A direct approach to fiber and membrane reinforced bodies. Part II. Membrane reinforced bodies

The paper deals with membrane reinforced bodies with the membrane treated as a two-dimensional surface with concentrated material properties. The bulk response of the matrix is treated separately in two cases: (a) as a coercive nonlinear material with convex stored energy function expressed in the small strain tensor, and (b) as a no-tension material (where the coercivity assumption is not satisfied). The membrane response is assumed to be nonlinear in the surface strain tensor. For the nonlinear bulk response in Case (a), the existence of states of minimum energy is proved. Under suitable growth conditions, the equilibrium states are proved to be exactly states of minimum energy. Then, under appropriate invertibility condition of the stress function, the principle of minimum complementary energy is proved for equilibrium states. For the no-tension material in Case (b), the principle of minimum complementary energy (in the absence of the invertibility assumption) is proved. Also, a theorem is proved stating that the total energy of the system is bounded from below if and only if the loads can be equilibrated by a stress field that is statically admissible and the bulk stress is negative semidefinite. Two examples are given. In the first, we consider the elastic semi-infinite plate with attached stiffener on the boundary (Melan’s problem). In the second example, we present a stress solution for a rectangular panel with membrane occupying the main diagonal plane.     

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.     

Rheological characterization of continuous fiber—reinforced viscous fluid

The present paper describes a micromechanical technique to determine rheological properties of viscous fluid reinforced with unidirectional continuous fibers. Fluid viscosity is described by a shear thinning model and high viscosity is considered for continuous fibers having considerable rigidity compared to net fluid. The microstructure is identified by a representative volume element that is subjected to equivalent macroscopic deformation fields. The energy balance and periodicity conditions are considered to relate deformation and stress in macro and micro-levels. It is shown that response of viscous fluid reinforced with rigid fibers depends on deformation history as well as rate-of-deformation in the transverse intraply shear and transverse squeeze flows. An orthotropic viscous constitutive equation is derived to describe response of such materials. The material viscosities are evaluated for viscous fluid reinforced with different fiber volume fractions during deformation applied in different rates of deformation. The results are used to derive the functions predicting effective anisotropic viscosities of reinforced fluid.     

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.     

Mechanics of an elastic solid reinforced with bidirectional fiber in finite plane elastostatics: complete analysis

A continuum-based model is presented for the mechanics of bidirectional composites subjected to finite plane deformations. This is framed in the development of a constitutive relation within which the constraint of material incompressibility is augmented. The elastic resistance of the fibers is accounted for directly via the computation of variational derivatives along the lengths of bidirectional fibers. The equilibrium equation and necessary boundary conditions are derived by virtue of the principles of virtual work statement. A rigorous derivation of the corresponding linear theory is developed and used to obtain a complete analytical solution for small deformations superposed on large. The proposed model can serve as an alternative 2D Cosserat theory of nonlinear elasticity.     

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.     

Mechanical response of neo-Hookean fiber reinforced incompressible nonlinearly elastic solids

The mechanical behavior of an incompressible neo-Hookean material, directionally reinforced by neo-Hookean fibers, is examined under homogeneous deformations. A composite model for this transversely isotropic material is developed based on a multiplicative decomposition of the deformation gradient which considers interaction between the fiber and the matrix. The so-called standard reinforcing model exhibits non-monotonic behavior in compression. The present composites-based approach leads to a modification of the standard reinforcing model in which monotonic behavior in compression is observed. This stems from the micromechanical basis of the model in which the fiber is treated as a neo-Hookean material. The conditions for loss of monotonicity and positivity in the stress-shear behavior in off-axis simple 2D shear are also obtained.     

Well-posedness of the problem of fiber suspension flows

The well-posedness of the equations governing the flow of fiber suspensions is studied. The fluid is assumed to be Newtonian and incompressible, and the presence of fibers is accounted for through the use of second- and fourth-order orientation tensors, which model the effects of the orientation of fibers in an averaged sense. The fourth-order orientation tensor is expressed in terms of the second-order tensor through various closure relations. It is shown that the linear closure relation leads to anomalous behavior, in that the rest state of the fluid is unstable, in the sense of Liapounov, for certain ranges of the fiber particle number. No such anomalies arise in the case of quadratic and hybrid closure relations. For the quadratic closure relation, it is shown that a unique solution exists locally in time for small data.     

On some connections between equivalent single material and mixture theory models for fiber reinforced hyperelastic materials

There are two approaches that can be used to model the large strain mechanical response of material systems in which elastic fibers are embedded in an elastic matrix. In the first approach, a fiber reinforced material undergoing large deformation is homogenized in the sense that it is assumed to act as an equivalent single material that is transversely isotropic and hyperelastic. Both constituents then share a common reference configuration, which is typically assumed to be a natural or stress-free configuration for the equivalent single material. The stress depends on a single deformation gradient defined with respect to the natural configuration.In the second approach, the fiber/matrix system is treated as a mixture, with the matrix and the fibrous constituents having their own reference configurations and material symmetries. The total stress depends on the deformation gradients and material symmetries for both constituents, defined with respect to their reference configurations.Under appropriate assumptions, the constitutive theory developed using mixture theory can coincide with the constitutive theory assuming an equivalent single material that is transversely isotropic and hyperelastic. This paper explores the connection between the two approaches by considering the various reference configurations and material symmetries.     

Macroscopic constitutive relations for elastomers reinforced with short aligned fibers: Instabilities and post-bifurcation response

This paper is concerned with the characterization of the macroscopic response and possible development of instabilities in a certain class of anisotropic composite materials consisting of random distributions of aligned rigid fibers of elliptical cross section in a soft elastomeric matrix, which are subjected to general plane strain loading conditions. For this purpose, use is made of an estimate for the stored-energy function that was derived by Lopez-Pamies and Ponte Castañeda (2006b) for this class of reinforced elastomers by means of the second-order linear comparison homogenization method. This homogenization estimate has been shown to lose strong ellipticity by the development of shear localization bands, when the composite is loaded in compression along the (in-plane) long axes of the fibers. The instability is produced by the sudden, collective rotation of a band of fibers to partially release the high stresses that develop in the elastomer matrix when the composite is compressed along the stiff, long-fiber direction. Consistent with the mode of the impending instability, a lower-energy, post-bifurcation solution is constructed where “striped domain” microstructures consisting of layers with alternating fiber orientations develop in the composite. The volume fractions of the layers and the fiber orientations within the layers adjust themselves to satisfy equilibrium and compatibility across the layers, while remaining compatible with the imposed overall deformation. Mathematically, this construction is shown to correspond to the rank-one convex envelope of the original estimate for the energy, and is further shown to be polyconvex and therefore quasiconvex. Thus, it corresponds to the “relaxation” of the stored-energy function of the composite, and can in turn be viewed as a stress-driven “phase transition,” where the symmetry of the fiber microstructures changes from nematic to smectic.     

Coupled twoscale analysis of fiber reinforced composite structures with microscopic damage evolution

The simultaneous twoscale analysis of unidirectionally fiber reinforced composite structures with attention to damage evolution is the objective of the contribution. The heterogeneous microstructure of the composite represents the microscale, whereas the laminate or the structural component are addressed as the macroscale. The macroscale is conventionally discretized by the finite element method (FEM). The generalized method of cells (GMC) in its efficient stress based formulation serves as the discrete microscale model. The stiff and brittle fibers behave linearly elastic. The epoxy resin is described by the nonlinear-elastic model of Ramberg–Osgood. By introducing microcrack models, the damage of the epoxy matrix under combined tensile and shear loading is taken into account. The cell boundaries of the micromodel are used to locate microscopic cracks deterministically. Interface models for the representation of damage in the matrix phase as well as for the weakening of the fiber–matrix-bond are used. This approach circumvents the need for the regularization, as it would be necessary for continuum damage models with softening characteristics. Hence, the micromodel is numerically stable and convergent. The GMC allows to obtain the consistently linearized constitutive tensor in the case of nonlinear material behavior in a simple and straight forward manner which is easily implemented in comparison to micromodels based on the finite element technique. The damage evolution on the microscale manifests itself macroscopically in the degradation of the homogenized stiffnesses.     

Debonding and fiber pull-out in reinforced composites

In order to evaluate the strength of fiber-reinforced composites, there is first the need to investigate the interfacial debonding and the pull-out of fibers in a fractured composite with intact fibers. This type of problem in crack bridging has been investigated by several authors based on different models and assumptions [1–7]. In this study, we will consider a three-dimensional model of a single fiber of finite length bonded by a finite cylindrical matrix with an initial crack existing in a portion of the interface. In the model, one end of the cylinder is so constrained that the axial component of displacement vanishes. A tensile stress is applied to the fiber at the other end. The aim is to determine the pull-out of the fiber and the critical condition for interfacial debonding. Both the fiber and the matrix are treated as elastic materials. Analysis is made based on a method using Papkovich-Neuber displacement potential functions for the problem of an elastic solid subjected to axisymmetrical boundary conditions. Solutions are found by means of the technique of trigonometrical series. Effects of initial misfit strains and frictional sliding between the fiber and the matrix over the interfacial crack are also included in the study.     

A continuum model for fiber reinforced materials with debonding

A continuum theory for a fiber-reinforced material with debonding between the constituents is presented. The debonding phenomenon is simulated by imposing the continuity of the normal displacements at the fiber-matrix interfaces while allowing free tangential slip there. The derived theory is of the lowest order and is obtained by using a first order expansion in the displacements in the fiber and matrix phases. The theory is applied to investigate the effect of debonding on the propagation of waves in a boron/epoxy fiber reinforced material. It is shown that an additional mode of propagation is obtained as compared with the usual case of perfect bonding     

Modeling oxidation damage of continuous fiber reinforced ceramic matrix composites

For fiber reinforced ceramic matrix composites(CMCs),oxidation of the constituents is a very important damage type for high temperature applications. During the oxidizing process,the pyrolytic carbon interphase gradually recesses from the crack site in the axial direction of the fiber into the interior of the material. Carbon fiber usually presents notch-like or local neck-shrink oxidation phenomenon,causing strength degradation. But,the reason for SiC fiber degradation is the aw growth mechanism on its surface. A micromechanical model based on the above mechanisms was established to simulate the mechanical properties of CMCs after high temperature oxidation. The statistic and shearlag theory were applied and the calculation expressions for retained tensile modulus and strength were deduced,respectively. Meanwhile,the interphase recession and fiber strength degradation were considered. And then,the model was validated by application to a C/SiC composite.     

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.     

Nonlinear stability of timber column strengthened with fiber reinforced polymer

By taking into account the effect of the bi-modulus for tension and compression of the fiber reinforced polymer (FRP) sheet in the reinforcement layer, a general mathematical model for the nonlinear bending of a slender timber beam strengthened with the FRP sheet is established under the hypothesis of the large deflection deformation of the beam. Nonlinear governing equations of the second order effect of the beam bending are derived. The nonlinear stability of a simply-supported slender timber column strengthened with the FRP sheet is then investigated. An expression of the critical load of the simply-supported FRP-strengthened timber beam is obtained. The existence of postbuckling solution of the timber column is proved theoretically, and an asymptotic analytical solution of the postbuckling state in the vicinity of the critical load is obtained using the perturbation method. Parameters are studied showing that the FRP reinforcement layer has great influence on the critical load of the timber column, and has little influence on the dimensionless postbuckling state.     

Nonlinear stability of timber column strengthened with fiber reinforced polymer

By taking into account the effect of the bi-modulus for tension and compression of the fiber reinforced polymer (FRP) sheet in the reinforcement layer, a general mathematical model for the nonlinear bending of a slender timber beam strengthened with the FRP sheet is established under the hypothesis of the large deflection deformation of the beam. Nonlinear governing equations of the second order effect of the beam bending are derived. The nonlinear stability of a simply-supported slender timber column strengthened with the FRP sheet is then investigated. An expression of the critical load of the simply-supported FRP-strengthened timber beam is obtained. The existence of postbuckling solution of the timber column is proved theoretically, and an asymptotic analytical solution of the postbuckling state in the vicinity of the critical load is obtained using the perturbation method. Parameters are studied showing that the FRP reinforcement layer has great influence on the critical load of the timber column, and has little influence on the dimensionless postbuckling state.     

Capillary instabilities in a thin nematic liquid crystalline fiber embedded in a viscous matrix

$m$ to take into account non-axisymmetric modes. Capillary instabilities in nematic fibers reflect the anisotropic nature of liquid crystals, such as the orientation contribution to the surface elasticity and surface bending stresses. Surface gradients of bending stresses provide additional anisotropic contributions to the capillary pressure that may renormalize the classical displacement and curvature forces that exist in any fluid fiber. The exact nature (stabilizing and destabilizing) and magnitude of the renormalization of the displacement and curvature forces depend on the nematic orientation and the anisotropic contribution to the surface energy, and accordingly capillary instabilities may be axisymmetric or non-axisymmetric, with finite or unbounded wavelengths. Thus, the classical fiber-to-droplet transformation is one of several possible instability pathways while others include surface fibrillation. The contribution of the viscosity ratio to the capillary instabilities of a thin nematic fiber in a viscous matrix is analyzed by two parameters, the fiber and matrix Ohnesorge numbers, which represent the ratio between viscous and surface forces in each phase. The capillary instabilities of a thin nematic fiber in a viscous matrix are suppressed by increasing either fiber or matrix Ohnesorge number, but estimated droplet sizes after fiber breakup in axisymmetric instabilities decrease with increasing matrix Ohnesorge number. Received November 26, 2001 / Published online May 21, 2002 Communicated by Epifanio Virga, Pavia     

Evolution of bond fractures in a randomly distributed fiber network

Fracture in a planar randomly ordered fiber network subjected to approximately homogenous macroscopic stress and strain field is considered. A theory describing material degradation on a macroscopic scale is derived via Griffith’s energy balance for an internal fractured area in the network assuming the active fracture process on the microscopic level is fiber–fiber bond breakage. Attention is confined to a purely mechanical theory assuming isothermal processes and the theory relies on equations commonly used in theories of statistical physics. In the theory, a bond breaking driving force is stated to be equal to the elastic strain energy density of a non-fractured network. A debond fraction can be coupled to a linearly decrease of the network’s macroscopic stiffness. The rate of the fracture processes is determined by the network’s inherent properties (bond and fiber density, bond strength, etc.). During the loading process, until onset of localization, the bond breaks occur at randomly distributed locations spread over the fiber network and the theory estimate material degradation on a macroscopic level. When localization takes place, the fracture process changes from a two-dimensional randomly distributed process to a one-dimensional process and other theories have to be included to describe post-localization behavior. An approximately in-plane isotropic low-density paper is used in tensile experiments while monitoring acoustic emission activity to evaluate the theory. The experimentally obtained results support the theory surprisingly well.