13th International Conference on Fracture June 16–21, 2013, Beijing, China -3- To determine the overall damage tensor D (Eq. 5), the spectral decomposition concept of stress tensor is adopted (Eqs. 1-4). * iΣ is the ith principal strain and pi the i th corresponding to the unit principal direction of eigenvalue and eigenvector of Σ*. The symbol ⊗ represents the tensor product. The 4th order positive spectral projection tensor + P given in (Eq. 4) is determined by equations (Eq. 2) and (Eq. 3). According to (Eq. 5), the damage is considered to be entirely active when all the eigenvalues are positive in the three principal directions; whereas, it becomes fully passive once the eigenvalues are negative, i.e., depending on the + P configuration. Hence, + P allows verifying naturally the complexity of the damage activation/deactivation phenomenon whatever the applied loading path. The overall rigidity tensor for a damaged material d R and its evolution d R& are defined respectively by equations (Eq. 6) and (Eq. 7), where o R is the classical 4 th order rigidity tensor for an initially isotropic material. As recently proposed [1], the overall stress tensor evolution coupled with damage activation/deactivation phenomenon is mathematically described by (Eq. 8). In (Eq. 9), the second term in the right-hand side depends explicitly on the eigenvectors variation during cyclic loading. Thus, when the loading is applied according to laboratory reference axes, the principal vectors coincide with the latter. In this case, these vectors are constant, i.e., their characteristics vary neither with respect to time, nor according to the deformation. Hence, the second and third terms in the right-hand side of (Eq. 9) vanishes. As a result, (Eq. 8) has the advantage to successfully treat a great number of loading types especially the multiaxial ones. 3. Algorithms of optimization The identification process is to find numerically a set of model coefficients, which correlates the best possible predictions and experimental results. It is based on minimizing the difference between the recorded model response and the given experimental result. Such a difference can never be zero. However, the rule states that when the difference is smaller, the set of coefficients is better. In this work, identification (calibration) of the model parameters is to find a search space where these values should minimize the gap between experimental results and predictions. Solving this problem is realized by minimizing the following function: ∑ = = N n n F P F P 1 ( ) ( ) , (10) ( ) ( ) ∫ − − − = 1 0 exp exp 1 0 1 ( ) t t sim T sim n V V DV V dt t t F P . (11) where, P: Model parameters, N: number of tests, [t0, t1]: time interval of the test n, Vexp-Vsim: difference between observed experiments and their simulations for the test n, D: weighting matrix of the test n. The complexities of search space are the minimum function using radically different methods of resolutions. As a first approximation, the deterministic method is suitable for search in small
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