Effective Elastic Moduli of Two-Dimensional Brittle Solids With Interacting Microcracks, Part I: Basic Formulations

1994 ◽  
Vol 61 (2) ◽  
pp. 349-357 ◽  
Author(s):  
J. W. Ju ◽  
Tsung-Muh Chen
Keyword(s):  
Elastic Moduli ◽  
Effective Elastic Moduli ◽  
Two Dimensional ◽  
Damage Models ◽  
Brittle Solids ◽  
Averaged Equations ◽  
Approximate Analytical Solutions ◽  
Special Cases ◽  
Microcrack Growth ◽  
Microcrack Interaction

Statistical micromechanical formulations are presented to investigate effective elastic moduli of two-dimensional brittle solids with interacting slit microcracks. The macroscopic stress-strain relations of elastic solids with interacting microcracks are micromechanically derived by taking the ensemble average over all possible realizations which feature the same material microstructural geometry, characteristics, and loading conditions. Approximate analytical solutions of a two-microcrack interaction problem are introduced to account for microcrack interaction among many randomly oriented and located microcracks. The overall elastic-damage compliances of microcrack-weakened brittle solids under uniaxial and biaxial loads are also derived. Therefore, stationary statistical micromechanical formulation is completed. Moreover, some special cases are investigated by using the proposed framework. At variance with existing phenomenological continuum damage models, the proposed framework does not employ any fitted “material parameters. ” “Cleavage 1” microcrack growth and “evolutionary damage models” within the proposed context will be presented in Part II of this series. It is emphasized that microstructural statistical informations are already embedded in the proposed ensemble-averaged equations and, therefore, no Monte Carlo simulations are needed.

Effective Elastic Moduli of Two-Dimensional Brittle Solids With Interacting Microcracks, Part II: Evolutionary Damage Models

1994 ◽  
Vol 61 (2) ◽  
pp. 358-366 ◽  
Author(s):  
J. W. Ju ◽  
Tsung-Muh Chen
Keyword(s):  
Elastic Moduli ◽  
Biaxial Tension ◽  
Interaction Model ◽  
Evolutionary Models ◽  
Two Dimensional ◽  
Damage Models ◽  
Brittle Solids ◽  
Biaxial Tests ◽  
Integration Algorithms ◽  
Microcrack Interaction

In Part I of this series, basic formulations of stationary micromechanical theory and overall responses are presented for two-dimensional brittle solids with randomly dispersed microcracks. The basic formulations hinge on an ensemble average approach which includes pairwise microcrack interactions. In this paper, statistical micromechanical evolutionary models are proposed to account for “cleavage 1” growth of randomly oriented and located microcracks under microcrack interaction effects. Biaxial tension/compression loadings are also considered to take into account mixed microcrack opening and closure effects. Efficient numerical integration algorithms for the proposed ensemble averaged constitutive equations are subsequently given. Further, uniaxial and biaxial tests are presented to illustrate the proposed models and procedures. Finally, a higher-order microcrack interaction model within the proposed micromechanical framework is discussed.

scholarly journals Effective elastic moduli of two-dimensional solids with multiple cracks

1998 ◽  
Vol 41 (10) ◽  
pp. 1114-1120 ◽  
Author(s):  
Shige Zhan ◽  
Ziqiang Wang ◽  
Xueli Han
Keyword(s):  
Elastic Moduli ◽  
Multiple Cracks ◽  
Effective Elastic Moduli ◽  
Two Dimensional

Analytical treatment on the effective material properties of a composite material with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix

2016 ◽  
Vol 01 (03n04) ◽  
pp. 1640012 ◽  
Author(s):  
Y. Mikata
Keyword(s):  
Thermal Conductivity ◽  
Effective Thermal Conductivity ◽  
Material Properties ◽  
Elastic Moduli ◽  
Effective Elastic Moduli ◽  
Applied Physics ◽  
Isotropic Matrix ◽  
Equivalent Inclusion Method ◽  
Special Cases ◽  
Effective Material Properties

Effective material properties of a composite with spheroidal and ellipsoidal inhomogeneities in an isotropic matrix are investigated analytically using the dilute approximation and the Mori–Tanaka approximation together with the Eshelby's equivalent inclusion method. Both uniaxially aligned and uniformly randomly oriented spheroidal and ellipsoidal inhomogeneities are treated. For a spheroid, both oblate and prolate spheroidal inhomogeneities are considered. It is analytically shown that a composite with uniaxially aligned anisotropic ellipsoidal inhomogeneities in an isotropic matrix is anisotropic in general in thermal conductivity. It is also analytically shown that a composite with uniformly randomly oriented anisotropic ellipsoidal inhomogeneities in an isotropic matrix is exactly isotropic in thermal conductivity. Various special cases are also treated for the effective thermal conductivity of a composite with ellipsoidal and spheroidal inhomogeneities. Similar results are also obtained for the effective elastic moduli. Newly obtained expressions for the effective elastic moduli of a composite with isotropic spheroidal inhomogeneities are rather involved. Conversely, an effective thermal conductivity of a composite with anisotropic ellipsoidal inhomogeneities is relatively simple. An effective thermal conductivity of a composite with isotropic spheroidal inhomogeneities reduces to a known result (Kerner, E. H. [1956] “The electrical conductivity of composite media,” Proceedings of the Physical Society London Section B 69, 802–807; Hashin, Z. and Shtrikman, S. [1962] “A variational approach to the theory of the effective magnetic permeability of multiphase materials,” Journal of Applied Physics 33, 3125–3131.) as the spheroid aspect ratio approaches 1 (i.e., a sphere). The effective thermal conductivity of a composite with uniformly randomly oriented isotropic spheroidal inhomogeneities in an isotropic matrix obtained in this paper as a special case is similar to the one obtained by Hatta and Taya (Hatta, H. and Taya, M. [1985] “Effective thermal conductivity of a misoriented short fiber composite,” Journal Applied Physics 58, 2478–2486.) in some respects, but is different. Numerical results are shown for a composite with oblate spheroidal voids in an isotropic matrix.

Elastic Moduli of Two Dimensional Materials With Polygonal and Elliptical Holes

1994 ◽  
Vol 47 (1S) ◽  
pp. S18-S28 ◽  
Author(s):  
I. Jasiuk ◽  
J. Chen ◽  
M. F. Thorpe
Keyword(s):  
Elastic Constants ◽  
Conformal Transformation ◽  
Elastic Moduli ◽  
Far Field ◽  
Effective Elastic Moduli ◽  
Two Dimensional ◽  
Effective Elastic Constants ◽  
Two Dimensional Materials ◽  
Elliptical Holes ◽  
General Relations

We study the effective elastic moduli of two-dimensional composite materials containing polygonal holes. In the analysis we use a complex variable method of elasticity involving a conformal transformation. Then we take a far field result and derive the effective elastic constants of composites with a dilute concentration of polygonal holes. In the discussion we use the recently-stated Cherkaev-Lurie-Milton theorem, which gives general relations between the effective elastic constants of two-dimensional composites. We also discuss known results for elliptical holes in the context of the present work.

Modes of Wave Propagation and Dispersion Relations in Inclusion Reinforced Composite Plates

2010 ◽  
Author(s):  
Yu Cheng Liu ◽  
Jin Huang Huang
Keyword(s):  
Composite Plate ◽  
Elastic Moduli ◽  
Lamb Waves ◽  
Plate Thickness ◽  
Composite Plates ◽  
Dispersion Relations ◽  
Elastic Behavior ◽  
Effective Elastic Moduli ◽  
Aspect Ratios ◽  
Reinforced Composite

This paper mainly analyzes the wave dispersion relations and associated modal pattens in the inclusion-reinforced composite plates including the effect of inclusion shapes, inclusion contents, inclusion elastic constants, and plate thickness. The shape of inclusion is modeled as spheroid that enables the composite reinforcement geometrical configurations ranging from sphere to short and continuous fiber. Using the Mori-Tanaka mean-field theory, the effective elastic moduli which are able to elucidate the effect of inclusion’s shape, stiffness, and volume fraction on the composite’s anisotropic elastic behavior can be predicted explicitly. Then, the dispersion relations and the modal patterns of Lamb waves determined from the effective elastic moduli can be obtained by using the dynamic stiffness matrix method. Numerical simulations have been given for the various inclusion types and the resulting dispersions in various wave types on the composite plate. The types (symmetric or antisymmetric) of Lamb waves in an isotropic plate can be classified according to the wave motions about the midplane of the plate. For an orthotropic composite plate, it can also be classified as either symmetric or antisymmetric waves by analyzing the dispersion curves and inspecting the calculated modal patterns. It is also found that the inclusion contents, aspect ratios and plate thickness affect propagation velocities, higher-order mode cutoff frequencies, and modal patterns.

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