Modified Eshelby Tensor Research Articles

A damage plasticity constitutive model for wavy CNT nanocomposites by incremental Mori-Tanaka approach

In this paper, a novel nonlinear micromechanics constitutive model is proposed to simulate elastoplastic and ductile damage behavior for wavy carbon nanotube (CNT) nanocomposites. A modified Eshelby tensor corresponding to wavy CNT is derived by integrating a rotational transformation of the effective composite stiffness tensor along the sinusoidal wavy CNT. The modified Eshelby tensor was utilized to update local and global strain concentration tensors, rebalancing local strains of inclusions and matrix caused by constantly changing algorithmic tangent operator of the matrix. They were integrated with a general framework for nonlinear MT homogenization. A 3D FE micromechanics model is developed to verify the feasibility and accuracy of the proposed model. Following the verification, effects of CNT waviness are closely investigated under various volume fractions, aspect ratios and loading directions.

Micromechanics-based prediction of the effective properties of piezoelectric composite having interfacial imperfections

We derive an analytical expression to predict the effective properties of a particulate-reinforced piezoelectric composite with interfacial imperfections using a micromechanics-based mean–field approach. We correctly derive the analytical formula of the modified Eshelby tensor, the modified concentration tensor, and the effective property equations based on the modified Mori–Tanaka method in the presence of interfacial imperfections. Our results are validated against finite element analyses (FEA) for the entire range of interfacial damage levels, from a perfect to a completely disconnected and insulated interface. For the facile evaluation of the nontrivial tensorial equations, we adopt the Mandel notation to perform tensor operations with 9×9 symmetric matrix operations. We apply the method to predict the effective properties of a representative piezoelectric composite consisting of polyvinylidene fluoride (PVDF) and SiC reinforcements.

Modified Eshelby tensor for an anisotropic matrix with interfacial damage

We derive a simple tensor algebraic expression of the modified Eshelby tensor for a spherical inclusion embedded in an arbitrarily anisotropic matrix in terms of three tensor quantities (the fourth-order identity tensor, the elastic stiffness tensor, and the Eshelby tensor) and two scalar quantities (the inclusion radius and interfacial spring constant), when the interfacial damage is modelled as a linear-spring layer of vanishing thickness. We validate the expression for a triclinic crystal involving 21 independent elastic constants against finite element analysis.

Effective Thermoelasticity of Polymer-Bonded Particle Composites with Imperfect Interfaces and Thermally Expansive Interphases

A micromechanical model for the thermoelasticity of polymer-bonded composites is presented. The model is particularly aimed at describing materials where the polymeric binder phase undergoes non-negligible thermal expansion affecting the overall thermoelastic response. Constitutive choices for modeling a mixed binder-void interphase layer are proposed, and an associated decomposition of total eigenstrains into classical, elastic imperfection (damage), and binder thermal expansivity parts is examined within the context of imperfect inter-particle interfaces. A novel temperature dependent modified Eshelby tensor is identified, making possible the development of a temperature dependent modified self-consistent homogenization scheme—what we call the M $\theta$ -SCH model. A method for distinguishing between dispersed and isolated parts of the binder and void phases in the model is also provided, along with a description of particle coating (or interphase) thickness derived from particle morphology and mesoscale effective properties. Although the theory is general, its development is motivated by the need to model anisotropic and highly nonlinear observed thermal expansion behavior of the polymer bonded explosive PBX 9502, for which model simulations are performed and compared with existing measurements.

A micromechanics-based analytical solution for the effective thermal conductivity of composites with orthotropic matrices and interfacial thermal resistance

We obtained an analytical solution for the effective thermal conductivity of composites composed of orthotropic matrices and spherical inhomogeneities with interfacial thermal resistance using a micromechanics-based homogenization. We derived the closed form of a modified Eshelby tensor as a function of the interfacial thermal resistance. We then predicted the heat flux of a single inhomogeneity in the infinite media based on the modified Eshelby tensor, which was validated against the numerical results obtained from the finite element analysis. Based on the modified Eshelby tensor and the localization tensor accounting for the interfacial resistance, we derived an analytical expression for the effective thermal conductivity tensor for the composites by a mean-field approach called the Mori-Tanaka method. Our analytical prediction matched very well with the effective thermal conductivity obtained from finite element analysis with up to 10% inhomogeneity volume fraction.

New micromechanics model for saturated porous media with connected pores

A new micromechanics method is proposed to investigate the effective properties of saturated porous media with connected pores. This topic is seldom discussed in the literature because it is difficult to describe the connected pores and skeleton using conventional micromechanics methods. A new micromechanics model (i.e., Model I) is suggested to characterize such saturated porous media in which the pores saturated by fluid are taken as the matrix, and the interconnected randomly oriented long fiber (ROLF)-like solid skeleton is taken as the inclusions. The proposed model is verified by numerical simulations; the simulation results indicate that the difference of the elastic constants calculated for media with interconnected pores and for those with dispersed ROLF solid inclusions is small. Thus, the elastic moduli of Model I can be treated as approximate values for porous media with connected pores. Further, a modified Eshelby tensor for spherical inclusions is derived based on the equivalency of the elastic moduli of Model I and a conventional micromechanics model in which spherical fluid inclusions are distributed randomly in a solid matrix. By means of the modified Eshelby tenor, conventional micromechanics methods can be utilized directly to calculate the effective mechanical and thermal properties of saturated porous media with interconnected pores. Some examples are presented to show that the macroscopic elastic moduli predicted by the proposed method are in good agreement with test data found in the literature.

Mathematical modeling of volumetric material growth

Growth (resp. atrophy) describes the physical processes by which a material of solid body increases (resp. decreases) its size by addition (resp. removal) of mass. In the present contribution, we propose a sound mathematical analysis of growth, relying on the decomposition of the geometric deformation tensor into the product of a growth tensor describing the local addition of material and an elastic tensor, which is characterizing the reorganization of the body. The Blatz-Co hyperelastic constitutive model is adopted for an isotropic body, satisfying convexity conditions (resp. concavity conditions) with respect to the transformation gradient (resp. temperature). The evolution law for the transplant is obtained from the natural assumption that the evolution of the material is independent of the reference frame. It involves a modified Eshelby tensor based on the specific free energy density. The heat flux is dependent upon the transplant. The model consists of the constitutive equation, the energy balance, and the evolution law for the transplant. It is completed by suitable boundary conditions for the displacement, temperature and transplant tensor. The existence of locally unique solutions is obtained, for sufficiently smooth data close to the stable equilibrium. The question of the global existence is examined in the simplified situation of quasistatic isothermal equations of linear elasticity under the assumption of isotropic growth.

Modified Eshelby tensor for an ellipsoidal inclusion imperfectly embedded in an infinite piezoelectric medium

The modified Eshelby tensor for predicting the effective moduli of particle-reinforced piezoelectric composites is derived for the problem of an ellipsoidal inclusion which is imperfectly bonded to the matrix. A linear interface relation is adopted, which involves discontinuities of the mechanical displacements and electric potential across the interface, and assumes that the corresponding jumps are proportional to the continuous stresses and electric displacements at the interface. The piezoelectric field induced by a uniform eigenstrain given only in the inclusion is deduced analytically. As the induced piezoelectric field is no longer uniform, the average strains and electric displacements are calculated, and the modified piezoelectric Eshelby tensor is evaluated by both an iterative method and a direct method. By comparison, it is shown that the iterative method yields rapidly convergent results.

Effective Elastic Moduli of Spherical Particle Reinforced Composites Containing Imperfect Interfaces

The effective elastic moduli of composite materials are investigated in the presence of imperfect interfaces between the inclusions and the matrix. The primary focus is on the spherical particle reinforced composites. By admitting the displacement jumps at the particle–matrix interface, the modified Eshelby inclusion problem is studied anew. To derive the modified Eshelby tensor, three approximate methods are presented and compared by emphasizing the existence of a unique solution and computational efficiency. Subsequently, the effective elastic stiffness tensor of the composite is formulated based on the proposed micromechanical framework and homogenization. Specifically, by incorporating imperfect interface, the modified versions of the Mori–Tanaka method, the self-consistent method, and the differential scheme are presented. By comparing these three methods, the effects of interfacial sliding and separation on the degradation (damage) of the effective elastic moduli of composites are analyzed and assessed. Finally, a critical aspect of the presented formulations is specifically addressed.

111 不完全界面が粒子強化金属複合材料の弾塑性挙動に及ぼす影響(OS1-3 微視構造を有する材料の変形と破壊 : シミュレーション)

As a predominant damage mode, the interfacial damage between the reinforcements and the mairix is matter of importance for the composite materials. Correspondingly, in this study, the effect of imperfect interface is investigated based on a spring interface model. By accounting for the presence of imperfect interface, the modified Eshelby tensor is computed by making use of different methods, and the uniqueness of the solution is discussed. Further, to admit a relatively large amount of interfacial damage, an equivalent model is proposed in the framework of micromechanics. Firstly, in the absence of interfacial damage, the predictive capability of proposed micromechanical framework is demonstrated by comparing with the available experimental data. Then, the effects of interfacial damage on the deformation behavior of a composite ara systematically investigated.

Modified Eshelby tensor modeling for elastic property prediction of carbon nanotube reinforced ceramic nanocomposites

A modified model using the Eshelby equivalent tensor is developed to evaluate overall elastic properties of carbon nanotube (CNT) reinforced aluminum oxide nanocomposites. This model accounts for the effect of carbon nanotube geometry and porosity on the effective elastic modulus. Experimental results are compared with the computed predictions. Computed results show higher elastic modulus values, which are attributed to the perfect bonding between the reinforcement and the matrix. It is concluded that the modified Eshelby model can predict the elastic property of CNT reinforced ceramic nanocomposites.

Probabilistic Modeling for Composites with Particle Interface Degrading

A popular type is of metal matrix reinforced by ceramics particles. It has been revealed that the composites are susceptible to interfacial degrading, which may be dramatically detrimental to the overall properties of composites. In present paper, an equivalent inclusion type constitutive model was developed for the composites with dispersed particles on which imperfections of the material interface are incurred. Two fundamental tensors are derived, the modified Eshelby tensor and the damage tensor of the weakened particles. By applying these tensors into a carefully schemed constitutive law, the effects of interfacial degrading on the overall properties of composite materials can be investigated. The interface degrading includes sliding and debonding. The numerical results show that even with the nil resistible sliding coefficient, its effect on the overall Young’s modulus is not notable unless the volume fraction of the particle is so high as more than 70%. For the global Poisson’s ratio, when there is the sliding on the interface, the Poisson’s ration rises irrespective of the constituent material values. It is noted that even in the elastic state, the global Poisson’s ratio rises greater than that of both the constituents. This phenomenon might indicate that even at the elastic state, the particle interfacial sliding would give somewhat a plasticity-like deformation behavior. The effect of the interfacial debonding on the overall properties of the composite is more conspicuous in comparisons of the sliding. The debonding parameter greatly affects both the properties for almost entire range of the particle volume fraction. Unlike the sliding effect case, the debonding decreases Poisson’s ratio at all cases, which represents the micro-damage effect occurring in the composite.

The influence of stochastic interfacial parameter distribution on the stiffness of composites with complex fibre orientation

This paper deals with the overall stiffness determination of a short-fibre-reinforced composite with complex fibre orientation and soft interfaces. The fibres are oriented differently in the matrix through a distribution function of the Euler angles. No fibre length distribution function is used, i.e. the fibres are of the same length. The approach is based upon the Mori–Tanaka mean-field theory. Furthermore, the effects of an interface with stiffness differing from those of the matrix and fibre are considered through a modified Eshelby tensor. The parameter describing the interfacial stiffness is considered as a stochastic variable described by a statistical distribution function. It is shown that when the fibres are oriented mainly towards the directions of the principal stresses the effects of the soft interface is diminishing. Furthermore, it is shown that when the interfacial stiffness parameter is described by a β distribution the resulting composite stiffness distribution resembles a uniform distribution.

Overall creep modelling of short fibre reinforced composites with weakened interfaces and complex fibre orientation distributions

This paper deals with the overall creep response of a short fibre reinforced composite. The fibres are oriented differently in the matrix through a distribution function of the Euler angles. No fibre length distribution function is used, i.e. the fibres are of the same length. The approach is based upon the Mori–Tanaka mean field theory in which the fibres are considered as inhomogeneities containing equivalent eigenstrains. Furthermore, the effect of an imperfect interface is considered through a modified Eshelby tensor. It is shown that for UD composites the interfacial parameter does not influence the elastic stiffness in the fibre direction. For various fibre distributions the weakened interface has the largest influence when applying shear loading to a nearly unidirectional composite.