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MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING

Modelling Simul. Mater. Sci. Eng. 15 (2007) S425–S434

doi:10.1088/0965-0393/15/4/S08

Simulation of laminate composites degradation using mesoscopic non-local damage model and non-local layered shell element Norbert Germain1 , Jacques Besson2 and Fr´ed´eric Feyel1 1 2

ONERA DMSE-LCME, 29 avenue de la Division Leclerc, BP 72, F-92322 Chˆatillon, France ENSMP/ARMINES, CdM, UMR CNRS 7633, BP 87, F-91003 Evry Cedex, France

E-mail: [email protected], [email protected] and [email protected]

Received 29 September 2006, in final form 17 January 2007 Published 30 May 2007 Online at stacks.iop.org/MSMSE/15/S425 Abstract Simulating damage and failure of laminate composites structures often fails when using the standard finite element procedure. The difficulties arise from an uncontrolled mesh dependence caused by damage localization and an increase in computational costs. One of the solutions to the first problem, widely used to predict the failure of metallic materials, consists of using non-local damage constitutive equations. The second difficulty can then be solved using specific finite element formulations, such as shell element, which decrease the number of degrees of freedom. The main contribution of this paper consists of extending these techniques to layered materials such as polymer matrix composites. An extension of the non-local implicit gradient formulation, accounting for anisotropy and stratification, and an original layered shell element, based on a new partition of the unity, are proposed. Finally the efficiency of the resulting numerical scheme is studied by comparing simulation with experimental results. 1. Motivations and aims Recent advances in the mechanics of materials have greatly improved descriptions of heterogeneous materials such as laminate polymer matrix composites (PMCs). New complex constitutive equations sets address the issue of initial and induced anisotropy caused by fibres’ orientation and damage growth. Using these equations, the finite element method makes it possible to predict the evolution of damage in complex structures and then to prevent their failures. However, apart from the computational cost, standard finite element procedures raise a lot of numerical problems. For instance, a strong dependence on the elements size and orientation appears which leads to meaningless results: the benefit of these new constitutive equations is lost. The robustness is not achieved and these models cannot be used in an industrial framework. In this paper, an original finite element procedure is proposed to overcome these difficulties. This one is based on a non-local implicit gradient formulation of the mechanical 0965-0393/07/040425+10$30.00

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Figure 1. Some scales to study composite structures.

problem. This method has been used with success in the case of homogeneous materials. In this work, an extension to laminate composites is proposed in order to account for the induced anisotropy and the stratification (section 3). In section 4, a non-local layered shell element is developed so as to decrease the computational costs. The discretization of the displacement is based on the local formulation of the element. To discretize the non-local fields (continuous layer by layer), a specific partition of the unity is used to account for the discontinuities at the layer boundaries. In the last part of this work (section 5), the scheme is validated by identifying a non-local behaviour (accounting for fibres rupture) and by comparing finite elements and experimental results.

2. Composite layered material: scales and mechanical behaviour In order to simulate the behaviour of composite layered structures, it is first necessary to choose a modelling scale. Three different scales are usually explored: the microscopic (i.e. fibre and matrix, figure 1(a)), the mesoscopic (i.e. layer, figure 1(b)) and the macroscopic (i.e. laminate, figure 1(c)) ones. Working at the macroscopic scale does not allow to take into account layers’ positions and orientations due to the homogenization of the laminate. The microscopic scale is also not well adapted to large structures because it needs important computational resources: it is necessary to explicitly mesh all the fibres and the matrix or to use multi-scale modelling methods such as TFA or FE2 [1]. A good trade-off consists of working at the mesoscopic scale. The layers are modelled as a homogeneous anisotropic material, allowing us to take into account the orientation and position of all constituents. Mechanical constitutive equations sets have been developed to take into account and describe all the main physical phenomena: the progressive rupture of the fibre [2], the decohesion [3], etc. In this work, damages are considered as scalar variables. This formulation is well suited for this kind of materials because the damage is oriented by the material (fibres’ orientation in the layers). In a general framework, three variables D i are defined corresponding to a fibre failure in the first direction and matrix damage in the second and third directions (see figure 2). Damage increase usually leads to a loss of stiffness and to a decrease in the limit load. Despite their physical backgrounds, these constitutive equations lead to serious numerical problems. One of them is damage localization which induces a strong mesh dependence when the finite element method is used to model structures [4]: meaningful results are then difficult to obtain [5]. A solution, to regularize mesh dependence, consists of working in a non-local framework. The mechanical state of a structure point depends not only on its own local history but also on the global structure history. Laminated materials require specific non-local damage models to account for their specific behaviour (anisotropy, layer confinement of the damage).

Mesoscopic non-local damage model and non-local layered shell element

(a)

(c)

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Figure 2. Scalar damages in a layer. (a) Healthy layer (D i = 0), (b) fibre failure (D 1 = 1), (c) in plane matrix failure (D 2 = 1) and (d) out of plane matrix failure (D 3 = 1).

3. Development of a mesosopic anisotropic non-local model The material mechanical behaviour is characterized by so called ‘state variables’. The time evolution of these variables is usually (i.e. in the case of standard local models) expressed as a function of the variables at each material point, x
. In the case of non-local models, the evolution of some of the variables at point x
depends not only on the local state but also on the variables in the neighbourhood of x
. In the following, one non-local variable, fnl , will be used which depends on a local variable fl . In the pioneering works by Pijauder-Cabot and Bazant [6], fnl at x
was defined as a weighted average of fl around x
. However, this technique is difficult to implement in a finite element software; the integral relation can be rewritten using the gradient of fl (explicit formulation) or fnl (implicit formulation). Following the work by Peerlings et al [7], an implicit formulation should be preferred (figure 3):  div(σ ) = 0 in  (a),     2 2  in  (b),  fnl − l ∇ fnl = fl   (σ , fl ) = F I (ε, fnl ) (c), (1)  σ · n
= f
d on f (d),      u
= u
d on u (e),    
∇fnl · n
= 0 on  (f ), where u
is the displacement field, f
d (respectively, u
d ) are the prescribed forces (respectively, displacements) and n
is the normal to the domain boundary. The material constitutive equation (σ , fl ) = F (ε, fnl ) relates the strain tensor ε and the non-local variables fnl to the stress tensor σ and the local variables fl which controls the non-local interaction between materials points. In the case of a laminate material, it is difficult to directly use this formulation with one non-local variable because damage is described by three scalar variables (see above). Each of these variables can induce softening and localization depending on the loading conditions. The non-local regularization must be applied to each of these variables. In this work, each

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Figure 3. Boundary value problem.

Figure 4. Fracture propagation and decohesion between two layers oriented at 0◦ and 90◦ .

damage mechanism is driven by a specific non-local variable fnli so that three scalar non-local variables are defined. This choice leads to a first modification of equations 1(b), (c) and (f )  i f − l i2 ∇ 2 fnli = fli in ; for i = 1 : 3 (b),   nl 3 3 (σ , fl1 , fl2 , fl ) = F I (ε, fnl1 , fnl2 , fnl ) (c), (2)  
i ∇f · n
= 0 on ; for i = 1 : 3(f ). nl

The material constitutive equation (σ , fl1 , fl2 , fl3 ) = F (ε, fnl1 , fnl2 , fnl3 ) relates the strain tensor ε and the non-local variables fnli to the stress tensor σ and the local variables fli . It is now necessary to define three internal lengths l i . This non-local model is well suited to describe the behaviour of one single layer or of a bulk anisotropic material. In a laminate, the interaction between layers is still an open question particularly with respect to the non-local damage variables. In the following, it is assumed that damage remains confined in each layer so that non-local damage variables must be defined for each individual layer (figure 4). From a finite element point of view, two sets of nodal damage variables are defined at all nodes lying on the interface between two layers (as shown in figure 5). From a mechanical point of view, the non-local relations are not written on the whole domain  and the domain boundary  but on the layer domains p and the layer boundaries  p (where p varies from 1 to P + 1 if the laminate have P + 1 layers):  i fnl − l i2 ∇ 2 fnli = fli in p ; for i = 1 : 3; for p = 1 : P (b), (3)
i · n
= 0 ∇f on  p ; for i = 1 : 3; for p = 1 : P (f ). nl

The interaction between layers only results from the continuity of the displacement field at interfaces. To identify the parameters of the mechanical equation set F i and the internal lengths l i , it is necessary to use complex structures exhibiting singularities. It leads to a large number of simulations: the computational cost can be important even if only one finite element is used

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Figure 5. Location of the degrees of freedom.

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node: displacements, non local variables and associated forces Gauss point for the layer i: stress, strain and internal variables

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node: displacements, non local variables and associated forces Gauss point for the layer i: stress, strain and internal variables

(a)

(b)

Figure 6. The local layered finite element. (a) Classical mesh and (b) layered mesh.

in the out of plane direction of each layer. That is why a specific non-local finite element is developed. It formulation is presented in the next section. 4. Development of a non-local layered finite element The use of one element in the thickness of each layer can induce too expensive simulations in the case of a large structure or in the case of a non-local identification procedure. To decrease the computation costs, a solution consists of using only one element in the out of plane direction of the laminate. But according to the discussion in section 2, the laminate cannot be homogenized and one has to explicitly account for the position and the orientation of the layers. To build this element, it is first assumed that the displacement through the laminate thickness is a linear function of the out of plane coordinate, following the classical laminate theory for thin structures [8]. To account explicitly for the position and the orientation of the layers, a specific layer-wise integration rule is used. In this quadrature rule, the number of integration points is proportional to the number of layers and does not differ from the rules used for a layer by layer mesh (with one element in the out of plane direction of each layer, figure 6). The main difficulties, in building this element in a non-local framework, come from the non-local fields. In fact, due to the material confinement of the damage, these fields are discontinuous across the layer boundaries. This discontinuity has to be preserved when the laminate is modelled using one element in the thickness. A mapping can be used for that purpose [9]. In fact, defining 3 unknown fields (fnli with i = 1 : 3) with P discontinuities (which describe the transition between the layers) is equivalent to defining a 3 × (P + 1) ij unknown continuous field (fnl with i = 1 : 3 and j = 1 : P + 1) and P + 1 known Heaviside

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Layer 3 x33 Layer 2 x23

2

Layer 1 x13

1

Figure 7. Position of the interface between the layers.

functions (H j with j = 1 : P + 1, which describe the transition between the layers):   ij  x ) = j =1,P +1 H j (
x )fnl (
x ), (a) f i (
  nl  j 0 for x3 < x3 , j  (b) x) =  H (
1 for else,

(4)

j >1

where x31 is the height of the first layer and x3 is the interface height between the layer j and the layer j + 1 (see figure 7). According to [10], this discretization keeps all the good convergence properties of the classical finite element method. Finally, the number of integration points and non-local degrees of freedom is the same for a layer by layer mesh (with one element in the out of plane direction) and a layered mesh. The benefit only results from the decrease in the number of displacement degrees of freedom. Like the local formulation of the element, the efficiency of the element only results from the linear assumption. If this is well verified, the induced error is likely to remain small. Otherwise, the induced error can be important but it can be decreased using more layer elements over the laminate thickness [11]. For extremely thin elements, this ‘solid shell’ element is certainly not as good as a shell formulation of non-local finite element. However, the strong advantage of this kind of element is that the finite element formulation is still a solid formulation, which can more easily be extended to the large displacements. 5. Applications The application aim is to perform a first comparison between model and experiments. Using the mesoscopic anisotropic non-local model and the layered non-local finite element, a nonlocal mechanical behaviour is identified to predict the limit load of different perforated plates with one, two or four holes (figure 8). Two stack sequences are used: [02 /90/02 ] and [02 /+602 /−602 ]s . The unidirectional layer is a T700M21 carbon-epoxy. A tension loading is applied parallel to the direction of the fibres oriented at 0◦ and classical measures are realized (load versus displacement, gauge and acoustic emission). 5.1. Example of experimental results for the [02 /900.5 ]s plate with one hole The global response of the different plates is similar to the [02 /900.5 ]s plate with one hole (figure 9). In fact, for this structure, the response is divided into three parts: (i) for a

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Figure 8. Geometry of the perforated plates and orientation of the fibres.

displacement lower than 0.6 mm, the behaviour is elastic, (ii) then, a first nonlinearity appears due to the failure of weak fibres and (iii) finally, when fibres cluster breaks at u = 2.72 mm and F = 65.7 kN, the maximum load is reached and the structure breaks. 5.2. Mechanical behaviour According to the experimental results, a specific non-local behaviour is used [2]. Rupture of the different structures is driven by the fibre failure in tension. So, the behaviour is assumed to be elastic in compression and in the matrix directions (numbered 2 and 3), even in tension loading cases. In the fibre direction (1), in the tensile state, the behaviour is damageable. Two damage kinetics are defined, one for the weak fibres3 :  ε11 − εawf +  wf  = Local rupture criterion, f    εbwf Weak fibres : (5)  wf nwf   Dτwf = α wf 1 − e−(fnl )     wf D = maxt