In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.

The surface/interface energy theory based on three configurations proposed by Huang et al. is used to study the effective properties of thermoelastic nanocomposites. The particular emphasis is placed on the discussion of the influence of the residual interface stress on the thermal expansion coefficient of a thermoelastic composite filled with nanoparticles. First, the thermo-elastic interface constitutive relations expressed in terms of the first Piola-Kirchhoff interface stress and the Lagrangian description of the generalized Young-Laplace equation are presented. Second, the Hashin's composite sphere assemblage (CSA) is taken as the representative volume element (RVE), and the residual elastic field induced by the residual interface stress in this CSA at reference configuration is determined. Elastic deformations in the CSA from the reference configuration to the current configuration are calculated. From the above calculations, analytical expressions of the effective bulk modulus and the effective thermal expansion coefficient of thermoelastic composite are derived. It is shown that the residual interface stress has a significant effect on the thermal expansion properties of thermoelastic nanocomposites.

A thermal-elastoplastic constitutive model is proposed for particle-filled composites in this paper. Particles are assumed to be linear thermoelastic while the matrix follows the thermal-elastoplastic responses with the generalized Ramberg-Osgood relation. Based on the micromechanics methodology and homogenization procedures, the effective thermal-mechanical constitutive functions are derived including the macroscopic Helmholtz free energy and the macroscopic yield function.First, it is assumed that in the case of plastic unloading or stress-strain state being in the macroscopic yield surface, the constitutive relation of the composites is linear thermoelastic expressed by the macroscopic Helmholtz free energy. The micromechanics-based thermoelastic properties of the composite are obtained including the effective elastic moduli, thermal expansion coefficients, and specific heats.Furthermore, with the concept of linear comparison composites, the variational principle is extended to consider the thermal effect, from which the lower bound of the macroscopic stress potential for the nonlinear composites can be computed. The associated macroscopic plastic strain is defined, and the macroscopic yield function in the temperature-strain space is therefore determined.Finally, the above two constitutive functions are combined with the thermal-elastoplastic constitutive theory proposed by Huang (1994) to develop the loading-unloading criterion in the temperature-strain space and the incremental thermal-elastoplastic constitutive relations for particulate composites. The results can be useful in the study of the thermomechanical behavior of particle-filled composites at elevated temperatures.

A thermal-elastoplastic constitutive model is proposed for particle-filled composites in this paper. Particles are assumed to be linear thermoelastic while the matrix follows the thermal-elastoplastic responses with the generalized Ramberg-Osgood relation. Based on the micromechanics methodology and homogenization procedures, the effective thermal-mechanical constitutive functions are derived including the macroscopic Helmholtz free energy and the macroscopic yield function.First, it is assumed that in the case of plastic unloading or stress-strain state being in the macroscopic yield surface, the constitutive relation of the composites is linear thermoelastic expressed by the macroscopic Helmholtz free energy. The micromechanics-based thermoelastic properties of the composite are obtained including the effective elastic moduli, thermal expansion coefficients, and specific heats.Furthermore, with the concept of linear comparison composites, the variational principle is extended to consider the thermal effect, from which the lower bound of the macroscopic stress potential for the nonlinear composites can be computed. The associated macroscopic plastic strain is defined, and the macroscopic yield function in the temperature-strain space is therefore determined.Finally, the above two constitutive functions are combined with the thermal-elastoplastic constitutive theory proposed by Huang (1994) to develop the loading-unloading criterion in the temperature-strain space and the incremental thermal-elastoplastic constitutive relations for particulate composites. The results can be useful in the study of the thermomechanical behavior of particle-filled composites at elevated temperatures.

In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.

The surface/interface energy theory based on three configurations proposed by Huang et al. is used to study the effective properties of thermoelastic nanocomposites. The particular emphasis is placed on the discussion of the influence of the residual interface stress on the thermal expansion coefficient of a thermoelastic composite filled with nanoparticles. First, the thermo-elastic interface constitutive relations expressed in terms of the first Piola-Kirchhoff interface stress and the Lagrangian description of the generalized Young-Laplace equation are presented. Second, the Hashin's composite sphere assemblage (CSA) is taken as the representative volume element (RVE), and the residual elastic field induced by the residual interface stress in this CSA at reference configuration is determined. Elastic deformations in the CSA from the reference configuration to the current configuration are calculated. From the above calculations, analytical expressions of the effective bulk modulus and the effective thermal expansion coefficient of thermoelastic composite are derived. It is shown that the residual interface stress has a significant effect on the thermal expansion properties of thermoelastic nanocomposites.