Numerical modelling of contact for low velocity impact damage in composite laminates J. Bonini, F. Collombet, J.L. Lataillade Esplanade des Arts et Metiers, INTRODUCTION A low velocity impact on a composite laminate can cause the appearance of damage invisible to the naked eye. The impact phenomenology shows a succession of multi-scale damage phenomena due to the structural heterogeneity of the material. The numerical modelling of contact and links enables us to represent in a dynamic explicit finite element code - Plexus [1] - two of those phenomena : - the application of impact forces to the structure versus time, essential step before the implementation of a representative local behaviour law. - the delamination, damage factor on a ply scale, which has an effect on the structural behaviour of the composite laminate.
454 Contact Mechanics CONTACT FORCE MODEL Two methods are commonly used to calculate boundary conditions in dynamic finite element codes : the "penalty method" and what we call the "direct method". These methods consist in adding a link force {Flink} in the equation of dynamic to reveal the condition we have to impose on the nodes of the mesh, and to make sure that the equilibrium will be conserved [1]: [M].{u} = {Fext} - {Pint} + {Flink} [M] is the mass matrix, {ti} the acceleration vector, {Fext} and {Fint} the external and internal force vector. We choose the "direct method" which considers the link force as : (Flink}={AFA with X : Lagrange multiplier The {A} vector is defined according to the conditions we want to impose on the nodes. In case of contact between two nodes (1 and 2), we can write for total displacements Ui and U%: Ui = U2 or Ui - Us = 0 or {A} (U}=0 With those hypothesis, we can calculate {Flink} : - {Fint}) CONTACT-IMPACT MODEL We apply this "direct method" to the calculation of impact forces on a macroscopic scale [2]. Our model, based on the impenetrability of the target, enables us to describe two kinds of impacts : a punctual node-on-node impact, and a surface impact by means of the sliding surfaces theory [3]. It falls into two parts : - 1 : the contact is not established : {A} vector is at zero. If the projectile p penetrates the target t during the time step At, a first contact force Fc is calculated from the local conservation of momentum. The nodes of the projectile are linked to the target, (A) vector is not at zero. If we suppose the continuity of contact (V'p=V't), we can write : mv = mv' + nitv't and F, = At At (nip + nit)
Contact Mechanics 455-2 : the contact is established : {A} vector is not at zero. A link force (Flink) is calculated to hinder separation until the contact rupture by using the "direct method". A displacement criterion of the nodes of the target under impact sets off the contact rupture. If the nodes exceed a zero position corresponding with the moment of shock [2], we decide the separation of the projectile (Fig 1) : the coefficients of the (A) vector are put at zero, {Flink} is cancelled. j x Beiore im pact Mp Impact Mp During impact After impact HJ V J 1 Mt 1 1 Mt 1 1 xt FC Flink q=mp.vp q'=mp.v'p+mt.v't xc=xt=xi xp>xt=xo Fig 1 : different steps of the impact projectile-target DELAMINATION MODEL Physical bonding and delamination: A composite laminate is a stacking of unidirectional layers. The process of baking causes the plies to join by matrix curing. The post-mortem observations show some impact damage in the laminate by propagation of mesocracks at the interface of plies of different angles. These cracks are propagated along the direction of the fibres of the lower ply to create a localised unsticking called delamination (Fig 2) [4].
456 Contact Mechanics fiber direction of the upper ply delarnination fiber direction of the lower ply global direction of the laminate Fig 2 : delamination propagation and shape [4] Numerical sticking and delamination: The "direct method" used to determine contact forces is here modelled on the ply scale. It allows a finite three-dimensional node-on-node joining of the laminate (one contact force per node in each space direction). This numerical modelling of sticking is more physical than the classical laminate theory. It allows an effective separation of layers. The separation criterion we have chosen is based on the interlaminar components of contact actions - forces and displacements in the fiber direction of the lower ply [5]. The determination of the maximal force remains arbitrary. interlaminar node Fa = FI.COS a + Fg.sin a < F^ax and
Contact Mechanics 457 This criterion sets off the separation of the interlaminar nodes and the appearance of delamination (Fig 3). The part of the (A) vector corresponding to the link is put at zero. The values of internal and external forces, (Pint) and (Fext), are put at zero for the corresponding nodes to respect the local lack of transmission of the load due to delamination. Flink Each layer is independant Sticking of each layer Delamination : no contact with contact forces force when the criterion is reached Fig 3 : numerical sticking and delamination of interfacial nodes ^ main program ^ Force and displacement criterion for interfacial nodes reached not reached Calculation of contact forces between layers k and k+1 Delamination Separation of interfacial nodes : {A} = 0 No contact force Fig 4 : delamination algorithm
458 Contact Mechanics APPLICATION TO THE IMPACT OF A LAMINATED PLATE Here is an example of the results we have obtained, namely the impact of an hemispherical projectile on the centre of a circular embedded composite plate. Projectile : steel, mass 835 g, diameter 25 mm, length 600 mm, impact velocity 5 m/s. Plate : glass-epoxy [03,904,03] 40% fibers, diameter 160 mm, thickness 1.9 mm. The plate is meshed with 250 fully integrated hexaedric elements to prevent hourglassing phenomena. There is one element in the thickness of each group of ply (Fig 6). During the test, the material is considered as elastic and the projectile as a massive material point. The results show a contact force versus time with a great number of high frequency oscillations (Fig 5). These oscillations, due to the finite numerical modelling, account for the propagation of waves in the thickness and in the plane of the plate. One millisecond after the beginning of the shock, a delamination appears at both interfaces (two delaminated nodes per interface). One delaminated node generates a four square delamination area (Fig 6). The transmission of the load is no longer observed between the layers because of the separation of the nodes. The result is a new distribution of interlaminar forces (Fig 7). Using a high-speed camera to determine the experimental moment of the first delamination would allow us to find the arbitrary value of the maximal force. This model provides a numerical plotting of damage into the laminate. We can measure the thickness and the area of the delamination zone. We record a 25 jam maximal thickness for an area of 4 cm^. A refined mesh linked with some post-mortem measurements on impacted plates would validate the numerical model.
1000-j 0-5 -loooo> u -2000- Contact Mechanics 459 S. -3000- -4000-5000 -1 1 2 3 time (ms) Fig 5 : impact force projectile-target versus time Fig 6 : maximal delamination area in a [63,904,03] glass-epoxy
460 Contact Mechanics -1000-777-555-333-111 111 333 555 777 1000 Forces (N) Fig 7 : distribution of interlaminar forces Fa at the bottom interface before and after delamination CONCLUSION Thanks to the contact forces, we have modelled numerical tools in good agreements with the phenomenological results to represent : - on a macroscopic scale, the impact forces. - on a mesostructural scale, the delamination damage. The implementation of a through ply cracking behaviour law linked with a delamination criterion would allow us to predict the impact damage in a composite laminate [5,6]. Comparaisons with experimental results would validate this model. ACKNOWLEDGEMENTS This work was supported by the Mechanical Analysis of Structure Laboratory of CEA/CEN Saclay and the french Ministery of Research and Space.
Contact Mechanics 461 BIBLIOGRAPHY [1] Verpeaux P., 'Programme Plexus - Formulation theorique et organisation generate - note n l', Rapport CEA EMT/78/60, 1978. [2] Bonini J., Collombet F., Lataillade J.L., 'Caracterisation numerique de la sollicitation d'impact localise transverse basse eneergie sur plaques stratifiees composites', Proceedings of the french 'Colloque National en Calcul de Structures', France, 1993. [3] Galon R., Bung H., Lepareux M., 'Programme Plexus - Algorithme de traitement des surfaces de glissement', Rapport CEA/DEMT/90/092, 1992. [4] Takeda N., 'Experimental studies of the delamination mechanisms in impactedfiber-reinforcedcomposite plates, PhD, University of Florida, 1980. [5] Bonini J, 'Prediction numerique tridimensionelle de I'endommagement de stratifies composites sous choc basse vitesse', These ENSAM Bordeaux, to be published Feb. 1994. [6] Collombet F, Espinosa Ch., 'Simulation of through ply cracking damage history sustained by laminated composite plates during transverse impact', Proceedings of ISIE, Japan, 1992.