Lucas Santana Moura
Numerical and experimental analysis
of bonded composites
Análise numérica e experimental
de compósitos colados
CAMPINAS 2019
Numerical and experimental analysis
of bonded composites
Análise numérica e experimental
de compósitos colados
Dissertation presented to the School of Mechanical Engineering of the University of Campinas in partial fulfillment of the requirements for the degree of Master in Mechanical Engineering, in the area of Solid Mechanics and Design.
Dissertação de Mestrado apresentada à Faculdade de Engenharia Mecânica da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestre em Engenharia Mecâ-nica na área de MecâMecâ-nica dos Solidos e Projeto Me-cânico.
Orientador: Prof. Dr. Paulo Sollero
Coorientador: Prof. Dr. Éder Lima de Albuquerque
ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA DISSERTAÇÃO DEFENDIDA PELO ALUNO LUCAS SANTANA MOURA, E ORIEN-TADO PELO PROF. DR. PAULO SOLLERO.
... ASSINATURA DO ORIENTADOR
CAMPINAS 2019
Biblioteca da Área de Engenharia e Arquitetura Luciana Pietrosanto Milla - CRB 8/8120
Santana Moura, Lucas,
1994-M856n Numerical and experimental analysis of bonded composites/ Lucas Santana Moura – Campinas, SP: [s.n.], 2019.
Orientador: Paulo Sollero.
Coorientador: Éder Lima de Albuquerque
Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica.
1. Métodos dos elementos de contorno. 2. Colagem. 3. Fadiga. 4.
Compósitos. I. Sollero, Paulo, 1950-. II. Albuquerque, Éder Lima de, 1972-. III. Universidade Estadual de Campinas. Faculdade de Engenharia Mecânica. IV. Título.
Informações para Biblioteca Digital
Título em outro idioma: Análise numérica e experimental de compósitos colados Palavras-chave em Inglês:
Boundary element methods Bonding
Fatigue Composites
Área de concentração: Mecânica dos Sólidos e Projeto Mecânico Titulação: Mestre em Engenharia Mecânica
Banca Examinadora: Paulo Sollero [Orientador] Luiz Carlos Wrobel Renato Pavanello
Data da defesa: 30-07-2019
Programa de Pós Graduação: Engenharia Mecânica
Identificação e informações acadêmicas do(a) aluno(a) - ORCID do autor: https://orcid.org/0000-0003-3814-116X
FACULDADE DE ENGENHARIA MECÂNICA
COMISSÃO DE PÓS-GRADUAÇÃO EM ENGENHARIA MECÂNICA
DEPARTAMENTO DE MECÂNICA COMPUTACIONAL
DISSERTAÇÃO DE MESTRADO ACADÊMICO
Numerical and experimental analysis
of bonded composites
Análise numérica e experimental
de compósitos colados
Autor: Lucas Santana Moura Orientador: Prof. Dr. Paulo Sollero
Co-orientador: Prof. Dr. Éder Lima de Albuquerque
A Banca Examinadora composta pelos membros abaixo aprovou esta dissertação:
Prof. Dr. Paulo Sollero, Presidente DMC/FEM/UNICAMP
Prof. Dr. Luiz Carlos Wrobel ED&PS/Brunel University London
Prof. Dr. Renato Pavanello DMC/FEM/UNICAMP
A Ata da defesa com as respectivas assinaturas dos membros encontra-se no processo de vida acadêmica do aluno.
To my family for their affection, my mother Salete, my father Cicero, my brother Robson.
To my MSc advisor: Professor Paulo Sollero for his unwavering support and friendship th-roughout this work.
To my MSc co-advisor: Prof. Éder Lima de Albuquerque for all discussions, friendship and dedication since graduation.
To my friend and counselor in numerical simulation and in writing of paper: PhD Andrés Galvis. He is one of the best people I have ever met.
To Professor Pablo Siqueira, Fernando Ortolano and Zé Luis for all discussions, collaboration and useful advice in the experimental tests.
To Isabella and her family for their love, affection and unconditional support.
To my friends from Brasilia Sales, Ciro, Arthur, Brandon, Pantu, João, Rodrigo and Bernardo.
To my Colombian friends Daniel, Edgar, Andrea, Juan Esteban.
To my friends in Barão Geraldo Ana, Marcelo, Bruno, Raimundo, Valter H. F. J. G. Bravim, Douglas, Raquel, Mari.
To Posto Ipiranga Av.1 for the psychological support.
To University of Campinas for the opportunity to carry out this work.
To Air Force Office of Scientific Research (AFOSR) for the financial support.
To the Center for Computational Engineering and Science - CCES/UNICAMP for the support and use of the computational facilities.
To CENAPAD for the support of computational facilities.
This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001
W.S.
The hardest choices require the strongest wills.
Nos últimos anos, o crescimento das indústrias aeronáutica e petrolífera tem elevado a demanda por técnicas mais eficientes de reparo, visando a menores períodos de manutenção, maior durabilidade e confiabilidade, e menor custo. Nesse contexto, reparos compósitos cola-dos têm se mostrado uma excelente ferramenta, uma vez que permitem a restauração rápida e podem ser implementados sem comprometer a integridade estrutural ou qualquer outra função do componente. O objetivo do presente trabalho é a avaliação e a validação de recursos compu-tacionais que permitem analisar a propagação de falhas em estruturas reparadas com compósito colado. Para isso, chapas de alumínio com uma trinca central são analisadas numérica e expe-rimentalmente. Na análise numérica, um estudo inicial da eficiência de algumas geometrias de reparo é realizado. Baseando-se nesses resultados, algumas geometrias são selecionadas para uma análise mais detalhada da evolução da falha. Nessa análise, os campos de deslocamento e tensão são calculados utilizando a formulação elastoestática 3D do método dos elementos de contorno (BEM). Um critério de falha generalizado baseado em densidade de energia de de-formação é considerado tanto para a estrutura danificada, quanto para o adesivo. Ademais, a propagação da falha também é analisada para juntas adesivas de sobreposição simples (SLJ) e cantilever dupla (DCB). Na análise experimental, um ensaio de fadiga é realizado nas estruturas selecionas, e a propagação da trinca é avaliada utilizando a técnica de correlação digital de ima-gem (DIC). O descolamento do reparado é medido utilizando acelerômetros e um vibromêtro laser unidirecional. Quando os resultados numéricos e experimentais são comparados, pode-se notar que a avaliação da falha do adesivo gera uma melhora significativa nos resultados quando comparados com modelos que consideram apenas a propagação da falha na chapa.
Palavras-chave: Reparo compósito colado; junta colada; análise de falha; método dos elemen-tos de contorno.
In recent years, the growth of the aeronautical and oil industries has raised the demand for more efficient repair techniques, aiming at shorter maintenance periods, greater durability and reliability, and lower cost. In this context, bonded composite repairs have proven to be an excellent tool. They allow the restoration of components and can be implemented without com-promising their structural integrity. The aim of this MSc thesis is the assessment and validation of computational resources that allow analyzing the crack propagation in structures with a bon-ded composite repair. For this, aluminum plates with a central crack are analyzed numerically and experimentally. In the numerical analysis, first, a study of the efficiency of some repair ge-ometries is performed. Based on these primary results, some gege-ometries are selected for a more detailed failure propagation analysis. The mechanical behavior of the displacement and stress fields is evaluated via the 3D elastostatic BEM formulation. A generalized failure criterion ba-sed on strain energy density is considered for both the damaged structure and the adhesive. In addition, failure evolution is also assessed for a bonded single lap joint (SLJ) and a bonded double cantilever beam (DCB). In the experimental analysis, a fatigue test is carried out on the selected structures, and the crack propagation is evaluated using the digital image correlation technique (DIC). A unidirectional laser vibrometer is used to measure the repair debonding. By comparing the numerical and experimental results, it is noticed that considering the evaluation of the adhesive failure, results were improved when compared with models that consider only the failure of the plate.
1.1 Basic joint geometries. . . 19
1.2 Composite repair system . . . 20
1.3 Full-field 3D displacement in a landing gear component. . . 21
1.4 Surface characterization using 3D Scanning Vibrometer . . . 22
2.1 Unidirectional fiber-reinforced composite layer . . . 33
2.2 Fabric weaves . . . 35
2.3 Unit cell of a fabric structure. . . 35
3.1 Linear three-node discontinuous element. . . 39
3.2 Multidomain example with a single lap joint. . . 44
3.3 Interfaces. . . 44
3.4 Multidomain analysis with a short beam. . . 46
3.5 3D BEM algorithm. . . 48
3.6 Parallel algorithm of interfaces. . . 49
3.7 Parallel algorithm: matrices H𝒟 and G𝒟 . . . 50
3.8 Skull mesh created in Autodesk 3ds Max. . . 52
4.1 Dimensions of square, elliptical and circular repairs. . . 58
4.2 Taper length schematic. . . 58
4.3 Wedding cake repair a) rectangular and b) elliptical . . . 59
4.4 Energy diagram for the tensile failure of a) Aluminum 2024-T3 and b) Epoxy AR260/AH260. . . 62
4.5 Failure algorithm. . . 63
4.6 Crack tip model . . . 64
4.7 composite plate subjected to uniaxial tension. . . 65
4.8 a) one-sided repairs subjected to membrane tension and bending moment and b) notations and boundary conditions . . . 67
4.9 Geometrically non-linear deformation of a one-sided repair. . . 69
4.10 Deflection of a one-sided repair accounting for geometrically non-linear defor-mation: a) center of overlap, and b) along the joint. . . 71
5.1 Single-lap-joint. a) model and b) manufactured. . . 73
5.2 Aluminum test panel: a) geometry and b) central notch. . . 74
5.3 Image showing the result of the procedure using the razor at the notch end. . . . 75
5.4 Composite repair during the curing process in a vacuum bag . . . 76
5.5 Composite repair during the curing process in a vacuum bag . . . 76
5.6 Result of surface treatment: a) no treatment, b) chemical and c) abrasive. . . 77
5.7 Proof bodies: a) in a vacuum bag and b) in MTS Tensile test machine. . . 78
5.11 Reference image showing the points for calculating the displacements. . . 82
5.12 Laser Doppler Vibrometer applied for debonding measurements. . . 83
5.13 Laser signal: a) outside the reflective tape and b) on the reflective tape. . . 84
5.14 Laser and accelerometer positions in the repair. . . 84
6.1 DCB geometry: 𝑙 = 356 mm, 𝑤 = 25.4 mm, 𝑡 = 12.70 mm, 𝑎 = 51 mm, 𝑏 = 25.4 mm, 𝑐 = 6.35 mm, and 𝑑 = 6.35 mm. . . 85
6.2 Convergence: a) total displacement and b) maximum von Mises stress. . . 86
6.3 DCB total displacement: a) 3D BEM and b) 3D FEM . . . 87
6.4 DCB: a) boundary conditions and b) mesh. . . 87
6.5 Static failure load for DCB . . . 88
6.6 DCB damage evolution: a) 1525 N b) 3575 N c) 4523 N . . . 88
6.7 Single lap joints: a) Al 2024-T3 adherents, b) Al 2024-T3 and carbon-epoxy adherents, c) carbon-epoxy adherents. . . 89
6.8 Convergence: a) total displacement and b) maximum von Mises stress. . . 90
6.9 SLJ total displacement: a) 3D BEM and b) 3D FEM . . . 91
6.10 SLJ: a) mesh and b) boundary conditions. . . 91
6.11 Shear stress distribution . . . 92
6.12 Peel stress distribution . . . 92
6.13 SLJ damage evolution: a) 5228 N b) 7838 N c) 10456 N . . . 93
6.14 Failure loads for SLJ. . . 93
6.15 Finite element model: a) mesh and b) boundary conditions. . . 94
6.16 Stress intensity factor: a) unpatched b) one-sided, c) two-sided and d) comparison. 95 6.17 Stress: a) shear, b) peel and c) von Mises . . . 96
6.18 Stress intensity factor along the thickness. . . 97
6.19 Convergence of the plate with rectangular repair: a) displacement 𝑥2axis and b) maximum von Mises stress. . . 97
6.20 Convergence of the plate with elliptical repair: a) displacement 𝑥2 axis and b) maximum von Mises stress. . . 98
6.21 Mesh: a) rectangular repair and b) elliptical repair. . . 98
6.22 Deflection along the plate. . . 99
6.23 Damage evolution on the plate with rectangular repair: a) Step 6, b) Step 20, c) Step 40 and d) Step 48 . . . 100
6.24 Damage evolution on the plate with elliptical repair: a) Step 6, b) Step 20, c) Step 40 and d) Step 48 . . . 101
6.28 Crack length evaluated in the fatigue test for plates with elliptical repair. . . 105
6.29 DIC Plate 2: Displacement 𝑥2. Dimensions in [mm]. . . 106
6.30 DIC Plate 2: Displacement 𝑥1.Dimensions in [mm]. . . 107
6.31 DIC Plate 2: Strain 𝜀22. . . 108
6.32 DIC Plate 5: Displacement 𝑥2. Dimensions in [mm]. . . 109
6.33 DIC Plate 5: Displacement 𝑥1. Dimensions in [mm]. . . 110
6.34 DIC Plate 5: Strain 𝜀22. . . 111
6.35 Absolute displacement values in the 𝑥3-axis measured by the vibrometer. . . 113
6.36 Absolute displacement values in the 𝑥3-axis measured by the accelerometer. . . 113
6.37 Maximum displacement envelopes measured by the vibrometer. . . 114
6.38 Maximum displacement envelopes measured by the accelerometer. . . 114
6.39 Comparison of experimental of repared and no repaired plates. . . 115
6.40 Crack propagation profiles for crack 1. . . 116
6.41 Crack propagation profiles for crack 2. . . 116
6.42 Crack growth rate versus stress intensity factor range. . . 117
6.43 Crack length for plates with elliptical repair. . . 118
1.1 Typical features of different connections . . . 18
2.1 Typical properties of fabric composites (VASILIEV AND MOROZOV, 2001). . . 37
3.1 Multidomain performance. . . 47
4.1 Patch material properties. . . 53
4.2 Carbon fiber biaxial woven fabric properties. . . 54
4.3 Adhesive properties. . . 55
4.4 Aluminum 2024-T3 properties. . . 55
4.5 Repairs dimensions . . . 60
5.1 Mechanical properties of the patch system . . . 72
5.2 Repair dimensions used in experimental analysis. . . 75
5.3 Processes applied for each specimen. . . 78
5.4 Main equipments used in the fatigue tests. . . 80
5.5 Results of horizontal displacements 𝑢. . . 82
5.6 Results of vertical displacements 𝑣. . . 82
6.1 Adhesive film CYTEC FM-73 properties. . . 86
6.2 Adhesive film AF 163-2K properties. . . 90
6.3 Crack length evaluated in the fatigue test for plates with rectangular repair. . . . 103
6.4 Crack length evaluated in the fatigue test for plates with elliptical repair. . . 104
6.5 Comparison of BEM and DIC results for some points around the central notch in the plate 2. . . 112
6.6 Comparison of BEM and DIC results for some points around the central notch in the plate 4. . . 112
Symbols
𝜎𝑖𝑗 - Cauchy stress tensor
𝑏𝑖 - Body forces 𝜌 - Density 𝑢𝑖 - Displacement vector 𝑡𝑖 - Traction vector 𝜀𝑖𝑗 - Strain components 𝐶𝑖𝑗𝑘𝑙 - Stiffness tensor 𝑆𝑖𝑗𝑘𝑙 - Compliance tensor 𝐸1, 𝐸2, 𝐸3 - Elasticity moduli 𝐺12, 𝐺13, 𝐺23 - Shear moduli 𝜈12, 𝜈13, 𝜈23 - Poisson’s ratios 𝛺𝑖𝑗 - Transformation matrix
K - General 3D transformation tensor
𝑈𝑖𝑘 - Fundamental solution for displacement
𝑇𝑖𝑘 - Fundamental solution for traction
H - Influence matrix of tractions G - Influence matrix of displacements 𝐻𝑖𝑘 - Barnett-Lothe tensor
𝜆(𝑚,𝑛)𝑖𝑘 - Fourier coefficients
𝐹𝑡 - Ultimate tensile strength
𝐹𝑦𝑡 - Tensile yield strength
𝐹𝑐 - Compresive strength
𝐹𝑠 - Shear strength
𝑆 - Stiffness ratio between the patch and the cracked body 𝑡𝑅 - Thickness of the repair
𝑡𝑠 - Thickness of the damaged structure
𝑙𝑝 - Plastic transfer length in the adhesive
𝑙𝑒 - Elastic transfer length in the adhesive
𝜏𝑝 - Plastic shear stress of the adhesive
𝑁 - Number of transfer zones 𝑃𝑤 - Patch width
𝑎 - Semi-major axis of the elliptical repair 𝑏 - Semi-minor axis of the elliptical repair
𝐸𝑐0 - critical energy for a unit volume of material under pure tension
𝐸𝑠0 - critical energy for a unit volume of material under simple shear
𝐸𝑑 - Dissipative energy density
𝐸𝑝 - Plastic energy density
𝐸𝑒 - Elastic energy density
𝐸𝑇 𝑦 - Elastic strain energy density for yielding under uniaxial tension
𝐾𝐼 - Stress intensity factor - mode I
𝐾𝑚𝑒𝑎𝑛 - Membrane stress intensity factor
𝐾𝑏 - Bending stress intensity factor
Acronyms
ASTM - American Society for Testing and Materials BEM - Boundary Element Method
CAD - Computer Aided Design DCB - Double Cantilever Beam DIC - Digital Image Correlation DOF - Degrees Of Freedom
DRBEM - Dual Reciprocity Boundary Element Method EDM - Electrical Discharge Machine
FDM - Finite Difference Method FEM - Finite Element Method LDV - Laser Doppler Vibrometer MPI - Message Passing Interface
MUMPS - Multifrontal Massively Parallel Sparse SIF - Stress Intensity Factor
1 Introduction 17
1.1 State of art . . . 22
1.2 Motivation . . . 24
1.3 Objectives . . . 25
1.4 Outline . . . 25
2 Constitutive model for anisotropic materials 26 2.1 Anisotropic elasticity . . . 26
2.2 Constitutive equations of a lamina . . . 31
2.3 Mechanics of woven fabric composites . . . 34
3 Elastostatic BEM formulation 38 3.1 Integral formulation . . . 38
3.2 Fundamental solution . . . 41
3.3 Multidomain assembly . . . 43
3.4 Computational aspects . . . 47
3.5 Meshing process using multimedia software . . . 51
4 Bonded composite repairs 53 4.1 Repair material selection . . . 53
4.2 Design of repair . . . 56
4.3 Failure analysis . . . 60
4.3.1 Generalized energy failure criterion . . . 60
4.3.2 Crack progation . . . 62
4.3.3 Fatigue life and stress intensity factor . . . 64
4.4 Analytical models . . . 65
5 Experimental setup and test procedure 72 5.1 Specimen design and fabrication . . . 72
5.1.1 Materials . . . 72
5.1.2 Proof bodies . . . 73
5.1.3 Surface treatment . . . 77
5.2 Testing Procedures . . . 79
5.3 Digital Image Correlation . . . 80
5.4 Laser vibrometry . . . 83
6.1.2 Single lap joint . . . 89
6.1.3 Bonded composite repair . . . 94
6.2 Experimental results . . . 102
6.2.1 Crack propagation . . . 103
6.2.2 Digital image Correlation . . . 105
6.2.3 Laser vibrometry . . . 113
6.3 Discussion . . . 115
7 Conclusions 119 References 121 Appendices 129 A – Composite patch design . . . 129
B – Strain-energy release rate . . . 131
1
Introduction
The use of adhesive bonding in the industry has increased significantly in the past years, especially in technically demanding applications, e.g., bonded repairs, and doublers in aircraft structures. Kinloch (1987) defines an adhesive as a material which, when applied to surfaces of materials joins them together and provides resistance to separation. This work focuses on structural adhesives, which are routinely capable to overlap shear strengths over 1000 psi (6.89 MPa) when metals are bonded and tested at room temperature.
In general, structural adhesives can be generally categorized by their chemical composi-tion as acrylics, cyanoacrylates, urethanes, and epoxies. Acrylics provide the highest bonding strength on plastics and also offer satisfactory bonds to metals. However, they tend to present lower resistance to vibration and impact. They also feature lower mechanical performance at extremes temperature. Cyanoacrylates have good shear strength on several plastics and rub-bers, but they are rigid and show low peel and impact resistance. Urethanes, on the other hand, are flexible but have lower strength in comparison with the other categories above mentioned. They can produce relatively good bonds to plastics, rubbers and composites. Generally, they are lower-priced than other categories of structural adhesives. Finally, epoxies display the widest range of properties and can have the best overall properties on metals and often on thermoset composites.
In addition to a correct selection of the adhesive material, the success of an adhesive de-pends directly on its adhesion. It is the attraction forces between two substances resulting from intermolecular forces. This concept is different from that of cohesion, which only involves inter-molecular forces inside one substance (DA SILVA ET AL., 2011). The use of surface treatment is usually recommended to optimize adhesion between the adhesive and the adherent. It can minimize the potentially damaging influence of external factors such as environmental aging, corrosion and applied loading.
In order to make connections between two or more components, different techniques can be used during the process, such as welding, brazing, riveting and bonding. The choice between bonded and mechanical joints comes with advantages and disadvantages. The increasing use of composite materials points up the disadvantages of mechanical joints, once a 60% decrease in the in-plane properties of laminated composite structures takes place when fasteners are used as the joining method (BAKER AND SCOTT, 2016). Table 1.1 shows typical features of different bonded and mechanical connections.
Table 1.1: Typical features of different connections
Bonded connections
Advantages Disadvantages
• Reduced average stress and stress concen-tration
• Strength affect thermal cycling and high humidity
• Minimum weight addition • Difficulty of inspection • High fatigue strength and corrosive
resis-tant
• Careful design needed for peel loadings
• Smooth aerodynamic remain surfaces • Requirement for special surface prepara-tion
• Corrosive resistant
• Assembly of dissimilar materials
Mechanical connections
Advantages Disadvantages
• Special surface preparation is not required • Disrupted aerodynamics surfaces
• Easily disassembled • Stress concentrations
• Strength not affected by thermal cycling or high humidity
• Special gaskets or sealants required for fluids and weather tightness
Adhesive joints can be subjected to any combination of the different types of loading (tensile, compression, cleavage, shear and peel). For designing the joint, it is necessary to mini-mize peel and cleavage forces as well as to optimini-mize the area over which the load is distributed. Peel and cleavage loadings induce very high stresses on the boundary line. Tension, shear and compression loadings provide a uniform distribution of the load on the adhesive. However, ten-sion loadings can induce significant peel or cleavage loadings very easily (JEANDREAU, 2006). Figure 1.1 shows joint designs typically used in the industry.
The main applications of adhesives in the aeronautical and aerospace industry are bonded doublers and bonded composite repairs of airframe structures. This works focuses on bonded composite repairs. They reduces the stress in the cracked region and prevents the crack from opening and growing. As a general rule, the repair should be simple and the least intrusive to the structure as possible, while restoring structural capability to the required level. The patch must be implemented without compromising other functions of the component or structure, such as clearance in moving parts, aerodynamic smoothness and balance (BAKER ET AL., 2002).
Single lap
Stepped lap
Double lap
Tapered double strap
Double strap
Tapered lap
Figure 1.1: Basic joint geometries.
Despite the methods and materials that are used for bonded composite repairs are fairly standard, tools and methods that allow a reliable analysis of bonded structures have improved with the advancement of industrial applications. Boundary Element Method (BEM) and Finite Element Method (FEM) are essential tools for this task. Although FEM is more common in industry, BEM could be considered as an effective alternative in several important areas of engineering analysis, especially those involving infinite or semi-infinite domains under stress concentration. Domain types of numerical methods, such as FEM and Finite Difference Method (FDM) require a discretization of the surface and inside the domain. On the other hand, BEM requires only a surface discretization. This allows the reduction of the spatial dimensions of the problem, and consequently, less time and data are required to run a program efficiently. Tang et al. (2006) shows an interesting comparison of FEM and BEM for interactive object simulation. In several problems, it is necessary to consider more than one region or material, e.g. bonded joints, polycrystalline materials and fiber-matrix problems. The multidomain approach is an extension of BEM to simulate several material domains. This methodology can further reduce the computational cost optimizing the disk storage requirements as well as eliminating unnecessary operations.
Even with the advancement of computational tools, the low safety factor industry, such as aeronautics and aerospace, require special care for the maintenance of their products. Thus, the demand for non-contact inspection methods has increased. New equipment and techniques using optical or laser systems have been developed to improve structural failure detection as
Figure 1.2: Composite repair system.1
well as reliability in measured results.
Digital Image Correlation (DIC) is an optical non-contact method used to measure spa-tial coordinates for the evaluation of deformation. This technique has become popular due to the amount of information gathered about the deformation during the tests, when compared with strain gages and extensometers. The impact of this method in engineering applications has increased with the technological advances of the computational sources. DIC uses image re-gistration algorithms to track the relative displacements of material points between a reference (undeformed) image and a current (deformed) image. In this work, DIC is applied for monito-ring the crack propagation dumonito-ring the experimental test using the software package Ncorr. It is an open-source subsetbased 2D DIC package that amalgamates modern DIC algorithms propo-sed in the literature with additional enhancements (BLABER ET AL., 2015). The location of the crack tip was placed taking into consideration the results of the image processing by the cor-relation algorithm, respect to the displacements and the strain. Figure 1.3 illustrates a full-field three dimensional displacement in a landing gear component evaluated using DIC.
Figure 1.3: Full-field three-dimensional displacement in a landing gear component.2
Another non-contact inspection technique that has evolved in recent years is the laser vibrometry, witch first introduced by Stoffregen and Felske (1985). The operating principle is based on the Doppler effect, which occurs when light is back-scattered from a vibrating surface. The velocity and the displacement can be determined by analyzing the optical signals in diffe-rent ways. Nowadays, vibrometry covers a huge amount of applications, such as the study of microstrucutures, health monitoring of structures and surface characterization (see Figure 1.4). The present study employs a unidirectional vibrometer laser to measure the displacement of the bonded composite repair.
Figure 1.4: Surface characterization using 3D Scanning Vibrometer.3
In this work, the failure evolution on bonded composites is numerically and experimen-tally analysed. The failure is evaluated using a generalized energy failure criterion. In the nu-merical analysis, a preliminary study was carried out on some repair geometries using FEM. From these results, the failure was assessed in selected geometries using the elastostatic 3D BEM formulation with a fundamental solution based on double Fourier series and the multido-main algorithm. In the experimental analysis, fatigue crack propagation is evaluated using DIC, and the displacement of the patch is measured with accelerometers and a unidirectional laser vibrometer.
1.1 State of art
Several methods for modeling bonded joints have been developed over the last century. The main analytical models for stress analysis in bonded joints can be found in Volkersen (1938), Goland and Reissner (1944), and Hart-Smith (1973). A summary of the main analy-ses is presented in da Silva et al. (2009). For modeling adhesive joints of complex shape, nu-merical techniques such as FEM and BEM are preferable. These methods make the analyses of the joint rotation and the adherent’s and adhesive’s plasticity easier. One of the first nume-rical analyses taking into account these aspects was carried out by Harris and Adams (1984). More recently, Liu et al. (2018) studied fatigue crack growth characterization in adhesive carbon fiber-reinforced plastics joints and Floros and Tserpes (2019) presented a finite element study of fatigue crack propagation in single lap bonded joints with process-induced disbond.
3
The first bonded composite repair analyses began simultaneously with the advance of the techniques for the analysis of bonded joints. The concept of bonded composite repair was established by the Defence Science and Technology Organisation, in Australia, in the early 1970s. Erdogan and Arin (1972), and Ratwani (1979) presented some analytical solutions for the study of bonded isotropic repairs for infinite cracked plates. One of the first FEM analyses of bonded overlays of composite material was done by Mitchell et al. (1975). Their analysis articulates separately the responses of the sheet, the overlays, and the adhesive. Soutis and Hu (1997) evaluated bonded composite patch repairs numerically and experimentally. They concluded that the bonded patch provides a considerable increase in the residual strength.
Many works on bonded composite repair were developed applying BEM. Salgado and Aliabadi (1998) presented a formulation based on the Dual Boundary Element Method (DBEM) and on the Dual Reciprocity Method (DRBEM) for the analysis of thin cracked metal sheets for which thin metal patches and stiffeners are adhesively bonded. The DRM was used to avoid the discretization of the patch attachment domains into internal cells. A new BEM formulation for the analysis of curved cracked panels was established by Wen et al. (2003). The effect of the adhesive layer was modelled by distributed body forces (e.g. two in-plane forces, two moments and one out-of-plane force). A coupled boundary integral formulation of a shear deformable plate and two dimensional plane stress elasticity is used to determine bending and membrane forces along the adhesive layer taking into consideration the compatibility conditions in the patched area. Recent works show that several academic centers all over the world are focusing on research on bonded composite repairs (ALBEDAH ET AL., 2018; NESREDDINE ET AL., 2018; SALEM ET AL., 2018; WANG ET AL., 2019).
An efficient computation of the fundamental solution is crucial in BEM analysis. For isotropic elasticity, the fundamental solution can be represented in relatively simple explicit forms. However, the Green’s function is significantly more complex for general anisotropic solids. The anisotropic fundamental solution was first presented by Lifshitz and Rozenzweig (1947), but not as a closed form. Since then, several efforts were made to produce an explicit form to evaluate a way to compute the Green’s function (WILSON AND CRUSE, 1978; TING AND LEE, 1997; SALES AND GRAY, 1998). Shiah and Tan (SHIAH ET AL., 2012B; TAN ET AL., 2013) reformulated this fundamental solution and represented it in terms of double Fourier series. The most significant advantage of using this Fourier series representation of the Green’s function and its derivatives is that Fourier’s series coefficients are evaluated just once for a given material, reducing the computational cost.
the years. Sutton et al. (1983) introduced an improved digital correlation method for obtai-ning the full-field in-plane deformations of an object. This approach was applied to deter-mine parameters of interest for a rigid body dynamics problem by Peters et al. (1983). Since it was established, the DIC has been improved by many scientific researchers ( BRUCK ET AL., 1989; CHENG ET AL., 2002; PAN ET AL., 2007). A wide range of successful engineering ap-plications has clearly demonstrated the versatility and effectiveness of this technique ( TEKIELI ET AL., 2017; LU ET AL., 2017). In this work, the digital image correlation is conducted using the Ncorr, an open-source subset-based 2D DIC package developed by Blaber et al. (2015). shows some solid mechanics applications for this software. Some successful applications of laser vibrometry can be found in literature (STASZEWSKI ET AL., 2004; MALLET ET AL., 2004; LEONG ET AL., 2005; OHLER, 2007).
In our research group, several works were previously developed that support this rese-arch. Initially, a fracture mechanics analysis of anisotropic plates by BEM was developed in Sollero and Aliabadi (1993). Albuquerque (2001) presented an analysis of dynamic problems in anisotropic materials. Sato (2009) modelled problems of fracture mechanics and fatigue crack propagation using DIC and DBEM. An analysis of bonded lap joints problems in composites using BEM and the subregions technique was developed in Souza (2009). Rodriguez (2011) studied and implemented several analytical models and failure criteria for a SLJ. Fanton (2012) studied aeronautical metal structural joints reinforced by bonded doublers emphasizing on stress analysis of adhesives. Moura (2016) presents a 2D analysis of single lap joints using BEM. The fundamental solution in terms of double Fourier series was applied in multiscale approach of a carbon-fiber composite by Rodriguez (2016) and Rodriguez et al. (2017) and in polycrystal-line materials by Galvis et al. (2018). The BEM algorithm applied in this work was developed by Galvis et al. (2018).
1.2 Motivation
The increasing use of composites in airframes of new generation commercial aircrafts, such as Airbus A350, and A380, and Boeing 787 Dreamliner has boosted the demand for com-posite repairs and structural adhesives. The comcom-posite repairs market was valued at USD 9.44 Billion in 2015; it was projected to reach USD 21.97 Billion by 2026. The structural adhesives market is estimated to grow from USD 10.41 Billion in 2016 to USD 15.72 Billion by 2022, as stated by MarketsandMarkets 4. Thus, computational advances and new methodologies for
adhesive and composite repair analyses are necessary to meet the industry demand. Bonded 4
composite repairs avoid the interruption of production during maintenance work and can be applied to other industry segments.
1.3 Objectives
The objective of this research is the assessment and validation of the capabilities to predict the failure propagation in a bonded composite repair. In the numerical analysis, BEM is applied using a fundamental solution based on double Fourier series. The structure is discretized using triangular boundary elements. In the experimental approach, the crack propagation is evalua-ted using DIC and repair debonding is measured using accelerometers and a laser vibrometer. Specific objectives are listed below.
∘ To apply the 3D BEM elastotastic formulation to the analysis of bonded composite re-pairs.
∘ To study the failure propagation in the adhesive and patched structure.
∘ To validate the numerical results using an experimental analysis.
∘ To measure the composite repair debonding.
∘ To analyze the failure of bonded joints.
1.4 Outline
The present dissertation is composed of seven chapters. The first chapter presents a brief state of art, the motivation, objectives and contributions of this work. In the second chapter, the constitutive equations and equilibrium for anisotropic materials are briefly exposed. The third chapter presents BEM formulation for elastostatics and the anisotropic fundamental solution based on doubled Fourier series. The approach to analyze and design the bonded composite re-pair is presented in the fourth chapter. This chapter show the generalized failure criteria applied to the patched plate. In the fifth chapter, the details of the experimental analyses are described. The numerical and experimental results are presented in the sixth chapter. The conclusions of this research work are presented in the last chapter.
2
Constitutive model for anisotropic materials
In this chapter, a review of the theory of elasticity applied to anisotropic materials is shown. The relations between stresses and strains in an anisotropic elastic material are determi-ned. As well as, the process to obtain the components of the compliance and stiffness matrices are presented. The mathematical formulation presented in this chapter will be used in the sub-sequent chapters.
2.1 Anisotropic elasticity
Considering an infinitesimal element in a domain Ω, the equilibrium of forces can be expressed by
𝜎𝑖𝑗,𝑗 + 𝜌𝑏𝑖− 𝜌¨𝑢𝑖 = 0 , (2.1)
and the conservation of angular momentum is established by
𝜎𝑖𝑗 = 𝜎𝑗𝑖 , (2.2)
where 𝜎𝑖𝑗 is the Cauchy stress tensor, 𝑏𝑖are the body forces, 𝜌 is the mass density and 𝑢𝑖is the
displacement vector. The traction vector 𝑡𝑖 in a point of the boundary Γ of a domain Ω is given
by
𝑡𝑖 = 𝜎𝑖𝑗𝑛𝑗 . (2.3)
In linear elasticity, the displacement vector and its derivatives are considered as infinite-simals. The strain tensor is calculated using the following equation
𝜀𝑖𝑗 =
1
where 𝜀𝑖𝑗 are the strain components. In order to ensure the uniqueness of displacements, the
components of the strain tensor 𝜀𝑖𝑗 cannot be designated arbitrarily. Thus, some conditions of
compatibility and integrability must be satisfied. These conditions are given by
𝜀𝑖𝑗,𝑘𝑙+ 𝜀𝑘𝑙,𝑖𝑗− 𝜀𝑖𝑘,𝑗𝑙 − 𝜀𝑗𝑙,𝑖𝑘 = 0 . (2.5)
In the most general case the stress and strain components for an anisotropic material under isothermal conditions are related by the generalized Hooke’s law as follows
𝜎𝑖𝑗 = 𝐶𝑖𝑗𝑘𝑙𝜀𝑘𝑙 , (2.6)
where 𝐶𝑖𝑗𝑘𝑙 is a fourth-order tensor (81 elements) known as tensor of elastic properties or
stiff-ness tensor. It satisfies the full symmetry conditions (TING, 1996),
𝐶𝑖𝑗𝑘𝑙 = 𝐶𝑗𝑖𝑘𝑙 , 𝐶𝑖𝑗𝑘𝑙= 𝐶𝑗𝑖𝑙𝑘, 𝐶𝑖𝑗𝑘𝑙 = 𝐶𝑘𝑙𝑖𝑗 . (2.7)
The first relation of Eq. 2.7 follows from the symmetry os the stress tensor 𝜎𝑖𝑗 = 𝜎𝑗𝑖. The
second relation follows from the 𝜀𝑖𝑗 = 𝜀𝑗𝑖. The third relation follows from the consideration of
strain energy.
In general, it would be required 81 elastic constants to fully characterize the material. However, the considerations presented in Equation 2.7 reduce the number of independent elastic constants from 81 to 21. Thus, Equation 2.6 can be rewritten in matrix form as,
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎11 𝜎22 𝜎33 𝜎23 𝜎13 𝜎12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝐶1111 𝐶1122 𝐶1133 𝐶1123 𝐶1113 𝐶1112 𝐶1122 𝐶2222 𝐶2233 𝐶2223 𝐶2213 𝐶2212 𝐶1133 𝐶2233 𝐶3333 𝐶3323 𝐶3313 𝐶3312 𝐶1123 𝐶2223 𝐶3323 𝐶2323 𝐶2313 𝐶2312 𝐶1113 𝐶2213 𝐶3313 𝐶2313 𝐶1313 𝐶1312 𝐶1112 𝐶2212 𝐶3312 𝐶2312 𝐶1312 𝐶1212 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜀11 𝜀22 𝜀33 2𝜀23 2𝜀13 2𝜀12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . (2.8)
The inverse of Equation 2.6 is rewritten as,
𝜀𝑖𝑗 = 𝑆𝑖𝑗𝑘𝑙𝜎𝑘𝑙, (2.9)
where 𝑆𝑖𝑗𝑘𝑙is a fourth-order tensor known as tensor elastic flexibility or compliance tensor. As
well as the stiffness tensor, the compliance tensor is also fully symmetry. Then, the number of independent elastic constants can be reduced from 81 to 21, as shown in the following equation.
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜀11 𝜀22 𝜀33 2𝜀23 2𝜀13 2𝜀12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑆1111 𝑆1122 𝑆1133 𝑆1123 𝑆1113 𝑆1112 𝑆1122 𝑆2222 𝑆2233 𝑆2223 𝑆2213 𝑆2212 𝑆1133 𝑆2233 𝑆3333 𝑆3323 𝑆3313 𝑆3312 2𝑆1123 2𝑆2223 2𝑆3323 4𝑆2323 4𝑆2313 4𝑆2312 2𝑆1113 2𝑆2213 2𝑆3313 4𝑆2313 4𝑆1313 4𝑆1312 2𝑆1112 2𝑆2212 2𝑆3312 4𝑆2312 4𝑆1312 4𝑆1212 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎11 𝜎22 𝜎33 𝜎23 𝜎13 𝜎12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (2.10)
Equation 2.10 is usually expressed using the reduced Voigt notation, as follows
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑎11 𝑎12 𝑎13 𝑎14 𝑎15 𝑎16 𝑎12 𝑎22 𝑎23 𝑎24 𝑎25 𝑎26 𝑎13 𝑎23 𝑎33 𝑎34 𝑎35 𝑎36 𝑎14 𝑎24 𝑎34 𝑎44 𝑎45 𝑎46 𝑎15 𝑎25 𝑎35 𝑎45 𝑎55 𝑎56 𝑎16 𝑎26 𝑎36 𝑎46 𝑎56 𝑎66 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (2.11) where, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜀11 𝜀22 𝜀33 2𝜀23 2𝜀13 2𝜀12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (2.12)
and, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎11 𝜎22 𝜎33 𝜎23 𝜎13 𝜎12 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . (2.13)
It is also possible to define the elastic coefficients of the compliance tensor in terms of engineering constants. They can be expressed by
𝑎11= 1/𝐸1 𝑎12= 𝜈12/𝐸1 = −𝜈21/𝐸2 𝑎13= −𝜈31/𝐸1 = −𝜈13/𝐸3 𝑎14= 𝜂23,1/𝐸1 = 𝜂1,23/𝐺23 𝑎15= 𝜂32,1/𝐸1 = 𝜂1,32/𝐺23 𝑎16= 𝜂12,1/𝐸1 𝑎22= 1/𝐸2 𝑎23= 𝜈32/𝐸2 = −𝜈23/𝐸3 𝑎24= 𝜂23,1/𝐸2 = 𝜈23,3/𝐺23 𝑎25= 𝜂31,2/𝐸2 = 𝜂2,31/𝐺13 𝑎26= 𝜂12,2/𝐸2 = 𝜂2,12/𝐺12 𝑎33= 1/𝐸3 𝑎34= 𝜂23,3/𝐸3 = 𝜂3,23/𝐺12 𝑎35= 𝜂31,1/𝐸3 = 𝜂3,31/𝐺13 𝑎36= 𝜂12,3/𝐸3 = 𝜂3,12/𝐺12 𝑎44= 1/𝐺23 𝑎45= 𝜁32,23/𝐺23= 𝜁23,31/𝐺13 𝑎46= 𝜁12,23/𝐺23= 𝜁23,12/𝐺12 𝑎55= 1/𝐺13 𝑎56= 𝜁12,31/𝐺13= 𝜁31,12/𝐺12 𝑎66= 1/𝐺12 , (2.14)
where 𝐸𝑘 are the longitudinal elasticity moduli along the coordinate axes 𝑥𝑘, 𝐺𝑖𝑗 is the
trans-verse elasticity modulus for planes defined by 𝑥𝑖𝑥𝑗 coordinate axes. 𝜈𝑖𝑗 are the Poisson’s ratios.
The constants 𝜂𝑖𝑗,𝑘 are known as the mutual influence coefficients of the first kind; they denote
the extension in the direction of 𝑥𝑘 coordinate produced by shearing stresses acting in the
co-ordinate plane 𝑥𝑖𝑥𝑗. The constants 𝜂𝑖,𝑗𝑘 are known as the mutual influence coefficients of the
second kind and denote the shear in the coordinate plane 𝑥𝑗𝑥𝑘 due to a normal stress acting
on the 𝑥𝑖 coordinate direction. Finally, 𝜁𝑖𝑗,𝑘𝑙 are called Chentsov’s coefficients, and they
repre-sent the shear deformation in planes parallel to the coordinates planes produced by the shearing stresses acting in the other planes parallel to the coordinate planes.
symmetry, the number of independent elastic constants is reduced to nine, as various terms are interrelated. An orthotropic material is called transversely isotropic when one of its principal planes is a plane of isotropy. In this case, the number of independent elastic constants is reduced to five. Many unidirectional composites can be considered transversely isotropic. If a material presents an infinite number of planes of material symmetry through a point, it is called isotropic and can be fully characterized by only two independent constants.
In engineering applications, the choice of the coordinates of the material orientation often dictated by requirements of the problem. Frequently, this orientation may not coincide with the symmetry planes of the material. Thereby, the transformation of the stiffness tensor 𝐶𝑖𝑗𝑘𝑙 and
compliance tensor 𝑆𝑖𝑗𝑘𝑙 to a different coordinate system becomes necessary. In this case, it is
applied the method described in Ting (1996). The transformation matrix 𝛺𝑖𝑗 depends on the
rotation reference axis. Thus, for a rotation on axis 𝑥1, 𝛺𝑖𝑗 is given by
[𝛺𝑖𝑗]1 = ⎡ ⎢ ⎣ 1 0 0 0 cos 𝜃 sin 𝜃 0 − sin 𝜃 cos 𝜃 ⎤ ⎥ ⎦ , (2.15)
for a rotation on axis 𝑥2,
[𝛺𝑖𝑗]2 = ⎡ ⎢ ⎣ cos 𝜃 0 − sin 𝜃 0 1 0 sin 𝜃 0 cos 𝜃 ⎤ ⎥ ⎦ , (2.16)
and for a rotation on axis 𝑥3,
[𝛺𝑖𝑗]3 = ⎡ ⎢ ⎣ cos 𝜃 sin 𝜃 0 − sin 𝜃 cos 𝜃 0 0 0 1 ⎤ ⎥ ⎦ , (2.17)
where 𝜃 is the direction of rotation around the selected axis. The general 3D transformation tensor K is expressed as
K = [︃ K1 2K2 K3 K4 ]︃ , (2.18) where K1 = ⎡ ⎢ ⎣ 𝛺112 𝛺122 𝛺132 𝛺212 𝛺222 𝛺232 𝛺312 𝛺322 𝛺332 ⎤ ⎥ ⎦ , (2.19) K2 = ⎡ ⎢ ⎣ 𝛺12𝛺13 𝛺13𝛺11 𝛺11𝛺12 𝛺22𝛺23 𝛺23𝛺21 𝛺21𝛺22 𝛺32𝛺33 𝛺33𝛺31 𝛺31𝛺32 ⎤ ⎥ ⎦ , (2.20) K3 = ⎡ ⎢ ⎣ 𝛺21𝛺31 𝛺22𝛺32 𝛺23𝛺33 𝛺31𝛺11 𝛺32𝛺12 𝛺33𝛺13 𝛺11𝛺21 𝛺12𝛺22 𝛺13𝛺23 ⎤ ⎥ ⎦ , (2.21) K4 = ⎡ ⎢ ⎣ 𝛺22𝛺23+ 𝛺23𝛺32 𝛺23𝛺31+ 𝛺21𝛺33 𝛺21𝛺32+ 𝛺22𝛺31 𝛺32𝛺13+ 𝛺33𝛺12 𝛺33𝛺11+ 𝛺31𝛺13 𝛺31𝛺12+ 𝛺32𝛺11 𝛺12𝛺23+ 𝛺13𝛺22 𝛺13𝛺21+ 𝛺11𝛺23 𝛺11𝛺22+ 𝛺12𝛺21 ⎤ ⎥ ⎦ , (2.22)
The transformed stiffness C* and compliance S* tensors are given by
C* = KCK𝑇
S* = (K−1)𝑇CK−1 (2.23)
2.2 Constitutive equations of a lamina
A lamina or ply is a typical sheet of composite material. It represents a fundamental building block. A fiber-reinforced lamina consists of many fibers embedded in a matrix material,
which can be a metal like aluminum, or a nonmetal like thermoset or thermoplastic polymer. The fiber can be continuous or discontinuous, woven, unidirectional, bidirectional or randomly distributed. Unidirectional fiber-reinforcement laminae exhibit the highest strength and modulus in the direction of the fibers and very low strength and modulus in the direction transverse to the fibers. Weak bonding between a fiber and matrix results in poor transverse properties and failures in the form of fiber pull out, fiber breakage, and fiber buckling. Often, coupling chemical agents and fillers are added to improve the bonding between fiber and matrix material increasing toughness.
A laminate is a collection of laminae stacked to achieve the desired stiffness and thickness. The lamination scheme and material properties of individual lamina provide and add flexibility to the designers to tailor the stiffness and strength of the laminate to match the structural stiffness and strength requirements. Composite laminates also have disadvantages. Due to the mismatch of material properties between layers, the shear stress produced in this region, especially at the edges of a laminate, may cause delamination. Also, during manufacturing, material defects such as interlaminar voids, delamination, incorrect orientation, damage fibers, and variation in thickness may be introduced.
Composite materials are inherently heterogeneous from the microscopic point of view. At the macroscopic scale, wherein the material properties of a composite are derived from a weigh-ted average of the constituent materials, fiber and matrix, composite materials are assumed to be homogeneous. A unidirectional fiber-reinforced lamina is treated as an orthotropic material whose symmetry planes are parallel and transverse to the fiber direction. The orthotropic mate-rial properties of a lamina are obtained either by the theoretical approach or through suitable la-boratory tests. The theoretical approach used to evaluate the engineering constants of a continu-ous fiber-reinforced composite material is based on the following assumptions (REDDY, 2003):
1. Perfect bonding exists between fibers and matrix.
2. Fibers are parallel and uniformly distributes throughout.
3. The matrix is free of voids or microcracks and initially in a stress-free state
4. Both fibers and matrix are isotropic and obey Hooke’s law.
5. The applied loads are either parallel or perpendicular to the fiber direction.
The material coordinates axis 𝑥1 is taken parallel to the fiber, 𝑥2-axis transverse to the
x1 x 2 x 3 Fiber Matrix
Figure 2.1: Unidirectional fiber-reinforced composite layer
The lamina engineering constants are given by
𝐸1 = 𝐸𝑓𝑣𝑓 + 𝐸𝑚𝑣𝑚 , 𝜈12 = 𝜈𝑓𝑣𝑓 + 𝑛𝑢𝑚𝑣𝑚 , 𝐸2 = 𝐸𝑓𝐸𝑚 𝐸𝑓𝑣𝑚+ 𝐸𝑚𝑣𝑓 , 𝐺12= 𝐺𝑓𝐺𝑚 𝐺𝑓𝑣𝑚+ 𝐺𝑚𝑣𝑓 , (2.24)
where the 𝑓 and 𝑚 subscripts indicate the fiber and matrix materials, respectively. 𝐺𝑓 and 𝐺𝑚
are the shear moduli are given by
𝐺𝑓 = 𝐸𝑓 2(1 + 𝜈𝑓) , 𝐺𝑚 = 𝐸𝑚 2(1 + 𝜈𝑚) . (2.25)
The Hooke’s law for a homogeneous orthotropic lamina which has its axes aligned with the reference axes may be written as
⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜎1 𝜎2 𝜎3 𝜎4 𝜎5 𝜎6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝑐11 𝑐12 𝑐13 0 0 0 𝑐12 𝑐22 𝑐23 0 0 0 𝑐13 𝑐23 𝑐33 0 0 0 0 0 0 𝑐44 0 0 0 0 0 0 𝑐55 0 0 0 0 0 0 𝑐66 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝜀1 𝜀2 𝜀3 𝜀4 𝜀5 𝜀6 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ , (2.26)
where 𝑐𝑖𝑗 are the components of the matrix of elastic properties or stiffness matrix and are given
by, 𝑐11= 1 − 𝜈23𝜈32 𝐸1𝐸3∆ 𝑐12= 𝜈23+ 𝜈31𝜈23 𝐸2𝐸3∆ 𝑐13= 𝜈31+ 𝜈21𝜈32 𝐸2𝐸3∆ 𝑐22= 1 − 𝜈31𝜈13 𝐸3𝐸1∆ 𝑐23= 𝜈32+ 𝜈31𝜈12 𝐸3𝐸1∆ 𝑐33= 1 − 𝜈12𝜈21 𝐸1𝐸2∆ 𝑐44= 2𝐺23 𝑐55= 2𝐺31 𝑐66= 2𝐺12 . (2.27) where ∆ = 1 − 𝜈12𝜈21− 𝜈23𝜈32− 𝜈31𝜈13− 2𝜈21𝜈32𝜈13 𝐸1𝐸2𝐸3 . (2.28)
2.3 Mechanics of woven fabric composites
Textile preforming plays an important role in composite technology. The main advantages of woven composites are their cost efficiency and high processability, particularly, in lay-up manufacturing of large-scale structures. However, on the other hand, processing of fibers and their bending in the process of weaving results in substantial reduction of material strength and stiffness (VASILIEV AND MOROZOV, 2001). Figure 2.2 shows some woven fabrics.
Plain weave Twill weave
Satin weave
Warp tow
Fill tow
Figure 2.2: Fabric weaves
Figure 2.3 shows a typical woven structure. The warp (lengthwise) and fill (crosswise) yarns forming the fabric make 𝛼 ≥ 0 with the plane of the fabric lay.
27,07 mm h/2 27,07 mm h/2 64,53 mmt 46,21 mm 55,98 mm 1 t2 t1 1 3 12,48 ° α 1 2 3 Warp Fill
Figure 2.3: Unit cell of a fabric structure.
The apparent modulus of elasticity of the structure is expressed by
𝐸𝑎𝐴𝑎= 𝐸𝑓𝐴𝑓 + 𝐸𝑤𝐴𝑤 , (2.29)
ℎ
4(4𝑡1 + 𝑡2) are the areas of the fill and warp yarns in the cross section. Substitution into Eq.
(2.29) yields 𝐸𝑎= 1 2 [︂ 𝐸𝑓 + 𝐸𝑤(4𝑡1+ 𝑡2) 2(2𝑡1+ 𝑡2) ]︂ . (2.30)
Once the fibers of the fill yarns are orthogonal to the loading direction, we can take 𝐸𝑓 =
𝐸2, where 𝐸2 is the transverse modulus of a unidirectional composite. The compliance of the
warp yarn can be decomposed into two parts corresponding to 𝑡1and 𝑡2,
2𝑡1+ 𝑡2 𝐸𝑤 = 2𝑡1 𝐸1 + 𝑡2 𝐸𝛼 , (2.31)
where 𝐸1 is the longitudinal modulus of a unidirectional composite and 𝐸𝑎 can be determined
from compliance coefficients as
1 𝐸𝑎 = 1 𝐸1 𝑐𝑜𝑠4𝜃 + 1 𝐸2 𝑠𝑒𝑛4𝜃 + [︂ 1 𝐺12 − 2𝜈12 𝐸1 ]︂ 𝑠𝑒𝑛2𝜃𝑐𝑜𝑠2𝜃 . (2.32)
The final result is as follows
𝐸𝑎= 𝐸2 2 + 𝐸1(4𝑡1+ 𝑡2) 4 (︂ 2𝑡1+ 𝑡2 [︂ 𝑐𝑜𝑠4𝛼 + 𝐸1 𝐸2 𝑠𝑒𝑛4𝛼 +(︂ 𝐸1 𝐺12 − 2𝜈21 )︂ 𝑠𝑒𝑛2𝛼𝑐𝑜𝑠2𝛼 ]︂)︂ (2.33)
Table 2.1: Typical properties of fabric composites (VASILIEV AND MOROZOV, 2001).
Property Glass fabric-epoxy Aramid fabric-epoxy Carbon fabric-epoxy
Fiber volume fraction 0.43 0.46 0.45
Density (g/cm3) 1.85 1.25 1.40
Longitudinal modulus (GPa) 26 34 70
Transverse modulus (GPa) 22 34 70
Shear modulus (GPa) 7.2 5.6 5.8
Poisson’s ratio 0.13 0.15 0.09
Longitudinal tensile strength (MPa) 400 600 860
Longitudinal compressive strength (MPa) 350 150 560
Transverse tensile strength (MPa) 380 500 850
Transverse compressive strength (MPa) 280 150 560
3
Elastostatic BEM formulation
In this work, the most explored numerical method is BEM, which consists in the applica-tion of the boundary integral equaapplica-tion only using the surface informaapplica-tion. This allows to reduce the degrees of freedom (DOF) required by the model. The multidomain algorithm is implemen-ted to simulate several material domains as well as to reduce computational cost. In order to obtain the mechanical response of the anisotropic media, the displacement fundamental solu-tion based on double Fourier series is incorporated in the formulasolu-tion (TAN ET AL., 2013). It is also capable to model isotropic solids.
3.1 Integral formulation
For a homogeneous elastic body, the boundary integral equation in the absence of body forces is given by 𝑐𝑖𝑘(x′)𝑢𝑖(x′) + ∫︁ Γ 𝑇𝑖𝑘(x′,x)𝑢𝑖(x) dΓ = ∫︁ Γ 𝑈𝑖𝑘(x′,x)𝑡𝑖(x) dΓ , (3.1)
where 𝑢𝑖 and 𝑡𝑖 are displacement and traction fields, respectively, on a surface Γ. The
funda-mental solutions are represented by 𝑈𝑖𝑘(x′,x) for displacement and 𝑇𝑖𝑘(x′,x) for traction. (x′)
is the source point, (x) is the field point. Finally, 𝑐𝑖𝑘(x′) is 𝛿𝑖𝑘/2 for a smooth surface boundary
at source point.
Equation 3.1 must be discretized into surface elements. The overall domain is represented by 𝑐𝒟𝑖𝑘(x′)𝑢𝑖𝒟(x′) + 𝑁𝑓𝒟 ∑︁ 𝑗=1 ∫︁ Γ𝑓 −𝒟 𝑇𝑖𝑘𝒟(x′,x)𝑢𝒟𝑖 (x) dΓ = 𝑁𝑓𝒟 ∑︁ 𝑗=1 ∫︁ Γ𝑓 −𝒟 𝑈𝑖𝑘𝒟(x′,x)𝑡𝒟𝑖 (x) dΓ , (3.2)
where the superscript 𝒟 represents the 𝒟th domain, the subscript 𝑓 refers to a 𝑓 th face and 𝑁𝑓𝒟 is the number of the element corresponding to the 𝒟th domain.
In this work, linear three-node discontinuous triangular boundary elements are used in the surface discretization. This type of element provides a drastic reduction in the DOF and in the computational cost required by the numerical integration, when compared with quadratic discontinuous elements. Moreover, discontinuous discretization offers advantages in the imple-mentation of the multidomain algorithm. As shown in Figure 3.1, the three nodes are a function of the two intrinsic parametric coordinates (𝜂,𝜉). Thus, the position of all nodes in the element is controlled by the parametric distance 𝜆.
ξ η ( ,1-2 )λ λ ( , )λ λ (1-2 , )λ λ 0 0 1 1
Figure 3.1: Linear three-node discontinuous element.
The known shape functions of the continuous three-node triangular element expressed by
𝑁(1) = 𝜉 ,
𝑁(2) = 𝜂 ,
𝑁(3) = 1 − 𝜉 − 𝜂 ,
(3.3)
and the interpolated field 𝑋 expressed in terms of the known node values 𝑋(𝑘)are
𝑋(𝜉,𝜂) =
3
∑︁
𝑘=1
𝑁(𝑘)(𝜉,𝜂)𝑋(𝑘) . (3.4)
The first derivative of the interpolated field, in terms of the shape functions can be written as
𝜕𝑋(𝜉,𝜂) 𝜕𝜉 = 3 ∑︁ 𝑘=1 𝜕𝑁(𝑘)(𝜉,𝜂) 𝜕𝜉 𝑋 (𝑘) , 𝜕𝑋(𝜉,𝜂) 𝜕𝜂 = 3 ∑︁ 𝑘=1 𝜕𝑁(𝑘)(𝜉,𝜂) 𝜕𝜂 𝑋 (𝑘) . (3.5)
Equation 3.1 can be discretized into linear three-node discontinuous triangular elements, where the 𝒟th domain is divided by the elements 𝑒 and the nodes 𝑛 in the element. The nume-rical integration is carried by using Gaussian quadrature, following the procedures described by Tan et al. (2013). 𝑐𝒟𝑖𝑘𝑢𝑖𝒟+ 𝑁𝒟 𝑒 ∑︁ 𝑒=1 (︃𝑁𝑒 𝑛 ∑︁ 𝑛=1 ∫︁ 1 0 ∫︁ 1−𝜂 0 𝑇𝑖𝑘𝑁𝑛𝐽 d𝜉d𝜂 )︃ 𝑢𝑖𝑛 = 𝑁𝑒𝒟 ∑︁ 𝑒=1 (︃𝑁𝑒 𝑛 ∑︁ 𝑛=1 ∫︁ 1 0 ∫︁ 1−𝜂 0 𝑈𝑖𝑘𝑁𝑛𝐽 d𝜉d𝜂 )︃ 𝑡𝑖𝑛 , (3.6)
in which, the Jacobian 𝐽 is given by
𝐽 = 0.5(︀𝐽12+ 𝐽22+ 𝐽32)︀0.5 = 0.5 (𝐽𝑘𝐽𝑘) 0.5 , (3.7) where 𝐽1 = 𝜕𝑥2 𝜕𝜉 𝜕𝑥3 𝜕𝜂 − 𝜕𝑥3 𝜕𝜉 𝜕𝑥2 𝜕𝜂 , 𝐽2 = 𝜕𝑥3 𝜕𝜉 𝜕𝑥1 𝜕𝜂 − 𝜕𝑥1 𝜕𝜉 𝜕𝑥3 𝜕𝜂 , 𝐽3 = 𝜕𝑥1 𝜕𝜉 𝜕𝑥2 𝜕𝜂 − 𝜕𝑥2 𝜕𝜉 𝜕𝑥1 𝜕𝜂 . (3.8)
𝑐𝒟𝑖𝑘+ 𝑁𝑒𝒟 ∑︁ 𝑒=1 (︃𝑁𝑒 𝑛 ∑︁ 𝑛=1 ∫︁ 1 0 ∫︁ 1−𝜂 0 𝑇𝑖𝑘𝑁𝑛𝐽 d𝜉d𝜂 )︃ = H𝒟 , (3.9)
and the integration of the displacement fundamental solution is
𝑁𝑒𝒟 ∑︁ 𝑒=1 (︃𝑁𝑒 𝑛 ∑︁ 𝑛=1 ∫︁ 1 0 ∫︁ 1−𝜂 0 𝑈𝑖𝑘𝑁𝑛𝐽 d𝜉d𝜂 )︃ = G𝒟 . (3.10)
Thus, Equation 3.6 in matrix form per each domain is represented by
H𝒟u𝒟 = G𝒟t𝒟 , (3.11)
where H𝒟 and G𝒟 are the matrices from the integration of the traction 𝑇𝑖𝑘 and displacement
𝑈𝑖𝑘 fundamental solution.
3.2 Fundamental solution
Considering a homogeneous infinite body, where a unit load is applied at the source point 𝑃 in the 𝑥𝑗 direction. The displacement fundamental solution 𝑈𝑖𝑘(x′,x), is defined as the
res-ponse in the 𝑥𝑖direction at the field point 𝑄 due to the perturbation applied in the source point.
The analytical expression of displacement fundamental solution or Green’s function 𝑈𝑖𝑘(x′,x)
and its derivatives was first derived by Lifshitz and Rozenzweig (1947). Thus, for the case of 3D generally anisotropic materials, this solution is
𝑈𝑖𝑘 = 1 8𝜋2𝑟 ∫︁ 2𝜋 0 Z−1𝑑𝜓 , (3.12)
where 𝑟 is the distance between the source and the field point. The Z can be written is terms of the elastic stiffness tensor of the anisotropic material 𝐶𝑖𝑗𝑘𝑙and Z−1 is its inverse (TING, 1996).
Ting and Lee (1997) presented the Green’s function in terms of the Barnett-Lothe tensor 𝐻𝑖𝑘. This tensor is expressed in terms of a contour integral around a unit circle |n*| on a oblique
plane at the field point x. Thereby, using a spherical coordinate system, the explicit form of Equation 3.12 is expressed as
𝑈𝑖𝑘(𝑟,𝜃,𝜑) =
1
4𝜋𝑟𝐻𝑖𝑘(𝜃,𝜑) . (3.13)
A more explicit method to evaluate the Equation 3.13 is expressed by using of Stroh’s eigenvalues (TING AND LEE, 1997),
𝐻𝑖𝑘(𝜃,𝜑) = 1 |𝜅| 4 ∑︁ 𝑛=0 𝑞𝑛Γˆ(𝑛) , (3.14)
where 𝜅 and ˆΓ(𝑛)are functions of the stiffness tensor, and equations to evaluate these quantities
can be seen in detail in Tan et al. (2013). In this work, the evaluation of the BEM fundamental solution is carried out using the approach proposed by Shiah et al. (2012a) and Tan et al. (2013). They expressed the Barnett-Lothe tensor in terms of double Fourier series, as shown in Eq 3.15.
𝐻𝑖𝑘(𝜃,𝜑) = 𝛼 ∑︁ 𝑚=−𝛼 𝛼 ∑︁ 𝑛=−𝛼 𝜆(𝑚,𝑛)𝑖𝑘 𝑒𝑖(𝑚𝜃+𝑛𝜑), (𝑢,𝑣 = 1, 2, 3) , (3.15)
being 𝛼 is an integer number, large enough to yield the desired accuracy. 𝑚 and 𝑛 are two mutually orthogonal vectors defined as
𝑛 = (cos 𝜑 cos 𝜃, cos 𝜑 sin 𝜃, − sin 𝜑) 𝑚 = (− sin 𝜃, cos 𝜃,0)
{︃
0 ≤ 𝜃 < 2𝜋
0 ≤ 𝜑 ≤ 𝜋 , (3.16)
and 𝜆(𝑚,𝑛)𝑖𝑘 are the Fourier coefficients, which can be numerically integrated by, e.g., Gaussian quadrature. The 𝑘 abscissa points, 𝜆(𝑚,𝑛)𝑖𝑘 , can be written as
𝜆(𝑚,𝑛)𝑖𝑘 = 1 4 𝑘 ∑︁ 𝑝=1 𝑘 ∑︁ 𝑞=1 𝑤𝑝𝑤𝑞𝑓 (𝑚,𝑛) 𝑖𝑘 (𝜋𝜉𝑝,𝜋𝜉𝑞) (3.17)
where 𝑤 and 𝜉 are the weights and positions of the Gauss points and 𝑓𝑖𝑘(𝑚,𝑛)(𝜃,𝜑) represents the integrand of 𝜆(𝑚,𝑛)𝑖𝑘 . The most significant advantage of using this Fourier series representation of the Green’s function and its derivatives is that Fourier’s series coefficients, 𝜆(𝑚,𝑛)𝑖𝑘 , are evaluated just one time for a given material what reduces the computational cost (TAN ET AL., 2013).
Finally, the fundamental solution 𝑈𝑖𝑘 is expressed by
𝑈𝑖𝑘(𝑟,𝜃,𝜑) = 1 2𝜋𝑟 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 𝛼 ∑︁ 𝑚=1 𝛼 ∑︁ 𝑛=1 [︃
( ˜𝑅(𝑚,𝑛)𝑖𝑘 cos 𝑚𝜃 − ˜𝐼𝑖𝑘(𝑚,𝑛)sin 𝑚𝜃) cos 𝑛𝜑 −( ˆ𝑅(𝑚,𝑛)𝑖𝑘 sin 𝑚𝜃 − ˆ𝐼𝑖𝑘(𝑚,𝑛)cos 𝑚𝜃) sin 𝑛𝜑
]︃ + 𝛼 ∑︁ 𝑚=1 (︃ 𝑅(0,𝑚)𝑖𝑘 cos 𝑚𝜑 − 𝐼𝑖𝑘(0,𝑚)sin 𝑚𝜑 +𝑅(𝑚,0)𝑖𝑘 cos 𝑚𝜃 − 𝐼𝑖𝑘(𝑚,0)sin 𝑚𝜃 )︃ + 𝑅 (0,0) 𝑖𝑘 2 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ . (3.18)
Details of Equation 3.18 and its derivatives can be found in Shiah et al. (2012a), Shiah et al. (2012b), and Tan et al. (2013).
The traction fundamental solution 𝑇𝑖𝑘 is expressed as
𝑇𝑖𝑘 = (𝜎𝑖𝑗𝑛𝑗)𝑘 , (3.19)
in which, 𝜎𝑖𝑗 is the stress fundamental solution and 𝑛𝑗 is the outward normal vector on the
surface at the field point. Using the generalized Hooke’s law,
(𝜎𝑖𝑘)𝑗 = 𝐶𝑖𝑘𝑙𝑛(𝑈𝑚𝑗,𝑛+ 𝑈𝑛𝑗,𝑚)/2 . (3.20)
3.3 Multidomain assembly
In engineering problems, it is often necessary to consider more than one region or mate-rial, e. g. bonded joints, polycrystalline materials, and fiber-matrix problems. The multidomain approach is a straightforward extension of BEM to simulate several material domains. This methodology also reduces the computational cost optimizing the disk storage requirements as
well as eliminating unnecessary operations. Figure 3.2 shows a small example of a multidomain assembly with a single lap joint.
Figure 3.2: Multidomain example with a single lap joint.
After, BEM is applied to each homogeneous subregion as they were separated from the others, the sub-domains must be coupled considering displacement compatibility and traction equilibrium between the interfaces, as expressed in the following equation.
𝑢(𝑎)𝑖 = 𝑢(𝑏)𝑖 ,
𝑡(𝑎)𝑖 = −𝑡(𝑏)𝑖 . (3.21)
where the (𝑎) and (𝑏) indices represent the subdomains.Figure 3.3 shows an interface between adhesive and adherent, Γ𝑎and Γ𝑏 are the interface surface in each subregion. Ω𝑎and Ω𝑏 define
the domains and 𝑛 is the outward normal vector.
!
a n n!
bΩ
aΩ
b Figure 3.3: Interfaces.For this illustrative case of two subdomain, Figure 3.3, the final system of equation after the application of the boundary conditions is
[︃ A𝑖 H (𝑎) 𝑖 −G (𝑎) 𝑖 0 0 H(𝑏)𝑗 G(𝑏)𝑗 A𝑗 ]︃ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ x𝑖 u(𝑎)𝑖 t(𝑎)𝑖 x𝑗 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ = [︃ B𝑖 0 0 0 0 0 0 B𝑗 ]︃ ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ k𝑏𝑐 𝑖 0 0 k𝑏𝑐𝑗 ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ . (3.22)
where A𝑖 and B𝑖 are the blocks corresponding to H𝑖 and G𝑖 that belong to the boundary of
the domain where the load conditions are applied. The blocks H(𝑎)𝑖 and G(𝑎)𝑖 are the interfaces between the (𝑎) and (𝑏) subdomains. Vectors x𝑖 represent all the traction and displacement
unknowns to be evaluated in the elements corresponding to the boundaries and the vectors k𝑏𝑐
𝑖 are the known boundary conditions applied, respectively. In the interfaces, displacement
u(𝑎)𝑖 and traction t(𝑎)𝑖 are evaluated. A general algorithm to assemble the system of equation in multidomain applications can be found in Kane (1994). The final general matrix equation for elastostatic problems is
𝒜x = ℬk𝑏𝑐 . (3.23)
In order to evaluate the computational efficiency of the multidomain approach, a short beam is analyzed. A stress was applied at one end and the other was fixed. The number of elements on the main faces is constant in all simulations while the number of subregions is increased. Figure 3.4 presents some of the analyzed structures and the results obtained.
Figure 3.4: Multidomain analysis with a short beam.
Looking at the Figure 3.4, it is possible to observe that the number of subregions did not affect the result of the simulation. Table 3.1 shows the multidomain performance for all structures analyzed. In summary, the subregions method helps to reduce the computational time. However, an excessive number of subregions could increase computational time, once several elements are created to discretize the interfaces.
Table 3.1: Multidomain performance.
Subregions Elements Solver time [s] Total Time [s]
1 384 17.48 29.48 2 448 7.45 15.79 5 640 7.21 14.37 10 800 6.60 12.51 20 960 6.53 11.14 40 1280 6.75 12.98 80 1600 7.18 13.54 160 1922 8.52 14.07 250 3000 14.18 21.60 3.4 Computational aspects
Several algorithms have been developed to analyze problems of anisotropic elasticity by using the 3D BEM formulation. In this work, a procedure for elastostatic condition implemen-ted by Galvis et al. (2018) is used. Figure 3.5 illustrates the 3D BEM algorithm. For input, mesh (nodes, elements, normal vectors), boundary conditions and Fourier coefficients are necessary. Firstly, the mesh is read and discretized.In the discretization, physical nodes, geometrical no-des, integration points, subregion, and interfaces matrix are created. After discretization, BEM matrices for each subregion (H𝒟 and G𝒟) are generated. In the multidomain BEM, a sparce matrix treatment is carried out in the general system. The final system of equations is solved by using a Multifrontal Massively Parallel Sparse (MUMPS) (AMESTOY ET AL., 2006). MUMPS implements a direct method based on a multifrontal approach which performs a Gaussian fac-torization. The code and user’s guide are available in http://mumps.enseeiht.fr/. Finally, with known stresses, displacements and strains a failure analysis is performed.