Influence of the Interaction between Two Neighboring Periodically Curved Fibers on the Stress Distribution in a Composite Material

A method is developed for a stress analysis in an infinite elastic body containing two neighboring periodically cophasaly curved fibers located along two parallel lines. The stress distribution is studied when the body is loaded at infinity by uniformly distributed normal forces in the fiber direction. The investigation is carried out within the framework of a piecewise homogeneous body model with the use of exact three-dimensional equations of elasticity theory. Numerical results related to the stress distribution considered and the influence of interaction between the fibers on this distribution are given.     

Stress distribution caused by two neighboring out-of-plane locally cophasally curved fibers in a composite material

In the present paper, within the framework of a piecewise homogenous body model, with the use of the exact three-dimensional equations of elasticity theory, a method proposed earlier is developed for investigating the stress distribution caused by two neighboring out-of-plane locally cophasally curved fibers located along two parallel planes in an infinite elastic body. The body is loaded at infinity by uniformly distributed normal forces in the direction of fiber location. The self-equilibrated normal and shear stresses caused by the curved fibers are analyzed, and the influences of interaction between the fibers and of the geometric nonlinearity on the distribution of these stresses are studied. Numerical results for this interaction are obtained.     

Stress Distribution in an Elastic Body with a Periodically Curved Row of Fibers

Within the framework of a piecewise homogeneous body model, with the use of exact three-dimensional equations of elasticity theory for anisotropic bodies, a method is developed for investigating the stress distribution in an infinite elastic matrix containing a periodically curved row of cophasal fibers. It is assumed that fiber materials are the same and fiber midlines lie in the same plane. The self-balanced stresses arising in the interphase in uniaxial loading the composite along the fibers are investigated. The influences of problem parameters on these stresses are analyzed. The corresponding numerical results are presented.     

Stress distribution in an infinite elastic body containing two neighboring locally curved fibers

A method is developed for analyzing the stresses in an infinite elastic body containing two neighboring inphase locally curved fibers located along two parallel lines. The body is loaded at infinity by uniformly distributed nor mal forces in the direction of fibers. The investigation is carried out within the frame work of a piecewise homogeneous body model with the use of the three-dimensional ex act equations of the elasticity theory. Numerical results for stress distributions in the body and for the influence of interaction between fibers on these distributions are given. Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 45, No. 3, pp. 457-478, May-June, 2009.     

The representation of the displacement gradient of isotropic elastic body in terms of the piola stress tensor : PMM vol. 40, n 6, 1976, pp. 1070–1077

The representation of the displacement gradient of an isotropic elastic body is analyzed. It is shown on the basis of a single controlling inequality and a polar expansion of the Piola tensor that such representation has generally four branches. The mechanical meaning and the nature of that ambiguity is explained. It is established that when the angles of turn of material fibers are not excessively large, only one of the four branches is obtained. Particular cases in which the nature of ambiguity is more complex are investigated. It is noted that in many practical problems the representation of the displacement gradient by the Piola stress tensor is unambiguous.The considered problem is associated with the variational principle of complementary energy in the nonlinear theory of elasticity, where the statistically feasible fields of the asymmetric Piola stress tensor is varied [1], A method was proposed there for expressing the displacement gradient in terms of the Piola stress tensor for an isotropic elastic body. Later the concept of complementary energy and the representation of the strain gradient in terms of the Piola stress tensor were considered in [2, 3]. Examples of the use of the complementary energy concept are given in [2] and the case of an anisotropic body is considered in [3], These investigations disclosed that the considered representation of the strain tensor leads to ambiguity, but the character and nature of the ambiguity were not fully investigated.     

Elastic constants and state of stress of glass-reinforced strip

The author examines the problem of the state of stress of glass-reinforced strip under short-time loading, when viscoelastic effects are unimportant. The proposed model is composed of a system of parallel identical glass fibers the space between which is filled with a matrix—resin. The change in the elastic constants is found as a function of the volume content of glass reinforcing. Further, the stress distribution between the elementary fibers is investigated for a glass-reinforced plastic in the three-dimensional stress state.Mekhanika Polimerov, Vol. 2, No. 4, pp. 593–602, 1966     

Stresses in plastics reinforced with anisotropic fibers and subjected to transverse normal loading

The question of the stress distribution in plastics reinforced with anisotropic fibers and subjected to transverse normal loading is considered. The stresses in the components are determined by the methods of the theory of elasticity using stress functions. The theoretical relations obtained are used to construct diagrams showing the distribution of the tangential, radial, and shear stresses in the composite and the isoclines of the concentration coefficient for a carbon-reinforced plastic. The results obtained for the carbon-reinforced plastic are compared with the analogous results for a glass-reinforced plastic.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, No. 2, pp. 244–252, March–April, 1973.     

Structural theory of elastomeric composites based on fiber systems — Invariant description

A three-dimensional theory of elastomeric composites with elastomeric matrices reinforced by systems of fibers is presented. The theory is based on a structural approach in which the matrix and the reinforcement of the composite are considered separately without reduction to a medium having continuously changing characteristics. The approach is based on the idea of a vector field of macroscopic displacements given by the positions of the axial lines of the fibers in the curret (deformed) configuration of the composite. The vector field determines the current macroscopic configuration, the tensor fields of the measures of macroscopic strain, and the field of the macroscopic stress tensor in the composite. The displacement, strain, and stress fields in the elastomeric matrix and the fibers of the reinforcing systems are regarded as derivatives of the field of macroscopic displacements of the medium. Relations are presented to describe the kinematics of the fibers in the current configuration of the composite, including the evolution of their orientation and the frequency of their planar and spatial distribution. Equations are obtained for the macroscopic motion of the fiber-reinforced matrix, and the dynamic variational principle that governs this motion is established. The elastic macroscopic potential of the matrix is found and related to the components of the macroscopic stress tensor. The procedure to be followed in constructing the constitutive equations of the composite is described. The proposed system of equations, relations, and algorithms is closed and can be used to solve problems involving the deformation of products made of fiber-reinforced elastomers and the creation of elastomeric composite products, based on fiber systems, that possess the requisite properties.     

Analysis of stresses in an infinite elastic body with a locally curved covered nanofiber

Within the framework of a piecewise homogeneous body model, with the use of the three-dimensional geometrically nonlinear exact equations of the theory of elasticity, the method developed for determining the stress distribution in nanocomposites with unidirectional locally curved covered nanofibers is used to investigate the normal stresses acting along nanofibers. The investigation is carried out for an infinite elastic body containing a single locally curved covered nanofiber in the case where there exists a bond covering cylinder of constant thickness between the nanofiber and the matrix material. It is assumed that the body is loaded at infinity by uniformly distributed normal forces in the fiber direction. Upon formulation and mathematical solution of the boundary value problem, the boundary form perturbation method is used. Numerical results for the stress distribution in the body and the influence of geometrical nonlinearity on this distribution are presented and interpreted.     

Stress Distribution in an Infinite Elastic Body with a Locally Curved Fiber in a Geometrically Nonlinear Statement

Within the framework of a piecewise homogeneous body model, with the use of three-dimensional geometrically nonlinear exact equations of elasticity theory, a method for determining the stress—strain state in unidirectional fibrous composites with locally curved fibers is developed for the case where the interaction between the fibers is neglected. All the investigations are carried out for an infinite elastic body containing a single locally curved fiber. Numerical results illustrating the effect of geometrical nonlinearity on the distribution of the self-balanced normal and shear stresses acting on the interface and arising as a result of local curving of the fiber are presented.__________Russian translation published in Mekhanika Kompozitnykh Materialov, Vol. 41, No. 4, pp. 433–448, July–August, 2005.     

圆管纤维悬浮湍流场中粒子运动的研究

利用从细长体理论出发得到的三维分段积分法和湍流简化方法模拟了大量纤维粒子在圆管湍流内的运动．统计了不同Re数下计算区域内的纤维的取向分布,计算结果与实验结果基本吻合,结果表明湍流的脉动速度导致纤维取向趋于无序,且随着Re数的增加,纤维取向的分布越来越趋于均匀．其后又考虑了纤维速度和角速度的脉动,二者都充分体现了流体速度脉动的影响,且纤维速度的脉动在流向上的强度大于横向,而其角速度的脉动在流向上的强度小于横向．最后统计了纤维在管道截面上的位置分布,说明Re数的增加加速了纤维在管道截面上的位置扩散．     

Strength and deformability of reinforced polymers in tension normal to the fibers

The strength and deformability of reinforced polymers in tension across the fibers is investigated. It is assumed that the polymer deforms as an ideal elastoplastic body. Relations are obtained for the nature of the deformation of the polymer between the fibers and the strength and deformability of the composite with allowance for the structural distribution of the components. Theoretical stress-strain diagrams are presented for composites with different reinforcement densities and resin elasticities. The theoretical values of the strength and deformation of reinforced polymers with the load applied across the fibers are compared with the results of experiments on model specimens of epoxy-Thiokol polymers.Leningrad Mechanical Institute. Translated from Mekhanika Polimerov, No. 4, pp. 682–687, July–August, 1970.     

Forced vibration of the pre-stressed bi-layered plate-strip with finite length resting on a rigid foundation

Within the scope of the piecewise homogeneous body model utilizing Three-Dimensional Linearized Theory of Elastic Waves in Initially Stressed Bodies the time-harmonic dynamical stress field in the pre-stressed bi-layered plate-strip with finite length resting on the rigid foundation is investigated. The materials of the layers are assumed to be isotropic. The FEM modeling is developed for the solution to the corresponding boundary-value-contact problem. The numerical results regarding the influence of the finiteness of the layers’ length on the stress distribution on the interface planes are presented and discussed. In particular, it is shown that with increasing the plate length the results obtained for the considered case approach to the corresponding ones attained for the bi-layered plate with infinite length.     

基于有限元的金属基短纤维复合材料MMC的一种统计蠕变模型

在有限元分析的基础上建立了一个单向应力状态下金属基短纤维复合材料(MMC)的统计蠕变模型.首先建立细胞模型并进行有限元分析,得到了单向应力状态下材料细观尺寸及载荷方向对宏观蠕变响应的影响规律.通过在细胞模型中增加一界面层(考虑材料特性和厚度)来研究基体和纤维的界面对MMC宏观蠕变响应的影响.基于细胞模型的数值结果,提出了一适用于纤维平面随机分布的随机统计模型,该模型考虑了纤维的断裂.通过试验获得纤维的统计分布规律.分析结果表明随机统计模型可以满意地描述试验结果.进一步讨论了材料细观尺寸,纤维的断裂特性以及界面层的材料特性和厚度对MMC宏观蠕变响应的影响.     

The Effects of Poisson Contraction on Matrix Cracking in Brittle-Matrix Composites

The effects of Poisson contraction on matrix cracking in unidirectional fiber-reinforced brittle-matrix composites are studied in this paper. The fibers, initially held in the matrix by a compressive pressure due to the thermal expansion mismatch, are subjected to frictional slipping over the matrix as soon as a fiber-bridged crack is formed. The friction between the fibers and the matrix is assumed to follow the Coulomb friction law. A shear-lag model, which includes the Poisson contraction and the friction due to the relative fiber/matrix slipping, is adopted to calculate the stress and strain fields in the fibers and matrix. Using the energy balance approach, a relation for the critical matrix cracking stress for propagating of a semi-infinite fiber-bridged crack is derived. The results obtained show that the Poisson contraction has a strong effect on the predicted matrix cracking stress in brittle-matrix composites, especially in composites with a stiff matrix.     

Singular integral equation method for contact problem for rigidly connected punches on elastic half-plane

In the contact problem of a rigid flat-ended punch on an elastic half-plane, the contact stress under punch is studied. The angle distribution for the stress components in the elastic medium under punch is achieved in an explicit form. From obtained singular stress distribution, the punch singular stress factor (abbreviated as PSSF) is defined. A fundamental solution for the multiple flat punch problems on the elastic half-plane is investigated where the punches are disconnected and the forces applied on the punches are arbitrary. The singular integral equation method is suggested to obtain the fundamental solution. Further, the contact problem for rigidly connected punches on an elastic half-plane is considered. The solution for this problem can be considered as a superposition of many particular fundamental solutions. The resultant forces on punches are the undetermined unknowns in the problem, which can be evaluated by the condition of relative descent between punches. Finally, the resultant forces on punches can be determined, and the PSSFs at the corner points can be evaluated. Numerical examples are given.     

Interaction of a polymer matrix and short reinforcing fibers

A model of a glass-reinforced plastic with short unidirectional fibers is proposed. The distribution of tensile stresses in the polymer matrix and the fibers and the shear stress distribution at the interface in uniaxial tension are investigated in the elastic formulation.Riga Polytechnic Institute. Translated from Mekhanika Polimerov, No. 6, pp. 1030–1035, November–December, 1971.     

Die Anlaufströmung eines Binghamschen Stoffes zwischen zwei parallelen Wänden

Summary The initial flow of a Bingham plastic between two parallel plates is considered. First the stress distribution by applying Laplace transforms to the equation of motion is obtained To find the corresponding velocity distribution the stress distribution is integrated.     

Effective elastic moduli and stress concentrator factors in random structure aligned fiber composites

We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a statistically uniform random set of aligned fibers. Effective elastic moduli as well as the stress concentrator factors in the components are estimated. The micromechanical approach is based on the Green’s function technique as well as on the generalization of the “multiparticle effective field method” (MEFM, see for references, Buryachenko [1]). The refined version of the MEFM takes into account the variation of the effective fields acting on each pair of fibers. The dependence of effective elastic moduli and stress concentrator factors on the radial distribution function of the fiber locations is analyzed. Received: October 20, 2004