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 Open Access
Influence of plasticity on the fatigue lifetime prediction of adhesively bonded joints using the stresslife approach
 Vinicius Carrillo Beber^{1, 2}Email author,
 Pedro Henrique Evangelista Fernandes^{1},
 Juliana Espada Fragato^{1},
 Bernhard Schneider^{1} and
 Markus Brede^{1}
 Received: 5 January 2016
 Accepted: 22 March 2016
 Published: 4 April 2016
Abstract
The use of adhesive bonding for designing lightweight loadbearing components has increased in recent decades. In this paper the influence of plasticity on the lifetime prediction of bonded joints using the stresslife approach was investigated. The adhesive was a toughened epoxy for structural applications. Stress calculations were performed using finite element analysis. Three material models were employed, a linearelastic model and two elastoplastic models: Von Mises (pressure independent) and Drucker–Prager (pressure dependent). Effective stress was calculated using the theory of critical distances. Lifetime predictions were based on SN curves from literature for scarf and singlelap joints at four different temperatures (−35, −10 °C, RT, +50 °C). The material properties were acquired from uniaxial tensile quasistatic experiments on bulk adhesive specimens. These experiments showed a reduction in the values of Young’s modulus and yield stress with increasing temperature. A model was proposed based on an Arrheniustype equation in order to fit the yield stress as a function of temperature. The model showed good agreement to the experimental findings. Regarding lifetime predictions (a) the influence of critical distance was higher for singlelap joints than scarf joints and (b) the prediction errors were lower for elastoplastic modelling than linearelastic modelling, especially for singlelap joints.
Keywords
 Fatigue
 Finite element analysis
 Plasticity
 Toughened epoxy structural adhesives
 Stresslife
 SN curve
 Lifetime prediction
Background
The use of adhesive bonding in the design of loadbearing components has increased in recent decades due to the advantages of this joining technique. These advantages include uniform load distribution, enhanced fatigue properties and the ability to join dissimilar materials [1]. These characteristics make adhesives very attractive for lightweight applications in a wide range of industries such as the automotive, aerospace and rail sectors [2, 3]. In this context, fatigue is one of the main issues to address when designing bonded joints because cyclic loads occur in almost all engineering structures and may cause failure under loads that are considerably smaller than the quasistatic strength of the materials [4]. The influence of temperature is another important factor to be accounted for in the modelling of components due to its effect on the mechanical properties of adhesives [5]. Marques et al. [6] presented a review of adhesives for low and high temperature applications. They showed the effects of adhesive shrinkage, thermal expansion and viscoelasticity on the final mechanical properties of bonded joints. Beber et al. [7] studied the effect of temperature on the fatigue behaviour of a toughened epoxy adhesive (same as used in the current investigation) at five different temperatures ranging from −35 to +80 °C and found that with increasing temperature the fatigue strength is reduced. Additionally, they modelled parameters of SN curves as a function of temperature using an Arrheniustype equation. Among the different modelling methods available for bonded joints, the stresslife approach is widely applied for fatigue modelling. In this approach, the prediction of the number of cycles to failure (N _{ f }) is made as a function of the stress amplitude (σ _{ a }) with the aid of SN curves, i.e. Woehler plots [4]. Therefore, the stress calculation is an important step for fatigue analysis of bonded joints. The methods for stress calculation can be divided in two main groups: analytical methods (closed formulations) and numerical methods (e.g. finite element analysis, FEA). The second group is more suitable for complex geometries, especially due to advances in computeraided simulation which have reduced the calculation times. This has facilitated the modelling of the variation of geometry, loads and material properties. Da Silva and Campilho [8] presented a review of advances in FEA, pointing out applications of this method for the modelling of bonded joints using continuum, fracture and damage mechanics. In fatigue design, the consideration of effective stress as the maximum (peak) stress often produces overconservative predictions, particularly when dealing with inhomogeneous stress distributions. In order to address this matter, Taylor [9] summarised a group of methodologies called the theory of critical distance (TCD), which take into account a characteristic length (i.e. critical distance) in the assessment of the effective stress. Several authors employed the TCD with success for analysis of specimens under quasistatic and/or cyclic loads and involving a wide variety of materials [10–12].
Schneider et al. [13] estimated the lifetime of scarf and single lap joints of structural adhesives using the stresslife approach. The stress calculations were performed using analytical methods and FEA. They applied the TCD and linearelastic material behaviour. They concluded that the homogeneity of stress distributions in the adhesive layer has an influence on the quality of the lifetime predictions. Frequently, when dealing with joints that present stress peaks, such as singlelap joints, it is possible to reach local stress levels that might cause a plastic response of the material, despite nominal stresses being within the elastic range. The effect of plasticity was already included in early closed formulations from Hart Smith [14] and Crocombe [15]. The consideration of elastoplastic material behaviour can influence calculations in several ways, for example stress peak relief as demonstrated by Hua [16]. Ward [17] gave an extensive review of the yield behaviour of polymers, highlighting the dependence on the hydrostatic component of the stress, in contrast to other materials such as metals. Xu [18] performed FEA calculations using hydrostatic stress dependent elastoplastic material behaviour (Drucker–Prager) to model double lap joints under quasistatic loads. The resulting predictions were more accurate than a hydrostatic independent elastoplastic model (Von Mises).
The objective of the present work is to investigate the effect of plasticity on the fatigue lifetime prediction of bonded joints using the stresslife approach. Stress distributions were calculated using FEA. Three material models were employed, one linearelastic model and two elastoplastic models: Von Mises (pressure independent) and Drucker–Prager (pressure dependent). The effective stress was determined using three different critical distances. For each critical distance, three different methodologies of the TCD were applied. Predictions were based on SN curves obtained from the literature [7] involving scarf and singlelap joints (using the same adhesive used in the current investigation) at four different temperatures (−35 °C, −10 °C, RT, +50 °C] under tensiontension cyclic loading with a stress ratio of R = 0.1. The input material properties for FEA were acquired from quasistatic experiments on bulk adhesive specimens at the same aforementioned temperatures. The adhesive under investigation was a toughened epoxy intended for structural applications. From the experimental findings, a model was proposed based on an Arrheniuslike equation in order to fit the adhesive yield stress as a function of temperature.
This paper is structured as follows; second section describes the methods used in this work: experimental procedures, FEA, material modelling and lifetime prediction procedure. Third section presents the results of the quasistatic tests, the modelling of the adhesive yield stress and the lifetime predictions. Finally, the last section summarises the main conclusions.
Methods
Fabrication of bulk adhesive specimens
Quasistatic experiments
In total 24 tests were performed, i.e. six tests at each temperature (−35 °C, −10 °C, RT, +50 °C). The tensile stress–strain curves were used to determine the linearelastic properties [Young’s modulus (E) and Poisson’s ratio (v)] and plastic properties of the adhesive. The stress–strain curves were approximated using a polynomial equation. Young’s modulus was calculated from the tangent to the curve at the point where deformation started to be plastic. The method of definition of the yield point is described in “Definition of yield stress” section. These properties were employed as input data for FEA calculations of the stress.
FEA—numerical conditions
FEA—material modelling
Linear elastic properties of the adhesive
T (°C)  E (MPa)  ν (−) 

−35  3546.8 ± 140  0.428 ± 0.046 
−10  2538.1 ± 110  0.417 ± 0.024 
RT  1571.9 ± 80  0.402 ± 0.041 
+50  1158.9 ± 90  0.423 ± 0.040 
Definition of yield stress
For glassy polymers, as in the case of the adhesive under investigation, this point of the maximum second derivative is caused by molecular rearrangement and damage at both molecular and macroscopic levels [21].
The procedure to define the yield stress was performed as follows: (i) the engineering stress–strain curve was transformed into a true stress–strain curve; (ii) the experimental measurements (discrete values) were transformed into continuous equations using a polynomial regression; (iii) the yield stress was defined by 3rd derivation using Eq. 1. The yield stress results are presented in “Tensile properties and temperature” section.
In order to model plasticity using the finite element models, a description of plastic behaviour of the materials after yielding was necessary. This was done using a plastic potential function, which assumes that the components of the plastic strain increment tensor are proportional to partial derivatives of the plastic potential, which is a scalar function of stress. In this work, it was assumed that the plastic potential function (g) has the same form as the yield function (f), for both Von Mises and Drucker–Prager modelling, in a socalled associated flow. Hence the direction of increment of plastic strain is the same as the normal vector of the yield surface.
Von Mises yield criterion
Drucker–Prager yield criterion
Lifetime calculation procedure
Variations applied for the predictions
Material model  Normalised critical distance  Methodology 

Linearelastic  L _{ 1 } = 0.03125  PM 
Von mises  L _{ 2 } = 0.0625  LM 
Drucker–Prager  L _{ 3 } = 0.125  Max 
 (a)
A base SN curve was chosen.
 (b)
Base SN curve: the nominal stress amplitudes for lifetimes of N = (10^{3}, 10^{4}, 10^{5}, 10^{6}) were selected.
 (c)
Base SN curve: those nominal stress amplitudes were used as input data for a FEA model of the base SN specimen.
 (d)
Base SN specimen: for each nominal stress amplitude, the stress distribution in the centre of the adhesive layer was calculated.
 (e)
Base SN specimen: a combination of critical distance and TCD methodology was chosen, and from the stress distribution the effective stresses for N = (10^{3}, 10^{4}, 10^{5}, 10^{6}) were determined.
 (f)
The base SN curve was transformed from nominal stress into an effective stress SN curve.
 (g)
Predicted specimen: nominal stress amplitudes for lifetimes of N = (10^{3}, 10^{4}, 10^{5}, 10^{6}) were selected.
 (h)
Predicted specimen: for each nominal stress amplitude, the stress distribution in the centre of the adhesive layer was calculated.
 (i)
Predicted specimen: using the same combination of critical distance and TCD methodology as the base SN curve, the effective stresses were determined.
 (j)
Predicted specimen: the calculated effective stresses were used in the related transformed base SN curve, the predicted number (N _{ calc }) of cycles was determined.
 (k)
N _{ calc } was compared to experimental results from the literature N _{ exp }.
 (l)
The deviation factor (A _{ dev }) was calculated on the basis of N _{ calc } and N _{ exp }.
Results and discussion
Tensile properties and temperature
The proposed model showed good agreement with the experimental results (maximum error of 13 %), which supports the assumption of yielding as a thermally activated process for the adhesive under investigation.
Lifetime calculation
Performed predictions
Pred.  Base  

P45  P56.6  TAST  
P13  X  X  X 
P56  X  –  X 
SLJ  –  X  X 
Predictions of scarf joint (P13)

HSD based predictions had smaller Dev _{ Total } than ISD based ones.

For HSD based predictions the major differences were due to the TCD methodology rather than material modelling and critical distance.

For HSD based predictions Dev _{ Total } was maximum for LM and a minimum for PM.

For ISD based predictions elastoplastic models gave the smallest Dev _{ Total }, especially DP.
Prediction of scarf joint (P56.6)

HSD based predictions had smaller Dev _{ Total } than ISD based ones.

HSD based predictions showed little effect of the chosen critical distance due to the lack of peak stresses.

ISD based predictions: marked effect of the material modelling and critical distance on Dev _{ Total } due to stress inhomogeneity.
Prediction of singlelap joint (SLJ)

There was a bigger difference between different combinations of material modelling, critical distance and TCD methodology when compared to P13 and P56.6 predictions.

The chosen critical distance length had a greater influence when compared to P13 and P56.6 predictions due to the stress inhomogeneity of SLJ specimens.

The best predictions were obtained with elastoplastic models.

PM based predictions gave the highest values of Dev _{ Total }.

VM and DP predictions gave differing results depending on the chosen base SN curve due to different hydrostatic stress states of each base specimen.
Conclusions
In this work the influence of plasticity on the lifetime prediction of bonded joints using the stress life approach was investigated. Stress calculations were performed using FEA. Three material models were employed, a linearelastic model and two elastoplastic models: Von Mises (pressure independent) and Drucker–Prager (pressure dependent). The effective stress was calculated using the theory of critical distances (TCD). Predictions were based on SN curves obtained from the literature for scarf and singlelap joints at four different temperatures (−35 °C, −10 °C, RT, +50 °C) under tensiontension cyclic loading with a stress ratio of R = 0.1. The input material properties for FEA were acquired from quasistatic experiments on bulk adhesive specimens. The adhesive was a toughened epoxy intended for structural applications.
 (a)
The tensile data obtained from bulk adhesive specimens showed a reduction in the value of Young’s modulus and yield stress with increasing temperature.
 (b)
A model was proposed based on an Arrheniustype equation in order to fit the yield stress as a function of temperature; the model showed good agreement (maximum error of 13 %) with experimental findings.
 (c)
The lifetime predictions for scarf joints were better using base SN curves with almost homogeneous stress distributions.
 (d)
The prediction errors were overall lower for elastoplastic modelling than for linearelastic modelling, especially for singlelap joints.
 (e)
This is likely due to stress inhomogeneity, since the critical distance length had a greater influence on predictions for singlelap joints than scarf joints.
Declarations
Authors’ contributions
Authors VCB, PHEF and JEF performed the finite element analysis. Authors BS and MB conducted the quasistatic experiments. Authors VCB and BS modelled the yield stress using Arrheniustype equations. All the authors analysed the lifetime predictions and plasticity effects. All authors read and approved the final manuscript.
Acknowledgements
The authors are grateful to the Science without Borders programme (Ciência sem Fronteiras, V.C. Beber 13458/132, P.H.E. Fernandes 88888.065050/201300, J.E. Fragato 88888.071558/201300,) and the programme for Coordination of Improvement of Higher Education Personnel (CAPES–Brazil).
The IGF project 428 ZN/2 of the Forschungsvereinigung automobiltechnik e.V. (FAT) was funded by the AiF under the program for the promotion of joint industrial research and development (IGF) of the Federal Ministry of Economics and Energy based on a decision of the German Bundestag.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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