PSI - Issue 42

T. Vandellos et al. / Procedia Structural Integrity 42 (2022) 50–57 C2 - Restricted Author name / Structural Integrity Procedia 00 (2019) 000 – 000

54

5

3.2. Description of the continuum damage model ODM-OxOx The model named ODM-OxOx, based on the Continuum Damage Mechanics, was firstly developed to describe the behavior of woven oxide/oxide ply laminates under static loading. The model, defined at the woven ply scale, is thermodynamically consistent and is relevant to predict the damage and the failure of oxide/oxide composite materials. The formulation of ODM-OxOx, is based on previous works performed at Onera for composites with polymer matrix or ceramic matrix (Marcin (2010), Hemon (2013), Ben Ramdane (2014), Rakotoarisoa (2014), Sally(2018)) but takes into account the specificities of the studied woven oxide/oxide composite material. Under an imposed strain, the CMC behavior presents a first non-linearity due to the matrix damage mechanisms. This type of damage can be in three directions (warp, weft and out-of-plan) and increases until saturation. Then, a second non-linearity occurs, corresponding to the yarn failure in two possible directions (warp and weft). The macroscopic behavior, expressed in Eq. 1, derives directly from the Helmholtz free energy. = : ( − ℎ ) − 0 : with = ( 0 + ∑ + ∑ ) −1 (1) Where is the stress tensor, the effective elastic stiffness tensor taking into account the effects of the matrix damage and the yarn failure mechanisms, the initial elastic stiffness tensor, the total strain tensor, and the thermal strain tensor. In the present work, only the effects of in-plane damage mechanisms ( i.e. in-plane transverse cracking in the matrix and yarn failure) were taken into account. These damages are translated by an increase of the initial elastic compliance with an additional term ( ∑ m ), for the effect of matrix damages, and an additional term (∑ f ), for the effect of yarn failures. Ben Ramdane (2014) demonstrated experimentally that the damages in the oxide/oxide CMC material are mainly oriented by the microstructure. Thus, these additional terms depend on (i) the in-plane matrix damage variables in the warp and weft directions ( dm1 , dm2 ), (ii) the yarn failure variables in the warp and weft directions, under tensile and compressive loading, ( df1t , df1c , df2t , df2c ) and (iii) the corresponding effect tensors, describing the effects of an open crack on the effective stiffness. The scalar damage variables dm1 (warp direction) and dm2 (weft direction) are defined by = ( 1 − (−( 〈√ −√ 0 〉 + √ ) ) ) with i=1,2 and ̇ ≥ 0 (2) Where are the saturation of the damage, in the i-direction ( i.e. warp or weft), which is currently observed for CMC materials, 0 the onsets of damage, and parameters which are linked to the damage evolution laws and the driving forces formulated as These driving forces depend on (i) the components of the initial elastic sti0ffness tensor, (ii) the parameters 16 , 15 , 26 and 24 and (iii) the specific positive strain tensor. The positive strain tensor corresponds to the positive part of the total strain tensor, as proposed by Rakotoarisoa (2014), where all the components are zeros except those inducing damage. The classical Macauley brackets <.> permits to consider the positive value of the term √ − √ 0 , zero otherwise. Moreover, it is important to note that the damage rate is positive in order to ensure the second principle of thermodynamics. { 1 = 1 2 ( 1+ : 1 0 1 : 1+ + 16 6+ : 60 6 : 6+ + 15 5+ : 50 5 : 5+ ) 2 = 1 2 ( 2+ : 20 2 : 2+ + 26 6+ : 60 6 : 6+ + 24 4+ : 40 4 : 4+ ) (3)