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Contribution o a micromechanics-based approach or reliability assessment Hocine Dehmous 1, Moussa Karama 2 and Hélène Welemane 2 1 Université Mouloud Mammeri, Campus Hasnaoua, Tizi-Ouzou CP 15 000, Algérie, dehmous hocine@yahoo.r 2 Université de Toulouse; INP/ENIT; LGP; 47, avenue d Azereix, 65016 Tarbes cedex, rance, Moussa.Karama@enit.r, Helene.Welemane@enit.r The paper intends to widely develop the use o a new methodology or the design and optimization o composite materials and structures. Based on the coupling o reliability methods and homogenization techniques, such approach allows the integration o uncertainties at dierent scales in the problem analysis (i.e. rom the microscopic scale o the composite material components up to the macroscopic scale o the structure) and the investigation o their consequences in the ailure prediction. The principles and implementation steps o such original method have already been described in details (Int. J. Mech. Sci. 53 (2011) 935-945; Eng. ail. Analysis 18 (2011) 988-998). This work ocuses on its application and signiicant progress allowed in the design phase or engineering composite materials. Illustrations are presented on the case o a civil engineering structure, namely the Laroin ootbridge (rance) with carbon epoxy stay cables, and highlight reliability-based innovations and exploitations regarding the structure optimization. Keywords: Reliability, composite materials, multi-scale analysis, design, optimization 1 Introduction Uncertainties on constituent properties, ibre distribution, structural geometry, manuacturing process parameters or loading conditions clearly aect the design and optimization o composites structures. The related variability o their mechanical perormances requires large saety actors in deterministic structural calculations and induces a signiicant loss in the weight/resistance ratio. In this context, reliability-based analyses have been developed or a ew years to provide a consistent and rational estimation o the risk by introducing the random character o design parameters (materials properties, loads, geometry,..). The quantitative evaluation o reliability indicators (such as the ailure probability) obtained by these methods leads then to an enriched representation o the composites behaviour that signiicantly helps or both the design and maintenance o structures, and more generally or the development o composite materials [5, 10]. Most o reliability-based approaches are primarily designed to establish the reliability o a structure as unction o the applied load and macroscopic parameters such as geometry and material strengths (or instance [1, 6, 11, 13, 14]). An original approach that simultaneously considers reliability and micromechanics has been recently proposed by the authors to improve this representation [4, 16]. or heterogeneous materials such as composites, homogenization techniques allow indeed the derivation o their overall behaviour rom microscopic eatures (components behavior and morphology). At the same time, local stress or strain within components can also be obtained, which is particularly relevant or the consideration o a local ailure criteria (see [8, 17]). The association o reliability methods with micromechanics oers then a better insight into the modelling o uncertainties aecting composites, including especially the luctuations at various scales (microscopic to macroscopic) and also a physical motivation or the deinition o the material reliability. These two points are important issues to understand the propagation o variability rom micro to larger scales [15] but, above all, to derive a robust ramework or reliability evaluation [3]. Interest o the coupled approach has been previously illustrated or material engineering through the identiication and validation o micromechanical models [16]. The ability to derive multi-scale analyses has also been
Hocine Dehmous, Moussa Karama and Hélène Welemane demonstrated [4]. The present work intends to ocus on the question o design and optimization, through an application case in civil engineering. The objective is here to highlight the innovative tools provided by the methodology in the context o reliability based design, or instance to get the best structure under a reliability constraint or to improve the manuacturing process. Ater a general recall o the methodology and description o the case study, we present in what ollows the design interest or an engineering material optimization, based on the integration o uncertainties related to the components properties and composite morphology. 2 General background and case study The originality o the approach suggested by the authors lies in the association o two classical approaches or the study o composite materials: Reliability methods that introduce in the mechanical modelling the uncertain character o design variables and allow the study o the related consequences on the structure response, Homogenization techniques which aim at determining rom the microstructure o a Representative Volume Element (RVE) o the material its overall (eective) mechanical behaviour and the local response within components or a given macroscopic load. Theoretical developments and implementation details can be ound in [4, 16]. We intend here to recall the key points o such coupling and assumptions used or the case study. 2.1 Coupling o reliability methods and micromechanics In the coupled approach, major stages encountered during any reliability study take into account the speciic aspects o micromechanics: 1. irst the selection o random variables {X i } i=1,n in order to capture most o inherent luctuations involved in the problem, and at the same time to oer at this starting point a broad analysis o design parameters; or heterogeneous materials such as composites, microstructural parameters clearly inluence the material eective behaviour and should thereore be considered as random variables; important requirements are the availability o the statistical representation o these random parameters and the limit size o the problem (number N) that should be compatible with reasonable calculation time; 2. The choice o the ailure scenario deined by a mathematical unction G: this step includes both an adequate representation o the structure mechanical behaviour and a physical deinition o the limit state (according either to strength achievement or serviceability); the ability provided by micromechanics to derive local stresses or strains induced by a macroscopic load is a major asset or a consistent deinition o G in relation with the physical mechanisms involved ; 3. The estimation o probabilistic indicators (e.g. probability o ailure P or reliability index β): in the context o composite structures investigation, one oten resorts to numerical approximation methods in view o their suitability with inite element simulations and their design interest through sensitivity analyses on random variables. Generally speaking, note that this approach should be conducted several times in order to select the signiicant random parameters o the problem considered and to ensure thereore the computational eiciency o reliability calculations. 2.2 Application to the Laroin ootbridge The demonstration o the potential o such approach or composites design and optimization is practically supported by a case study in civil engineering. The considered structure is a pedestrian ootbridge located in Laroin (Pyrénées Atlantiques, rance, ig. 1(a)). Built in 2002, this is the irst structure in this country conceived with composite stay cables [7]. It is composed o a steel deck, two steel reversed V-shaped pylons o 20.60 m height,
Contribution o a micromechanics-based approach or reliability assessment composite stay cables steel stending stay cable deck pylon anchorage (a) (b) ig. 1: Laroin ootbridge: Picture (a) and structure (b). one high-strength steel stending cable or each pylon and its single span o 110 m length is maintained at each side by eight composite stay cables. Each stay cable is composed o two or three strands o seven unidirectional cylindrical composite rods (ig. 1(b)). Composite rods are made o Torayca high-strength carbon ibres T700SC-12K and Bostik indley epoxy resin Eponal 401. They have been manuactured by pultrusion by the Toray Carbon ibers Europe company (mean ibre volume raction o 67%). The mechanical behaviour o the composite material is described by means o the Mori-Tanaka ormulation [2, 12] with a slight adjustment to relect more eectively the manuacturing process. Based on the experience o Toray Carbon ibers Europe, it seems indeed that only a raction o ibres in the composite actually contributes to the composite response (due or instance to ibres misalignment, impregnation deects, etc.). Accordingly, the ibre volume raction appearing in the classical expressions o the Mori-Tanaka constitutive law has been changed in this way: = pact (1) with pact 100% the amount o ibres really active rom the mechanical point o view. This parameter, which characterizes the quality o the manuacturing process, has been identiied on the pultrusion production line. In this study, reliability assessment corresponds to the mechanical strength achievement o these rods under monotonic tensile load. The ailure scenario is deined to account or the microstructural origin o the ailure, namely by the ibres brittle behaviour through the ollowing limit state unction: G = σ (n) σi ( ) (2) where σ (n) denotes the ibre axial ailure strength (axial direction denoted by unit vector n) and σi ( ) the maximum principal value o the average local stress over the ibre phase. I the irst term is a material data provided by manuacturers, the second is derived through the micromechanical model. Reliability calculations have been carried out with the probabilistic code ERUM (inite Element Reliability Using Matlab [9]) by means o a direct coupling with the micromechanical model. ORM (irst-order Reliability Method [5, 10]) approximation method has been used since it exhibits a remarkable computational eiciency and provides valuable data or design issues (such as elasticities, see section 3.1). 3 Micromechanical-based reliability analysis The present work aims at studying the reliability-based design o composite rods used or the stay cables o the Laroin ootbridge, subjected to monotonic longitudinal tension. 3.1 Reerence point Based on the micromechanical model developed in [4, 16], the ive random variables {Xi }i=1,5 taken into account in this study concerns both micro and macro scale data o the problem:
Hocine Dehmous, Moussa Karama and Hélène Welemane components mechanical properties : ibre yield strength σ (n), manuacturing process parameters : ibre volume raction, active ibre raction p act, rod diameter ϕ, load condition : axial load, since their variability clearly aects the composite reliability. It has been shown in [16] that components elastic properties (ibre and resin) could be chosen as deterministic due to their weak inluence. Regarding the considered case study, manuacturers (o ibre materials and composite materials) have provided the statistical distribution o the irst our parameters that ollow a Normal law (mean value and standard deviation are respectively denoted by X i and S X. We assume also a Normal distribution or the load and the study is centred around the operating point = 76 kn that corresponds to reliability index β = 3.0514 (such security level is approved in civil engineering structures or which β should be greater than 3 or, equivalently, ailure probability P should be greater than 10 3 ). All these data are detailed in Table 1 and will be considered as the reerence point in what ollows. Random variables re. mean value X re i re. standard deviation SX re i ibre yield strength (MPa) σ (n) 4870 162 ibre volume raction (%) 67 0.333 Active ibre raction (%) p act 95 0.667 Rod diameter (mm) ϕ 6 0.03 Axial load (kn) 76 2.533 Tab. 1: Random variables Reerence point. The sensitivity analysis o the ORM method provides elasticities {e ri } i=1,n according to the distribution parameters o each random variable (here the mean value r i = X i or the standard deviation r i = S X deined by: e ri = r i β (P ), i [1..N] (3) β r i with P the design point (most probable ailure point) [10]. Such analysis gives the global tendency regarding the origin o the scatter in the material response. This helps to discriminate variables that could be considered as deterministic (such as components elastic properties [16]) and to compare variables between them. As an illustration, igure 2 gives the reliability elasticities or the composite rod regarding respectively the variables mean value (ig. 2(a)) and their standard deviation (ig. 2(b)). We can distinguish strength variables with e ri 0 (respectively loading variables with e ri 0) or which an increase in their mean value improves (resp. alters) the structure reliability. Whatever the variable, an increase o the standard deviation obviously leads to an increase in the ailure probability. Here, we note that the mean value o the geometric parameter ϕ and the deviation scatter on the ibre yield strength σ (n) play the most crucial role in the structure reliability. Yet, such analysis does not provide practical arguments or designers to take quantitative decisions regarding the composite structure. Our aim is then to develop in what ollows the micromechanics-based analysis rom another point o view in order to really quantiy the inluences detected beore and help or design choices. 3.2 Evolution range In this way, we need irst to consider allowable evolutions o the distribution parameters o the dierent variables. In particular, these evolutions have to be chosen according to the potentialities o materials and manuacturing machines and should also be compatible with assumptions o the mechanical model. To illustrate the approach, we have considered in table 2 an acceptable evolution range or the mean value X i o each random variable (range ratio relative to their reerence value is indicated or clearness). Without speciic data, an extended and uniorm range ratio has been taken or the standard deviation S Xi (S Xi = 0 corresponding to the deterministic case). Several simulations have then been carried out in which one distribution parameter o each random variable evolves independently within its interval deined in table 2, the other being ixed to their value given in table 1. or each calculation coniguration, explicit ormulation provided by the Mori-Tanaka scheme and the small number
Contribution o a micromechanics-based approach or reliability assessment e ri σ (n) p act φ e ri p act φ σ (n) (a) (b) ig. 2: Elasticities e ri according to the distribution parameters o random variables at the reerence point: mean value r i = X i (a) and standard deviation r i = S Xi (b). standard deviation S Xi Random variables mean value X i ([0, 150%] o SX re i ) ibre yield strength (MPa) σ (n) [4384, 5356] [0, 243.5] (±10% o X re [64, 70] ibre volume raction (%) (±5% o X re [90, 100] Active ibre raction (%) p act (±5.2% o X re Rod diameter (mm) Axial load (kn) ϕ [5.8, 6.2] (±3% o X re [64.6, 87.4] (±15% o X re [0, 0.5] [0, 1] [0, 0.05] [0, 3.8] Tab. 2: Evolution range o the distribution parameters o random variables around the reerence point. o variables allow to derive a precise value o reliability index β with a reasonable number o calls to unction G (around 50). 4 Results and discussion igure 3 shows the evolution o index β according to the mean value o each random variable. An increase (respectively decrease) o β with X i is obviously obtained or strength variables (resp. loading variables). or the present model and application case, we obtain a quasi linear evolution o β regarding mean values X i o random variables. In agreement with the model correction (1) and elasticities described on igure 2(a), we ind a similar behaviour or and p act variables and a major inluence o ϕ through the steepest slope. or designers, such result allows to deine either changes o parameters or the maximum load amplitude to respect a given reliability level. or instance, the reliability index β = 3.0514 at the reerence point can be raised to β = 3.5 (corresponding to an increase o 15%) either by an increase o 2% o the ibre yield strength (that is with σ (n) = 4970 MPa) or with an increase o 1% o the rod diameter (that is with ϕ = 6.07 mm). On the other hand, we note that a reliability level o β = 2 remains satisied with an increase o 6% o the mean load (that is with = 80.6 kn).
Hocine Dehmous, Moussa Karama and Hélène Welemane β σ (n) p act φ X i (%) re X i ig. 3: Evolution o reliability index β with respect to the mean value X i/x re i or each random variable X i. Regarding the eect o the standard deviation, igure 4 quantitatively highlights the inluence on the reliability o the uncertainties on variables. As expected, more (resp. less) the variables are reproducible, more reliability increases (resp. decreases). The tendency observed on igure 2(b) is also reinorced with a weak inluence o the scatter on manuacturing process parameters (, p act and ϕ). This indicates where to put the eort to get better reliable structures, namely on the ibre strength σ (n). or the same mean value as the reerence point, an highly reproducible σ (n) (with S X i 0) can increase the reliability index up to β = 4.4 (increase o 44%). The variability o the load plays also an important role on reliability, but the control over it is much more diicult, especially or civil engineering cases. Moreover, the evolution o β regarding the standard deviation o these two variables is o polynomial orm with a crossing o the curves at the reerence point: a better (resp. worse) reproducibility o σ (n) induces a greater increase (decrease) in β than the load. This conirms the main importance o the ibre strength in the composite perormance. β σ p act φ (n) S S X i re X i (%) ig. 4: Evolution o reliability index β with respect to the standard deviation S Xi /S re X i or each random variable X i.
Contribution o a micromechanics-based approach or reliability assessment (a) (b) ig. 5: Cross-analyses o the evolution o reliability index β with respect to the mean values o couple o parameters: (, p act) (a) and (σ (n), ) (b). The micromechanical-based study allows also to implement quantitative cross-analyses between random variables. This provides to engineers various combined solutions or design changes. We illustrate on the igure 5 the coupled study o manuacturing process parameters (namely the ibre content and active ibre raction) and material parameters (namely the ibre yield strength and ibre content). As shown o igure 5(a), the reliability index β can be improved to 3.5 either: by enhancing the ibre content (with = 68.5%) while keeping the present manuacturing process, or by keeping the ibre content and improving the manuacturing process (with p act = 100%). Regarding material eatures investigated on igure 5(b), same reliability improvement can be obtained either : with higher strength ibre (σ (n) = 5180 MPa) and a reduced ibre content ( = 64%), or with higher ibre content ( = 70%) and lower strength ibre (σ (n) = 4770 MPa). Obviously, such kind o cross-analysis can be done also on standard deviations o variables or even between mean value and deviation o dierent parameters according to the needs and possible actions o designers. 5 Conclusion The development o composite materials and composite structures requires a convenient consideration o the stochastic variability o the data entering their mechanical behaviour. The combination o reliability analysis and micromechanics allows to adress such issue. The use o classical and well-known rameworks (reliability methods and homogenization techniques) associated with the broad integration o uncertainties at dierent scales (micro to macro) and on various eatures (constituents properties, microstructural morphology, geometry, load) leads to a relevant tool or structural applications. This paper has illustrated some innovations provided by this approach or the reliability assessment o composite materials. Through various examples, we have investigated the consequences o design choices or composites in a probabilistic context. The quantitative description o dierent solutions at dierent scale or regarding dierent parameters such as manuacturing process, geometry, materials properties indicates which designs are qualiied to achieve a given reliability level and helps or the optimization o mechanical systems. rom this basic results, urther work will now be conducted in order to show others design applications provided by the approach, and especially to extend it at a structure scale through the global study o the Laroin ootbridge.
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