Gurses, Ercan; Lambrecht, Matthias; Miehe, Christian

Application of Relaxation Techniques to Nonconvex Isotropic Damage Model

A new energy relaxation technique for nonconvex inelasticity is applied to an isotropic damage model. The increasingdamage yields nonconvex incremental potentials indicating the instability of the material and the development of fine-scale microstructures. The microstructures are resolved as first-order laminates by convexifiaction techniques.

1. Variational Formulation of Damage Model in Principal Shear Value Representation

We point out a compact representation of the isochoric constitutive functions in 2–D framework in terms of theprincipal shear values that serves as framework for the variational formulation of an isochoric damage model. Thenwe derive an analytical closed form solution of the minimization problem of first–order rank–1 convexification. Weconsider isochoric–volumetric split of the strain where deviatoric part is computed as ε = ε − 1

2e1 with e = tr[ε] in2–D framework. The isochoric strain tensor can be represented in the space of principal shear values

ε = q(m1 ⊗ m2 + m2 ⊗ m1) with q := ‖ε‖/√2 (1)

where the orthogonal vectors m1, m2 represent the principal shear–directions and q the shear intensity. In case ofisotropy the incremental stress potential W can be written as W (ε) = Wvol(en+1)+Wiso(qn+1) that depends on thedilatation and the shear intensity only and specific form will be specified later in (5). We refer to work Miehe et al.[1] for the details of incremental variational formulation of inelasticity. Use of the spectral representation yields thefollowing form of the stresses and the moduli

σn+1 = Wvol, e 1 + 12Wiso, q (m1 ⊗ m2 + m2 ⊗ m1) (2)

Cn+1 = Wvol, ee 1 ⊗ 1 + 14Wiso, qq (m1 ⊗ m2 + m2 ⊗ m1) ⊗ (m1 ⊗ m2 + m2 ⊗ m1)

+ 14Wiso, q/qn+1 (m2 ⊗ m2 − m1 ⊗ m1) ⊗ (m2 ⊗ m2 − m1 ⊗ m1).

(3)

The following specific form of the free energy is considered for the model problem with internal variable α

ψ(ε, α) = 12κe

2 + [1 − d(α)] 2µq2 with d(α) = d∞[1 − (1 + ϑα)−ν ] (4)

where κ, µ denote the bulk and the shear modulus and d∞, ϑ and ν damage parameters. Note, that the damagefunction d(α) affects only the isochoric contribution of ψ. The conjugate internal force reads as β = d,α 2µq2. Weconsider the decoupled form of level set function f = f + f and the constant threshold c, i.e. f = β, f = −α d,α andc = 0. Then the internal variable is identified as α = 2µq2. As a consequence the normal direction has the simplerepresentation ∂β f = 1. The inelastic multiplier then coincides with the evolution of the internal variable λ = α.By use of these results the dissipation function can be expressed as φ = ααd,α. The discrete form of potential Wh

obtained by integration of potenial W over a typical time step and has the contributions

Wvol(en+1) = 12κ(e2n+1 − e2n)

Wiso(qn+1) = minγ{2µ [1 − dn+1(γ)] q2n+1 − 2µ [1 − dn] q2n + Φn,n+1(γ)}

}. (5)

with incremental parameter γ. Integration of the dissipation function over time increment [tn, tn+1] yields

Φn,n+1 = dn+1αn+1 − dnαn − d∞[αn+1 − αn − ϑ−1(1 − ν)[(1 + ϑαn+1)1−ν − (1 + ϑαn)1−ν ] (6)

and the update of the internal variable reads αn+1 = 2µq2n+1 if 2µq2n+1 > αn and αn+1 = αn if 2µq2n+1 ≤ αn.

2. Analytical Solution of the Two–Phase Relaxation Analysis

In this section we derive an analytical solution of the two–phase relaxation analysis that bases on the principalshear-value presentation of the isochoric strain tensor. Goal is a reduction of variables in convexification problemin order to obtain a robust and efficient solution algorithm. Application of the principal shear value representation

PAMM · Proc. Appl. Math. Mech. 3, 222–223 (2003) / DOI 10.1002/pamm.200310385

Figure 1: Visualization of micro–structures in the form of first order laminates.

and exploitation of the necessary conditions, i.e. vanishing of first derivatives with respect to laminate orientationslead to a–priori identification of the laminate orientations m = m1 and n = m2 with the principal shear directions.As a consequence, the convexification analysis is restricted to the space of principal shear values. In other wordesrank–1 convexified potential can be expressed as WR1(εn+1) = Wvol(en+1) + WR1(qn+1). WR1 is defined by

WR1(qn+1) = minc

{Wh(qn+1, c)} with Wh = ξWiso(q+) + (1 − ξ)Wiso(q−) (7)

in terms of the two principal shear values q+ = qn+1+(1−ξ)d, q− = qn+1−ξd, volume fraction ξ and micro intensity dand c := [ξ, d]T with C := {c | 0 ≤ ξ ≤ 1, d ≥ 0}. After having determined an appropriate starting value, the Newtoniteration is initialized. For the damage model considered it can be shown that the isochoric intensities q− and q+

are constant denoting the first and last points of the rank–1 convex hull constant during deformation. The solutionalgorithm of the two–phase relaxation analysis reads in two steps: We (i) a priori determine the micro–intensitiesq±, the internal variables α± numerically and (ii) set up a case–distinction scheme depending on the volume fractionξn of the previous time interval and the current micro–intensity qn+1. In order to show the performance of thealgorithm a boundary value problem is considered for four different discretizations where a square specimen with ahole loaded under tension. Development of micro–structures is plotted in Figure 1 and load–displacement curves ofboth relaxed and unrelaxed solutions are visualized in Figure 2 to show mesh–independency of the proposed method.

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 0.02 0.04 0.06 0.08 0.1

non−relaxedsolutions

Displacement u [mm]

Loa

d F

[N

]

0

500

1000

1500

2000

2500

3000

3500

4000

4500

0 0.02 0.04 0.06 0.08 0.1

relaxedsolutions

Displacement u [mm]

Loa

d F

[N

]

a. b.

Figure 2: Load–displacement curves for four different finite–element meshes based on a.) the non–relaxed (non–objective) formulation b.) the proposed relaxation technique.

Acknowledgements

Support for this research was provided by the Deutsche Forschunggemeinschaft (DFG) under grant SFB404/A8 and /C11.

3. References

1 Miehe, C.; Schotte, J.; Lambrecht, M.: Homogenization of Inelastic Materials at Finite Strains Based on IncrementalMinimization Principles: Application to the Texture Analysis of Polycrystals. J. Mech. Phys. Solids 50, 2123–2167, 2002.

2 Lambrecht, M.; Gurses, E.; Miehe, C.: Relaxation Analysis of Material Instabilities in Damage Mechanics Based OnIncremental Convexification Techniquess. COMPLAS VII (2003), E. Onate, D.R.J. Owen (eds.), Barcelona, Spain.

Ercan Gurses, Dr.-Ing Matthias Lambrecht, Prof. Dr.-Ing. habil. Christian Miehe, Institut furMechanik (Bauwesen), Lehrstuhl I, Universitat Stuttgart, Pfaffenwaldring 7, 70550 Stuttgart, Germany

Section 7: Damage and fracture 223