Dynamic fracture behaviour in fibre-reinforced cementitious composites

The object of this work is to simulate the dynamic fracture propagation in fibre-reinforced cementitious composites, in particular, in steel fibre reinforced concrete (SFRC). Beams loaded in a three-point bend configuration through a drop-weight impact device are considered. A single cohesive crack is assumed to propagate at the middle section; the opening of this crack is governed by a rate-dependent cohesive law; the fibres around the fracture plane are explicitly represented through truss elements. The fibre pull-out behaviour is depicted by an equivalent constitutive law, which is obtained from an analytical load–slip curve. The obtained load–displacement curves and crack propagation velocities are compared with their experimental counterparts. The good agreement with experimental data testifies to the feasibility of the proposed methodology and paves the way to its application in a multi-scale framework.     

碳纳米管增强Nb-Si基复合材料界面应力传递的研究

基于均匀化理论,建立了碳纳米管增强Nb-Si基复合材料的代表体积元模型,并采用剪切滞后模型对碳纳米管增强Nb-Si基复合材料界面上的应力分布和传递机制进行了研究,探讨了分子间作用力、杨氏模量比β、长径比α、体积分数?等对其应力分布和传递机制的影响。结果表明,复合材料界面应力分布的变化主要集中在碳纳米管的两端,最大的应力都是分布在加载端或拔出端,然后向另一端传递;分子间作用力、杨氏模量比、长径比、体积分数等参数对界面应力的传递均有一定的影响,其中长径比和体积分数的影响最明显,体积分数为0.02时拔出端的界面剪切应力值相对于体积分数为0.0025时增大幅度达到近7倍,而长径比从200减小到50时,其应力传递距离增大了近8倍。     

Analytical study of the load transfer in fibre-reinforced 2D composite materials

The phenomenon of the load diffusion from a fibre to a surrounding matrix is analysed for the 2D case. We use an approximate analytical approach based on the asymptotic reduction of the governing biharmonic problem into two harmonic problems. The comparison of the obtained solutions with known results of other authors shows an acceptable accuracy of the proposed asymptotic simplifications. All solutions are obtained in closed analytical form.The case of perfect bonding between fibre and matrix for a single fibre and for a periodic system of fibres is firstly considered. Then we study the influence of the interface stiffness. In the case when only a single fibre is loaded, the influence of all other fibres is predicted by means of a three-phase model. The proposed approach gives a possibility to solve the problems for a broken fibre and for a broken matrix, as well as for partly debonded fibres. The important problem of infinite matrix cracks is also solved in the present paper.The obtained results can be used for the calculation of pull-out and push-out tests, as well as for the investigation of the fracture of composite materials.     

Continuum-mechanical modelling of kink-band formation in fibre-reinforced composites

A kink is a singular surface across which the displacement is continuous but the deformation gradient and the fibre direction suffer a discontinuity. A kink band is a highly deformed or even damaged region bounded by two kinks. The objective of modelling kink-band formation, within the framework of finite elasticity theory, is to find a suitable strain–energy function, guided by results from a finite number of simple experiments, that can be used to predict what have been observed and what might be possible under other loading conditions. In this paper, we explain a theoretical basis for choosing such strain–energy functions. More precisely, for a given strain–energy function that allows formation of kinks and a given deformation field, we characterize all possible deformation fields that can join the given deformation field through a kink and explain a procedure that can be used to assess the stability properties of any kink solution that is mathematically possible. In contrast with most previous studies in the engineering community where, for instance, the kink orientation angle is undetermined, the present theory completely determines the kink propagation stress, the kink orientation angle and the fibre direction within the kink band.     

Damping in ceramic matrix composites with matrix cracks

The paper presents an analytical solution capable of predicting the effect of matrix cracking in ceramic matrix composites (CMC) on damping. The cracking scenarios considered in the paper include through-the-thickness cracks and cracks terminating at the layer interfaces. The increase in damping associated with matrix cracking is mostly due to the frictional energy dissipation along the damaged fiber–matrix interfaces adjacent to the bridging cracks whose plane of propagation intersects the fiber axis. Damping increases with a higher density of matrix cracks. The loss factor is affected by the angle of the lamina relative to the direction of the applied load. The loss factor is also influenced by the frequency and magnitude of local dynamic stresses. Examples of distributions of the local loss factor along the axis of a CMC beam subject to pulsating loads of various frequencies are shown in the paper.     

Dynamics of vorticity stretching and breakup of isolated viscoelastic droplets in an immiscible viscoelastic matrix

The effect of droplet elasticity on transient deformation of isolated droplets in immiscible polymer blends of equal viscosity was investigated. In terms of the deformation parameter, Def*=a*–c/a*+c where a* and c are apparent drop principal axes, it undergoes two cycles of positive oscillations before reaching a negative value, followed by one cycle of oscillation before attaining a steady state negative value. This behavior was observed when Capillary number, Ca, was varied between 3 and 9 at a fixed Weissenber number, Wi, of 0.31, and when Ca number was held fixed at 8 and Wi number was varied between 0.21 and 0.40. In another blend of relatively lower Wi number of 0.21, one cycle of oscillation in Def* was observed before reaching steady state negative values when Ca number was varied between 3 and 14. The steady state Def* varies inversely with Ca number, with a stronger dependence for the blend with higher Wi number. The magnitude of oscillation increases with increasing Ca and Wi numbers. The critical Ca was found to be 12 and 14 for the two blends studied; these values are about 30 times greater than that of Newtonian blends.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with radially-graded fibres

This paper presents a closed-form expression for the homogenized longitudinal shear moduli of a linear elastic composite material reinforced by long, parallel, radially-graded circular fibres with a periodic arrangement. An imperfect linear elastic fibre-matrix interface is allowed. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the fibre domain is solved in closed form by applying the theory of hypergeometric functions, for new wide classes of grading profiles defined in terms of special functions. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. A parametric analysis investigating the influence of microstructural and material features on the effective moduli is presented. The feasibility of mitigating the shear stress concentration in the composite by tuning the fibre grading profile is shown.     

Modeling long-term creep rupture by debonding in unidirectional fibre-reinforced composites

Delayed fracture due to debonding can be observed in many unidirectional fibre-reinforced composites when the fibre/matrix interface experiences creep. The aim of this work is to describe such a phenomenon within the recently proposed modeling framework of transverse isotropy that allows for a neat decomposition of the mechanical behavior into fibre-directional, transverse, and pure shear parts. Specifically, debonding is here chosen to be governed by the tension transverse to the fibres. One can then speak of a mode-I debonding if use is made of the terminology adopted in fracture mechanics. On another hand, the time-dependent response is attributed to the matrix constituent. As the role of this latter is to deform and support stresses primarily in shear, a viscoelastic behavior is introduced that affects solely the pure shear part of the behavior. We show that both characteristics can be easily embedded into the aforementioned formulation. Among others, the occurrence of tertiary creep is made possible to predict. It is otherwise found that the predicted debonding path always propagates along the direction of the fibres in agreement with many experimental observations found in the literature. On the numerical side, the algorithmic treatment of debonding is independent of the one for viscoelasticity. This renders the implementation within the context of the finite element method very easy.     

Effective longitudinal shear moduli of periodic fibre-reinforced composites with functionally-graded fibre coatings

This paper presents a homogenization method for unidirectional periodic composite materials reinforced by circular fibres with functionally graded coating layers. The asymptotic homogenization method is adopted, and the relevant cell problem is addressed. Periodicity is enforced by resorting to the theory of Weierstrass elliptic functions. The equilibrium equation in the coating domain is solved in closed form by applying the theory of hypergeometric functions, for different choices of grading profiles. The effectiveness of the present analytical procedure is proved by convergence analysis and comparison with finite element solutions. The influence of microgeometry and grading parameters on the shear stress concentration at the coating/matrix interface is addressed, aimed at the composite optimization in regards to fatigue and debonding phenomena.     

Development and relaxation of internal stresses in granular composites with an epoxy matrix

The development and relaxation of stresses in epoxy matrix-filler systems were studied experimentally under isotropic thermal and baric effects. Data on the nuclear quadrupole resonance of the filler crystals were analyzed using an empirical model for the relaxation processes in the matrix of the composite material. The effective stress-relaxation time was found to depend on the temperature, external pressure, and the magnitude and type of deformations of the polymer matrix. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 104–114, July–August, 2006.     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part I

Summary From the continuum mechanical point of view, most of practical fibre-reinforced composites may be considered to be transversely isotropic, orthotropic, tetratropic, hexatropic, or octotropic according to material symmetries. Engineering applications of fibre-reinforced materials have already been well established. Based on tensor function representations, the formulation of constitutive equations for fibre-reinforced composites constitutes a consistent mathematical basis for modelling the mechanical behaviour of these oriented materials. Although we have good knowledge of transversely isotropic and orthotropic tensor function representations, only integrity bases for tetratropic and hexatropic polynomials could be learnt up to now in very limited cases. In this paper, we present the complete and irreducible representations for tetratropic, hexatropic and octotropic scalr-, symmetric tensor-, skew-symmetric tensor- and vector-valued functions (not only polynomials) of any finite number of symmetric tensors, skew-symmetric tensors and vectors in plane problems; a generalization of these results is also given. By use of these representations we shall perform the formulation of constitutive equations of fibre-reinforced composites in plane problems of elasticity, heat conduction, asymmetric elasticity, plasticity and continuum damage mechanics in a continued paper (Part II).

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Formulierung von Stoffgleichungen für faserverstärkte Verbundwerkstoffe bei ebenen Problemen: Teil I

Übersicht Im Hinblick auf Materialsymmetrien kann man für den praktischen Einsatz transversal-isotrope, orthotrope, tetratrope, hexatrope oder auch oktotrope faserverstärkte Verbundwerkstoffe unterscheiden. Für Anwendungen im Ingenieurbereich sind sie von größter Bedeutung. Die Darstellungstheorie tensorwertiger Funktionen bildet die Grundlage zur Formulierung von Stoffgleichungen anisotroper Stoffe, zu denen auch die oben erwähnten Verbundwerkstoffe gehören. Tensorfunktionen speziell für transversal-isotrope und orthotrope Stoffe sind hinreichend bekannt. Hingegen fehlen entsprechende Untersuchungen zum tetratropen und hexatropen Verhalten. Daher sollen im folgenden vollständige und irreduzible Darstellungen für derartige Materialorientierungen entwickelt werden. Dazu müssen vektor- und tensorwertige Funktionen mit unterschiedlichen Tensor- und Vektorargumenten in Betracht gezogen werden. Eine weiterführende Untersuchung (Teil II), in der u. a. auch anisotrope Werkstoffschädigungen berücksichtigt werden, ist in Vorbereitung.

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Paper presented at the XVIIIth ICTAM Congress in Haifa, Israel, August 1992.     

The formulation of constitutive equations for fibre-reinforced composites in plane problems: Part II

Summary From the continuum mechanics points of view, most of commercial fibre-reinforced composites (FRCs) can be considered to be anisotropic materials with one of the five material symmetries: transverse isotropy, orthotropy, tetratropy, hexatropy and octotropy, as illustrated in the preceding paper (Part I) [1]. No properly general formulation of constitutive equations for tetratropic, hexatropic and octoctropic types of FRC has been found in the literature. In this paper, the restriction to the admissible deformation of a FRC imposed by the failure strains of the fibres is investigated. The complete and irreducible two-dimensional tensor function representations regarding tetratropy, hexatropy and octotropy derived in Part I are applied to formulate constitutive equations for FRCs in plane problems of elasticity, yielding and failure in the present work, and of heat conduction, continuum damage and asymmetric elasticity in a continued work (Part III, forthcoming).The supports from the Alexander von Humboldt Foundation, Germany and the Research Foundation of Tsinghua University, P. R. China are acknowledged by the first author.     

Particle-liquid mass transfer in viscoelastic media

The problem of mass transfer from a solid sphere to a viscoelastic fluid has been examined theoretically. It is shown that fluid elasticity increases marginally the mass transfer rate in the creeping flow regime. This will have serious implications on the mass transfer from bubbles if impurities are present. Some conclusions on mass transfer at high Reynolds numbers are also offered.     

ISPH modeling of natural convection heat transfer with an analytical kernel renormalization factor

The objective of this study is to extend the attention of the incompressible smoothed particle hydrodynamics method (ISPH) in the heat transfer field. The ISPH method for the natural convection heat transfer under the Boussinesq approximation in various environments: pure-fluid, nanofluid, and non-Darcy porous medium is introduced. We adopted the improved analytical method for calculating the kernel renormalization factor and its gradient based on a quintic kernel function for the wall boundary treatment in the ISPH method. The proposed method requires no dummy particle layer to meet the impermeability condition and makes the heat flux over the wall boundary easy to implement. We performed four different numerical simulations of natural convection in cavities with increasing complexity in modeling and implementation: the natural convection in a square cavity with constant differentially heated wall temperature, natural convection with the heat flux from the bottom wall for a wide range of Rayleigh numbers, natural convection in a non-Darcy porous cavity fully filled with nanofluid in different flow regimes, and natural convection in a partially layered porous cavity. The results showed excellent agreement with results from literatures and the in-house P1–P1 finite element method code.     

Experimental study of an impact oscillator with viscoelastic and Hertzian contact

Modeling an impact event is often related to the desired outcome of an impact oscillator study. If the only intent is to study the dynamic behavior of the system, numerous researchers have shown that simpler impact models will often suffice. However, when the geometric contours and material properties of the two colliding surfaces are known, it is often of interest to model the contact event at a greater level of complexity. This paper investigates the application of a finite time impact model to the study of a parametrically excited planar pendulum subjected to a motion-dependent discontinuity. Experimental and numerical studies demonstrate the presence of multiple periodic attractors, subharmonics, quasi-periodic motions, and chaotic oscillations.