

Finite Element Modeling of an Adhesive in a Bonded Joint
A practical finite element approach has been developed to model the adhesive in a bonded joint. Numerical examples have shown good agreement with classical solutions.
The method uses a gap the thickness of the adhesive, two rigid elements, and three zerolength spring elements between coincident nodes. One rigid element stretches from one adherend to a node at the center of the gap, while the second rigid element stretches from the other adherend to a coincident node also at the center of the gap. The spring elements connect the two rigid elements between the coincident nodes. The figure show the layout. For clarity, the coincident nodes are shown separated. No rotational springs are used in this modeling technique.
Forces, stresses, strains, etc., can be recovered directly for the adherend elements. Recovery of spring forces and deformations can in turn be used to determine the stresses and strains in the adhesive.
The various spring stiffness values are determined using the following equations. Note that there are different values depending on whether the spring elements are internal to, along the edge of, or at a corner of the finite element mesh.
A_{el} = Element area
G = Shear modulus of the adhesive
= Adhesive thickness
E = Young's modulus of the adhesive
= Poisson's Ratio
Shear spring constant, k_{s}, in both lateral directions,
Peel spring constant, k_{p}, through the thickness,
where
where
where
Output results for the adhesive can be calculated using the following equations:
where
i = Grid number
x, y = Shear directions
z = Peel direction
= Spring deflection
= Peel strain
= Spring force
= Shear strain
= Peel stress
= Shear stress
I have written two FORTRAN programs which obtain stresses and strains
in bonded joints using the above formulations. For a given bonded
joint model, these programs read the corresponding NASTRAN input and
output files, use the spring forces or deformations to obtain the
adhesive stresses or strains at the midplane grid points, sort the
stresses and strains in descending order, and generate Mathematica
plot files for 3D visualization of the stress and strain fields.
The following files are source codes and must be compiled in order to run them.
 adhsv_strs.f  FORTRAN source code for extracting adhesive stresses
 adhsv_strn.f  FORTRAN source code for extracting adhesive strains
You can view or download a memo, Software Tools for Analysis of Bonded Joints in .pdf format that describes the two programs in detail.
NOTE:
The grids and springs can be numbered in any order. The programs
determine the pairs of grids which are initially coincident, and the
spring triplets which correspond to such grid pairs. In the earlier
attempts at NASA/GSFC to model adhesives in bonded joints, the analyst
had to number the grids and/or springs in specific orders so that the
XYZ spring triplets acting between pairs of initiallycoincident
adhesive grids could be identified. Such ordering of grids and/or
springs can be very timeconsuming. For more information, refer to the
above memorandum.
REFERENCES:
 K.R. Loss and K.T. Keyward, Modeling and Analysis of Peel and Shear
Stresses in Adhesively Bonded Joints, AIAA paper 840913.
 L.J. HartSmith, AdhesiveBonded DoubleLap Joints, Technical Report NASA CR112235, Contract NAS111234, McDonnell Douglas/Douglas Aircraft Co., Jan. 1973.
 L.J. HartSmith, Design Methodology for BondedBolted Composite Joints, Final Technical Report AFWALTR813154, Vol. I, Contract F3361579C3212, McDonnell Douglas/Douglas Aircraft Co., Feb. 1982.
Farhad Tahmasebi, Ph.D.  Farhad.Tahmasebi@nasa.gov
