On the effect of fiber creep-compliance in the high-temperature deformation of continuous fiber-reinforced ceramic matrix composites

Creep models for unidirectional ceramic matrix composites reinforced by long creeping fibers with weak interfaces are presented. These models extend the work of Du and McMeeking (1995) [Du, Z., McMeeking, R. 1995. Creep models for metal matrix composites with long brittle fibers. J. Mech. Phys. Solids 43, 701–726] to include the effect of fiber primary creep present in the required operational temperatures for ceramic matrix composites (CMCs). The effects of fiber breaks and the consequential stress relaxation around the breaks are incorporated in the models under the assumption of global load sharing and time-independent stochastics for fiber failure. From the set of problems analyzed, it is found that the high-temperature deformation of CMCs is sensitive to the creep-compliance of the fibers. High fiber creep-compliance drives the composite to creep faster, leading however to greater lifetimes and greater overall strains at rupture. This behavior is attributed to the fact that the greater the creep-compliance of the fibers, the higher the creep rate but the slower the matrix stress relaxation – since the matrix must deform with a rate compatible with the more creep-resistant fibers – and therefore the less the load carried by the main load-bearing phase, the fibers. As a result, fewer fibers fail and less damage is accumulated in the system. Moreover, the greater the creep-compliance of the fibers, the slower the matrix shear stress relaxation – and thus the lower the levels of applied stress for which this effect becomes important. The slower the shear stress relaxes, the slower the “slip” length increases. Due to the Weibull nature of the fibers, the fiber strengths at the smaller gauge length of the slip length are stronger; therefore fewer fibers undergo damage. Hence, high fiber creep-compliance is desirable (in the absence of any explicit creep-damage mechanism) in terms of composite lifetime but not in terms of overall strain. These results are considered of importance for composite design and optimization.     

A variational method for unidirectional fiber-reinforced composites with matrix creep

A variational method is developed for analyzing the matrix creep induced time-dependent change in fiber stress profiles in unidirectional composites. A functional of admissible profiles of fiber stress rate is presented by supposing a fiber broken in matrix as well as a fiber pulled out from matrix. The functional is shown to have the stationary function satisfying an incremental differential equation based on the shear lag assumption. Then, the stationary function is approximately determined by assuming bilinear profiles of fiber stress and a power law of matrix creep, leading to analytical solutions for the time-dependent change in fiber stress profiles. The solutions are verified on the basis of an energy balance equation and a finite difference computation. Moreover, it is shown that the solution for the fiber pull-out model agrees well with an experiment on a single carbon fiber/acrylic model composite if the initial slip at fiber/matrix interface is taken into account. In addition, the solution for the fiber breakage model is used for evaluating the characteristic time in long-term creep rupture of unidirectional composite.     

纤维增强金属基复合材料整体蠕变性能的细观力学分析

高温下金属基复合材料的蠕变主要由基体蠕变和界面扩散蠕变两部分构成,以往的研究中常常只考虑其中一种蠕变机理,从而导致得到的规律具有较大的局限性．本文提出了一种可预测金属基复合材料整体蠕变性能的细观力学方法,同时考虑了基体蠕变和界面扩散蠕变两种蠕变机理,导出了具有张量形式并满足不可压缩性的界面扩散蠕变应变表达式．采用Mori-Tanaka法和自洽法二者结果的平均以便更准确地计算纤维中的应力,揭示了两种蠕变机理相互影响的竞争关系．研究了恒定双轴荷载下的总体蠕变和固定位移约束下的应力松弛这两种常见蠕变问题,探究了基体蠕变与界面扩散蠕变两种蠕变机理在总蠕变中发挥的作用,考察了不同加载条件和不同纤维体积分数对复合材料整体蠕变行为的影响．     

Creep-rupture lifetime simulation of unidirectional metal matrix composites with and without time-dependent fiber breakage

A numerical simulation for predicting the axial creep-rupture lifetime of continuous fiber-reinforced metal matrix composites is proposed, based on the finite element method. The simulation model is composed of line elements representing the fibers and four-node isoparametric plane elements representing the matrix. While the fibers behave as an elastic body at all times, the matrix behaves as an elasto-plastic body at the loading process and an elasto-plastic creep body at the creep process. It is further assumed in the simulation that the fibers are fractured not only in stress criterion but time-dependently with random nature. Simulation results were compared with the creep-rupture lifetime data of a boron-aluminum composite with 10% fiber volume fraction experimentally obtained. The simulated creep-rupture lifetimes agreed well with the averages of the experimental data. The proposed simulation is further carried out to predict a possibility of creep-rupture for the composite without time-dependent fiber breakage. It is finally concluded that the creep-rupture of a boron-aluminum composite is closely related with the shear stress relaxation occurring in the matrix as well as time-dependent fiber breakage.     

A periodic unit-cell simulation of fiber arrangement dependence on the transverse tensile failure in unidirectional carbon fiber reinforced composites

The effect of fiber arrangement on transverse tensile failure in unidirectional carbon fiber reinforced composites with a strong fiber-matrix interface was studied using a unit-cell model that includes a continuum damage mechanics model. The simulated results indicated that tensile strength is lower when neighboring fibers are arrayed parallel to the loading direction than with other fiber arrangements. A shear band occurs between neighboring fibers, and the damage in the matrix propagates around the shear band when the interfacial normal stress (INS) is sufficiently high. Moreover, based on the observation of Hobbiebrunken et al., we reproduced the damage process in actual composites with a nonuniform fiber arrangement. The simulated results clarified that the region where neighboring fibers are arrayed parallel to the loading direction becomes the origin of the transverse failure in the composites. The cracking sites observed in the simulation are consistent with experimental results. Therefore, the matrix damage in the region where the fiber is arrayed parallel to the loading direction is a key factor in understanding transverse failure in unidirectional carbon fiber reinforced composites with a strong fiber/matrix interface.     

Uncertainties of unidirectional composite strength under tensile loading and variation of environmental condition

A probabilistic strength model is developed for unidirectional composites with fibers in hexagonal arrays. The model assumes that, a central core of broken fibers surrounded by unbroken fibers which are subjected to unidirectional tensile loading. The proposed approach consists in using a modified shear lag model to calculate the ineffective lengths and stress concentrations around fiber breaks. The main feature in the model lies in incorporating the variation of composite properties due to temperature and moisture, in order to predict degradation of fibers and matrix characteristics. The strength degradation is often seen as a result of changes in ineffective lengths at fiber breaks, leading to stress concentrations in intact neighboring fibers. As fiber breaks are intrinsically random, the variability of input data allows us to describe the probabilistic model by using the Monte-Carlo method. The sensitivities of the mechanical response are evaluated regarding the uncertainties in design variables such as Young’s modulus of fibers and matrix, fiber reference strength, shear yield stress, fiber volume fraction and shear parameter defining the shear stress in the inelastic region.     

A homogenization theory for time-dependentnonlinear composites with periodic internal structures

A homogenization theory for time-dependent deformation such as creep andviscoplasticity of nonlinear composites with periodic internal structures is developed. To beginwith, in the macroscopically uniform case, a rate-type macroscopic constitutive relation betweenstress and strain and an evolution equation of microscopic stress are derived by introducing twokinds of Y-periodic functions, which are determined by solving two unit cell problems.Then, the macroscopically nonuniform case is discussed in an incremental form using thetwo-scale asymptotic expansion of field variables. The resulting equations are shown to beeffective for computing incrementally the time-dependent deformation for which the history ofeither macroscopic stress or macroscopic strain is prescribed. As an application of the theory,transverse creep of metal matrix composites reinforced undirectionally with continuous fibers isanalyzed numerically to discuss the effect of fiber arrays on the anisotropy in such creep.     

Influence of fiber architecture on the inelastic response of metal matrix composites

This three-part paper focuses on the effect of fiber architecture (i.e. shape and distribution) on the elastic and inelastic response of unidirectionally reinforced metal matrix composites (MMCs). The first part provides an annotated survey of the literature; it is presented as an historical perspective dealing with the effects of fiber shape and distribution on the response of advanced polymeric matrix composites and MMCs. A summary of the state of teh art will assist in defining new directions in this quickly reviving area of research. The second part outlines a recently developed analytical micromechanics model that is particularly well suited for studying the influence of these effects on the response of MMCs. This micromechanics model, referred to as the generalized method of cells (GMC), can predict the overall inelastic behavior of unidirectional, multiphase composites, given the properties of the constituents. The model is also general enough to predict the response of unidirectional composites that are reinforced by either continuous or discontinuous fibers, with different inclusion shapes and spatial arrangements, in the presence of either perfect or imperfect interfaces and/or interfacial layers. Recent developments on this promising model, as well as directions for future enhancements of the model's predictive capability, are included. Finally, the third part provides qualitative results generated by using GMC for a representative titanium matrix composite system, SCS-6/TIMETAL 21S. The results presented correctly demonstrate the relative effects of fiber arrangement and shape on the longitudinal and transverse stress-strain and creep behavior of MMCs, with both strong and weak fiber/matrix interfacial bonds. Fiber arrangements included square, square-diagonal, hexagonal and rectangular periodic arrays, as well as a random array. The fiber shapes were circular, square, and cross-shaped cross-sections. The effect of fiber volume fraction on the stress-strain response is also discussed, as is the thus-far poorly documented strain rate sensitivity effect. In addition to the well-documented features of the architecture-dependent behavior of continuously reinforced two-phase MMCs, new results are presented about continuous multiphase internal architectures. Specifically, the stress-strain and creep responses of composites with different size fibers and different internal arrangements and bond strengths are investigated; the aim was to determine the feasibility of using this approach to enhance the transverse toughness and creep resistance of titanium matrix composites (TMCs).     

金属基复合材料的蠕变与松弛

本文利用微观力学方法研究了金属基复合材料的常温蠕变和应力松弛,连续纤维在弹性粘塑性基体内单向铺设。本文的结果与实验结果符合较好。研究表明,纤维在轴向对基体的蠕变起到明显阻止作用,而在横向和剪切变形下的作用较小。在低应力水平下,复合材料的蠕变变形很小,在高应力水平时,蠕变变形明显甚至引起蠕变破坏。     

界面特性对短纤维金属基复合材料蠕变行为的影响

基于短纤维增强金属基复合材料(MMC)的单纤维三维模型(三相)，利用粘弹性有限元分析方法对影响金属基复合材料的蠕变行为的因素进行了较为系统的分析。研究中主要讨论了界面特性和纤维取向角对金属基复合材料的蠕变性能的影响。研究结果发现，界面特性诸如厚度、模量和应力指数都对纤维最大轴应力和稳定蠕变率产生影响：稳态蠕变率随界面模量的增大而逐渐减小，当高于基体模量时基本保持不变；纤维轴应力的变化与蠕变率正好相反。稳态蠕变率随界面厚度、应力指数的增加而增大；而轴应力则随之减小。同时不同的纤维取向也影响金属基复合材料蠕变时的轴应力分布和稳态蠕变率。     

用蒙特卡罗方法模拟单向复合材料的拉伸断裂过程

通过采用改进的剪滞模型，假定界面不发生破坏，求得不同基体纤维刚度比μ（μ＝ＥｍＶｍ／ＥｆＶｆ）不同基体韧性情况下断裂纤维附近的应力集中系数，然后通过蒙特卡罗方法模拟单向复合材料的拉伸断裂过程，比较若干情况下复合材料的拉伸强度，找出提高复合材料拉伸强度的方法。     

STATISTIC MODELING OF THE CREEP BEHAVIOR OF METAL MATRIX COMPOSITES BASED ON FINITE ELEMENT ANALYSIS

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Micromechanical prediction of the two-way shape memory effect in shape memory alloy composites

The two-way shape memory effect in monolithic shape memory alloys has been widely investigated both theoretically and experimentally. In the present study, this effect is analyzed for shape memory alloy composites by employing a micromechanical model. To this end, the responses of polymeric matrix and metal matrix unidirectional composites with embedded shape memory alloy fibers are determined. For the polymeric matrix composite, the effect of axial, transverse and shear loadings as well as the fiber volume fraction on the resulting two-way shape memory behavior are studied. The local distributions of stresses among the shape memory alloy fiber and epoxy matrix in the low- and high-temperature shapes of the composite are also investigated. Two training procedures that generate the two-way shape memory effect in the metal matrix composite are offered. The present analysis shows that the two-way shape memory effect in the chosen type of metal matrix composite is not as useful as in the polymeric matrix one. Finally, for a polymeric matrix composite that is subjected to a transverse normal loading, the effect of imperfect bonding between the shape memory alloy fibers and the neighboring matrix is investigated.     

Analysis of elastic crack bridging in ceramic matrix composites

A micromechanics analytical model based on the consistent shear lag theory is developed for predicting the failure modes in fiber reinforced unidirectional stiff matrix composites. The model accounts for a relatively large matrix stiffness and hence its load carrying capacity. The fiber and matrix stresses are established as functions of the applied stress, crack geometry, and the microstructural properties of the constituents. From the predicted stresses, the mode of failure is established based on a point stress failure criterion. The role of the microstructural parameters of the constituents on the failure modes such as self-similar continuous cracking, crack bridging and debonding parallel to the fibers is assessed.     

高温环境下纤维复合材料蠕变损伤的细观机理研究

首先利用复合材料纤维断裂单胞模型，编制蠕变损伤子程序，对单胞模型进行蠕变损伤分析。分析了纤维／基体弹性模量比对蠕变变形、蠕变损伤以及应力场的影响。从计算结果发现，蠕变损伤首先在纤维断裂尖端起始，然后沿着一定的角度向基体外围延伸，直至完全损伤，而且纤维／基体模量比对高温环境下的复合材料蠕变损伤产生很大的影响；纤维与基体的模量相差越大，复合材料越容易变形，抵抗蠕变变形的能力就越小，蠕变损伤越严重。经过对不同韧性的基体材料进行研究，发现基体韧性低的复合材料蠕变损伤明显高于高韧性基体复合材料，表明低韧性基体复合材料抵抗蠕变破坏的能力较低。     

单向纤维增强复合材料中剪切波多重散射问题的复变函数法

基于弹性波的多体散射理论和复变函数方法,利用波函数展开法和保角映射方法,研究了任意形纤维增强复合材料中剪切波的传播。根据相应的边界条件确定弹性波模式系数。给出了复合材料中基体区和纤维核区的波场。作为算例,分析了相同入射频率,不同纤维核的尺寸和形状的情况下,弹性波的传播特性。通过分析发现,低频长波区,弹性波以近似正弦函数的形式传播,纤维核的尺寸和形状对波场影响不明显;随着频率的增加,影响变得明显,大于一定的频率,纤维核的形状可明显地辨别出。最后对结果进行了分析讨论。     

End-shaped copper fibers in an epoxy matrix—predicted versus actual fracture toughening

End-shaped copper fibers are placed in a brittle thermoset epoxy matrix at 10 vol% and tested in four-point bending to determine the fracture toughness of the composite. Results from four-point bend tests agree well with the theoretical predictions of the fracture toughness increment ‘ΔG’ of a metal fiber/brittle thermoset matrix composite based on single fiber pullout (SFP) tests. This close agreement demonstrates that SFP testing, along with the theoretical model, can be used as an effective end-shape screening tool for ductile fibers before full scale composite testing. The model predicts that the composite’s fracture toughness will be 46% higher with flat end-impacted fibers and 4% lower with rippled fibers compared to straight fibers at a 0° orientation. Four-point bend results show the actual composite’s fracture toughness is 49% higher with flat end-impacted fibers and 5% lower with rippled fibers compared to straight fibers. Further, four-point bend results show that end-shaped copper fibers improve both the flexural strength and modulus of the composite, demonstrating that end-shaped ductile fibers provide a good stress transfer to the fibers by anchoring the fibers into the matrix. Lastly, experimental validation of the model also indicates that at low fiber volume fractions, fiber–fiber interaction has only a minor influence on the fracture toughness for the tested ductile fiber/brittle matrix composite.     

Effect of fiber geometry on stress in fiber-reinforced composite materials

Eleven two-dimensional photoelastic models of fiber-reinforced composite-material configurations were designed for the purpose of investigating the effects of various fiber-end shapes, various fiber arrangements and broken fibers on the shear-stress distribution in an elastic matrix. Maximum shear stresses in the matrix in the vicinity of fiber discontinuities are presented. The peak stress resulting from a gradually tapered fiber was found to be slightly higher than that from a square-ended fiber, and a round-ended fiber produced a peak stress which was slightly lower than that from the square-ended fiber. Peak stresses resulting from two square-ended fibers butted closely together were considerably higher and decreased with increasing gap length between fiber ends. It made little difference whether the gap between the butted fiber ends was open, simulating a void resulting from a broken fiber, or filled with matrix material.Paper was presented at 1966 SESA Spring Meeting held in Detroit, Mich., on May 4–6.     

Finite-volume direct averaging micromechanics of heterogeneous materials with elastic–plastic phases

In this communication, we extend the recently re-constructed micromechanics model called high-fidelity generalized method of cells (Bansal, Y., Pindera, M.-J., 2005. A second look at the higher-order theory for periodic multiphase materials. J. Appl. Mech. 72 (2), 177–195.) by incorporating inelastic response capability for the individual phases. The re-construction, based on the local/global stiffness matrix approach, has simplified the model’s theoretical framework and substantially increased its computational efficiency as well as implementability, enabling analysis of unit cells with realistic multiphase microstructures previously unattainable in the original formulation developed by Aboudi et al. (Aboudi, J., Pindera, M-J., Arnold, S.M., 2001. Linear thermoelastic higher-order theory for periodic multiphase materials. J. Appl. Mech. 68 (5), 697–707; Aboudi, J., Pindera, M-J, Arnold, S.M., 2003. Higher-order theory for periodic multiphase materials with inelastic phases. Int. J. Plasticity 19 (6), 805–847.) with an accuracy approaching finite-element solutions. Just as importantly, the re-construction has revealed the model to be based on a finite-volume, direct averaging approach with clearly discernible similarities to, and differences with, the finite-element method and the finite-volume technique used in computational fluid mechanics. Herein, easily programmable closed-form expressions have been derived for the thermo-inelastic contributions to the local stiffness matrix equations that facilitate incorporation of different inelastic constitutive theories for the phase response. The re-constructed model is then employed to investigate orientational and architectural effects in unidirectional metal matrix composites characterized by multi-inclusion unit cells. Classical incremental plasticity theory with isotropic hardening is employed for the matrix response for consistency and comparison with previously reported results by Aboudi et al. (2003). Unit cells representative of a square array of fibers rotated by an angle about the fiber axis, which lack planes of material symmetry in the rotated coordinate system in which the micromechanical analysis is performed, belong in the first category. New results are presented for such rotated unidirectional porous composites which suggest guidelines for optimizing stiffness and ductility of this class of light-weight materials relative to dominant loading directions. Strengthening effects due to fiber clustering, which require highly discretized multi-inclusion unit cells, fall in the second category. It is demonstrated that the previously documented results for particulate composites, explained by the clustering-induced alteration of stress invariants which govern plastic strain evolution, are recovered for unidirectional composites as well.     

Theoretical prediction of stiffness and strength of three-dimensional and four-directional braided composites

Based on unit cell model, the 3D 4-directional braided composites can be simplified as unidirectional composites with different local axial coordinate system and the compliance matrix of unidirectional composites can be defined utilizing the bridge model. The total stiffness matrix of braided composites can be obtained by the volume average stiffness of unidirectional composites with different local axial coordinate system and the engineering elastic constants of braided composites were computed further. Based on the iso-strain assumption and the bridge model, the stress distribution of fiber bundle and matrix of different unidirectional composites can be determined and the tensile strength of 3D 4-directional braided composites was predicted by means of the Hoffman＇s failure criterion for the fiber bundle and Mises＇ failure criterion for the matrix.