Issue 10

G. Bolzon et alii, Frattura ed Integrità Strutturale, 10 (2009) 56-63; DOI: 10.3221/IGF-ESIS.10.07 58 2 1 ( ) ( ) N Ci Mi i i d d w            z z (1) where i w represent weights on the displacement components Mi d , assumed to be proportional to the variances which quantifies the measurement discrepancies, also visualised in Fig. 1. The optimum value of the sought parameters is represented by the entries of vector opt z , which corresponds to the minimum discrepancy. These values can be returned by a number of numerical methods implemented in widely available optimisation tools [22]. In this kind of application, the discrepancy function ( )  z is expected to be non-convex and to admit multiple minimum points, often almost equivalent from the engineering point of view, in the sense that the corresponding parameter values approximate to the same extent the available experimental data and return comparable representation of the real material behaviour, consistently with the selected constitutive model. The optimisation algorithm is hence run several times, starting from different initial parameter sets or, alternatively, evolutionary search techniques like genetic algorithms or soft computing methodologies are exploited; see, e.g., the review paper [20]. The computational efficiency of the simulation tool represents, hence, an important issue in this context. B ULK M ATERIAL P ROPERTIES ccording to a popular approach developed by Tamura et al. [23] f or metal alloys and applied to different composite systems by a proper calibration of a characteristic ‘stress transfer’ parameter, the overall mechanical response of metal-ceramic composites is governed by the metal phase. Pressure-insensitive elastic-plastic laws, like the classical Hencky–Huber–Mises (HHM) model, are therefore mostly considered at the macro-scale [24-26] and supplemented by constitutive parameters (elastic modulus, yield limit, hardening coefficients), evaluated on the basis of some mixture theory that depends on volume fractions. In the proposal by Bocciarelli et al. [15], volume fractions govern the transition from HHM model toward Drucker–Prager (DP) constitutive law, capable to describe the mechanical behaviour of ceramics [14]. The elastic domain defined by the traditional DP yield criterion with linear isotropic hardening is represented by: 1 1 ' ' 0 2 ij ij f I k h        (2) where: ' ij  denote the components of the deviatoric stress tensor; 1 I represents the first stress invariant, i.e. the trace of the tensor collecting the stress components ij  ; λ (>0) is the cumulative multiplier of the plastic deformations, which develop as 0 f  and 0 f   (the superimposed dot denotes a rate quantity); α, k and h are constitutive parameters. Internal friction α and initial cohesion k depend on the initial tensile and compressive yield limits, 0 t  and 0 c  , respectively, as follows:     0 0 0 0 0 0 0 0 2 3 , 3 c t c t c t c t k           (3) while parameter h governs material hardening. HHM criterion can be recovered from DP model as the value of the internal friction coefficient α is set equal to zero. The plastic rate components p ij   of the strain tensor are assumed to develop orthogonally to a potential surface ( ) ij g  as follows: 1 1 , ( ) ' ' 2 p ij ij ij ij ij g g I            (4) where β represents the dilatancy coefficient. A