General formulation of macro-elements for the simulation of multi-layered bonded structures

ABSTRACT

Adhesively bonded joints are often addressed through Finite Element (FE). However, analyses based on FE models are computationally expensive, especially when the number of adherends increases. Simplified approaches are suitable for intensive parametric studies. Firstly, a resolution approach for a 1D-beam simplified model of bonded joint stress analysis under linear elastic material is presented. This approach, named the macro-element (ME) technique, is presented and solved through two different methodologies. Secondly, a new methodology for the formulation of ME stiffness matrices is presented. This methodology offers the ability to easily take into account for the modification of simplifying hypotheses while providing the shape of solutions, which reduced then the computational time. It is illustrated with the 1D-beam ME resolution and compared with the previous ones. Perfect agreement is shown. Thirdly, a 1D-beam multi-layered ME formulation involving various local equilibrium equations and constitutive equations is described. It is able to address the stress analysis of multi-layered structures. It is illustrated on a double lap joint (DLJ) with the presented method.

KEYWORDS:

• Macro-element
• multi-layered
• ordinary differential equation procedure
• analytical models
• stress analysis
• aeronautical

Nomenclature and Units

Aj	=	extensional stiffness (N) of adherend j
Bj	=	extensional and bending coupling stiffness (N.mm) of the adherend j
Dj	=	bending stiffness (N.mm²) of the adherend j
Ea	=	Young’s modulus (MPa) of the adhesive
Ej	=	Young’s modulus (MPa) of the adherend j
Ga	=	Coulomb’s modulus (MPa) of the adhesive
Gj	=	Coulomb’s modulus (MPa) of the adherend j
kI,i	=	peel stiffness (MPa/mm) of the adhesive i
kII,i	=	shear stiffness (MPa/mm) of the adhesive i
kv	=	peel stiffness (MPa/mm) of the spring of the adhesive in the SLJ geometry
ku	=	shear stiffness (MPa/mm) of the spring of the adhesive in the SLJ geometry
kvi	=	peel stiffness (MPa/mm) of the spring of the adhesive i in the DLJ geometry
kui	=	shear stiffness (MPa/mm) of the spring of the adhesive i in the DLJ geometry
K	=	stiffness matrix
U	=	vector of nodal displacements
F	=	vector of nodal forces
C	=	vector of integration constants
Y	=	vector of differential equations solution
S	=	peel stress (MPa) of the adhesive
Si	=	peel stress (MPa) of the adhesive i
T	=	shear stress (MPa) of the adhesive
Ti	=	shear stress (MPa) of the adhesive i
Vj	=	shear force (N) of the adherend j in the y direction
Nj	=	normal force (N) of the adherend j in the x direction
Mj	=	bending moment (N.mm) of the adherend j around the z direction
b	=	width (mm) of the adherends
ej	=	thickness (mm) of the adherend j
tj	=	thickness (mm) of the adhesive j
lj	=	length (mm) of the out-bonded adherend j
Lj	=	length (mm) of the bonded adherend j
uj	=	displacement (mm) of the adherend j in the x direction
vj	=	displacement (mm) of the adherend j in the y direction
θj	=	angular displacement (rad) of the adherend j around the z direction
P(x)	=	characteristic polynomial
λi	=	eigenvalues i
Vi	=	eigen vectors i
P	=	basis change matrix
$\oplus$	=	direct sum
Ji	=	Jordan block i
$\delta$	=	Kronecker delta
det	=	determinant of a matrix
dim	=	dimension of a matrix or vector
ker	=	kernel of a matrix
Re(x)	=	real part of x
Im(x)	=	imaginary part of x
(x,y,z)	=	system of axes
FE	=	Finite Element
ME	=	macro-mlement
ODE	=	ordinary differential equation
SLJ	=	single lap joint
DLJ	=	double lap joint

Acknowledgements

The author affiliated to Sogeti High Tech gratefully acknowledges the engineers and the managers involved in the development of JoSAT (Joint Stress Analysis Tool), which is an internal research program. The authors warmly acknowledge Mr Salah Seddiki² for the supplying of FE predictions.