Thickness-dependent selection of adhesive joints

ABSTRACT

The fracture resistance of adhesive joints depends on the bondline thickness, especially in the toughest systems. While increasing the adhesive thickness to operate at peak fracture toughness is anticipated to be attractive, it leads to a mass increase and to a decrease of the effective shear stiffness and strength, calling for a trade-off. Here, we follow a rational materials selection approach for mechanical design to address different sets of objectives and constraints representative of different classes of applications. The approach is applied to select the best conventional adhesive based on a novel database. Mechanical analysis with two different degrees of complexity is considered. The elementary level only accounts for the influence of thickness on toughness, while more advanced models introduce the dependence of shear strength on thickness as well as the stiffness mismatch with the adherends. The best adhesive choice strongly depends on the objectives, constraints, and loading configuration. However, the ranking is not drastically affected by the adherends’ stiffness. Gaps found for stiff/strong and tough adhesives motivate the development of architected joints. Inserting stop holes improves damage tolerance and density but reduces the shear strength and stiffness, requiring an optimum design.

KEYWORDS:

• Adhesive joint
• materials selection
• fracture toughness
• strength
• thickness
• architected joints

List of abbreviations and symbols

$a$	=	Half crack length [mm]
$a_c$	=	Half critical crack length [mm]
$a_{edge}$	=	Length of edge crack [mm]
$\alpha$	=	First Dundur’s parameter
$\beta$	=	Second Dundur’s parameter
$C$	=	Cost [€]
$C_a$	=	Mass cost of the adhesive [€/kg]
$C_m$	=	Joint manufacturing cost [€/m3]
D	=	Stop hole length [mm]
$E_a$	=	Adhesive Young’s modulus [GPa]
$E_s$	=	Adherends Young’s modulus [GPa]
$F_{\sigma }$	=	Tensile force [N]
$F_{\tau }$	=	Shear force [N]
$G_{Ic}$	=	Adhesive fracture toughness [J/m2]
$h_s$	=	Adherends thickness [mm]
$K_{Ic}$	=	Critical stress intensity factor[$MPa\sqrt{m}$]
$K_f$	=	Interface corner stress intensity factor[$MPa\sqrt{m}$]
${\kappa }_i$	=	Shear coefficient of material i (a or s)
$L_a$	=	Adhesive length [m]
$L_{s1}$	=	Top adherend length [m]
$L_{s2}$	=	Bottom adherend length [m]
$l_s$	=	Process zone length [mm]
$l_p$	=	Process zone length for architected joints [mm]
$\lambda$	=	Singularity order [-]
${\lambda }_w$	=	Elastic foundation wave number [1/mm]
m	=	Joint mass [kg]
$m_a$	=	Adhesive mass [kg]
$m_{TC}$	=	Objective function: mass and tensile constraint [kg]
$m_{SC1}$	=	Objective function: mass and shear stiffness constraint [kg]
$m_{SC2}$	=	Objective function: mass and shear strength constraint [kg]
${\mu }_a$	=	Adhesive shear modulus [MPa]
${\mu }_s$	=	Adherend shear modulus
${\rho }_a$	=	Adhesive density [kg/m3]
${\rho }_s$	=	Adherend density [kg/m3]
$S$	=	Surface area of the joint [m2]
$S_{archi}$	=	Spacing between stop holes [mm]
$S_{holes}$	=	Surface area occupied by stop holes when orthogonally projected [m2]
${\sigma }_a^f$	=	Adhesive tensile strength [MPa]
${\sigma }^{f\ast }$	=	Effective tensile strength [MPa]
${\sigma }^{\mathrm{\infty }}$	=	Remote tensile stress [MPa]
${\sigma }_{archi}^f$	=	Tensile strength of architected joint [MPa]
${\sigma }_{0a}$	=	Yield strength of the adhesive [MPa]
${\tau }_a^f$	=	Adhesive shear strength [MPa]
${\tau }^{\mathrm{\infty }}$	=	Remote shear stress [MPa]
${\tau }^{f\ast }$	=	Effective shear strength [MPa]
${\tau }_{archi}^f$	=	Shear strength of architected joint [MPa]
$u_{\tau }$	=	Shear displacement [mm]
$u_{\tau }^{max}$	=	Maximum shear displacement [mm]
${\nu }_a$	=	Adhesive Poisson ratio [-]
W	=	Joint width [m]
Y	=	Geometric factor

Acknowledgments

CvI is a FRIA grantee of the Fonds de la Recherche Scientifique de Belgique - FNRS and gratefully acknowledges their support. The authors gratefully acknowledge Frederik Van Loock, Michal K. Budzik, and Norman A. Fleck for interesting discussions and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

All the data used for this work can be found in the supplementary material.

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/00218464.2023.2294131.

Notes

¹ The term “failure” means that at least macroscopic damage is initiated in the adhesive. Under some conditions, unstable crack propagation can take place leading to complete adhesive fracture, while in other cases, the crack may start propagating without complete failure.

² This is an assumption that considers ${\sigma }_a^f$ as the “intrinsic” strength of a bulk adhesive material with no defect, and disregards the crack tip constraint effect (e.g.^[79])

³ When only considering the crack length with respect to the adhesive thickness, Van Loock et al. suggests as simplified analytical treatment ${\sigma }^{f\ast }=\sqrt{\left(E_a^{\ast }t\right)/\left(E_{s}2\pi a\right)}$ which is valid for large ${\sigma }_{peak}$, falling back on one parameter despite of the fracture resistance.

⁴ This limit is set to ensure that the influence of t on $G_{Ic}$ corresponds to the two types of influences shown in Figure 1 to not consider the fact that at very small thickness, a slight increase of $G_{Ic}$ when decreasing t has been sometime reported due to the change of failure mechanism from propagation at the center of the bondline to propagation near the interface and failure by shear yielding.^[10,80,81]

⁵ The process zone denoted by $l_p$ is different from the $l_s$ process zone. The $l_p$ process zone refers to the length corresponding to the zone that extends from the crack tip to the zone where no deformations take place in the adhesive and the DCB arms. For more information, the reader is referred to.^[82]

Additional information

Funding

The work was supported by the Fonds De La Recherche Scientifique – FNRS [1.E.128.21F].