Theory of thermal expansion of composites

The problem discussed is to find the overall thermal expansion of a composite consisting of inclusions in a matrix of material with different expansion coefficients and elastic moduli. The volume mismatch causes strain fields. The total strain energy must be a minimum. This problem was solved previously for inclusions which are either spheres or randomly oriented long cylinders. In these simple cases the matrix strain field consists of a short-range shear component and a uniform expansion; the inclusion suffers uniform strain. The matrix is replaced by an effective medium having the average properties of the composite. The overall expansion coefficient could be obtained in closed form. This separation of the strain field into short-range shear and long-range uniform dilation is valid, at least to a good approximation, for all inclusion shapes. Simple expressions can thus be obtained in terms of coefficients which, although not calculated exactly, can be deduced approximately or can be determined empirically. Plasticity can be accounted for by allowing the shear modulus to depend on the temperature and on the maximum shear strain. The size of the inclusions does not enter the theory except through the yield strain, which depends on the extent of the strain field.Paper presented at the Ninth International Thermal Expansion Symposium, December 8–10, 1986, Pittsburgh, Pennsylvania, U.S.A.     

The role of inclusions in the fracture of ceramic materials

The stress concentrations that occur at inclusions due to thermal expansion and elastic modulus mismatch are discussed and the stress intensity factors at interface cracks that result from these stresses are calculated. It is shown that conservative failure prediction based on an equivalence between inclusion size and crack size is usually acceptable if the shear modulusμ or thermal expansion coefficientα for the inclusion is larger than the matrix values. If, however,μ andα are smaller for the inclusion than the matrix, extensive cracking can develop at the inclusions which may lead to premature failure. For this case the only effective methods for failure prediction are techniques which give directly the maximum stress intensity factor, i.e., proof testing and/or acoustic emission.     

Closed-form solution for a coated circular inclusion under uniaxial tension

A coated circular inclusion embedded in an infinite matrix is analyzed in the framework of two-dimensional isotropic linear elasticity. A closed-form solution is obtained for the case of far-field uniaxial tension using Muskelishvilis complex potential method. The solutions for the stress and strain distributions for all three regions, that is, matrix, coating, and inclusion, were obtained for various coating-to-matrix shear modulus ratios, while keeping the fiber and matrix shear moduli the same. Test cases for an inclusion without the coating and hollow inclusion were also studied. The energy release rate was evaluated using the path-independent M-integral, which is used to calculate the energy release rate for the self-similar expansion of defects surrounded by the closed contour of the integral. The results for the stress and strain concentrations along with the energy release rate due to this material inhomogeneity were analyzed to yield a better understanding of the mechanics of materials with circular inclusions. This can be helpful in designing intelligent composite structures with embedded optical fiber sensors.     

Elevated temperature X-ray measurement of residual stresses in a fibre reinforced Al alloy

Thermal residual stresses have been measured using X-ray diffraction in an Al-2% Mg matrix with 10, 20 or 26 vol % Al₂O₃ short fibres. Stress measurements were made at room temperature as well asin situ at elevated temperatures up to 250?C. The thermal stresses arise due to the difference in coefficient of thermal expansion (CTE) between the matrix and the reinforcement. The largest CTE is found in the matrix, resulting in tensile residual stresses after a temperature drop, e.g. after processing or annealing. A high fraction of reinforcement implies higher matrix stresses than a low fibre content. The stresses decrease with increasing temperature for all fibre volume fractions. Measurements are compared with calculations using a modified Eshelby model for equivalent inclusions. Parameters taken into account in the model are coefficient of thermal expansion, Young's modulus, and volume fraction and geometric shape of the reinforcing phase. A good correlation between calculations and experimental results has been found, bearing in mind that no plasticity is taken into account in the Eshelby model. The plastic behaviour of the composites has been described using a model based on a rigid spherical cavity in an elastic-plastic matrix.     

A model for particle cavitation in rubber-toughened plastics

An energy-balance criterion for cavitation of rubber particles, which was proposed in an earlier paper [A. Lazzeri and C. B. Bucknall, J. Mater. Sci. 28 (1993) 6799], is developed by including a term for the energy stored in the matrix and released during expansion of the voids. The model relates the critical volume strain at cavitation to the radius of the rubber particle, and to the shear modulus, surface energy and failure strain of the rubber. The effects of temperature, strain rate and type of stress field upon cavitation behaviour and the resulting toughness of the two-phase polymer are discussed in terms of the model.     

Volume integral equation method for multiple circular and elliptical inclusion problems in antiplane elastostatics

A Volume Integral Equation Method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing interacting multiple isotropic and anisotropic circular/elliptical inclusions subject to remote antiplane shear. This method is applied to two-dimensional problems involving long parallel cylindrical inclusions. A detailed analysis of the stress field at the interface between the matrix and the central inclusion is carried out for square and hexagonal packing of isotropic and anisotropic inclusions. The effects of the number of isotropic and anisotropic inclusions and various fiber volume fractions on the stress field at the interface between the matrix and the central circular/elliptical inclusion are also investigated in detail. The accuracy of the method is validated by solving single isotropic and orthotropic circular/elliptical inclusion problems and multiple isotropic circular and elliptical inclusion problems for which solutions are available in the literature.     

含有扁长椭球形增强物复合材料的弹性常数

研究了含有扁长椭球形增强物复合材料的弹性过程并得到了材料整体的弹性常数.当材料基体和增强材料均为各向同性材料,增强材料的体积相对于基体非常地小并与基体紧密相联时,含有扁长椭球形增强物的材料具有各向异性,提高这种材料的杨氏模量是以牺牲剪切模量为代价的.     

复合材料的椭圆夹杂模型线性温变问题的界面热应力分析

利用处理平面多连通域热弹性问题的一种有效方法,获得了椭圆夹杂模型线性温变问题的热弹性场解答,并讨论了夹杂和基体材料的热膨胀系数、热传导系数以及剪切模量对界面热应力的影响规律,所获得的结论为增强复合材料的设计与应用提供了有价值的参考依据.     

Stress analysis and material tailoring in isotropic linear thermoelastic incompressible functionally graded rotating disks of variable thickness

We analyze axisymmetric deformations of a rotating disk with its thickness, mass density, thermal expansion coefficient and shear modulus varying in the radial direction. The disk is made of a rubberlike material that is modeled as isotropic, linear thermoelastic and incompressible. We note that the hydrostatic pressure in the constitutive relation of the material is to be determined as a part of the solution of the problem since it cannot be determined from the strain field. The problem is analyzed by using an Airy stress function φ. The non-homogeneous ordinary differential equation with variable coefficients for φ is solved either analytically or numerically by the differential quadrature method. We have also analyzed the challenging problem of tailoring the variation of either the shear modulus or the thermal expansion coefficient in the radial direction so that a linear combination of the hoop stress and the radial stress is constant in the disk. For a rotating annular disk we present the explicit expression of the thermal expansion coefficient for the hoop stress to be uniform within the disk. For a rotating solid disk we give the exact expressions for the shear modulus and the thermal expansion coefficient as functions of the radial coordinate so as to achieve constant hoop stress. Numerical results for a few typical problems are presented to illuminate effects of material inhomogeneities on deformations of a hollow and a solid rotating disk.     

Statistical model of internal energy dissipation in composition materials

A model of the theoretical evaluation of normalized losses of elastic-vibration energy in matrix-type composition materials with spherical inclusions as a function of the probability of the development of plastic deformations in microvolumes of the matrix is proposed. The distribution function of maximum shear stresses in the matrix in the vicinity of an inclusion is constructed on the basis of Muskhelishvili's relationships with allowance for the effect of neighboring inclusions. The effect of the volume percentage of inclusions and the ratio of the shear moduli of the inclusions and matrix on the level of energy losses associated with microlocalized plastic deformations of the matrix is analyzed.Translated from Problemy Prochnosti, No. 7, pp. 94–98, July, 1990.     

Some elastic properties of reinforced solids,with special reference to isotropic ones containing spherical inclusions

Based on Mori and Tanaka's concept of “average stress” in the matrix and Eshelby's solutions of an ellipsoidal inclusion, an approximate theory is established to derive the stress and strain state of constituent phases, stress concentrations at the interface, and the elastic energy and overall moduli of the composite. Both “stress-free” strain (polarization strain) and “strain-free” stress (polarization stress) are employed in these derivations under the traction- and displacement-prescribed conditions. The theory was developed first for a general multiphase, anisotropic composite with arbitrarily oriented anisotropic inclusions; explicit results are then given for a suspension of uniformly distributed, multiphase isotropic spheres in an isotropic matrix. Numerical results for stress concentrations in the spherical inclusions and at the interface are given for a 2-phase composite. Further, it is shown that the derived moduli are related to the Hashin-Shtrikman bounds and that, when the shear moduli are equal, the overall bulk modulus of a 2-phase composite reduces to Hill's exact solution. As compared with experimental data, the theory also provides reasonably accurate estimates for the Young's modulus of some 2- and 3-phase composites.     

On the axial and interfacial shear stresses due to thermal mismatch in hybrid composite sheets

A simple analytical model has been developed which allows the determination of the axial and the interfacial shear stresses which can occur in hybrid fiber composites as a consequence of the mismatch in coefficients of thermal expansion and Young's modulus. The configuration considered is a finite-width hybrid composite monolayer with alternating high- and low-modulus fibers. To account properly for the interfacial shear between fiber and matrix, a modified shear-lag model is used which permits extensional deformation due to thermal expansion of the matrix in the fiber direction. Typical stresses due solely to temperature changes are calculated, and these show steep boundary-layer edge stresses at free corners.     

Fracture initiation at a sphrerical inclusion in a matrix plastically deformed by shear

The critical strain for fracture initiation of a metallic material with a spherical inclusion has been analysed using EaheIby,s inclusion method forthree types of fracture initiation models including the recovery effect by diffusion of atoms. When the elastic constant of inclusion approaches that of the matrix, the critical strain for fracture initiation becomes large in the case of uniform shear deformation of the matrix. It was found that the critical strain becomes large due to the diffusion of atoms, especially for inclusions of small size and a large elastic constant. The model in which the inclusion is cracked by the localized shear deformation can explain the observed inclusion size dependence of the strain for fracture initiation. The inclusion size dependence of the critical strain for fracture initiation by uniform shear deformation of the matrix is different from that by localized shear deformation. Therefore, it is important to know which mechanism governs the fracture.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

湿热环境对碳纳米管复合材料界面应力传递的影响

基于碳纳米管的热膨胀系数及弹性模量分别为温度变化的函数,基体的热、湿膨胀系数及弹性模量分别为温度变化和湿度变化的函数,应用连续介质力学的经典弹性壳理论及传统纤维拉拔模型,分析了湿热环境对碳纳米管复合材料界面应力传递的影响.数值计算表明,湿度、温度的效应及碳纳米管的层数等参数对界面应力的传递均有显著影响.     

Successive iteration method applied to composites containing sliding inclusions: effective modulus and anelasticity

The effective shear modulus and Poisson's ratio of a body containing spherical sliding inclusions are calculated. First, the amount of sliding on the surface of a single spherical sliding inclusion is determined so that the tangential traction of the stresses due to the sliding, the Somigliana dislocations, cancels that of the external stresses on the surface of the inclusion. Next, the influence of other inclusions is accounted for by using a successive iteration method based on the average field theory. The successive iteration converges into closed forms, leading to analytical forms of the effective elastic constants. It is shown that the sliding occurs in a first order kinetics, the relaxation time of which is proportional to the radius of the inclusions with a constant depending on the volume fraction of the inclusions. The two-dimensional problem of a body containing aligned cylindrical fibers is also solved.     

Experimental and numerical study of the Young’s modulus vs temperature for heterogeneous model materials with polygonal inclusions

This work is related to the thermomechanical behaviour of industrial refractory materials. The microstructural complexity of such materials and the strong influence of the elastic properties on the resistance to thermomechanical sollicitations, lead us to study first of all the Young’s modulus of heterogeneous model materials with a simplified microstructure. The studied materials are composed of a glass matrix surrounding polygonal alumina inclusions. These two materials exhibit a dilatometric dissension sufficiently large to induce, during a thermal cycle, thermal stresses able to damage the matrix/inclusions interfaces. The present study deals with the Young’s modulus variations of the studied model materials according to the temperature. The numerical simulation of the matrix/inclusions interfaces behaviour was carried out using the Abaqus FEM code whose contact tool “Debond” allows to account for the interface matrix damage during a thermal cycle. The results show the ability of this tool to well describe the evolution of damage in function of the temperature. The Young’s modulus of the model materials was also measured using ultrasonic technique. A good agreement of the numerical and experimental results is obtained.     

Experimental evaluation of residual stresses in single fibre composites by means of the fragmentation test

The residual stresses in both thermosetting and thermoplastic single-fibre composites have been experimentally evaluated by means of an original technique based on the continuous monitoring of the fragmentation test performed at various temperatures. The difference between the strain at the break of a single fibre in air and one embedded in a polymeric matrix has been measured as a function of temperature. By considering the compressive fibre modulus this strain difference has been converted into fibre compressive stresses related to the matrix thermal shrinkage after curing of the samples. In fact, as the test temperature increased, the thermal compressive stresses decreased until a zero value was obtained, corresponding to a so called stress free temperature, equal to the curing temperature for amorphous thermosetting matrix composites or equal to the matrix melting temperature for semicrystalline-thermoplastic matrix composites. The experimental results have been compared with data obtained from a theoretical model and a good agreement was found especially if the temperature dependence of the matrix Young's modulus and matrix thermal expansion coefficient are accounted for in the computation.     

Bounds on the effective thermal conductivity of composites with imperfect interface

A new model is developed to bound the effective thermal conductivity of composites with thermal contact resistance between spherical inclusions and matrix. To construct the trial temperature and heat flux fields which satisfy the necessary interface conditions, the transition layer for each spherical inclusion is introduced. For the upper bound, the trial temperature field needs to satisfy the thermal contact resistance conditions between spherical inclusions and transition layers and the continuous interface conditions between transition layers and remnant matrix. For the lower bound, the trial heat flux field needs to satisfy the continuous interface conditions between different regions. It should be pointed out that the continuous interface conditions mentioned above are absolutely necessary for the application of variational principles, and the thermal contact resistance conditions between spherical inclusions and transition layers are suggested by the author. According to the principles of minimum potential energy and minimum complementary energy, the bounds of the effective thermal conductivity of composites with imperfect interfaces are rigorously derived. The effects of the size and distribution of spherical inclusions on the bounds of the effective thermal conductivity of composites are analyzed. It should be shown that the present method is simple and does not need to calculate the complex integrals of multi-point correlation functions. Meanwhile, the present method provides an entirely different way to bound the effective thermal conductivity of composites with imperfect interface, which can be developed to obtain a series of bounds by taking different trial temperature and heat flux fields. In addition, the present upper and lower bounds are finite when the thermal conductivity of spherical inclusions tends to ∞ and 0, respectively.     

The transverse coefficient of thermal expansion of a unidirectional composite

Various analytical models of the effective thermal expansion coefficients of unidirectional fibre-reinforced composite materials predict for certain fibre-matrix combinations an increase in the transverse coefficient of thermal expansion over that of its constituents at low fibre volume content. This effect is especially noticeable if the composite is fabricated with fibres of high modulus and low thermal expansion coefficient in matrices of low modulus and high thermal expansion coefficient. An experimental investigation was therefore conducted to study this behaviour in Textron fibre (SCS-6)-reinforced Hercules 3501 -6 epoxy matrix. Numerical calculations for this material system have shown that increases of the order of 20% over the matrix expansion coefficient is possible for fibre volume fraction in the range 3%–4%. Experimental measurements of the effective thermal expansion coefficients are seen to be in favourable agreement with the theoretical predictions. A parametric study is also undertaken to examine the influence of constituent properties on the effective composite behaviour. It is shown that the axial restraint of the fibre is responsible for a peak in the behaviour of the transverse expansion coefficient.