dependence on ε We study the asymptotic behavior of the functional E(e, p, α) when the minimum problem forces the damage to be concentrated along sets of codimension one. This means that α must be close to 1 on great part of the domain and can be close to 0 just on sets of codimension one. To force this behavior we assume that the three constants α min, b, l depend on a small parameter ε > 0 in a precise way: α min = δ ε, b = 1/ε, and l = ε, with δ ε /ε 0 as ε 0. Then E(e, p, α) becomes E ε (e, p, α) := Q(e, α) + H(p, α) + W ε (α) where Q(e, α) := 1 α e 2 dx, H(p, α) = κ(α) p dx, and 2 W ε (α) := 1 W(α) dx + ε α 2 dx. ε R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 4 / 10

The reduced functional We extend H to p M b (; R n ) which has better compactness properties, by setting H(p, α) := κ(α)d p. Therefore u BV() and e + p = Du = ul n + D s u. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 5 / 10

The reduced functional We extend H to p M b (; R n ) which has better compactness properties, by setting H(p, α) := κ(α)d p. Therefore u BV() and e + p = Du = ul n + D s u. We consider the functional F ε (u, α) defined for u BV() and α H 1 () by F ε (u, α):= min e,p {E ε(e, p, α):e L 2 (; R n ), p M b (; R n ), e + p=du} if δ ε α 1 and F ε (u, α):= + otherwise. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 5 / 10

The reduced functional We extend H to p M b (; R n ) which has better compactness properties, by setting H(p, α) := κ(α)d p. Therefore u BV() and e + p = Du = ul n + D s u. We consider the functional F ε (u, α) defined for u BV() and α H 1 () by F ε (u, α):= min e,p {E ε(e, p, α):e L 2 (; R n ), p M b (; R n ), e + p=du} if δ ε α 1 and F ε (u, α):= + otherwise. F ε represents the energy of the optimal additive decomposition of the displacement gradient (the minimum is achieved at a unique pair (e, p)). R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 5 / 10

The reduced functional The functional F ε (u, α) can be written in an integral form as F ε (u, α) := f ε (α, u ) dx + κ(α)d D s u + W ε (α), { 1 where f(α, t) := min 0 s t 2 αs2 + κ(α)(t s)} and f ε (α, t) := f(α, t) if δ ε α 1 f ε (α, t) := + otherwise. The asymptotic behavior of the functionals F ε (u, α) is described by their Γ -limit as ε 0 in the space L 1 () L 1 (). We set F ε (u, α) = + if u / BV() or α / H 1 (). R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 6 / 10

The limit problem The limit problem is given by the functional F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1, J u defined for u in the space GBV() of generalized functions of bounded variation, where J u is the jump set of u, [u] is the difference between the traces of u on both sides of J u, D c u is the Cantor part of Du, 1 2 f(1, t) = t2 if t κ(1) for every t 0, and κ(1)t κ(1)2 2 if t κ(1) R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 7 / 10

The limit problem The limit problem is given by the functional F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1, J u defined for u in the space GBV() of generalized functions of bounded variation, where J u is the jump set of u, [u] is the difference between the traces of u on both sides of J u, D c u is the Cantor part of Du, 1 2 f(1, t) = t2 if t κ(1) for every t 0, and κ(1)t κ(1)2 Ψ(t) := min 2 if t κ(1) { γ W (0), min 0<α 1 with γ W (α) defined by γ W (α) := 4 1 α [ κ(α)t + γ W (α)] }, W(s) ds for every α [0, 1]. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 7 / 10

The purely elastic problem Under the constraint p = 0 (i.e., e = u), which corresponds formally to κ(α) = + for every α min α 1, this problem has been studied by Ambrosio and Tortorelli. In this case F ε (u, α) := 1 2 α u 2 dx + W ε (α). R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 8 / 10

The purely elastic problem Under the constraint p = 0 (i.e., e = u), which corresponds formally to κ(α) = + for every α min α 1, this problem has been studied by Ambrosio and Tortorelli. In this case The limit functional is F ε (u, α) := 1 2 F(u) := 1 2 α u 2 dx + W ε (α). u 2 dx + γ W (0)H n 1 (J u ), if u GSBV() (i.e., D c u = 0), while F(u) = + if D c u 0. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 8 / 10

The fracture term In our case F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1 with J [ u Ψ(t) := min {γ W (0), min 0<α 1 κ(α)t + γ W (α)] } for every t 0. γ W (0) κ(1)t Ψ(1)t 0 1 Ψ is concave; R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 9 / 10

The fracture term In our case F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1 with J [ u Ψ(t) := min {γ W (0), min 0<α 1 κ(α)t + γ W (α)] } for every t 0. γ W (0) κ(1)t Ψ(1)t 0 1 Ψ is concave; Ψ(1)t Ψ(t) t if t is small; R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 9 / 10

The fracture term In our case F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1 with J [ u Ψ(t) := min {γ W (0), min 0<α 1 κ(α)t + γ W (α)] } for every t 0. γ W (0) κ(1)t Ψ(1)t 0 1 Ψ is concave; Ψ(1)t Ψ(t) t if t is small; Ψ (0) = κ(1); R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 9 / 10

The fracture term In our case F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1 with J [ u Ψ(t) := min {γ W (0), min 0<α 1 κ(α)t + γ W (α)] } for every t 0. γ W (0) κ(1)t Ψ(1)t 0 1 Ψ is concave; Ψ(1)t Ψ(t) t if t is small; Ψ (0) = κ(1); Ψ(t) = γ W (0) if κ(0)t γ W (0); R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 9 / 10

The fracture term In our case F(u) := f(1, u ) dx + κ(1) D c u () + Ψ( [u] ) dh n 1 with J [ u Ψ(t) := min {γ W (0), min 0<α 1 κ(α)t + γ W (α)] } for every t 0. γ W (0) κ(1)t Ψ(1)t 0 1 Ψ is concave; Ψ(1)t Ψ(t) t if t is small; Ψ (0) = κ(1); Ψ(t) = γ W (0) if κ(0)t γ W (0); lim t + Ψ(t) = γ W (0). R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 9 / 10

Theorem [Dal Maso-Orlando-T. 2015] The functionals F ε Γ -converge in L 1 () L 1 () to the functional F 0 : L 1 () L 1 () [0, + ] defined by { F(u) if u GBV() and α = 1 L n -a.e. in, F 0 (u, α) = + otherwise. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 10 / 10

Theorem [Dal Maso-Orlando-T. 2015] The functionals F ε Γ -converge in L 1 () L 1 () to the functional F 0 : L 1 () L 1 () [0, + ] defined by { F(u) if u GBV() and α = 1 L n -a.e. in, Moreover, F 0 (u, α) = F(u) = min e,p + otherwise. { 1 2 } e 2 dx + κ(1) p ( \ J u ) + Ψ( [u] ) dh n 1, J u where the infimum is taken among all e L 2 (; R n ), p M b (; R n ) such that Du = e + p as measures on \ J u. Hence F can be interpreted as the total energy of an elasto-plastic material with a cohesive fracture. R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 10 / 10

Theorem [Dal Maso-Orlando-T. 2015] The functionals F ε Γ -converge in L 1 () L 1 () to the functional F 0 : L 1 () L 1 () [0, + ] defined by { F(u) if u GBV() and α = 1 L n -a.e. in, Moreover, F 0 (u, α) = F(u) = min e,p + otherwise. { 1 2 } e 2 dx + κ(1) p ( \ J u ) + Ψ( [u] ) dh n 1, J u where the infimum is taken among all e L 2 (; R n ), p M b (; R n ) such that Du = e + p as measures on \ J u. Hence F can be interpreted as the total energy of an elasto-plastic material with a cohesive fracture. G. Dal Maso, G. Orlando, R. T.: Fracture models for elasto-plastic materials as limits of gradient damage models coupled with plasticity: the antiplane case. http://cvgmt.sns.it/paper/2677/ R. Toader (Univ. Udine) Fracture in elasto-plastic materials December 17, 2015 10 / 10