Analytical solution for the effective elastic properties of rocks with the tilted penny-shaped cracks in the transversely isotropic background

Seismic prediction of cracks is of great significance in many disciplines,for which the rock physics model is indispensable.However,up to now,multitudinous analytical models focus primarily on the cracked rock with the isotropic background,while the explicit model for the cracked rock with the anisotropic background is rarely investigated in spite of such case being often encountered in the earth.Hence,we first studied dependences of the crack opening displacement tensors on the crack dip angle in the coordinate systems formed by symmetry planes of the crack and the background anisotropy,respectively,by forty groups of numerical experiments.Based on the conclusion from the experiments,the analytical solution was derived for the effective elastic properties of the rock with the inclined penny-shaped cracks in the transversely isotropic background.Further,we comprehensively analyzed,according to the developed model,effects of the crack dip angle,background anisotropy,filling fluid and crack density on the effective elastic properties of the cracked rock.The analysis results indicate that the dip angle and background anisotropy can significantly either enhance or weaken the anisotropy degrees of the P-and SH-wave velocities,whereas they have relatively small effects on the SV-wave velocity anisotropy.Moreover,the filling fluid can increase the stiffness coefficients related to the compressional modulus by reducing crack compliance parameters,while its effects on shear coefficients depend on the crack dip angle.The increasing crack density reduces velocities of the dry rock,and decreasing rates of the velocities are affected by the crack dip angle.By comparing with exact numerical results and experimental data,it was demonstrated that the proposed model can achieve high-precision estimations of stiffness coefficients.Moreover,the assumption of the weakly anisotropic background results in the consistency between the proposed model and Hudson's published theory for the orthorhombic rock.

Topology and Shape Optimization of 2-D and 3-D Micro-ArchitecturedThermoelastic Metamaterials Using a Parametric Level Set Method

2-D and 3-D micro-architectured multiphase thermoelastic metamaterials are designed and analyzed using a parametric level set method for topology optimization and the finite element method.An asymptotic homogenization approach is employed to obtain the effective thermoelastic properties of the multiphase metamaterials.Theε-constraint multi-objective optimization method is adopted in the formulation.The coefficient of thermal expansion(CTE)and Poisson’s ratio(PR)are chosen as two objective functions,with the CTE optimized and the PR treated as a constraint.The optimization problems are solved by using the method of moving asymptotes.Effective isotropic and anisotropic CTEs and stiffness constants are obtained for the topologically optimized metamaterials with prescribed values of PR under the constraints of specified effective bulk modulus,volume fractions and material symmetry.Two solid materials along with one additional void phase are involved in each of the 2-D and 3-D optimal design examples.The numerical results reveal that the newly proposed approach can integrate shape and topology optimizations and lead to optimal microstructures with distinct topological boundaries.The current method can topologically optimize metamaterials with a positive,negative or zero CTE and a positive,negative or zero Poisson’s ratio.

Interactors and a Third-order Interacting Model for Effective Elastic Properties of Damaged Solids

Interactors introduced in another recent paper by the authors are fourth order tensors which describes the interacting effects on the effective elastic properties of damaged solids due to spatial distribution of defects. Based on the concept of interactors, an interacting model for the effective elastic properties of anisotropically damaged solids is proposed. As an application of the proposed model, the anisotropic effective elastic properties of an isotropic matrix material with circular holes in periodically rectangular distribution are theoretically estimated up to third order terms of the porosity and the estimation is compared with numerical results.

Submicron size-scale mapping of carbonate effective elastic properties from FIB-SEM images and finite element method

In this paper, an automatic unstructured focused ion beam （FIB） and scanning electron microscopy （SEM） images induced representative volume element （RVE） finite element （FE） method is developed to predict submicron scale carbonate rock effective Young＇s and bulk moduli and Poisson＇s ratio on parallel CPU-GPU platform. Based on high resolution-contrast surface morphology and internal fabric-texture structure images from carbonate rock specimen （covered 0.12-64 μm2 area and 8000 μm3 domain）, the cubic RVE FE models are constructed from different sites through Avizo with user-defined parameters Matlab coding. The effective Young＇s and bulk moduli and Poisson＇s ratio of the different RVEs and porosity and pore size are computed by using periodic boundary condition in the well-known FE software Abaqus. FE mesh sensitivity analysis has been conducted where all moduli converge to a certain constant value at larger FE mesh density. The effect of fabric-texture （pore size, shape, and distribution） on the elastic properties is discussed. The correlations between the computed effective elastic properties and pore size, porosity, RVE size have been established. The simulation results show that the effective Young＇s and bulk moduli and Poisson＇s ratio have strong anisotropic behavior and depend on RVE size, porosity and pore size. The RVE size, porosity and pore size are three independent factors in affecting of the effective elastic moduli, the effect mechanism of porosity and pore size is same while the effect mechanism of RVE size is difference.

Direct Numerical Simulation of Effective Elastic Properties of 2-D Solids with Circular Holes

Approximate micro mechanically based theoretical schemes for predicting the effective elastic properties of particulate reinforced composites were evaluated using numerical simulations of a sheet containing holes. The boundary element method was used because it is very efficient. Two models were analyzed, one was a sheet containing randomly distributed holes while the other had normally distributed holes. The effectiveness of the existing theoretical models was investigated by comparing the computational results with the theoretical solutions.

Modeling of interphases in multiple heterogeneities reinforced composites using Voronoi cell finite elements

In this paper,a Voronoi cell finite element model is developed to study the microscopic and macroscopic mechanical behaviors of heterogenous materials,including arbitrary distributed heterogeneity(inclusions or fibers)coated with interphase layers,based on linear elasticity theory.The interphase between heterogeneity and a matrix are regarded as in the third phase(elastic layers),in contrast to the perfect interface of the spring-like Voronoi cell finite element model(VCFEM)in the literature.In this model,both stress and the displacement field are assumed to be independent in an element.Formulations of stress are derived for each of the three phases in an element,as is the type of functional.Numerical examples were used to study the microscopic and macroscopic properties,such as the effective modulus,of the composites.The results of the proposed VCFEM were compared with analytical solution and numerical results obtained from a standard finite element analysis to confirm its effectiveness.

A NOTE ON MORI-TANAKA'S METHOD

Explicit expressions of Mori-Tanaka＇s tensor for a transversely isotropic fiber rein- forced UD composite are presented. Closed-form formulae for the effective elastic properties of the composite are obtained. In a 3D sense, the resulting compliance tensor of the composite is symmetric. Nevertheless, the 2D compliance tensor based on a deteriorated Mori-Tanaka＇s tensor is not symmetric. Nor is the compliance tensor defined upon a deteriorated 2D Eshelby＇s tensor. The in-plane effective elastic properties given by those three approaches are different. A detailed comparison between the predicted results obtained from those approaches with experimental data available for a number of UD composites is made.