NUMERICAL MODELLING OF FIBRE METAL LAMINATES UNDER THERMO-MECHANICAL - PDF document

18 TH INTERNATIONAL CONFERENCE ON COMPOSITE MATERIALS NUMERICAL MODELLING OF FIBRE METAL LAMINATES UNDER THERMO-MECHANICAL LOADINGS M. Hagenbeek 1* , S.R. Turteltaub 2 1 INHolland Composites Lab, INHolland University of Applied Sciences, Delft, The Netherlands, 2 Koiter Institute Delft, Delft University of Technology, Delft, The Netherlands * Corresponding author (michiel.hagenbeek@inholland.nl) Keywords : thermo-mechanical behaviour, solid-like shell element, fibre metal laminates 1 General Introduction therefore the application of Glare in the fuselage A thermo-mechanical finite element model, based on skin means a major improvement in aircraft safety. a solid-like shell element, has been developed. The 3 Numerical modelling with the solid-like shell use of standard continuum elements to model thin- walled structures, such as a fuselage skin, may lead Finite element simulations can be of assistance to investigate thin-walled Glare structures under to problems as they tend to show Poisson-thickness thermo-mechanical loading. In the present paper the locking for high aspect ratios. development of a mesoscopic model is discussed. Therefore a solid-like shell element has been The uncoupled thermo-mechanical 3D- extended to include the temperature field and analysis process of composite structures was shown thermal expansion. The coupled system of equations by Rolfes et al [5], where a shell finite element is solved simultaneously. This numerical model is used to characterise the behaviour of fibre metal model was used throughout. The mechanical part in laminates under thermo-mechanical loadings. their research consists of thermally induced stresses, A bi-material strip subjected to a heat source which are also calculated in transverse direction [6]. is presented as a benchmark test to demonstrate the The use of standard continuum elements to model thin-walled structures, as the fuselage skin, may lead performance of the thermo-mechanical solid-like to problems. They tend to show Poisson-thickness shell element. With a minimum amount of elements locking when their aspect ratios (i.e. the ratio of and a high aspect ratio the results are accurate and in element length over its width) are too high. As a agreement with the analytical solution. result, the elements become overly stiff. 2 The development of fibre metal laminates An alternative method discussed in this The development of fibre metal laminates resulted in paper is the so-called solid-like shell element, which an improved fatigue performance and higher can describe the behaviour of fibre metal laminates residual strength [1, 2]. However, the use of in a fully three-dimensional state [7, 8] and which different constituents also raises new questions can handle failure mechanisms like cracking and especially regarding the thermo-mechanical delamination in connection with interface elements properties. Differences in thermal expansion as shown by Remmers et al [9]. coefficients cause residual stresses after curing of The 8 or 16 external nodes have three the laminate. And in service, when the temperature degrees of freedom in the case of only mechanical can vary between -55 up to 70  C due to solar loading, since only the displacements are radiation and convection, internal stresses can be considered. For the thermo-mechanical solid-like expected as well. For asymmetric lay-ups this will shell element a temperature field and thermal expansion is included. Consequently, each external lead to secondary bending. node has four degrees of freedom, the three On the other hand, the combination of constituents appears to possess unexpectedly good ˆ , ˆ , and ˆ , and the temperature displacements, u u u y x z thermal insulation [3]. This property leads to a ˆ . at the node  relative low temperature on the inside of Glare By adding the temperature degree of panels in burn-through tests [4]. Moreover, the final freedom only at the corner nodes of the sixteen-node burn-through time increases significantly and element eventual numerical instability, due to a

difference in order for mechanical and thermal strain, can be avoided. The coupled system of 1  GL  g  ( G ) . (5) equations is solved simultaneously. ij ij ij 2 4. Constitutive relations Similarly, the thermal expansion strain tensor The stress tensor consists of a mechanical induced   can be written as: part and a thermal expansion part. The mechanical stress tensor can be written as the relation between     ij  G  i j the second Piola-Kirchhoff stress and the Green- G ; (6)  GL Lagrange strain tensor . The thermal stress tensor is written in a similar way with the thermal where   strain tensor . Thus:     G , (7) ij ij   GL  GL     D D , (1) and  is the relative temperature. The thermal GL D where is the tangent stiffness matrix for the expansion in composites is in general orthotropic;  the expansion is different in fibre direction and D Green-Lagrange strains and is the thermal transverse to the fibre direction. expansion matrix, which consists of the thermal    expansion coefficients times the bulk modulus: The thermal strain tensor is a function of the ij virtual temperature field in the element and will be    D 0 0 0 0 0 derived similar to the derivation of the virtual Green 1 11      0 D 0 0 0 0 GL tensor as function of the virtual displacement   2 22 ij    0 0 D 0 0 0  u ˆ as performed by Hashagen [8]. The thermal   3 33 D   , (2) expansion provides the coupling between the 0 0 0 0 0 0   temperature field and the displacement field. Vice   0 0 0 0 0 0 versa, strong deformations could cause thermal heat    0 0 0 0 0 0  in the structure. However in the current derivation   this effect is not considered as it plays a minor role in an aircraft structure. The Green-Lagrange strain tensor is conventionally For the 16 node element only at the 8 corner nodes written in terms of the deformation gradient F : the temperature is included. For the displacement field second-order shape functions are thus used and 1  GL  T  for the temperature field first-order functions. In this (F F I ) . (3) 2 way both the mechanical strain, due to mechanical loading, and the thermal strain, due to expansion, are The deformation gradient F can be written as a of the same order. The mechanical strain follows function of the covariant base vector in the deformed from the displacement variation and the thermal g and the contravariant base vector strain follows directly from the temperature configuration i distribution times the thermal expansion coefficient G . This in the undeformed reference configuration i in a given direction. Hence, both the displacements lead to the following expression for the mechanical due to mechanical loading and due to thermal  GL strain tensor : loading have a constant distribution over the element. Same order shape functions for the temperature and displacement field can lead to    G  GL GL i j G ; (4) ij slightly different values at the integration points within one element, and causes a so called where "checkerboard" pattern in the calculation field which