A CONTINUUM DAMAGE MODEL FOR COMPOSITE LAMINATED STRUCTURES - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS A CONTINUUM DAMAGE MODEL FOR COMPOSITE LAMINATED STRUCTURES SUBMITTED TO STATIC AND FATIGUE LOADINGS P. Nimdum 1* , J. Renard 1* 1 Mines-Paris Tech, CNRS UMR 7633, BP 87, F-91003 Evry, Cedex * Corresponding authors ( pongsak.nimdum@ensmp.fr, jacques.renard@ensmp.fr ) Keywords : thick composite, damage model, matrix cracking, delamination onset coupled behaviour model has been proposed based Abstract on an understanding of physical mechanisms [1-3]. This study is mainly focused on two major types of As this type of damage is not so critical for the damage: intra-laminar ply cracking and inter-ply structure, we consider its possible evolution during delamination in fiber-reinforced composite under calculation and prediction of life-time. We can then static and fatigue loading. Off-axis and angle-ply simulate the influence of ply cracking on the non woven laminate have been used to study matrix mechanical properties of the structure, (ii) cracking. Damage evolution based on continuum considering delamination an onset criterion has been damage mechanics is considered for prediction intra- proposed [4]. As this damage type could be critical laminar ply cracking. While, angle-ply woven for a structure, the objective in this case is rather to laminate is used to study the delamination. The design a structure by avoiding delamination. delamination onset criterion based on average stress The objective in this study is to investigate the has been proposed. Identification of the different matrix cracking using damage mechanics and to parameters of damage evolution model of matrix predict initiation of delamination in woven and non cracking and delamination onset has been made with woven laminate during static and fatigue loading. classical rectangular specimen. Validation was made 2. Experimental procedure with static and fatigue tests performed on non woven and woven laminates with drilled circular hole. The Static and fatigue tension-tension loading were numerical predictions are in good agreement with performed in woven and non-woven carbon fiber- experimental results. reinforced epoxy laminates. In fatigue case, the σ different maximum applied stress, , which are 1. Introduction max σ ) is applied. less than the ultimate tensile stress ( R Thick composite laminated structures able to support The load ratio, R , and the frequency (f) is 0.1 and 1 significant efforts, are more and more used for Hz, respectively. All tests are performed at room engineering structural parts. Then it is necessary to temperature. The off-axis and angle-ply non-woven consider the ability of such laminates to resist from laminates ((0° 3 ,90° n ) s , (0° 3 ,±45° 3 ) s and (0° 3 ,±55° 3 ) s ) damage development, the consequence of which is have been tested in order to study different initiation mechanical degradation of properties (stiffness and evolution of mode of matrix cracking. Whereas, decrease). Thus, it requires appropriate design tools the angle-ply woven laminate ((0°,±20°) s , to prevent from damage evolution and predict the (0°,±20° 2 ) s , (0°,±30°) s and (0°, ±30° 2 ) s ) is to study influence of damage on mechanical properties . the delamination onset. Damage mechanisms up to failure are rather 3. Damage mechanism complex in composites laminates. One reason is that several damage phenomena (matrix cracking, 3.1 Matrix cracking delamination, fiber breaking, fiber/matrix interface debonding …) are acting alone or coupled. In the similar way to the previous study, the These last remarks explain the different philosophy experimental results show that matrix cracking is a associated with these two types of damage: (i) diffuse damage on off-axis laminate (Fig.1). Crack considering ply and matrix cracking, a damage density evolution is strongly dependent on

maximum applied stress. However, for a given stacking sequence, the difference of maximum applied stress level gives the same saturation density. We found that a good correlation is observed between static and fatigue loading for matrix damage. Fig.2. Stiffness degradation as a function of number of cycles In order to propose the criterion for prediction the onset of delamination under fatigue loading, the (a) σ relationship of , onset of delamination stress max σ under tensile loading test ( ) and the number of onset cycles onset ( N ) is investigated as shown in Fig.3. a This nonlinear relation can be expressed as: σ ( ) ( ) = − − K 2 N 1 max K (1) σ 1 R K are�two�constant�parameters,�these� where K and 1 2 parameters� are� independent� on� i.e.� the� stacking� (b) sequence�and�number�of�plies�(thickness)�but�depend� on�the�composite�material�study. Fig.1. (a) Transverse crack and (b) crack evolution density in (0° 3 ,90° n ) s laminate under cyclic loadings. 3.2 Delamination onset We now consider the case of the woven angle-ply laminates. The experimental results illustrate that the delaminations are not straight (plan) but bended. After the interlaminar delamination appeared, we investigate on stiffness degradation and find that the modulus decrease can be divided into three stages (Fig.2): (i) initial region (stage I) with a slightly decrease stiffness reduction of about 2.5%, and then, (ii) an intermediate region (stage II), in which an Fig.3. A number of cycles to delamination onset additional about 10% stiffness reduction occurs in an ( N ) in different stacking sequences a approximately linear fashion and (iii) the final 4. Numerical region (stage III) with a rapid decrease of stiffness (about 40%), and then the stiffness reduction 4.1 Damage mechanic for matrix cracking become unstable and lead to the final failure of specimen. In the same manner with matrix cracking, In this study, we choose to develop a damage model the delamination mechanism is a good correlation within the framework of the Damage Mechanics. between static tensile and fatigue loading. The model is written at the scale of the ply and describes the multiplication of cracks and loss of

A CONTINUUM DAMAGE MODEL FOR COMPOSITE LAMINATED STRUCTURES SUBMITTED TO STATIC AND FATIGUE LOADINGS rigidity caused by damage. The ply cracking is In fatigue problem, we assume that the damage described by two internal variables: the first one is a phenomenon is physically and geometrically scalar which characterizes the damage state of the identical under quasi-static. Then, the damage materials, f . The second one is a vector which evolution law (Eq. (6)) can be rewritten to take into describes the direction of damage, U ,[3,5]: account both quasi-static and fatigue loading case as in equation (7). α     V ( , m , r ) U ( m , r ) �  T   T  = α = α ×     V V ( , m , r ) f ( ) U ( m , r ) (2) N N         0 0 In the quasi-static loading, the free energy state function ψ is written in order to describe the damage evolution law: � ψ = ψ ε α = ϕ ε + ϕ ε 0 T ( , V ( , m , r )) ( ) ( , V ) T (3) + ϕ ε + ϕ ε N NT ( , V ) ( , V , V ) N N T with, A is dual variable of α : � ∂ ψ ε α ( , V ( , m , r )) = (4) A ∂ α (a) And threshold in static loading is written as: = α − ε α ≤ C c A ( , m , r ) A ( , , m , r ) 0 (5) Finally, we can simplify the damage evolution law in static case as:   ∂ ψ ∂ 2 c ∂ ψ A 2   − − ∂ ε dm   ∂ α ∂ ∂ ∂ ε ∂ α   m m α = + d ∂ ψ ∂ ∂ ψ ∂ 2 c 2 c A A − − ∂ α ∂ α ∂ α ∂ α 2 2 (6)   ∂ ψ ∂ 2 c A   − dr   ∂ α ∂ ∂   r r + ∂ ψ ∂ 2 c A − (b) ∂ α ∂ α 2 Fig.4. Numerical results for identification of static ∂ ψ ∂ 2 c A − ∂ ε (a) and fatigue loading (b) dN ∂ ε ∂ α ∂ N α = + d ∂ ψ ∂ ∂ ψ ∂ 2 c 2 c A A − − As the previous study [3], we suppose that m has a ∂ α ∂ α ∂ α ∂ α 2 2 more significant influence than r on the damage (7)  ∂ ψ ∂   ∂ ψ ∂  2 c 2 c C A A evolution. As a result, the critical threshold, A , is  −   −    dm   dr ∂ α ∂ ∂ ∂ α ∂ ∂ written as:     m m r r + + ∂ ψ ∂ ∂ ψ ∂ 2 c 2 c A A − − ∂ α ∂ α ∂ α ∂ α 2 2 3