efficient modelling of complex adhesively bonded structures

127 Pages·2010·5.65 MB·English

University of Modena e Reggio Emilia

E
FFICIENT MODELLING OF COMPLEX
ADHESIVELY BONDED STRUCTURES BY

STANDARD FINITE ELEMENT TECHNIQUES

A Dissertation
Submitted to the PhD school of High Mechanics and Automotive
Design and Technology, Modelling and Mechanical Design Methods

for the Degree of Doctor of philosophy
by
Ing. Andrea Spaggiari

Tutor: Prof. Antonio Strozzi
Co-Tutor Prof. Eugenio Dragoni
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

2 Andrea Spaggiari
Research is to see what everybody else has seen, and to think

what nobody else has thought

Albert Szent-Györgyi

There is nothing like looking, if you want to find something. You

certainly usually find something, if you look, but it is not always

quite the something you were after.

J.R.R. Tolkien

For a successful technology, reality must take precedence over

public relations, for Nature cannot be fooled.

Richard P. Feynman

The important thing is not to stop questioning. Curiosity has its
own reason for existing.
Albert Einstein
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

4 Andrea Spaggiari
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

1. SUMMARY

This thesis describes the method and the results of my research in the adhesively bonded
constructions. The main objectives of my research were the development of methodology for the
efficient analysis of the bonded constructions and the validation of the proposed method.
In particular there is a lack of simple design tool which guide the adhesive joint designer in using
Finite Element Method, especially for the industrial world. These limitations can be overcome
with the proposed Tied Mesh (TM) method, which is the core of my thesis. The TM method relies
on structural elements to model the adherends and on solid elements for the adhesive modelling,
linked with kinematic constraints (the Tied Mesh). These elements are present in the vast
majority of the commercial FE software. The proposed method was extensively analyzed and
compared with other solution retrieved in literature: analytical, numerical and experimental
methods. Each of this approaches imply some drawbacks. The analytical approach can analyze
only 2D simple joints and implies a lot of approximation, the numerical solution via full FE model
(FM) draws to complex model, long time of analysis and often cannot solve complex 3D
structures, the experimental test are time consuming and expensive. The TM method overcomes
these problems combining efficiency and accuracy.
The TM method is compared with the other solutions in terms of stresses, displacement, stiffness,
peak force and deformation energy. The elastic analysis shows excellent prediction of the TM
model both versus the analytical methods and FM, with a speed up ratio of 50 times. The post
elastic analysis show a good prediction of TM in terms of force displacement curves both against
FM and experimental. These analyses show the capability of the TM method to support both
stress based criteria for the failure of the adhesive and cohesive zone model approach. The TM
method in the post elastic field and in 3D analyses may be 2000 times faster than the full FE
model. The results of the TM method were presented during the three years of my PhD in many
international congresses and accepted as papers both in national and international journals.
The overall evaluation of the TM method is that this simplified approach is able to obtain high
numerical efficiency and good results compared with the other methods, and when the
complexity of the problems increases it became the only valid option for the analysis of the
bonded structures.
Andrea Spaggiari 5
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

6 Andrea Spaggiari
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

2. SOMMARIO
Il presente lavoro di tesi riporta la sintesi dei risultati della ricerca sulle problematiche delle
costruzioni incollate affrontate durante il periodo di dottorato. I principali obiettivi del lavoro
sono la ricerca e lo sviluppo di una metodologia efficiente ed efficace per l’analisi delle strutture
incollate e la verifica del metodo proposto.
Si intende colmare la mancanza di strumenti progettuali per le costruzioni incollate in campo
industriale, ciò, infatti, limita l’utilizzo degli adesivi strutturali come metodo di giunzione leggero
ed efficace. In particolare lo strumento principe per l’analisi strutturale, cioè il metodo degli
elementi finiti (FEM), presenta problemi di implementazione che ne rendono difficile
l’applicabilità nelle strutture incollate.
Queste limitazioni possono essere superate adottando il metodo proposto, denominato “Tied
Mesh”. Questo metodo è basato sull’utilizzo di elementi strutturali per la modellazione degli
aderendi e su elementi solidi per la modellazione dell’adesivo, uniti uno con l’altro per mezzo di
vincoli cinematici (in inglese appunto Tied Mesh). Questi elementi sono normalmente già presenti
nei programmi FEM più diffusi.
Il metodo proposto è confrontato con diverse soluzioni: con modelli analitici di letteratura, con il
metodo agli elementi finiti classico e con prove sperimentali. Ognuno di questi presenta alcuni
difetti peculiari: l’approccio analitico è applicabile solo a semplici geometrie bidimensionali e
implica alcune approssimazioni, il metodo FEM classico porta modelli computazionalmente molto
pesanti, con tempi di calcolo lunghi ed è spesso inapplicabile a strutture 3D mentre le prove
sperimentali sono costose e complesse da gestire.
Il metodo Tied Mesh si propone di risolvere questi problemi combinando efficienza di calcolo e
accuratezza dei risultati. Il confronto con le altre soluzioni è fatto in termini di tensioni,
spostamenti, rigidezza, forza massima sopportabile dalla struttura ed energia di deformazione.
L’analisi elastica mostra una previsione eccellente del metodo Tied Mesh sia in confronto a
modelli analitici sia rispetto al FEM classico, con tempi di analisi mediamente 50 volte più brevi.
L’analisi numerica al di là del limite elastico evidenzia che il confronto in termini di curve forza-
spostamento è buono sia rispetto alle prove sperimentali che al metodo FEM classico. Il metodo
Tied Mesh in campo post-elastico se applicato ad analisi 3D arriva però ad essere anche 2000
volte più veloce del metodo FEM puro. I risultati mostrano anche la versatilità del metodo
proposto di supportare sia criteri di cedimento alle tensioni sia basati su approcci energetici come
il modello di zona coesiva, non essendo legato ad alcuna specifica implementazione, ma basato su
elementi standard.
Andrea Spaggiari 7
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

Il metodo proposto ed i risultati ottenuti sono stati presentati durante tutto il periodo di
dottorato in diversi congressi nazionali ed internazionali e pubblicati su libri e riviste
internazionali.
La valutazione complessiva del metodo Tied Mesh è molto positiva. Questo approccio semplificato
è in grado di ottenere grande efficienza di calcolo e risultati accurati e può diventare l’unico
metodo numerico applicabile in caso di analisi di complesse strutture incollate tridimensionali.
8 Andrea Spaggiari
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

CONTENTS
1. SUMMARY .......................................................................................................................................... 5
2. SOMMARIO ........................................................................................................................................ 7
3. Introduction ..................................................................................................................................... 13
3.1 Motivation ..................................................................................................................................... 13
3.2 Structure of the thesis .................................................................................................................... 14
3.3 Original contribution of the author ................................................................................................ 15
3.3.1 Papers published on International Journals .............................................................................. 15
3.3.2 Papers published on National Books ......................................................................................... 15
3.3.3 Papers presented at International Congress ............................................................................. 15
3.3.4 Papers presented at National Congress .................................................................................... 16
4. Literature review .............................................................................................................................. 17
4.1 Stress analysis of bonded joints ..................................................................................................... 17
4.1.1 Analytical methods .................................................................................................................... 18
4.1.1.1 Volkersen shear analysis ................................................................................................... 18
4.1.1.2 Goland and Reissner ......................................................................................................... 20
4.1.1.3 Hart-Smith ......................................................................................................................... 23
4.1.1.4 Bigwood and Crocombe .................................................................................................... 24
4.1.2 Numerical methods ................................................................................................................... 26
4.1.2.1 Classic finite elements methods ....................................................................................... 26
4.1.2.2 Special purpose 2D elements............................................................................................ 26
4.1.2.3 Special purpose 3D elements............................................................................................ 29
4.2 Failure criteria for bonded joints .................................................................................................... 31
4.2.1 Stress based criteria .................................................................................................................. 31
4.2.1.1 Maximum principal stress ................................................................................................. 31
4.2.1.2 Von Mises equivalent stress ............................................................................................. 31
4.2.1.3 Independent mode stresses ............................................................................................. 31
4.2.2 Strain based criteria ................................................................................................................... 32
4.2.2.1 Maximum shear or peel strain criterion ........................................................................... 32
4.2.2.2 Peak maximum principal strain criterion .......................................................................... 32
4.2.3 Energy based criteria ................................................................................................................ 32
4.2.3.1 Maximum plastic energy density criterion ....................................................................... 32
Andrea Spaggiari 9
Doctoral school in High Mechanics and Automotive Design & Technology - Advanced Methods for Mechanical Design

4.2.3.2 Fracture Mechanics ........................................................................................................... 33
4.2.3.3 Cohesive zone model approach ........................................................................................ 33
4.2.4 Concluding remarks .................................................................................................................... 34
5. Development of efficient FE methods for bonded structures ............................................................ 37
5.1 Preliminary finite element analysis ................................................................................................. 38
5.1.1 Full Model .................................................................................................................................. 39
5.1.2 Stiffness correction method ....................................................................................................... 40
5.1.3 Tied mesh method ..................................................................................................................... 41
5.1.4 Preliminary analysis results and discussion................................................................................ 41
5.2 Systematic analyses of a 2D single lap joints.................................................................................. 44
5.2.1 Design of numerical campaign ................................................................................................... 44
5.2.2 Systematic analysis results and discussion ................................................................................. 46
6. Validation of the tied mesh method in the elastic range ................................................................... 51
6.1 Wavy single lap 2D joint ................................................................................................................. 51
6.1.1 Full FE method ............................................................................................................................ 52
6.1.2 Tied mesh method ..................................................................................................................... 52
6.1.3 Wavy single lap joint results and discussion .............................................................................. 53
6.2 Square bracket 3D joints ................................................................................................................. 54
6.2.1 Full FE method ............................................................................................................................ 55
6.2.2 Tied mesh method ..................................................................................................................... 56
6.2.3 3D square bracket results and discussion .................................................................................. 56
6.3 Tubular single lap joint ................................................................................................................... 57
6.3.1 Full FE method ............................................................................................................................ 58
6.3.2 Tied mesh method ..................................................................................................................... 59
6.3.3 Tubular joint results and discussion ........................................................................................... 59
7. Validation of the tied mesh method in the post elastic range ........................................................... 61
7.1 T-peel joint 2D................................................................................................................................. 61
7.1.1 Design of the test campaign ....................................................................................................... 61
7.1.2 Full FE method ............................................................................................................................ 66
7.1.3 Tied mesh method ..................................................................................................................... 66
7.1.4 2D T-peel joint results and discussion ........................................................................................ 67
7.2 Post elastic analysis of a 3D T-peel joint ......................................................................................... 76
7.2.1 Failure criterion of the adhesive ................................................................................................ 77
10 Andrea Spaggiari