High fidelity simulations of failure in fiber-reinforced composites

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TÉCNICA SUPERIOR DE INGENIEROS DE CAMINOS, CANALES Y PUERTOS. High fidelity simulations of failure in fiber-reinforced composites. TESIS DOCTORAL. SERGIO SÁDABA CIPRIAIN Ingeniero Industrial. 2014.

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(3) Departamento de Ciencia de Materiales Escuela Técnica Superior de Ingenieros de Caminos, Canales y Puertos Universidad Politécnica de Madrid. High fidelity simulations of failure in fiber-reinforced composites. TESIS DOCTORAL. Sergio Sádaba Cipriain Ingeniero Industrial Directores de Tesis Carlos Daniel González Martı́nez Dr. Ingeniero de Caminos, Canales y Puertos Javier LLorca Martı́nez Dr. Ingeniero de Caminos, Canales y Puertos 2014.

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(5) A Blanca y Mikel a mis padres Mateo y Nieves, a mi hermana Silvia..

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(7) Agradecimientos. A pesar de figurar como único autor de esta tesis, muchas más personas han contribuido significativamente. En primer lugar me gustarı́a agradecer a mis directores, Carlos y Javier, la confianza depositada en mı́ para formar el embrión del Instituto IMDEA Materiales. A Carlos por su disponibilidad para analizar y discutir cualquier aspecto de la investigación dentro y fuera del trabajo. A Javier, quien a pesar de no estar tan pegado al fragor de la batalla, ha aportado consejos e indicaciones determinantes para llegar a buen término en un tema tan complejo, farragoso y lleno de trampas como es la mesoescala. Ambos me han dado todo el apoyo profesional, pero sobre todo personal, para llegar a finalizar la tesis. Sin sus consejos, orientaciones, apoyo y ánimos nunca hubiera podido llegar tan lejos. Algunas partes fundamentales de la presente tesis se han apoyado en el trabajo de otras personas: Rocı́o Seltzer, Alejandro Enfedaque (UPM) y Airbus España realizaron parte de los ensayos de la tesis. Tambien en el Departamento de Ciencia de Materiales de la E.C.C.yP., donde además de ayudarme con los ensayos me acogieron como un compañero más. Otras personas sin cuya colaboración este trabajo no habrı́a sido posible, son Federico Sket y Juan Carlos Rubalcaba con el tomógrafo, Joaquim Vila con las mediciones de C-Scan, Emili Gonzales (Amade) suministrándome el código de elementos cohesivos para Abaqus –utilizado para implementar ideas con menos esfuerzo–, Claudio Lopes con el acceso a códigos de mecánica del daño continuo en composites –lo que ha permitido confirmar mis conclusiones con independencia de la implementación–, Andreas Zilian y Thomas-Peter Fries (RWTH Aachen) distribuyeron amablemente el código del “stable X-FEM” en Matlab, lo que me permitió concentrarme en los aspectos metodológicos y crear un demostrador del nuevo concepto “stable cohesive X-FEM” en un tiempo mucho más breve. El apoyo financiero, imprescindible para llevar a cabo esta investigación, ha llegado del proyecto europeo MAAXIMUS, del séptimo programa marco, liderado por Airbus. El resultado de la tesis no habrı́a sido de la misma calidad sin las importantes contribuciones del profesor Ignacio Romero al Capı́tulo 4, principalmente su visión distinta y complementaria, que me permitió afrontar dificultades fundamentales con técnicas novedosas en el campo de la simulación de daño en compuestos. Pero sin duda su contribución fundamental fue hacerme pasar del pesimismo al optimismo..

(8) En el plano profesional, y también en el personal, ha sido importante la ayuda de Javier Segurado, Álvaro Ridruejo y Luis Pablo Canal con quienes he intercambiado innumerables ideas, ası́ como la de Raúl Muñoz al que también debo parte del trabajo de esta tesis. También deseo expresar mi agradecimiento a mis compañeros de mesa Marcos Rodrı́guez y Francisca Martı́nez, con quienes tuve interesantı́simas discusiones en la pantalla del ordenador, y a mis compañeros de grupo: Diego, Miguel, Alejandro, Fernando, Vanesa, Silvia, Mohammad. Tampoco puedo olvidar a mis otros compañeros, tanto en mis inicios en la UPM como en las diferentes etapas del Instituto IMDEA Materiales, quienes son demasiados para figurar aquı́. Un investigador me dijo una vez que donde un ingeniero ve problemas, él ve oportunidades. El escollo fundamental de la tesis no han sido ni las dificultades del daño continuo, ni Linux, ni Fortran, ni X-FEM, ni los espacios discretos, ni Babuska-Brezzi, ni Korn, sino la grave enfermedad que padecı́ coincidiendo con el ecuador de la tesis, que me apartó de la investigación y la ingenierı́a durante mucho tiempo. La superé gracias al equipo médico liderado por los doctores Vila-Costa y Maroto del Hospital Clı́nico en la fase aguda, y el proceso de rehabilitación en la Clı́nica Ubarmin, Gorostiaga y Tirapu, durante largos meses. De no haberse dado tan duras circunstancias, no hubiera podido sentir hasta este punto el cariño y el apoyo de los mı́os. En todos estos años también he aprendido que para realizar una tesis se requiere de mucho más que buenas ideas y un poco de suerte. En mi caso han sido fundamentales el apoyo de mis padres, del que he gozado siempre, también a la hora de trasladarme a Madrid, el de mi hermana Silvia que siempre estaba ahı́ cuando la necesitaba, en Madrid y en Pamplona. Por supuesto, Blanca y Mikel, que han tenido que soportar un esposo y padre que ha tenido que conciliar el tiempo dedicado a la tesis con el dedicado a la familia. Mis amigos, sin cuyo apoyo jamás habrı́a podido terminar: Alberto, Mikel, Juan Pablo, Miguel, Jesús, Eva, David, Josetxo, Joaquı́n, Esther, Paco, Fernando, Bea, Noemı́, Konstantina y muchı́simos otros que no tengo espacio para mencionar aquı́. Sin todos ellos, esta tesis nunca habrı́a sido posible. Sergio Sádaba Madrid, 4 de Junio de 2014.

(9) Resumen. Los ensayos virtuales de materiales compuestos han aparecido como un nuevo concepto dentro de la industra aeroespacial, y disponen de un vasto potencial para reducir los enormes costes de certificación y desarrollo asociados con las tediosas campañas experimentales, que incluyen un gran número de paneles, subcomponentes y componentes. El objetivo de los ensayos virtuales es sustituir algunos ensayos por simulaciones computacionales con alta fidelidad. Esta tesis es una contribución a la aproximación multiescala desarrollada en el Instituto IMDEA Materiales para predecir el comportamiento mecánico de un laminado de material compuesto dadas las propiedades de la lámina y la intercara. La mecánica de daño continuo (CDM) formula el daño intralaminar a nivel constitutivo de material. El modelo de daño intralaminar se combina con elementos cohesivos para representar daño interlaminar. Se desarrolló e implementó un modelo de daño continuo, y se aplicó a configuraciones simples de ensayos en laminados: impactos de baja y alta velocidad, ensayos de tracción, tests a cortadura. El análisis del método y la correlación con experimentos sugiere que los métodos son razonablemente adecuados para los test de impacto, pero insuficientes para el resto de ensayos. Para superar estas limitaciones de CDM, se ha mejorado la aproximación discreta de elementos finitos enriqueciendo la cinemática para incluir discontinuidades embebidas: el método extendido de los elementos finitos (X-FEM). Se adaptó X-FEM para un esquema explı́cito de integración temporal. El método es capaz de representar cualitativamente los mecanismos de fallo detallados en laminados. Sin embargo, los resultados muestran inconsistencias en la formulación que producen resultados cuantitativos erróneos. Por último, se ha revisado el método tradicional de X-FEM, y se ha desarrollado un nuevo método para superar sus limitaciones: el método cohesivo X-FEM estable. Las propiedades del nuevo método se estudiaron en detalle, y se concluyó que el método es robusto para implementación en códigos explı́citos dinámicos escalables, resultando una nueva herramienta útil para la simulación de daño en composites..

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(11) Abstract. Virtual testing of composite materials has emerged as a new concept within the aerospace industry. It presents a very large potential to reduce the large certification costs and the long development times associated with the experimental campaigns, involving the testing of a large number of panels, sub-components and components. The aim of virtual testing is to replace some experimental tests by high-fidelity numerical simulations. This work is a contribution to the multiscale approach developed in Institute IMDEA Materials to predict the mechanical behavior of a composite laminate from the properties of the ply and the interply. Continuum Damage Mechanics (CDM) formulates intraply damage at the the material constitutive level. Intraply CDM is combined with cohesive elements to model interply damage. A CDM model was developed, implemented, and applied to simple mechanical tests of laminates: low and high velocity impact, tension of coupons, and shear deformation. The analysis of the results and the comparison with experiments indicated that the performance was reasonably good for the impact tests, but insufficient in the other cases. To overcome the limitations of CDM, the kinematics of the discrete finite element approximation was enhanced to include mesh embedded discontinuities, the eXtended Finite Element Method (X-FEM). The X-FEM was adapted to an explicit time integration scheme and was able to reproduce qualitatively the physical failure mechanisms in a composite laminate. However, the results revealed an inconsistency in the formulation that leads to erroneous quantitative results. Finally, the traditional X-FEM was reviewed, and a new method was developed to overcome its limitations, the stable cohesive X-FEM. The properties of the new method were studied in detail, and it was demonstrated that the new method was robust and can be implemented in a explicit finite element formulation, providing a new tool for damage simulation in composite materials..

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(13) Contents. 1 Introduction.. 1. 1.1. Fiber-reinforced polymers . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1. 1.2. Failure in polymer-matrix composites . . . . . . . . . . . . . . . . . . . . .. 5. 1.2.1. Intralaminar and interlaminar failure modes . . . . . . . . . . . . .. 6. 1.2.2. Failure in laminate coupons . . . . . . . . . . . . . . . . . . . . . .. 9. 1.2.3. Mesomechanical size effects . . . . . . . . . . . . . . . . . . . . . .. 11. Mesomechanical failure models . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 1.3.1. Strength of unidirectional laminae . . . . . . . . . . . . . . . . . . .. 15. 1.3.2. Damage in composite materials . . . . . . . . . . . . . . . . . . . .. 16. 1.3.3. Computational CDM for multiple failure modes . . . . . . . . . . .. 19. 1.3.4. Interlaminar failure models . . . . . . . . . . . . . . . . . . . . . . .. 21. 1.3.5. Strong discontinuity approaches . . . . . . . . . . . . . . . . . . . .. 22. Summary of the thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 1.4.1. Objectives. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 25. 1.4.2. Organization of the thesis. . . . . . . . . . . . . . . . . . . . . . . .. 25. 1.3. 1.4. 2 The Continuum Damage Approach. 27. 2.1. Introduction to Continuum Damage Mechanics . . . . . . . . . . . . . . . .. 27. 2.2. CDM for unidirectional laminates . . . . . . . . . . . . . . . . . . . . . . .. 30. I.

(14) Contents. 2.3. 2.4. 2.5. 2.6. 2.2.1. Intralaminar failure: CDM with LaRC04 . . . . . . . . . . . . . . .. 30. 2.2.2. Interlaminar failure: the cohesive element. . . . . . . . . . . . . . .. 43. Low velocity impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 2.3.1. Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . .. 48. 2.3.2. Numerical simulation of drop weight impact . . . . . . . . . . . . .. 56. 2.3.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61. High velocity impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. 2.4.1. Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . .. 69. 2.4.2. Numerical simulations of ice impact . . . . . . . . . . . . . . . . . .. 73. 2.4.3. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 75. On CDM scope and validity . . . . . . . . . . . . . . . . . . . . . . . . . .. 79. 2.5.1. The continuum problem: from plasticity to localization . . . . . . .. 79. 2.5.2. Locking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 81. 2.5.3. The microstructure governs the post failure behaviour . . . . . . . .. 82. 2.5.4. Behaviour under tension: the plain tension test . . . . . . . . . . .. 85. 2.5.5. Behaviour under shear: the V-notch test . . . . . . . . . . . . . . .. 90. Conclusions and way forward . . . . . . . . . . . . . . . . . . . . . . . . .. 95. 3 The strong discontinuity approach. 3.1. 97. The eXtended Finite Element Method, X-FEM . . . . . . . . . . . . . . .. 97. 3.1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 97. 3.1.2. Basics of the FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . 100. 3.1.3. Partition of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104. 3.1.4. Enrichment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105. 3.1.5. The level set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110. 3.1.6. The X-FEM method: LEFM crack tip vs cohesive crack . . . . . . 110 II.

(15) Contents 3.1.7 3.2. 3.3. 3.4. 3.5. The cohesive X-FEM problem statement . . . . . . . . . . . . . . . 112. An implementation of X-FEM cohesive approach in explicit dynamics . . . 119 3.2.1. A Phantom Node integration scheme . . . . . . . . . . . . . . . . . 120. 3.2.2. Mass matrix integration . . . . . . . . . . . . . . . . . . . . . . . . 125. Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 3.3.1. The explicit time incrementation scheme . . . . . . . . . . . . . . . 127. 3.3.2. Integration in Abaqus/Explicit . . . . . . . . . . . . . . . . . . . . 128. 3.3.3. Model preprocessing, solution and postprocessing . . . . . . . . . . 132. 3.3.4. Benckmark example: the Double Cantilever Beam test . . . . . . . 135. Application: tensile test of [±45]S laminates . . . . . . . . . . . . . . . . . 138 3.4.1. Laminate model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138. 3.4.2. Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141. 3.4.3. Some remarks on X-FEM results obtained . . . . . . . . . . . . . . 146. Limitations of X-FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.5.1. Time continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148. 3.5.2. Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151. 3.5.3. The discrete space: mixed locking . . . . . . . . . . . . . . . . . . . 153. 4 A stable cohesive X-FEM for the transition from closed to open crack. 155 4.1. 4.2. Introduction to embedded discontinuity methods . . . . . . . . . . . . . . . 155 4.1.1. Discrete variational strategies in the embedded cohesive interaction. 156. 4.1.2. The embedded interface in X-FEM . . . . . . . . . . . . . . . . . . 157. 4.1.3. Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160. Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 4.2.1. The cohesive crack . . . . . . . . . . . . . . . . . . . . . . . . . . . 160. 4.2.2. The strong form of the problem . . . . . . . . . . . . . . . . . . . . 162 III.

(16) Contents 4.2.3 4.3. 4.4. 4.5. The potential energy . . . . . . . . . . . . . . . . . . . . . . . . . . 162. The stable cohesive X-FEM . . . . . . . . . . . . . . . . . . . . . . . . . . 163 4.3.1. Description of the method . . . . . . . . . . . . . . . . . . . . . . . 163. 4.3.2. Variational formulations . . . . . . . . . . . . . . . . . . . . . . . . 166. 4.3.3. Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167. 4.3.4. Relationship with known formulations . . . . . . . . . . . . . . . . 170. Numerical examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 4.4.1. Embedded methods: the closed crack . . . . . . . . . . . . . . . . . 178. 4.4.2. The stable cohesive X-FEM . . . . . . . . . . . . . . . . . . . . . . 182. 4.4.3. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185. 4.4.4. Condition number and Courant stability criterion . . . . . . . . . . 187. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189. 5 Conclusions. 191. 5.1. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191. 5.2. Original contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193. 5.3. Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194. A Periodic Boundary Conditions in Abaqus/Explicit. 195. B Abaqus/Explicit technical notes. 201. B.1 Lamina modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 B.2 Cohesive interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 B.2.1 The cohesive element . . . . . . . . . . . . . . . . . . . . . . . . . . 203 B.2.2 The modified cohesive law . . . . . . . . . . . . . . . . . . . . . . . 205 B.2.3 The cohesive surface . . . . . . . . . . . . . . . . . . . . . . . . . . 206 B.3 Large shear strain in Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . 206 IV.

(17) Contents C Code of X-FEM implementation in Abaqus. 209. C.1 Python scripts: preprocessing . . . . . . . . . . . . . . . . . . . . . . . . . 209 C.2 Input file: model organization . . . . . . . . . . . . . . . . . . . . . . . . . 210 C.3 Fortran codes: solution time . . . . . . . . . . . . . . . . . . . . . . . . . . 213 C.3.1 User material routine: VUMAT . . . . . . . . . . . . . . . . . . . . 213 C.3.2 User element routine: VUEL . . . . . . . . . . . . . . . . . . . . . . 215 D Analytical solution of a circular inclusion with compliant interface under uniaxial traction. 221. D.1 Compatibility, constitutive, and equilibrium relations . . . . . . . . . . . . 222 D.2 Hydrostatic pressure loading . . . . . . . . . . . . . . . . . . . . . . . . . . 223 D.3 Shear strain loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 Bibliography. 246. V.

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(19) Chapter. 1. Introduction.. “If a man will begin in certainties he shall end in doubts; but if he will be content to begin in doubts he shall end in certainties.” Sir Francis Bacon. 1.1. Fiber-reinforced polymers. Fiber-reinforced polymers (FRP) stand out among structural materials because of their unique combination of properties. The reinforcement with high-performance fibers provides a stiffness and strength comparable to those of metallic alloys, the structural materials par excellence, and very often higher if the comparison is carried out in terms of specific properties, figure 1.1a. In addition, structural composites can dissipate a large amount of energy upon fracture (in the range ≈10–50 kJ/m2 ), leading to a flaw-insensitive, damage-tolerant behaviour that is not prone to catastrophic failure, figure 1.1b. This combination of stiffness, strength, and toughness is only found in metallic alloys, in which stiffness comes from 1.

(20) 1.1 Fiber-reinforced polymers the strong metallic bonding, whereas energy dissipation during fracture is ensured by plas216. tic deformation and different strategies (alloying, second-phase precipitation, and grain-size reduction) can be used to increase strength while maintaining acceptable toughness values. Polymers and rubbers can also dissipate large amounts of energy during deformation, but their strength and stiffness are limited, whereas ceramics are stiff and strong, but brittle, as a result of their ionic or covalent bonding, which hinders dislocation motion. The combination of high specific stiffness and strength as well as toughness in the absence of plastic deformation can also be found in structural biomaterials (Elices (2000)), such as tendons, bones, seashell nacre, and spider silk. In all cases, the superior mechanical properties of these materials in terms of stiffness, strength, and particularly, of toughness can be traced to their hierarchical microstructure, which leads to the simultaneous activation of different deformation and energy dissipation mechanics at length scales from nm to 216 component level, dramatically enhancing the damage LLorca, González,Espinosa Molina-Aldareguı́a, and Lópes the resistance, et al. (2012).. of a crack perpendicular to the tensile axis (Fig. 3a),8 whereas if the ply is loaded in compression perpendicular to the fibers, then final failure occurs by formation of a shear band of localized plastic deformation in the matrix across the ply (Fig. 3b).9 Tensile loading parallel to the fibers leads to brittle fracture controlled by the fibers, whereas a compression along the fibers induces failure by fiber kinking.10 These failure modes control the mechanical behavior of unidirectional plies, but other energy-dissipation mechanisms develop during deformation of multidirectional laminates made up by stacking unidirectional plies in different orientations. They include multiple cracking of the plies parallel to the fibers, as well as fiber bridging, followed by fiber fracture and pull out (Fig. 4), dramatically increasing the energy dissipated during fracture of multidirectional laminates. In addition, interply decohesion often develops from the laminate edges as a result of the stress concentration caused by the mismatch in the elastic properof adjacent plies or from the bifurcation Fig.ties 1. Comparison of the mechanical properties of composites along with the interply of intraply cracks. Finally, Figure 1.1: Comparison of the mechanical properties of composites with other materials. other materials. (a) Specific stiffness versus specific strength andother (b) strength versus fracture toughness.with CFRP,the carbon-fiber reinforced mechanisms associated structural defor(a) Specific stiffness versus specific strength and (b) strength versus fracture toughness. polymers; GFRP, glass–fiber reinforced polymers. crushing, and mation of the component (buckling, CFRP, carbon-reinforced polymers; GFRP, glass-fiber reinforced polymers. LLorca to et al. large deformations) also contribute the total energy absorption of composite structures. (2013) Obviously, the presence of different energy dissilaminate thickness,atrespectively) in for a a pation mechanics various lengtharranged scales calls hierarchical fashion (Fig. 2). Carbon fiber diameters multiscale modeling Although thisin challenge Structural components manufactured with are FRP areorder made up strategy. of laminates that turn 12 of the of 5–10 lm, whereas ply thicknesses was clear to scientists and engineers, it was only arevery in the range 100–300 lm,This and leads standard lami- to are obtained by stacking individual plies with different fiber orientation. to three recently that a coherent multiscale approach nates are out several mm in simulations thickness and greater. carrying high-fidelity of the mechanUpon mechanical loading, different deformation and different entities (ply, laminate, and component) with three different characteristic ical performance of composite laminates waslength proposed 7 failure mechanisms take simultaneously and validated. This newplace strategy comes about at as a theresult three scales depicted 2. Within scales (fiber diameter, ply, and laminate thickness, respectively) arranged ininaFig. hierarchical of length recent advances in different areas, including each ply, the main damage mechanisms on micromechanical characterization of depend constituent theproperties, load orientation with respect to the fibers. If thefor development of accurate modeling tools 2 plycomposites is subjected to tensile stresses perpendicular to at the microscale and mesoscale, coherent thestrategies fibers, then matrix failure leads to thelength formation to pass information between scales, and the use of parallelization techniques to increase the power of digital computers. As a result, ‘‘virtual tests’’ are starting to be used in engineering applicaFig. 1. Comparison of the mechanical properties of composites with tions to limit the number of costly experimental tests other materials. (a) Specific stiffness versus specific strength and (b) to certify the safety of composite structures, and to strength versus fracture toughness. CFRP, carbon-fiber reinforced reduce development time. In addition, this strategy polymers; GFRP, glass–fiber reinforced polymers. makes structural design with nonconventional lami-. LLorc. of a crack pe whereas if pendicular t formation o deformation Tensile load fracture con pression alo kinking.10 mechanical other energy ing deforma up by stacki entations. T plies paralle followed by dramatically ing fracture tion, interpl laminate ed tion caused ties of adjac the interply mechanisms mation of th large deform energy abso Obviously, pation mech multiscale m was clear to very recently carrying out ical performa and validate result of rece micromechan properties, d composites a strategies to and the use the power of tests’’ are sta tions to limit to certify th reduce devel makes struc nates possib nontrivial lo tures of this presented in future direc engineering. MULTISCA. Multiscale been attemp approach. I.

(21) 1.1 Fiber-reinforced polymers fashion, figure 1.2. Carbon fiber diameters are of the order of 5-10 µm, whereas ply thicknesses are in the range 100-300 µm, and standard laminates are several mm in thickness and larger. Upon mechanical loading, different deformation and failure mechanisms take place simultaneously at the three length scales depicted in figure 1.2. FRP are nowadays extensively used in applications where outstanding mechanical properties areModeling necessary in combination withTesting weight savings. in Multiscale of Composites: Toward Virtual … and Beyond Good examples can be found 217 the A380 and A350XWB, the last civil Airbus aircrafts containing up to 25% and 53% in performance of the whole structure is analyzed. point, the phenomenological nature of the composite. identify the critical regions in which damage is. sults to design optimized lay-up configurations or to. refined analyses using phenomenological models for. that present very different damage mechanisms. weight of composite materials, respectively for wings, tailthe surmaterial modelsfuselage limits itssections ability toand extend re(normally using the finite-element method) to (used faces) while the These Boeingregions 787 Dreamliner to be out the‘‘virtual first airliner with fully composite carry tests’’ on structural components likely to develop. are subjectedclaims to fuselage manufactured advanced carbon component upon loading.technologies. However, despite the composite behavior thatwith include damage. These. As opposed to this strategy, a new approach that models contain a number of parameters whose valall existing information and current knowledge about these materials, the accurate predicfollows the hierarchical structure of composite ues are chosen to reproduce the actual material 7. materials has been validated. behavior as afailure result ofstress experience and costly materials testing The tion of the of composite and structures hasproposed been anand elusive problem 13 campaigns.. overall multiscale simulation scheme is depicted in advantage of the natural separa-. Although this strategy has proven. Fig. 5 and takes very useful the structuralofengineering viewbecause of from the complexity their failure micromechanisms.. (a). ply fiber diameter ≈ 10 µm. matrix shear yielding interface decohesion fiber fracture fiber kinking. (b). (c) Component. Laminate ply thickness 150-300 µm. laminate thickness 2-20 mm. crack bridging fiber pull-out matrix cracking interply decohesion. buckling crushing. Fig. 2. Hierarchical structure of composite materials, showing the three entities, their relevant length scale and their dominant energy dissipation Figure 1.2: Hierarchical structure of composite materials, showing the three entities, mechanisms. (a) ply, (b) laminate, and (c) component.. their relevant length scale and their dominant energy dissipation mechanisms. (a) ply, (b) laminate, and (c) component.. In absence of accurate models to predict the failure strength of composites, the burden of testing is immense so as to prove the safety in composite structures upon whose integrity human lives depend: a typical large airframe, for example, currently requires up to 104 tests of material specimens, along with tests of components and structures up to entire tails, 3. Fig. 3. Scanning electron microscopy of ply damage by loading perpendicular to the fibers. (a) Tensile damage by interface decohesion and matrix fracture at the notch root in epoxy/glass FRP.8 (b) Compressive damage by the formation of plastic shear band in the matrix in an epoxy/C.

(22) 1.1 Fiber-reinforced polymers wing boxes, and fuselages, to achieve safety certification, Cox & Yang (2006), see figure 1.3. This structure, called the ‘building block’ approach, MIL-HDBK-17-1F (2002), relies on a pyramid where the complexity of the test specimens, and the subsequent cost, drastically increases on the next level, although the number of specimens is reduced. Such approach has proven its validity and efficiency in the last decades, being still applied for aircraft development. However, the certification process involves huge costs associated with the large number of tests required and with the long time spent from the definition of a new material to its application. To overcome these limitations, two different multiscale strategies have been developed to predict failure of composite materials. The top-down approach (also known as the global-local), begins by the analysis of the composite structure, that is enriched in those regions where failure is expected to occur. In this strategy, the loads applied to the global model are transmitted down the scale to hot-spots where stresses are expected to trigger damage and fracture in the composite material. Experimental Testing Simulation. FULL‐SCALE. CURVED PANEL. STRUCTURAL COMPONENTS. STRUCTURAL DETAILS. COUPON TESTS. Figure 1.3: Traditional pyramid of physical tests performed during a Technology Development, the ‘top-down’ approach.. On the contrary, the bottom-up approach is aimed to predict failure of composite materials and their structures starting from the scale of the constituents, fiber, matrix 4.

(23) 1.2 Failure in polymer-matrix composites Multiscale Modeling of Composites: Toward Virtual Testing … and Beyond. 219. and interfaces, moving updispersion to the a single laminate, component region,ply, and this leads to the development of damageand the final containing a random of a scale few dozenof fibers is enough to accurately reproduce the ply behavior. in front of the notch tip upon loading (Fig. 6b). The. mation and damage mechanism experimentally ob-. in Fig. 3a. From the macroscopic load–displacement. simulations with interface elements (to model. possible to determine the ply toughness (as well as. crack propagation from the notch accurately repro- the accurate general loading conditionsetin al. the standard structure, seeunder figure 1.4. LLorca (2011) claim that, this approach allows duces the experimental crack path, which is shown case unidirectional reinforcement. The main defor-. prediction, free of inadjustable behaviour until failure of 6c), composite coupon of this ‘‘virtual’’ fracture test (Fig. it is served the constituentsparameters, are introduced in of thethecurve the increase in fracture resistance with crack interface decohesion) or the appropriate constitutive specimens and simple components. The bottom-up approach relies on the analysis of length) using the standard expressions available in models to take into account the plastic deformation and fracture of matrix and fibers. This strategy has. the fracture mechanics textbooks.. unidirectional plies subjected to various in-plane. Computational Mesomechanics. beenmaterial successfully used predict ply, the failure locus of the composite atto the laminate and component level using computational 17–24. and out-of-plane loading conditions. Finally, Computational mesomechanics,mechanics, the second step respectively micromechanics, computational mesomechanics and computational computational micromechanics can also be used to in the multiscale modeling strategy, also uses the determine the fracture resistance of a ply,8,25 although the modeling strategy is slightly different because damage is localized and propagates from the tip of a sharp notch (Fig. 3a). The simulations are carried out within the framework of an embedded cell model (Fig. 6a) in which the full details of the composite microstructure are resolved in the fracture region. The remaining ply material is represented as a homogeneous, transversally isotropic solid whose behavior is given by any suitable homogenization model. Of course, matrix fracture and interface decohesion are included in the corresponding constitutive equations in the embedded. finite-element method to determine the mechanical. (figure 1.4). Each simulation strategy providesresponse the information to carry out the of laminates (Fig. 5).necessary The virtual laminate is built by stacking plies with different fiber orien-. tation (h), and the geometrical model explicitly insimulation at the next level. This thesis is focused on the computational mesomechanics. cludes each ply as well as the interfaces between plies (Fig. 7a). Meshing of the laminate is carried. out with solid elements for the plies, while cohesive The mesomechanical modeling strategy assumes ply(orand interply properties (moduli, interfacethe elements cohesive surfaces) are used to take into account the ply interfaces. In this way,. intraplysequence and interply damage can be introduced strength and toughness), and the laminate stack as data. The output of the separately together with the complex interaction between them.. mesomechanical simulations is the laminate behaviour. input matrix, fiber, interface properties & spatial distribution. Multiscale. Computational micromechanics. output ply behavior. input ply & interply properties laminate stack. Computational mesomechanics input laminate behavior component geometry. output laminate behavior. Computational mechanics. output Component behavior Fig. 5. Local-to-global multiscale simulation strategy to carry out virtual mechanical tests of composite materials.. Figure 1.4: Multiscale simulation strategy: from the fiber-matrix microstructure to the component.. 1.2. Failure in polymer-matrix composites. The multiscale approach depicted above considers three failure modes at the ply level (fiber breakage, matrix failure, fiber-matrix decohesion) that lead to the failure at the laminate level: the intralaminar and interlaminar failure modes. 5.

(24) Fig. 5 and takes advantage of the natural separa-. very useful from the structural engineering view-. (a). ply. (b). fiber diameter ≈ 10 µm. (c) Component. Laminate ply thickness 150-300 µm. laminate thickness 2-20 mm. 1.2 Failure in polymer-matrix composites. 1.2.1. Intralaminar and interlaminar failure modes. The possible failure modes at the microscopic level are fiber breakage, matrix cracking and fiber-matrix decohesion, which lead to the ply failure. For example, fiber-matrix decohesions appear in the 100 µm image, which lead to an apparent crack in the 400 µm image (in the plane perpendicular to fiber direction) emerging from a notch, figure 1.5a. In matrix shear yielding. buckling. crack bridging. crushing figure 1.5b, matrix plasticity appears in the 10 µm image, which give rise to an apparent interface decohesion fiber pull-out fiber fracture. matrix cracking. shear fiber bandkinking in the 100 µm image. observed failure modes at the lamina level are interply The decohesion enumerated next.. Fig. 2. Hierarchical structure of composite materials, showing the three entities, their relevant length scale and their dominant energy dissipation mechanisms. (a) ply, (b) laminate, and (c) component.. Fig. 3. Scanning electron microscopy of ply damage by loading perpendicular to the fibers. (a) Tensile damage by interface decohesion and Figure Scanning electron of ply damage byplastic loading perpendicular Compressive damage by the formation of shear band in the matrix in anto epoxy/C matrix fracture at the1.5: notch root in epoxy/glass FRP.8 (b)microscopy FRP9. the fibers. (a) Tensile damage by interface decohesion and matrix fracture at the notch. root in epoxy/glass FRP, Canal et al. (2012). (b) Compressive damage by the formation of plastic shear band in the matrix in an epoxy/carbon FRP, González & LLorca (2007).. Matrix tensile cracking Tensile matrix cracks appear perpendicular to the loading direction due to fiber-matrix decohesion, figure 1.5a. The intact neighbour plies create a supporting effect: the matrix cracking may not lead to catastrophic failure, Talreja & Singh (2012), and characteristic microcracking patterns arise. 6.

(25) 1.2 Failure in polymer-matrix composites S.T. Pinho et al. / Composites Science and Technology 66 (2006) 2069–2079. 2077. (c). (d). Figure 8. Sequence of events during kink-band formation. The laminate is being loaded in compression in the vertical direction.. Misalignment introduced by a matrix crack in(b) an adjacent layer. (b) Matrix-fibre splitting fiber exists throughout (see zoom); the first fibre (a) Tensile fiber failure(a)failures Compressive failure are indicated. (c) Further fibre failure. (d) Final kink band. Legend. K Matrix cracking; L Fibre failure.. Figure 1.6: Scanning electron microscopy of ply damage by loading parallel to the fibers. (a) Fiber failure upon tensile loads in a carbon cross-ply laminate: fiber pull-out and fiber breakage are observed in 0o plies, Pinho et al. (2006). (b) Fiber failure by compression parallel to the fibers by the formation of a kink band, Pinho et al. (2012). Matrix compressive failure Under compression perpendicular to the fibers, failure occurs by the propagation of a plastic shear band. The shear band for compression-dominated loads forms an angle of. Fig. 10. (a) SEM micrograph of the CT specimen’s fracture surface; (b) the magnitude of fibre pull out depends on the distance to the 90 layers.. 53-56o with the normal to the compression axis (figure 1.5b) in epoxy matrices, Totry et al. (2008b).. In-plane shear failure 11. C-scan of a (a) CT specimen and (b) CC specimen. DuringFig.shear deformation, the composite material exhibits large deformations before. localization occurs, Totry et al. (2008a). In addition, “fiber scissoring” occurs due to the matrix degradation: the material loses integrity and the fibers rotate towards the load direction, Wisnom (1995), Herakovich et al. (2000).. Fiber pullout and fracture Tensile stresses along the fiber direction lead to fiber breakage and matrix-fiber decohesion, leading to fiber pull-out. In many cases, the development of fiber breakage determines the maximum load supported by the laminate, figure 1.6a. 7.

(26) Table 2 Mode III interlaminar fracture toughness of glass/epoxy Vf. 0.26 0.56. GIIIC (kJ/m2 ) NL. 5% Offset. 3.79 0.79 1.23 0.09. 4.20 0.56 1.48 0.18. Fig. 7. Optical micrographs of fracture surfaces. (a) Vf ¼ 0:26; (b). Vf ¼ 0:56. 1.2 Failure in polymer-matrix composites. / Composites Science and Technology 64 (2004) 1279–1286. 1285. that the toughness, GIIIC , evaluated from the point at onset of nonlinearity and the load corresponding to a 5% compliance offset are quite similar reflecting the small extent of nonlinear response prior to crack propagation, see Fig. 3. There is a substantial decrease of GIIIC with increasing fiber volume fraction. Table 3 lists GIIIC determined here and previously reported mode I, II, and III interlaminar fracture toughness data for E-glass/epoxy and carbon/epoxy composites [10,17]. It should be pointed out that the data in Table 3 can be used only as a reference, not for quantitative comparison purposes because the carbon/ Fig. 8. Optical of specimen edge, specimen (a) Micrography of a shear delamination (b) micrograph Fiber bridging between laminashalves held open to show fiber bridging. Vf ¼ 0:26. epoxy and E-glass/epoxy composites examined in [10,17] Figure 1.7: (a) Scanning electron microscopy of a delamination surface, with matrixdominated shear failure. (b) Two composite laminas with fiber bridging at the interply Table 3 Mode I, II, and III interlaminar fracture toughnesses E-glass and carbon fiberLi reinforced epoxya crack, observed during for a delamination test. et al. (2004). Composite. Vf. Glass/epoxy [17] Carbon/epoxy [10] Fiber a b. 0.53 0.60 kinking. GIC (kJ/m2 ). GIIC (kJ/m2 ). GIIIC (kJ/m2 ). 0.25 0.015 0.32 0.03. 1.79 0.21 1.06 0.10. 1.23 0.09b 1.42 0.12. Data were determined using the load at onset of nonlinearity (NL). Determined in this study. Vf ¼ 0:56.. e surfaces, initiation ¼ 0:56.. mena are mainly opagation. The values, obtained ad–displacement. large difference gh and low fiber ary to more clothe starter film. ron microscopy h materials, and ble on the lower For the material sin is apparent, order of 100 lm the fiber–matrix on specimens are nsive matrix deareas of matrix les or cusps seen. Compressive stresses along the fiber direction lead to failure by the appearance of a kink band, consisting of two sliding planes of fracture, figure 1.6b. Fig. 10. Scanning electron micrographs of fracture surfaces, higher magnification of shear cracks in region at end of starter film. Vf ¼ 0:26.. Out-of-plane crushing on mode II specimen fracture surfaces are observed, but the mode III cracks appear shallower and are oriented at 45° to the crack propagation direction. Mode II hackles Out-of-plane crushing typically occurs under pre-stressed bolts or at tend to be perpendicular to this direction (see, for example [18]). There is a thick matrix region in front of the during thespecimens, large out-of-plane shear and compressive stresses lead starterimpact: film in these which allows the plastic zone to develop without encountering fiber/matrix infollowed by lamina piercing. terfaces and dissipate strain energy before macroscopic crack propagation (and nonlinearity on the load–displacement plot) occurs. This appears to be the main cause of the higher measured values of GIIIC .. the contact point to fiber breakage,. Interlaminar 6. Conclusions failure: delamination Mode III fracture toughness has been characterized for a glass/epoxy composite with two widely differing Thevolume interply regions are regarded as the fiber fractions. A large increase in the mode III weakest part of toughness was observed with decreased fiber volume casefraction. of a unidirectional (UD) composite, the ply interfaces Fracture surfaces of broken specimens were. a composite laminate. In the suffer additional stresses due. to the stiffness mismatch between plies. The interplies are resin-rich regions, and failure occurs as the normal or shear stresses overcome the strength of the resin, figure 1.7a. 8.

(27) 1.2 Failure in polymer-matrix composites. 1.2.2. Failure in laminate coupons. The multiscale approach considers that the failure modes indicated above (matrix tensile failure, matrix compressive faiure, fiber fracture and kinking, in-plane shear failure, crushing and delamination) lead to the failure at the laminate level. Additionally, the ply failure modes described above are rarely presented alone and different failure patterns appear depending on the laminate sequence and loading conditions; that is, the individual failure of each ply influences the delamination and the stress distribution in the other plies. Before approaching a complex laminate, the problem has to be understood in simpler laminate configurations. Unless otherwise specified, the examples below are loaded in uniaxial tension, so that the applied strain is the same in all plies. The load direction is 0o .. % applied stress Figure 1.8: Failure stages in a cross ply laminate: first, the crack density increases with the load, typically to a crack density saturation (CDS) value. Subsequent loading causes initiation of cracks transverse to the primary intralaminar cracks in the adjacent plies. Interlaminar shear crack are nucleated — the crack-induced delamination. Finally, the fiber fracture load is reached. The three failure modes compete: 0o lamina fiber breakage, 90o lamina matrix failure and interface shear delamination. Talreja (1987).. Cross-ply laminates [0/90]S In cross-ply laminates, the fibers in the 0o plies sustain the load (until fiber breakage), while the 90o plies maintain the material integrity, figure 1.8. Meanwhile, the 90o plies support tensile stresses, and tensile cracks appear in the matrix. The stress is redistributed 9.

(28) 1.2 Failure in polymer-matrix composites and a crack pattern develops, with an initial crack spacing. The crack density increases as the specimen is further loaded. The cracks may nucleate delaminations (known as crackinduced delaminations) because the shear stress rises close to the intralaminar crack tip. These mechanisms of microcracking and crack-deflection have been extensively studied in the literature, as in Nairn & Hu (1992), Crossman & Wang (1982) or Talreja & Singh (2012).. Angle ply laminates [±θ/90X ]S. The fracture process in angle ply laminates (Johnson & Chang (2001a), Crossman & Wang (1982)), namely [±θ, 90X ]S , includes different competing fracture modes: edge cracking/delamination, matrix microcracking (opening and shear), ply delamination and fiber fracture.. Figure 1.9: X-radiograph image obtained at increasing load levels showing microcrack propagation in a [603 / − 603 /0]S laminate, from Johnson & Chang (2001a). They observed that matrix cracks developed uniformly as the stress increased. These cracks also acted as weak points that initiated delaminations. The delaminations propagated if the available energy exceeded the interply toughness. The opening mode I was dominant at the specimen edge, Pipes & Pagano (1970), Mittelstedt & Becker (2007), while the shear mode II governed the crack-induced delaminations. 10.

(29) 1.2 Failure in polymer-matrix composites Laminate [±45]2S The tensile test of [±45]2S laminate is used to measure the shear properties because the lamina stress state (in absence of damage) is of simple shear. Figure 1.10 shows the characteristic shear stress-strain curve of a [±45]2S loaded in tension. Matrix plasticity, microcracking, fiber rotation (“scissoring” in Herakovich et al. (2000), Wisnom (1995)), edge effects and delaminations are observed. The stress state at the edges produces microcracking that does not propagate into the specimen until the load is close to failure. (6) (5). (1). (2). (3). (4). (1). (4). (2) (5). ( ) (3). (6). Figure 1.10: Microcracking evolution in a plain [±45]2S laminate as a function of the applied load. Adapted from Sket et al. (2014). 1.2.3. Mesomechanical size effects. The behaviour of an isolated ply differs from the behaviour within the laminate; also, the specimen size may determine the appearance of one or another failure mode when they occur with similar (proportional) stress. Such scaling of the failure modes has an effect on the laminate final failure, since the competition is won by a different failure mode at each scale. The failure modes of a composite material do not scale with size as a monolithic material. Rather, the individual failure modes scale in a fashion, but the laminate fails as the first catastrophic failure mode appears, Wisnom et al. (2010). 11.

(30) 1.2 Failure in polymer-matrix composites The in-situ ply strength The in-situ effect, originally detected in Parvizi et al. (1978), is characterized by the higher transverse tensile and shear strengths of a ply when it is constrained by plies with different fiber orientations within a laminate, compared with the strength of the same ply in a unidirectional laminate. The in-situ strength also depends on the number of plies clustered together, figure 1.11, and on the fiber orientation of the constraining plies. The orientation of the constraining plies and the number of plies clustered together also affect the crack density and the stiffness reduction of the cracked ply.. Figure 1.11: In-situ strength: the shear strength of the 90o ply is modified by the constraint of surrounding 0o plies. The graph shows the effect of ply thickness scaling, n. Adapted from Camanho et al. (2006).. Scaling of individual failure modes According to Bažant (2002), the “size” of a monolithic structure has to be compared with the size of the fracture process zone (i.e. ≈1 mm for an epoxy) to determine the. failure type: ductile, brittle or quasi-brittle. Typically, smaller structures are more ductile and tough. These considerations on monolithic size effects are also relevant in polymer composites if ply failure is caused by multiple mechanisms: tensile failure in the fiber direction (with fiber pull out and fiber breakage) and mode I delamination (with interply failure and fiber bridging), Dávila et al. (2009) and Gutkin et al. (2011). 12.

(31) 1.2 Failure in polymer-matrix composites. Figure 1.12: Failure modes in an open hole tensile test (OHT) in a carbon fiber-epoxy composite: brittle (left), pull-out (center), delamination (right). Wisnom et al. (2010).. Competition between failure modes Scaling effects have been studied by Green et al. (2007) in quasi-isotropic symmetric laminates [45m /90m / − 45m /0m ]nS in open hole specimens loaded in tension. m and n represent two different ways of scaling the thickness, ply scaling and sublaminate scaling, respectively. Three types of laminate failure modes appear upon laminate scaling: brittle failure, pull-out failure and delamination failure, figure 1.12. Tensile test have been performed by Wisnom et al. (2010) to study the effect of thicker specimens, with in-plane scaling (all the in-plane dimensions, diameters and lengths, are multiplied by the same scale factor). The ply scaling consists in stacking plies with the same orientation to obtain a laminate with thicker laminas, and the laminate scaling stacks sublaminates. The effect of geometric scaling in strength and failure mode is depicted in figure 1.12. These results show that the failure mode interactions at the mesoscale lead to a very complex transition in failure modes, very different to those found in monolithic materials. Similar analyses were made for tensile unnotched specimens in Wisnom et al. (2008), for unnotched compression in Soutis & Lee (2008), and notched compression in Lee & Soutis (2008).. 13.

(32) 1.2 Failure in polymer-matrix composites. 600. P. 400. P. P. D. 300. D 200. Ply scaling. 100. 0. D. Sublaminate scaling. 2. 4. P. D. 300. 8. 10. B. D. D 200. Sublaminate scaling Ply scaling. 100. 6. D B. 400. 0. 0. P. 500. Strength (MPa). 500. Strength (MPa). 600. P. 0. 5. 10. 15. 20. 25. Hole diameter (mm). Thickness (mm). Figure 1.13: Scaling of the laminate strength with size when varying the specimen dimensions: the nominal ply thickness, the ratios of width-to-hole and length-to-hole size were kept constant at 0.125 mm, 5 and 20. Failure modes: brittle (B), pull-out (P) and delamination (D). Adapted from Wisnom et al. (2010).. 14. 30.

(33) the failure locus of FRP plies uniaxially reinforced with either C or glass fibers.[33–38] The failure locus (Figure 5) indicates the maximum load-bearing capability of the ply under any stress 1.3 Mesomechanical failure models state and it is made up of the intersection of the various smooth Mesomechanical models surfaces,1.3which correspondfailure to the different physical failure modes depicted Figure 1a.theSo far,behaviour the offailure The multiscalein approach assumes that mechanical the lamina islocus–basic known. transferred to the mesoscale includes the (transversally isotropic) elastic elementThe toinformation predict the onset of intraply damage–was obtained constants of the lamina and the failure locus in the stress space, figure 1.14. Some of the by means of phenomenological models, whose accuracy has to modeling approaches below also require the knowledge of the post-failure behaviour (fracture energies associated with each failure mode). This information is used by computational mesomechanics to predict the failure of the laminate under general loading conditions.. Figure 1.14: LaRC04 failure locus in the stress space of a unidirectional FRP ply subjected to normal stresses parallel (σ1 ) and perpendicular (σ2 ) to the fibers and to in-plane shear (τ12 ). The axis arrows indicate the positive direction of each axis. The failure surface is symmetric with respect to the τ12 = 0 plane and only the upper surface is plotted for the sake of clarity. 1 2. Figure 5. Failure locus in the stress space of a unidirectional FRP ply subjected to normal stresses parallel (σ ) and perpendicular (σ ) to the fibers and to in-plane shear (τ12). The axis arrows indicate the positive direction A comprehensive survey of the micromechanical and mesomechanical models can be of each axis. The failure surface is symmetric with respect to the τ12 = 0 found in Talreja & Singh (2012). plane and only the upper surface is plotted for the sake of clarity. 1.3.1. Strength of unidirectional laminae. Phenomenological failure (strength) criteria use experimental data to determine the. wileyonlinelibrary.com bH & Co. KGaA, material Weinheim constants that determine the lamina failure, following the traditional approach for metals. According to this approximation, the ply failure occurs when the local stress or strain reaches a critical value. Several approaches have been used historically: the maximum stress and strain theories, the Tsai-Hill and Tsai-Wu invariant criteria, and the 15.

(34) 1.3 Mesomechanical failure models Hashin’s multiple failure criteria. The world-wide failure exercise of Soden et al. (2004) was conducted to evaluate the validity of the different models. The LaRC criteria, Davila et al. (2005) and Dávila & Camanho (2003) provided best results, in the spirit of Puck & Schürmann (1998). The LaRC failure surface is represented in figure 1.14. Once failure has been reached, the “ply-discount” approach assumes that no stiffness remains after one lamina reaches the failure criterion. Nevertheless, this approach is unrealistic and might not be conservative, as explained in the next section.. 1.3.2. Damage in composite materials. A strength criterion approximation is well suited for predicting failure in monolithic materials, such as ceramics, because initiation of cracking and unstable growth occur simultaneously. However, the formation of the crack and the instability are determined by different conditions when the constraint to crack growth is imposed by the presence of reinforcements (i.e. the neighbour plies with different orientations). Energy considerations have to be included under these conditions to predict failure. Following the discussion in Talreja & Singh (2012), two widely different approaches to failure emerged historically: “micro-damage mechanics” (MIDM) and “macro-damage mechanics”(MADM), the later also known as“Continuum Damage Mechanics”(CDM). MIDM evolved from micromechanics, where microcracks are treated as voids and the geometries are simpler. The main objective of CDM approach is to characterize the consitutive stiffness tensor of the homogenized continuum body containing damage. Recently, both MIDM and MADM approaches have been combined in the synergistic damage mechanics (SDM).. Macro-damage theories (MADM): Continuum Damage Mechanics The CDM model has the required capabilities of a physically-based constitutive framework that can be incorporated into a structural analysis scheme. Classical CDM is based on two main ingredients, section 2.1: • The laminate stiffness is calculated by classical laminate theory (CLT), Barbero (2011). The laminae have variable properties as a function of internal parameters, namely, damage variables dM associated with damage modes M . 16.

(35) 1.3 Mesomechanical failure models • The damage model is based on the plasticity damage theories and includes a state variable rM , a damage criterion φM , the corresponding thermodynamical force YM , and a damage evolution law. In a similar spirit to section 2.1, a damage tensor involving the quantities in figure 1.15 is used. Also, following Talreja & Singh (2012), the crack density is assumed to govern the damage (rM ≡ ρcrack ) and the strain state drives the crack formation (YM ≡ εM ), ρcrack (εM ).. ⎡ N ⎤ ⎡A B ⎤ ⎡ ε ⎤ ⎢ M ⎥ = ⎢ B D ⎥ ⎢κ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ CLASSICAL LAMINATE THEORY. σ ijL = ( (1 − Ω L )^ L )ijkl ε klL L Ωijkl ∝. ε klL. κ tc 2 st. f (θ ). s, κ. MIDM or experiments. Figure 1.15: Mesomechanical MADM approach for laminate damage from microcracking. The matrices from Classical Laminate Theory are obtained from damaged lamina properties. Damage in the lamina is calculated according to the proposed damage theory. The internal parameter evolution, crack spacing s, is calculated from the thermodynamic force, the lamina strain εij or measured in experiments. The constraint parameter κ is also an output of the MIDM model.. The effect of the crack density as a function of strain is typically an output of the previous microdamage theories, or suitable (and case-dependent) experimental data, Nairn & Hu (1992). However, an strength-based approach has limited applicability. The energy-based approach in Nairn (2000), using finite fracture mechanics, gives significant improvement. Nairn’s approach is essentially a Hashin’s model where the difference 17.

(36) 1.3 Mesomechanical failure models between the complementary energy of the lamina with N and N + 1 cracks (the new crack formed midways) are compared for the given stress state. If reaching the critical energy release rate, GM , then the new crack is formed. We can therefore rearrange the equations to obtain the crack density as a function of the external load and the fracture toughness. Energy-based formulations for the MIDM theories above can be found in Parvizi et al. (1978), Joffe & Varna (1999). The stochastic nature of the strength is also taken into account in Vinogradov & Hashin (2005), Berthelot & Le Corre (2000), Wang (1984). Johnson & Chang (2001b) develops an analytical model to predict the interference between modes, including delamination and edge effects, based on finite fracture mechanics considerations. A recent sophisticated approach for multidirectional laminates is found in Singh & Talreja (2010). Additional examples are found in Maimı́ et al. (2011) and Maimı́ et al. (2011).. Micro-damage theories (MIDM) Typically, one fracture surface is generated in CFRP in the fiber direction (leading to failure) while multiple cracking in the transverse direction. Our interest is mainly focused on the second case, figure 1.15. The simplest lamina postfailure model is the ACK theory for a fibrous composite loaded in tension in the fiber direction, Aveston & Kelly (1973). The model assumes that one of the constituents (matrix or fibers) has lower strength than the other, and when the weaker constituent fails, the stronger is able to carry the additional load, with perfect debonding. In the case of matrix microcracking, the shear lag models are able to capture the shear stress transfer at the interface. The key assumption is that the transverse ply does not carry the axial load in the plane of a transverse crack, while a part of this load is transferred back to the transverse ply away from the crack and the adjacent uncracked ply. The shear lag models are dependent on the “shear lag parameter”, usually fitted experimentally. Several variants are surveyed in Berthelot (2003). Variational theories account for crack interactions in simplified crack 2D stress fields (Hashin (1985), Varna & Berglund (1994) and McCartney (1992)). The methods based on the crack opening displacement, COD-methods, consider the overall strain of the representative laminate volume contributed by the crack surface displacements of the individual 18.

(37) 1.3 Mesomechanical failure models. Figure 1.16: Comparison of variation of Young’s modulus (a) and Poisson’s ratio (b) with different microdamage models for cross-ply laminates. Joffe & Varna (1999). cracks, with stiffness relations in terms of average COD (Gudmundson & Weilin (1993), Lundmark & Varna (2006)). A comparison of different methods is shown in figure 1.16. Obviously, all these MIDM approaches can only be applied to very simple laminate stacking sequences (cross-ply laminates, sometimes extended to angle ply laminates under strong assumptions) and cannot be extended to build a generic framework for damage in composite materials.. 1.3.3. Computational CDM for multiple failure modes. It is obvious that the interaction between failure modes, including delamination, is difficult to take into account. Moreover, the models above are designed for uniform far-field stresses, and a spatial discretization is needed for more general loadcases. The assumption of plane stress in Classical Laminate Theory, and the large surface-to-thickness ratio of common composite structures make discretizations based on shell elements well-suited for most practical applications. The CDM in the framework of the Finite Element Method (FEM) (Lemaitre (1992)), developed for concrete in Bažant & Planas (1998), was first applied to anisotropic materials by Matzenmiller et al. (1995), and is able to assess multiple failure modes. Significant progress has been made since then. For example, the phenomenological model of Ladevèze et al. (2000) has been extensively used in commercial software applications. The model of Chang & Chang (1987) has been adapted to codes such as LS-DYNA, Hallquist et al. 19.

(38) 1.3 Mesomechanical failure models (2010). Similar approaches, like Tay et al. (2005) or Basu et al. (2003), are surveyed in Sleight (1999). The CDM has attracted the attention of the experimentalists, which have included physically-based failure criteria, such as the LaRC of Davila et al. (2005), in the computational models. Then, Maimı́ et al. (2007a), Maimı́ et al. (2007b) implemented the LaRC04 criteria in the commercial code Abaqus, Dassault Systemes (2010), figure 1.17. The shell elements (Bathe (1996)) are suitable for large surface-to-thickness ratios. But their kinematics is unable to represent delamination and structural details. To overcome these limitations, the 2D-1/2 models include a discretization of one solid 3D element per lamina thickness, which captures the in-plane stiffness and failure modes, and interlaminar cohesive elements to capture interlaminar failure.. Figure 1.17: Maimi’s CDM model in an open hole test with shell elements. Camanho et al. (2007).. These models have been extensively applied to several load cases. For instance, the brittle failure of a laminate was successfully analyzed with shell elements, Camanho et al. (2007). Convergence difficulties for the quasi-equilibrium solutions were observed. Impact loads were studied in explicit dynamics using 2D-1/2 models of González et al. (2011), Lopes et al. (2009), which do not suffer from convergence pathologies. However, the applicability of the solid models has been questioned in van der Meer & Sluys (2009a), figure 1.18. Van der Meer overcomes the quasi-equilibrium convergence 20.

(39) 1.3 Mesomechanical failure models by applying an energy-based load, and also analyzes the post-failure in some benchmark problems. They concluded that, regardless of the plasticity approach used (Jirásek (2007)), the kinematics of the CDM is insufficient to describe some relevant aspects of the mesomechanical failure of a composite laminate, as will be further explained in section 2.5.. Spurious secondary crack. Experimental crack. Figure 1.18: CDM model of a shear test. It is observed that an spurious secondary crack perpendicular to the load direction appears in the simulations and controls the final 11/30/2013 presentation title 1 failure. van der Meer & Sluys (2009a).. 1.3.4. Interlaminar failure models. Interlaminar failure models take advantage of the fact that the location of the fracture surface is known in advance. Approaches based on fracture mechanics, such as the Virtual Crack Closure Technique (VCCT) of Krueger (2002), or the cohesive zone approach, are well-suited. The cohesive zone model, CZM, is an approximation to fracture mechanics by a cohesive crack which is able to transmit stresses between the crack surfaces, Elices & Planas (1996). It was first applied to model concrete fracture, Bažant & Planas (1998). It was subsequently refined for more general finite element applications in Xu & Needleman (1993), de Borst (2003), Ortiz & Pandolfi (1999), and applied to fracture in composites, Segurado & LLorca (2004), Yang & Cox (2005). The CZM for modeling delaminations can be either embedded in a shell element, Remmers et al. (2003), or discretized independently with cohesive elements. In a cohesive element, as further explained in section 2.2.2, the cohesion between crack faces is modelled with a two-surface element and a traction law dependent on the relative 21.

(40) 1.3 Mesomechanical failure models displacement between crack faces. The elements are placed in the a-priori known crack trajectories, figure 1.19. Cohesive elements. Figure 1.19: Finite element simulation of a double cantilever beam, the peel test. Two laminas (in white) at 0o are glued with an interface made of cohesive elements (in red). The traction-separation law governs the lamina decohesion. Castro et al. (2010). A cohesive formulation for mixed-mode fracture in composite materials was developed in Allix et al. (1995) and Turon et al. (2006), and the later was implemented in commercial software packages such as Abaqus, Dassault Systemes (2010). An implementation for explicit dynamics is found in González et al. (2009).. 1.3.5. Strong discontinuity approaches. Contrary to the CDM intralaminar failure formulated at constitutive level (stressdeformation relationship), several damage approaches are formulated at the kinematic level (jump of displacement-traction relationship). Hallett et al. (2008) included the intraply matrix cracks as open (non transmitting stresses) from the beginning of the analysis. The intraply cracks are combined with interply cohesive elements. Of course, this approach is not predictive because the information about the dominant matrix cracks has to be known a-priori. Bouvet et al. (2012) also introduced bond interactions (active/inactive) at several intralaminar locations to simulate matrix cracking, and cohesive-like elements at the interplies to model delaminations. However, this approach has important limitations for general loadcases: either the initial stiffness of the cohesive interfaces affects the overall laminate stiffness, or the initially-rigid cohesive interfaces require a tracking algorithm, explained in section 3.5.2. The limitations above can be overcome with the use of embedded discontinuity techniques, further detailed in chapters 3 and 4. The embedded discontinuity introduces dis22.

(41) 1.3 Mesomechanical failure models continuous fields at arbitrary locations of the mesh, inside the elements. The enhanced strain of Simó & Rifai (1990) has been widely used to include the displacement jump field [[u]] as the enhanced strain field εenh = (δ ⊗ [[u]])sym , where δ stands for Kronecker’s delta. Following the results indicated in section 3.1.1, it is obvious that an enhanced strain approach has some limitations that encourage the use an embedded strong discontinuity at the kinematic level, the displacement field u.. (a). (b). (c). Figure 1.20: The Phantom Node method and applications to coupons of composite materials. (a) Phantom Node discretization. (b) Load-displacement curve of [±45] specimen with snap-back, at monotonically growing fracture energy. (c) Fracture of the specimen with mesh-independent intralaminar cracks and interlaminar cohesive elements. van der Meer & Sluys (2009b).. The Partition of Unity method of Melenk & Babuska (1996), further explained in section 3.1.3, constitutes a suitable framework to include enhanced displacement fields, such as discontinuous displacement [[u]], at arbitrary places of a discretization. A particular case, coined the X-FEM method (Moës et al. (1999)), has gained popularity. Variants of the X-FEM, such as the Phantom Node, were applied by van der Meer & Sluys (2009b), van der Meer et al. (2012) or Ling et al. (2009) to model failure in composite laminates, figure 3.10. The regularized version of X-FEM (Rx-FEM) in Iarve et al. (2011) was able to capture important aspects of failure in composites, figure 1.21. 23.

(42) 1.3 Mesomechanical failure models The X-FEM approaches include the matrix cracks as enrichments representing the displacement discontinuity. Fiber-dominated damage modes are still represented with a CDM approach. The appearance and growth of matrix cracks is tracked by a suitable “tracking algorithm”. However, the introduction of a cohesive law results in unstable (oscillating) crack tractions and difficulties of implementation. As it will be shown in Chapters 3 and 4, 770. E. V. IARVE ET AL.. the Phantom Node method can be studied as a non-conforming method, where the mesh is not a subset of the solid domain (or subspace of the exact solution space). Recently, different discretizations are proposed to avoid the traction oscillations, trying to approach the interpolation properties of the conformal methods. The Discrete Strong Discontinuity al. (2009), the cohesive segment method 770 Approach of da Costa E. V. IARVEet ET AL. in Kawashita et al. (2012) or the floating node method of Pinho et al. (2013) belong to this category. However, the trade off is to loose some of the characteristics that made the Phantom Node method attractive. In particular, the simple mesh strategy of X-FEM (with additional degrees of freedom with the same location as the original nodes) is not 770. preserved.. E. V. IARVE ET AL.. Delamination -45/45. Intralaminar cracks. Delamination -45/90. Figure 13. Predicted damage at increasing load levels in a [45/−45/90] laminate: (a) delamination. s Figure 1.21: Predicted damage with increasing load in (b)a intermediate carbon-epoxy laminate initiation; stage; and (c) immediately preceding final failure. [45/ − 45/90]s with a symmetric Rx-FEM model: delamination initiation (left), interCONCLUSIONS mediate stage (center) and immediately before final failure (right). Iarve et al. (2011).. A fully three-dimensional analysis methodology is proposed for modeling complex matrix cracking and delamination networks in laminated composites. The proposed methodology is based on a regularized x-FEM formulation [23] for MIC modeling of arbitrary transverse matrix cracks, and a cohesive formulation to model delaminations between plies. Verification studies include simulation of delamination initiation from matrix cracks, modeling of delamination jumps from one ply interface to another, prediction of the effects of ply thickness on delamination shape and transverse Figure 13. Predicted damage at increasing load levels in a [45/−45/90] laminate: (a) delamination crack densitys and examination of delamination initiation variations due to varying ply orientations. initiation; (b) intermediate stage; and (c) immediately preceding final failure.studies, good agreement between experimental observations and/or other In each of the verification computational techniques and the MIC modeling methods described above was shown. Delamination evolution emanating from transverse cracking was examined in detail by modeling CONCLUSIONS a mode II fracture specimen called the transverse crack tension specimen. Comparison between. Finally, it is clear that a kinematic enrichment of Partition of Unity methods (sec-. tion 3.1.3), or any of the continuous-discontinuous approaches above, requires a careful choice of the discrete spaces, Braess (1992), Brezzi & Fortin (1991), Chapelle & Bathe (1993). Unless the formulation and spaces are carefully chosen, a formulation working for A fully three-dimensional analysis methodology is proposed for modeling complex matrix cracking. Published 2011. This article is a US Government Int. J. Numer. Meth. Engng 2011; 88:749–773 a given discretization and situation longer beandapplicable other discretizations or DOI: 10.1002/nme is in the public domain in the USA. and delamination networks in may laminatedno composites. Thework proposed methodology is to based on a. regularized x-FEM formulation [23] for MIC modeling of arbitrary transverse matrix cracks, and a formulation to model delaminations between plies. Verification studies include simulation loadcases, chapter 4.cohesive of delamination initiation from matrix cracks, modeling of delamination jumps from one ply interface to another, prediction of the effects of ply thickness on delamination shape and transverse density and examination of delamination initiation variations due to varying ply orientations. Figure 13. Predicted damage at increasing load levels in crack a [45/−45/90] s laminate: (a) delamination In each of the verification studies, good agreement between experimental observations and/or other initiation; (b) intermediate stage; and (c) immediately preceding final failure. computational techniques and the MIC modeling methods described above was shown. Delamination evolution emanating from transverse cracking was examined in detail by modeling a mode II fracture specimen called the transverse crack tension specimen. Comparison between CONCLUSIONS. Published 2011. Thiscomplex article is matrix a US Government A fully three-dimensional analysis methodology is proposed for modeling cracking work and is in the public domain in the USA. and delamination networks in laminated composites. The proposed methodology is based on a regularized x-FEM formulation [23] for MIC modeling of arbitrary transverse matrix cracks, and a cohesive formulation to model delaminations between plies. Verification studies include simulation of delamination initiation from matrix cracks, modeling of delamination jumps from one ply interface to another, prediction of the effects of ply thickness on delamination shape and transverse crack density and examination of delamination initiation variations due to varying ply orientations. In each of the verification studies, good agreement between experimental observations and/or other computational techniques and the MIC modeling methods described above was shown. Delamination evolution emanating from transverse cracking was examined in detail by modeling a mode II fracture specimen called the transverse crack tension specimen. Comparison between Published 2011. This article is a US Government work and is in the public domain in the USA.. Int. J. Numer. Meth. Engng 2011; 88:749–773 DOI: 10.1002/nme. Int. J. Numer. Meth. Engng 2011; 88:749–773 DOI: 10.1002/nme. 24.