Chia, Julian Yan Hon (2002) A micromechanics-based continuum damage mechanics approach to the mechanical behaviour of brittle matrix composites. PhD thesis. http://theses.gla.ac.uk/2856/ Copyright and moral rights for this thesis are retained by the author A copy can be downloaded for personal non-commercial research or study, without prior permission or charge This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Glasgow Theses Service http://theses.gla.ac.uk/ [email protected] Continuum Damage Mechanics A Micromechanics-Based Approach to the Mechanical Behaviour Brittle Matrix of Composites. by Chia Yan Hon, Julian Submitted to the University Glasgow of for degree the of Doctor Philosophy of in Mechanical Engineering © Chia Yan Hon, Julian, Sept 2002 I Abstract The describes development thesis the damage of a new continuum mechanics (hereafter, CDM) for the deformation failure brittle model and of matrix composites fibres. The CDM is large reinforced with continuous model valid over sizes scales compared to the spacing of the fibres and the dimensions of the damage. The composite is damage in form to the delamination, allowed sustain of matrix micro-cracking, shear tensile delamination fibre failure. The developed by and constitutive equations are decomposing the composite compliance into terms attributable to the fibre and matrix, the failure by intersecting failure based and modelling competing modes surfaces on maximum stress theory. The fibres are treated as being weakly bonded to the matrix so that the fibres transmit loads, fail in tension. The is only axial and matrix modelled as isotropic linear is treated transversely-isotropic damage has initiated. elastic and as after The the determined from effect of multiple matrix cracking on stiffness was data, failure by decay in load bearing the experimental while was modelled a rapid Although the is largely to loading, capacity. model motivated proportional matrix damage has been by damage During unloading and closure modelled elasticity. is identical the to the the compression, matrix stiffness undamaged state with exception that the fibres to transmit loads. The are assumed not compressive model was implemented FORTRAN interfaced through the computationally a subroutine with ABAQUS/Standard finite element solver. The CDM by model was validated comparing experimental and computational balanced 0°-90° fibres test results of specimens with unidirectional and woven of a brittle fabricated from fibres in This matrix composite, polyester a polyester matrix. low between fibres has composite system exhibits elastic mismatch and matrix, and SiC/SiC for to similar non-dimensionalised stress-strain response a composite proposed diffuser Rolls-Royce EJ200 Test the the exhaust unit of aero-engine. specimens fibres have been tensioned to the reinforced with aligned and misaligned uniaxially damage failure To tensile to axis produce a range of mechanisms and processes. demonstrate the the to ability of model analyse engineering structures, a range of idealised from diffuser Rolls-Royce EJ200 the the parts exhaust unit of aero-engine in bending. The included tested were and analysed sub-structural specimens a simple bar, bar T-shaped These thickened rectangular a with a cross-section, and a component. ii full damage the sub-structures showed range of mechanisms, which often occurs in from brittle simultaneously engineering structures made matrix composites. The behaviour the fibre-matrix interface is to the behaviour fibre of central of It is that full interfacial be reinforced composites. argued a range of properties can by treating the interface imperfect be formalised infinite modelled as such can as an Interfacial developed to the periodic array of cracks. elements were model properties of imperfect interface. The the to the fibres an result support simplification used represent load bearing in the CDM insight into the as simple axial components model, and gave behaviour imperfectly bonded interfaces. of iii Acknowledgements I to formally to the following wish express my sincere gratitude persons and for help their to this thesis. organisations and support complete My deepest to Professor John W. Hancock, gratitude goes my academic You have invaluable taught supervisor. me skills concerning research and personal I indebted for to the management. am especially you encouragement, guidance, support in this thesis to to the Committee and grace making come past, recommendation of Vice-Chancellors Principles (CVCP) the Universities the United Kingdom to and of of the Overseas Research Student (ORS) Award, to the receive recommendation University Glasgow to the Mechanical Engineering Departmental of receive Scholarship, in department. the and provision of a pleasant work environment I like Professor Mike Cowling for to thank would acknowledge and supporting for the ORS Departmental Scholarship; Dr. David V. Philips for my application and finite Dr. Ronald D. Thomson for techniques; advice on element modelling advice on Dr Bill Broughton for Dr. Neil L. testing; continuum mechanics; advice on composite McCartney for Mr Ian Peden for technical correspondence on composite modelling; detection Dr. Ian Watson for the technical work on acoustic emission system; resources for detection Mr. Jamie Cunningham for the acoustic emission system; advice on Mr Alexander Tory for thermodynamics; the test making mechanical rigs and fabrication jigs for the experimental work on composites and sharing an office with me; Mr Alan Birkbeck, Mr George Falconor, Mr John Davidson Mr Brian Robb for and Panagiotis Tsouchnikas, Christos Kastritseas, the shaping composite specimens; Lubrano Andrea, Niall MacLeod for in the fabrication the assisting and analysis of Mr Kenny Stevenson, Ms Yassamine Mather, Walter Robinson composite specimens; for Ms Fiona Downie, Ms Careen Fraser, Ms Pauline Kyriacou, Ms computing support; Elaine McNamara, Ms Jane Livingstone for in the department; administrative support Ms Marion Richardson for in the faculty; Bostjan administrative support engineering Bezensek, Anuradha Banerjee, Kirsteen Lowe Moshiur Rahman for being and in fracture Dr Paul Molly, Ms Margaret the supportive colleagues mechanics group; Smith, Dr Maureen Joan Perry Dr Muhammad Shahid for being and supportive in department; Dr Seat Han Cheng, Dr Wu Jia Jang, Dr Yeo Chiew Beng the colleagues iv Allen, Dr Chong Boon Keat Dr Mark Müller for being and supportive colleagues and friends faculty. the within and outwith engineering I like to thank brothers in Glasgow Chinese Christian would also my and sisters Church (GCCC), St. Silas Church, Faith Assembly God Church (Singapore) for and of Thanks to GCCC for the their encouragement and support. also go accommodation and during in Glasgow the in Refiner's Fire to be sustenance my stay and allowing youths a life. Also, I like to thank Matty Chan Yiu Tung for being special part of my would my flatmate Deacon Paul Watson for his Scottish friendship. and My heartfelt to dad, David, Irene, special and appreciation goes my and mum, for their unfailing love to see me complete the thesis. Also to my brother Michael, Stella, Angela Caroline, John, Grace, sister-in-law sisters and nephew niece grandma Cheong Foong Yin, Chia Weng Kok & family for their during encouragement my uncle I like to late Chin Koon Seng late research. would also remember my grandpa and uncle Chia Weng Fatt for their kind influence life. Finally, I like to on my would specially Saviour Lord, Jesus Christ, for hope, faith love during this thank grace, and my and thesis. V Nomenclature Stresses, Strains Elasticity and C Stiffness tensor of a material A., Unit three el, e2, e3 vectors of mutally perpendicular co-ordinate axis x1, x2, and x3. Strain tensor sl, Resultant strain vector Normal cn strain vector Shear ss strain vector Mean strain sSý Hydrostratic tensor strain Volumetric dilatational or strain -C(cid:30) Deviatoric tensor eu strain E Young's modulus FI Body force G Shear modulus G Transformation for the tensor matrix stress and strain H Transformation for matrix engineering strain Engieering y shear strain Principal 71 73 engineering shear strain 172 ' SZ Complementary denisty the energy or complementary energy per unit volume ll; Direction between the 'and cosine co-ordinate axes x, xj Ii 13 Invariants tensoro of stress I]' ', 13 ' Invariants tensors of strain -b Jl, J2, J3 Invariants the deviatoric tensor of stress s1; J J2 'I J3' Invariants the deviatoric tensor of strain e in body. Unit n normal of a plane a continuous between Direction cosines n and el, e2, e3- ni, n; (1) Principal directions nn(2), n(3) Poisson's v ratio Cauchy's tensor. stress 6f Normal stress vector ßn Shear stress vector 6S Principal stress 6 & Mean stress 6(5; Hydrostratic tensor stress j Deviatoric tensor stress Sy vi Principal deviatoric tensor s stress S,, Compliance tensor of material Jk/ 93 Principal shear strain 191 091 , T, T; Cauchy's (or traction) stress vector. Displacement Ui vector W Strain density the energy or strain energy per unit volume bW Rate the density W of change of strain energy Damage Mechanics Tf Kachanov (1958) damage scaler variable Robotnov (1968) damage CO scaler variable 0 Murakami & Ohno (1981) damage tensor second-order D Cordebois & Sidoroff (1981) damage tensor second-order Dyk, Lemaitre & Chaboche (1978) fourth-order damage tensor 1 Void density in wo area a plane A Net area A/z Apparent area 6 Net stress or effective stress Entropy density s S Entropy (where Internal state variables a1..., n) Internal density energy u Density p Body heating and radiation r Heat flow q vector T Absolute temperature F Internal entropy production Helmholtz free denisty energy yr Gibb's free density energy 77 Internal entropy production per unit mass y Composites Terminology E,,, V(cid:30) EfVf a ratio of over failure Emu ultimate strain of matrix failure fibres £ ultimate strain of y applied shear strain fracture to form work a matrix crack surface debond fibres bridging to a unit area of matrix crack 7db energy used vii fibre bundle 6b stress failure fibre bundle a-bu ultimate stress of o, composite stress failure 6Cu ultimate stress of composite fibre 6f stress failure fibres 6fu ultimate stress of failure fibres brr mean ultimate stress of fibre 0f stress when matrix cracks fibre bundle 6b stress failure fibre bundle 6bu ultimate stress of Um matrix stress initiation 6mc matrix crack stress 6mc(sat) matrix crack saturation stress 6MS matrix softening stress failure 6(cid:30)ZU ultimate stress of matrix in to fibre 007M radial stress matrix with respect axis delamination 6Sd shear stress of composite delamination tensile stress of composite a-td far field stress a_ interfacial shear stress in fibre dUf change strain energy per unit area in AU7z change matrix strain energy per unit area done dW work to create a steady state matrix crack per unit area dl displacement fibres during Aw or additional of matrix cracking length a crack length transient crack ao E, Young's modulus of composite fibres Ef Young's modulus of Em Young's modulus of matrix by Voigt Evoigt Young's modulus of a composite by Ruess ERuess Young's modulus of a composite G, Shear modulus of composite fibres Gf Shear modulus of Shear G(cid:30)Z modulus of matrix by Voigt Gvoigt Shear modulus of a composite by Ruess GRuess Shear modulus of a composite G1 fracture energy release rate of matrix per unit area fibre-matrix interface G11 debond energy release rate of per unit area fibre length L intensity factor K' of composite effective stress viii K intensity factor effective stress of matrix intensity factor KID critical stress of composite K; ' intensity factor critical stress of matrix Weibull m modulus density failure function Pf probability failure Pf probability of PS probability of survival fibre r0 radius p closure pressure at matrix crack surface distance fibres R centre-to centre separation of by bridging fibres T traction closure displacement at crack surface u U matrix crack surface energy per unit area US dissipate due to frictional energy sliding per unit area Vf fraction fibres volume of fibre fraction forms Vf critical volume at which multiple matrix crack crit fraction V. volume of matrix load distance transfer x' critical load distance x transfer mean distance X normalised at crack surface Computational Model Subscripts stress 6 - strain c composite £ - - f fibre S compliance - - C stiffness m matrix - - E Young's modulus me matrix cracking - - G shear modulus mc(sat) matrix crack saturation - - Poisson's ratio ms matrix softening - v - delamination T transformation td tensile - - delamination d increment an of sd shear - - failure V fraction volume u ultimate - - A fraction area - {} Supers cripts vector -3x1 increment matrix n current - -3x3 increment magnitude of n-1 previous - - c- composite position system f- fibre position system m- matrix principal stress position system