18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS COMPARISON OF MODELS FOR THE SIMULATION OF FATIGUE - DRIVEN DELAMINATION USING COHESIVE ELEMENTS K. Kiefer * , P. Robinson & S.T. Pinho Department of Aeronautics, Imperial College London, UK * Corresponding author(K.kiefer09@imperial.ac.uk) Keywords : delamination, high-cycle fatigue, cohesive elements, numerical modeling delamination numerically so as to understand how Abstract delaminations might develop in a particular structural design and how the design could be In this pape,r two formulations for introducing improved to be less susceptible to delamination. One fatigue damage into cohesive elements have been way of doing this in finite element (FE) models is examined to determine their mesh sensitivity and through the use of cohesive (or interface) elements numerical stability in mode I. A simple, non-FE [1] ,which have been used extensively for model has been used to quickly assess a wide range delamination growth under quasi-static loading. of parameters associated with the fatigue More recently cohesive elements have been adapted degradation routines developed for cohesive to also model fatigue-driven delamination [2,4]. In this paper, two of these formulations for introducing elements. The model has been used to explore the fatigue damage into cohesive elements have been sensitivity of two fatigue degradation strategies to examined to determine their mesh sensitivity and changes in key parameters. For small element and numerical stability in mode I. cycle increments, both models predict the fatigue 2 Cylinder Model crack growth rate accurately for various applied loads where the Paris law is valid. However the For the investigations undertaken here, Finite Element Analyses are relatively time-consuming. models exhibit significant mesh sensitivities. The Therefore, a mathematical non-FE model was used analysis has shown that in order to obtain a which was previously developed to enable a more numerically stable algorithm, the models need to be rapid assessment of potential fatigue modeling improved. strategies [3]. 1 Introduction This model, referred to as the cylinder model, consists of two zero-thickness layers of width W . Laminated carbon fibre reinforced polymer The bottom layer is attached to the ground and the composites can fail in a variety of modes. One key top one to a cylinder of sufficiently large radius R failure mode is delamination, in which the layers in (see Figure 1). The interface response between the the laminate separate. This can result in a layers is modelled through initially unstressed considerable loss of structural stiffness and strength. vertical springs with an equidistant spacing Δ l with Delaminations occur at locations in a structure the first spring at x=0 . The role of the springs in this where interlaminar stresses are high, for example at model is to replicate the behaviour of the cohesive free edges, ply drops and corners. These stresses can elements in the FE implementation. The springs can be caused by static loads as well as dynamic ones be associated with a constitutive law and a damage (such as impact) but the case of interest here is model to account for static and fatigue loading. The delamination due to cyclic loading. A particularly springs are assumed to remain vertical as the concerning aspect of delamination is that it often cylinder rotates. When a constant clockwise moment develops without any readily visible external sign M a <M c is applied to the cylinder, it rotates and the and requires the use of non-destructive testing point of contact moves from x=0 initially to a new methods (such as ultrasound-based techniques) to equilibrium position x C where the resistance of the establish the full extent of the damage. It is therefore springs is equal to the applied moment. M c is the of considerable interest to be able to model
COMPARISON OF MODELS FOR SIMULATING FATIGUE-DRIVEN DELAMINATION critical moment at which the springs fail statically, In the static case, the spring force can be calculated so no equilibrium position can be found anymore. as the product of the area that is represented by the spring (Δ lW ) and the traction σ i (δ i ) . An exception is Applying a moment equal to or greater than M c leads to static crack propagation. When the cylinder is in the first spring where the represented area is 1 2 W l equilibrium, the externally applied moment M a must .The total moment that is exerted by the springs is be equal and opposite to the moment M t applied by N 1 S the springs, which are resisting the rotation of the ( ) (4) M l W x l W x x t 1 C i C i cylinder, about the contact point. In case the rotation 2 i 2 θ or contact point x C is known, this moment can be If an external moment M a is applied, the equilibrium calculated in a straightforward way by first position of the cylinder is unknown and has to be calculating the spring extensions and from those the approximated through iteration algorithm. In the spring traction. From the tractions, a spring force fatigue models discussed here, the load is applied in can be calculated and from the forces the moment two steps. Firstly a static moment M a is applied to exerted by each spring. Summing up the moment the cylinder and the equilibrium position x C is found. over all the springs about the contact point then Since the aim is to model high-cycle-fatigue, it is too gives the total moment M t . The x -coordinate of the computationally expensive to model every loading contact point x C of the cylinder for a certain angle θ cycle. The numerically applied load in the second is x C = θR . The number of active springs N S in the step is therefore kept at the constant level M a interface is then throughout the algorithm until the final number of cycles N max has been reached. R N 1 , (1) The degradation of the interface elements is S l controlled by a damage variable D, which is a where x denotes the largest integer smaller than x . function of the displacement of the spring as well as the number of cycles, for each element. In the The springs are numbered from 1 to N S beginning fatigue degradation, the remaining traction σ is with the spring at the origin; x i is the x -coordinate of computed as a function of the damage and the the i th spring (see Figure 2). The extension δ i of the displacement σ(δ,D) . In order for the models to be i th spring then is computationally feasible, in each step a number of x x 1 cycles Δ N are processed and the displacements are C i R R cos(sin ). (2) i R taken as the maximum values. The traction-separation response σ( δ ) is expressed 3 Models through a static constitutive law (see Figure 3). For the fatigue degradation algorithms implemented in 3.1 Peerlings model this paper, the bilinear law was used, which is shown This model was originally developed by Robinson et in Figure 3 and is described analytically as al. [2]. The static constitutive relationship used is the bilinear law mentioned in equation (3). The traction K for 0 0 with respect to the damage is expressed as follows c (1 D K ) , (5) ( ) for , (3) 0 0 c c 0 where the damage is divided into a static and a 0 for c fatigue part, D s and D f respectively for which where K= σ 0 /δ 0 . The parameters for the cylinder as D=D s +D f . The static damage can be written in terms well as the bilinear law are given in Table 1 and are of the displacement using equations (3) and (5) as chosen, so that the area under the cohesive law is 0 0 for equal to G c . 0 D ( ) c 0 for (6) s 0 c Table 1 Cylinder and constitutive law properties c 0 1 for Cylinder R=100 mm W=1 mm c σ 0 = 30 N / mm 2 δ 0 = 1.e-6 mm Constitutive The change in static damage between N cycles and G c = 0.26 N/mm δ c = 0.0173 mm law N+ΔN can then be computed from the respective displacements and equation (6) as 2
Recommend
More recommend
