PAMM · Proc. Appl. Math. Mech. 14, 93 – 94 (2014) / DOI 10.1002/pamm.201410034 Calculation of Optimal Bounds on the Probability of Failure of Soft Biological Tissues Daniel Balzani 1, * , Thomas Schmidt 2, ** , and Michael Ortiz 3, *** 1 Dresden University of Technology, Faculty of Engineering, 01069 Dresden, Germany 2 University of Duisburg-Essen, Institute of Mechanics, Department of Civil Engineering, 45141 Essen, Germany 3 Engineering and Applied Sciences Division, California Institute of Technology, Pasadena, CA 91125, USA In this contribution, a methodology for the calculation of optimal bounds on the probability of failure of soft biological tissues is presented. Two potential rupture criteria are considered and an uncertainty quantification method [1] is applied to a virtual experimental data set. The results for both criteria are compared in a finite element example. c  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim 1 Introduction Undesired rupture of soft biological tissues, as it may for example occur during clinical interventions or in aneurysms, is usually accompanied by fatal consequences for the affected patient such as bleeding to excess, heart attack or stroke. Several criteria are proposed in the literature to analyze the tissues’ behavior in terms of rupture events. However, a big challenge is posed by the parametric variability of soft biological materials with respect to different specimens. In this contribution, we propose the application of an uncertainty quantification method [1], in order to compute optimal bounds on the probability of failure (PoF) of the tissue, here computed for a virtual experimental data set. The notion of failure is hereby related to the rupture event. The PoF is defined as the probability, that a mechanical quantity g exceeds its maximum admissible value g max , hence PoF:= P [g>g max ]. The quantity g is evaluated as a response of a constitutive model [2] with damage function [3], that is utilized here to characterize the tissues’ material behavior. 2 Calculation of Failure Bounds and Numerical Example In this contribution, due to lack of real experiments, an artificial experimental data set is created. Therefore, a real cyclic tension test in circumferential direction of the media of a human carotid artery (see [2]) is taken as a basis. In [3] the constitutive model for soft fibrous tissues [2] based on the strain-energy function Ψ=Ψ iso (I 1 ,I 3 )+ 2  a=1 α 1  (1 - D (a) )  κI 1 + ( 1 - 3 2 κ ) I 1 J (a) 4 - J (a) 5  - 2  α2 (1) is adjusted to the real experiment. The above strain energy contains an isotropic contribution Ψ iso and two transversely isotropic terms, wherein the material parameters α 1 ,α 2 ,κ and a damage variable D (a) occur. I 1 = trC, I 3 = detC, J 4 = tr[CM] and J 5 = tr[C 2 M] represent invariants of the right Cauchy-Green tensor C and a structural tensor M accounting for transverse anisotropy. The invariant J (a) 4 = λ 2 fib,a is associated to the stretch λ fib,a of a fiber family a. In the considered boundary value problem, the stretch is equal for both fiber families, hence λ fib,1 = λ fib,2 = λ fib . An analogous relation holds for D (a) . The real experiment from [2] was carried out until rupture of the specimen. For the creation of a set of artificial experiments, we introduce a maximum admissible fiber stretch λ max fib and prescribe a beta distribution for this parameter, see Fig. 1a. Furthermore, a functional dependence α 1 := α 1 (λ fib ) is defined providing, that all experiments within the artificial set fail at about the same stress level, when a cyclic protocol is applied until λ max fib is reached, see Fig. 1b. Next, we analyze rupture criteria formulated in terms of the fiber stretch as well as of the damage variable. For the maximully admissible damage variable D max an associated distribution to the same experimental data base is obtained by performing a Monte-Carlo- Simulation on 10 7 samples of the artificial set, Fig. 1c. Thereby, always the same cyclic loading protocol is applied until the respective λ max fib is reached. At this point the current value of the damage variable is stored and interpreted as the respective maximum admissible damage value. The motivation for this virtual experimental data set is that by prescribing a specific distribution function of some material parameters the resulting PoF can be calculated and compared with the optimal bounds which are computed for a reduced data set. It is emphasized that for the real problem only such reduced information will be available. For the computation of optimal bounds on the probability of failure, the uncertainty quantification framework [1] is considered. In the latter publication, a reduction theorem is derived, which enables the representation of a failure probability as a finite convex combination of Dirac masses as μ[g>g max ]= n  i=0 α i χ(g max i ) with n  i=0 α i =1 and χ(g max i )=  1 if g>g max i , 0 else . (2) ∗ e-mail daniel.balzani@uni-due.de, phone +49 (0)201 183 3603 , fax +49 (0)201 183 2680 ∗∗ e-mail t.schmidt@uni-due.de, phone +49 (0)201 183 2674, fax +49 (0)201 183 2680 ∗∗∗ e-mail ortiz@aero.caltech.edu, phone +1 (626) 395 4530, fax +1 (626) 449 6359 c  2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim