An elastic–plastic crack bridging model for brittle-matrix fibrous composite beams under cyclic loading

A fibrous composite beam with an edge crack is submitted to a cyclic bending moment and the crack bridging actions due to the fibers. Assuming a general elastic-linearly hardening crack bridging model for the fibers and a linear-elastic law for the matrix, the statically indeterminate bridging actions are obtained from compatibility conditions. The elastic and plastic shake-down phenomena are examined in terms of generalised cross-sectional quantities and, by employing a fatigue crack growth law, the mechanical behaviour up to failure is captured. Within the framework of the proposed fracture mechanics-based model, the cyclic crack bridging due to debonding at fiber–matrix interface of short fibers is analysed in depth. By means of some simplifying assumptions, such a phenomenon can be described by a linear isotropic tensile softening/compressive hardening law. Finally, numerical examples are presented for fibrous composite beams with randomly distributed short fibers.     

Residual strength of a fiber reinforced metal matrix composite with a crack

The residual strength of a cracked unidirectional fiver reinforced metal matrix composite is studied. We propose a bridging model based on the Dugdale strip yielding zones in the matrix ahead of the crack tips that accounts for ductile deformations of the matrix and fiber debonding and pull-out in the strip yielding zone. The bridging model is used to study the fracture of an anisotropic material and its residual strength is calculated numerically. The predicted results for a SiC/titanium composite agree well with the existing experimental data. It is found that a higher fiber bridging stress and a larger fiber pull-out length significantly contribute to the composite's residual strength. The composite's strength may be more notch-insensitive than the corresponding matrix material's strength depending on several factors such as fiber-matrix interface properties and the ratio of the matrix modulus to an ‘effective modulus’ of the composite.     

Ductile reinforcements for enhancing fracture resistance in composite materials

Ductile reinforcements can supply fracture toughness to a polymer matrix by pulling out and by plastically deforming. In the case of metal reinforcements that are not in a toughened condition, there may be more toughening to be gained when the fibers remain in the matrix and plastically deform rather than pulling out. These fibers can be said to have an unused plastic potential. When these fibers bridge a crack, their plastic deformation causes a rise in the force which is trying to pull out the fiber. Because of this, the shape of the fiber must be adjusted along its length if it is to remain anchored and contribute its plastic work. The use of anchored, ductile fibers provides a new design axis that brings new possibilities not achievable by the current research focus on the fiber–matrix interface. This paper describes the experimental pullout of aligned ductile fibers from a polymer matrix, and indicates the effect of the shape and embedded length of the fiber on the toughness increase of the composite. Anchored, plastically deforming fibers are shown to provide a major improvement to the toughening. Even for unoptimized ductile fibers, the calculated toughening improvement equals or exceeds the toughening available from current short glass or graphite fibers. In addition, pullout values are obtained for fibers that are embedded at an angle, simulating fiber bridging from fibers not perpendicular to the crack surface. These results further demonstrate the toughening efficiency of ductile fibers.     

Helical Fiber Pull-out in Biological Materials

Many biological materials, such as wood and bone, possess helicoid microstructures at microscale, which can serve as reinforcing elements to transfer stress between crack surfaces and improve the fracture toughness of their composites. Failure processes, such as fiber/matrix interface debonding and sliding associated with pull-out of helical fibers, are responsible mainly for the high energy dissipation needed for the fracture toughness enhancement. Here we present systemic analyses of the pull-out behavior of a helical fiber from an elastic matrix via the finite element method(FEM) simulation, with implications regarding the underlying toughening mechanism of helicoid microstructures. We find that, through their uniform curvature and torsion, helical fibers can provide high pull-out force and large interface areas, resulting in high energy dissipation that accounts, to a large extent, for the high toughness of biological materials. The helicity of fiber shape in terms of the helical angle has significant effects on the force-displacement relationships as well as the corresponding energy dissipation during fiber pull-out.     

短纤维增强三元乙丙橡胶横观各向同性黏-超弹性本构模型

短纤维增强三元乙丙橡胶包覆薄膜,是一种应用于固体火箭发动机缠绕包覆装药的新型复合材料.为了描述其在工作过程中受振动、冲击等载荷作用时的力学行为,基于黏弹性理论和纤维增强连续介质力学理论,提出了一种考虑应变率强化效应的横观各向同性黏-超弹本构模型.模型中应变能函数被分解为超弹性应变能和黏性应变能,其中超弹性应变能包括表征各向同性的橡胶基体应变能和表征各向异性的纤维拉伸应变能,黏性应变能采用表征橡胶和纤维黏性响应的宏观唯象模型.选取表征各应变能的函数形式,经过数学变换、替代、叠加,求解确定最终的应力应变形式,明确模型参数获取的具体步骤,将预测结果与实验结果对比分析,准确性较高.研究表明:该模型能有效预测材料在低应变率下纤维方向为0?～45?的非线性率相关力学特性;模型形式易于实现有限元开发,对固体火箭发动机装药结构完整性分析具有参考价值.     

Mechanical response of a short fiber-reinforced thermoplastic: Experimental investigation and continuum mechanical modeling

The present work can be regarded as a first step toward an integrated modeling of mold filling during injection molding process of polymer composites and the resulting material behavior under service loading conditions. More precisely, the emphasis of the present paper is laid on how to account for local fiber orientation in the ground matrix on the prediction of the mechanical response of the composite at its final solid state. To this end, a set of experiments which captures the mechanical behavior of an injection molded short fiber-reinforced thermoplastic under different strain histories is described. It is shown that the material exhibits complex response mainly due to non-linearity, anisotropy, time/rate-dependence, hysteresis and permanent strain. Furthermore, the relaxed state of the material is characterized by the existence of an equilibrium hysteresis independently of the applied strain rate. A three-dimensional phenomenological model to represent experimentally observed response is developed. The microstructure configuration of the material is simplified and assumed to be entirely represented by a distributed fiber orientation in the ground matrix. In order to account for distributed short fiber orientations in a continuum sense, a concept of (symmetric) generalized structural tensor (tensor of orientation) of second order is adopted. The proposed model is based on assumption that the strain energy function of the composite is given by a linear mixture of the strain energy of each constituent: an isotropic part representing Phase 1 which is essentially related to the ground matrix and an anisotropic part describing Phase 2 which is mainly related to the fibers and the interphase as a whole. Hence, taking into account the fiber content and orientation, the efficiency of the model is assessed and perspectives are drawn.     

Interaction of fiber and transformation toughening

B small scale bridging (SSB) assumption, a theoretical study is made of the interaction between the fracture toughening effects of transforming particles and crack bridging fibers which are aligned in the direction perpendicular to the crack surfaces. The fibers bridging the crack are assumed to undergo large amounts of slipping inside the matrix. It is found that the interaction can be synergistic over a parametric range of material properties of the ceramic composite system. The Mode-1 plane-strain fracture toughness of the ceramic composite system is determined in terms of fiber strength, transformation toughening parameters and interaction parameters. The results obtained here are qualitatively similar to those for the interaction between the fracture toughening effects of transforming particles and crack-bridging ductile particles by A and B (1988a, J. Mech. Phys. Solids 36, 581).     

Asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials

An elastic analysis of an internal crack with bridging fibers parallel to the free surface in an infinite orthotropic anisotropic elastic plane is studied, and asymmetrical dynamic fracture model of bridging fiber pull-out of unidirectional composite materials is presented for analyzing the distributions of stress and displacement with the internal asymmetrical crack under the loading conditions of an applied non-homogenous stress and the traction forces on crack faces yielded by the bridging fiber pull-out model. Thus the fiber failure is ascertained by maximum tensile stress, the fiber ruptures and hence the crack propagation should also appear in the modality of self-similarity. The formulation involves the development of a Riemann-Hilbert problem. Analytical solution of an asymmetrical propagation crack of unidirectional composite materials under the conditions of two increasing loads given is obtained, respectively. In terms of correlative material properties, the variable rule of dynamic stress intensity factor was depicted very well. After those analytical solutions were utilized by superposition theorem, the solutions of arbitrary complex problems could be gained.     

Influence of fiber’s shape and size on overall elastoplastic property for micropolar composites

A general micromechanical method is developed for a micropolar composite with ellipsoidal fibers, where the matrix material is idealized as a micropolar material model. The method is based on a special micro–macro transition method, and the classical effective moduli for micropolar composites can be determined in an analytical way. The influence of both fiber’s shape and size can be analyzed by the proposed method. The effective moduli, initial yield surface and effective nonlinear stress and strain relation for a micropolar composite reinforced by ellipsoidal fibers are examined, it is found that the prediction on the effective moduli and effective nonlinear stress and strain curves are always higher than those based on classical Cauchy material model, especially for the case where the size of fiber approaches to the characteristic length of matrix material. As expected, when the size of fiber is sufficiently large, the classical results (size-independence) can be recovered.     

含空心纤维热电复合材料的能量转换效率及力学性能

空心纤维常用于热电复合材料的结构设计。纤维附近产生的不均匀温度场会引起局部热应力集中,威胁材料的可靠性并可能导致结构失效。本文采用圆环夹杂模型,研究了含空心纤维热电复合材料在均匀远场电流和能流作用下的力学响应。基于非线性全耦合的热电本构方程,利用复变函数中的级数法得到了纤维和基体中热电场和应力场的解析解。通过数值算例,分析了空心纤维的传导能力和几何尺寸对温度场、应力场和局部热电转换效率的影响。结果表明：随着空心纤维内径和界面热阻的增大,界面周围的应力场增大,但并不改变应力场的分布趋势。此外,我们发现：温度分布和应力场对几何参数比对界面热阻更为敏感。     

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.     

A discrete-fibers model for bridged cracks and reinforced elliptical voids

A model of crack bridging and reinforced elliptical voids is proposed, in which the fibers joining the surfaces of the void or crack are modelled as discrete, linear elastic bars. We show that a theory recently developed by us to analyze structural interfaces permits analytical attack and solution of multiple important previously unsolved problems of stress concentration and fracture. In particular, an analytical solution is provided for a reinforced elliptical void, which, by superposition, allows treatment of arbitrary fiber distributions, which can be even randomly distributed and oriented. In the special case of small or null ratio between a void's axes, new stress intensity factor expressions are obtained, which account for fibers’ inclination and geometry.     

Effective elastic moduli of composites reinforced by particle or fiber with an inhomogeneous interphase

The solution of the strain energy change of an infinite matrix due to the presence of one spherical particle or cylindrical fiber surrounded by an inhomogeneous interphase is the basis of solving effective elastic moduli of corresponding composites based on various micromechanics models. In order to find out the strain energy change, the composite sphere or cylinder, i.e., the spherical particle or cylindrical fiber together with its interphase, is replaced by an effective homogeneous particle or fiber. Independent governing differential equations for each modulus of the effective particle or fiber are derived by extending the replacement method [J. Mech. Phys. Solids 12 (1964) 199]. As far as the strain energy changes of the infinite matrix subjected to various far-field stress systems are concerned, the present model is simple. Meanwhile, FEM analysis is carried out for a verification, which shows that the model can lead to rather accurate results for most practical interphases. Besides, to check the validity of the model further when the interactions among composite cylinders exist, the two problems of an infinite matrix containing two composite cylinders and the effective moduli of composites with the equilateral triangular distribution of composite cylinders are analyzed using FEM. The FEM results show that the model is still rather accurate, especially for the case of interphase properties varying between those of fiber and matrix. Therefore, composite spheres or cylinders are assumed as the effective homogeneous particles or fibers and simple expressions of the effective moduli of composites containing the composite spheres or cylinders are obtained. Furthermore, the present model is compared with some existing models that are based on very complicated derivations.     

Homogenization of long fiber reinforced composites including fiber bending effects

This paper presents a homogenization method, which accounts for intrinsic size effects related to the fiber diameter in long fiber reinforced composite materials with two independent constitutive models for the matrix and fiber materials. A new choice of internal kinematic variables allows to maintain the kinematics of the two material phases independent from the assumed constitutive models, so that stress–deformation relationships, can be expressed in the framework of hyper-elasticity and hyper-elastoplasticity for the fiber and the matrix materials respectively. The bending stiffness of the reinforcing fibers is captured by higher order strain terms, resulting in an accurate representation of the micro-mechanical behavior of the composite. Numerical examples show that the accuracy of the proposed model is very close to a non-homogenized finite-element model with an explicit discretization of the matrix and the fibers.     

Analysis on small scale bridging in fibrous monolithic ceramics

Based on shear lag model of interface between fiber and matrix, a new formula that relates the crack opening displacement and bridging force in fibrous monolithic ceramics was constructed under the framework of small scale bridging. This formula was applied to predict the fracture resistance orR-curve response of a three-point bending specimen made of fibrous monolithic ceramics. A parametric investigation on the influences of fiber volume fraction, fiber radius, characteristics of constituents, BN's fracture toughness and specimen's geometry on the bridging forces and fracture resistance in Si₃N₄/BN composite was carried out. The upper and lower limits of theR-curve of Si₃N₄/BN in small scale bridging were derived. This research revealed the role played by the above parameters in the fracture toughness of fibrous monolithic ceramics. Supported by National Natural Science Foundation of China (59632090).     

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．     

Effects of matrix inelasticity on the overall hysteretic behavior of TiNi-SMA fiber composites

The effects of the inelastic deformation of the matrix on the overall hysteretic behavior of a unidirectional titanium–nickel shape-memory alloy (TiNi-SMA) fiber composite and on the local pseudoelastic response of the embedded SMA fibers are studied under the isothermal loading and unloading condition. The multiaxial phase transformation of the SMA fibers is predicted using the phenomenological constitutive equations which can describe the two-step deformation due to the rhombohedral and martensitic transformations, and the inelastic behavior of the matrix material using the standard nonlinear viscoplastic model. The average behavior of the SMA composite is evaluated with the micromechanical method of cells. It is observed that the inelastic deformation of the matrix due to prior tension results in a compressive stress in the matrix after unloading of the SMA composite and this residual stress impedes the complete recovery of the pseudoelastic strain of the SMA fibers. This explains that a closed hysteresis behavior of the SMA composite is no longer observed in contrast with the case that an elastic behavior of matrix is assumed. The predicted local stress–strain behavior indicates that the cyclic response of matrix is crucial to the design of the hysteretic performance of the SMA composite under the repeated loading conditions.     

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.     

平纹编织陶瓷基复合材料面内剪切细观损伤行为研究

采用约西佩斯库(Iosipescu)纯剪切试件,研究了平纹编织SiC/SiC和C/SiC复合材料的面内剪切应力-应变行为和细观损伤特性.通过试验获得了材料不同方向上的单调和迟滞应力-应变行为,对比分析了两种材料的剪切损伤特性,结果表明材料的剪切损伤演化规律受热残余应力水平影响严重.由试件断口电镜扫描结果发现剪切加载状态下桥连纤维承受显著的弯曲载荷和变形,据此提出了纤维弯曲承载机制,并结合裂纹闭合效应分阶段阐释了材料的剪切迟滞环形状.基于材料的剪切细观损伤机制,通过两个损伤变量表征了材料的剪切损伤演化进程,得到了材料的面内剪切细观损伤演化模型.对比发现2D-C/SiC复合材料45°方向基体裂纹的起裂应力明显小于2D-SiC/SiC复合材料,而两者0°/90°方向裂纹的起裂应力基本相同.