Mean stress correction and fatigue failure criteria for hyperelastic adhesive joints

ABSTRACT

The work investigates failure criteria and mean stress correction approaches for the fatigue lifetime prediction of two hyperelastic adhesives (a polyurethane, PU, and a silicon-modified polymer, SMP). Fatigue experiments are carried under constant amplitude cyclic loading at RT and 40°C/60% r.h with butt- and the thick-adherend-shear-test-joints at three stress ratios R = −1, 0.1 and 0.5. Three mean stress correction approaches are evaluated: Goodman (static strength based), Schütz (mean stress sensitivity based) and Kujawski & Ellyin (parameter optimisation based). Fatigue failure criteria considered are: Drucker-Prager (linear-relation with the hydrostatic stress), Beltrami (quadratic relationship with the hydrostatic stress), and a multivariable nominal shear and tensile stresses criterion (data-based). The comparison is based on prediction accuracy (R-squared of master SN curves) and complexity of parameter determination. The highest values of R-squared were obtained by the Kujawski and Ellyin correction, followed by Schütz and then Goodman. However, the complexity of parameter determination follows an opposite trend with Goodman being the lightest approach. Failure criteria yielded comparable results with the multivariable criterion having the advantage of not dealing with FEA, but being limited to joints with nearly uniform stress distribution. Finally, compared to Drucker-Prager, the Beltrami criterion had a more robust parameter determination.

KEYWORDS:

• Hyperelastic adhesives
• polyurethane
• SMP
• fatigue
• mean stress correction
• hydrostatic stress

Nomenclature

Roman Language	=
$\mathit{A}$	=	bonding surface
$\mathit{b}$	=	material parameter for failure criterion
BJ	=	butt-joint
${\mathit{d}}_{\mathit{a}\mathit{d}\mathit{h}}$	=	adhesive layer thickness
F	=	force
E	=	tensile modulus
${\mathit{L}}_{\mathit{a}\mathit{d}\mathit{h}}$	=	overlap length
${\mathit{I}}_1$	=	first invariant of the principal stress tensor
${\mathit{J}}_2$	=	second invariant of the deviatoric stress tensor
${\mathit{L}}_{\mathit{a}\mathit{d}\mathit{h}}$	=	overlap length
$\mathit{K}$	=	bulk modulus
${\mathit{k}}_{\mathit{w}}$	=	parameter related to the slope of the SN curve
${\mathit{M}}_{\mathit{\sigma }}/{\mathit{M}}_{\mathit{\tau }}$	=	mean stress sensitivity
${\mathit{N}}_{\mathit{e}\mathit{x}\mathit{p}}$	=	experimentally obtained number of cycles to failure
${\mathit{N}}_{\mathit{p}\mathit{r}\mathit{e}\mathit{d}}$	=	predicted number of cycles to failure
${\mathit{n}}_{\mathit{\sigma }}$/${\mathit{n}}_{\mathit{\tau }}$	=	material parameter for Kujawsky and Ellyin correction
${\mathit{p}}_{\mathit{\sigma }}/{\mathit{p}}_{\mathit{\tau }}$	=	material parameter for Kujawsky and Ellyin correction
R	=	stress ratio
${\mathit{R}}_{\mathit{m}}^{\hspace{0.17em}}$	=	static strength
R2	=	R-Squared (coefficient of determination)
r.h	=	relative humidity
RT	=	room temperature
TAST	=	thick adherend shear test
Greek Language	=
$\mathit{\alpha }$	=	intercept of multivariable model
${\mathit{\beta }}_1,\hspace{0.17em}{\mathit{\beta }}_2$	=	coefficients of multivariable model
$\mathit{\sigma }$	=	tensile stress
${\mathit{\sigma }}_{\mathit{a}}$	=	tensile stress amplitude
${\mathit{\sigma }}_{\mathit{m}}$	=	tensile mean stress
${\mathit{\sigma }}_{\mathit{a}}^{\ast }$	=	transformed tensile stress amplitude
${\mathit{\sigma }}_{\mathit{n}\mathit{o}\mathit{m}}^{\hspace{0.17em}}$	=	nominal tensile stress
${\mathit{\sigma }}_{\mathit{B}\mathit{E}}$	=	Beltrami failure criterion
${\mathit{\sigma }}_{\mathit{D}\mathit{P}}$	=	Drucker-Prager failure criterion
${\mathit{\sigma }}_{\mathit{H}}$	=	hydrostatic stress
${\mathit{\sigma }}_{\mathit{V}\mathit{M}}$	=	Von Mises stress
$\mathit{\tau }$	=	shear stress
${\mathit{\tau }}_{\mathit{a}}$	=	shear stress amplitude
${\mathit{\tau }}_{\mathit{m}}$	=	shear mean stress
${\mathit{\tau }}_{\mathit{a}}^{\mathbf{\ast }}$	=	transformed shear stress amplitude
${\mathit{\tau }}_{\mathit{n}\mathit{o}\mathit{m}}^{\hspace{0.17em}}$	=	nominal shear stress
$\nu$	=	Poisson’s ratio

Acknowledgments

The IGF project No. 20655 N “Nachweisführung für die Beanspruchbarkeit von hyperelastischen Klebverbindungen unter betriebsrelevanten Bedingungen II” of the Research Association for Welding and Allied Processes of DVS was funded by the AiF within the framework of the programme for the promotion of Industrial Collective Research (IGF) of the Federal Ministry for Economic Affairs and Climate Action BMWK on the basis of a resolution of the German Bundestag. V.C. Beber acknowledges the funding from CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) through the Science without Borders program under the grant BEX 13458/13-2.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The work was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior [BEX13458/13-2]; Federal Ministry of Economic Affairs and Climate Action BMWK [IGF-Project No. 20655 N].