X.-L. Gao

Bernoulli–Euler beam model based on a modified couple stress theory

A new model for the bending of a Bernoulli–Euler beam is developed using a modified couple stress theory. A variational formulation based on the principle of minimum total potential energy is employed. The new model contains an internal material length scale parameter and can capture the size effect, unlike the classical Bernoulli–Euler beam model. The former reduces to the latter in the absence of the material length scale parameter. As a direct application of the new model, a cantilever beam problem is solved. It is found that the bending rigidity of the cantilever beam predicted by the newly developed model is larger than that predicted by the classical beam model. The difference between the deflections predicted by the two models is very significant when the beam thickness is small, but is diminishing with the increase of the beam thickness. A comparison shows that the predicted size effect agrees fairly well with that observed experimentally.

Micromechanics model for three-dimensional open-cell foams using a tetrakaidecahedral unit cell and castigliano's second theorem

A micromechanics model for three-dimensional open-cell foams is developed using an energy method based on Castiglianos second theorem. The analysis is performed on a tetrakaidecahedral unit cell, which is centered at one lattice point of a body-centered cubic lattice and is subjected to compression on its two opposite square faces. The 36 struts of the unit cell are treated as uniform slender beams undergoing linearly elastic deformations, and the 24 vertices as rigid joints. All three deformation mechanisms of the cell struts (i.e., stretching, shearing and bending) possible under the specified loading are incorporated, and four different strut cross section shapes (i.e., circle, square, equilateral triangle and Plateau border) are treated in a unified manner in the present model, unlike in earlier models. Two closed-form formulas for determining the effective Youngs modulus and Poissons ratio of open-cell foams are provided. These two formulas are derived by using the composite homogenization theory and contain more parameters than those included in existing models. The new formulas explicitly show that the foam elastic properties depend on the relative foam density, the shape and size of the strut cross section, and the Youngs modulus and Poissons ratio of the strut material. By applying the newly derived model directly, a parametric study is conducted for carbon foams, whose modeling motivated the present study. The predicted values of the effective Youngs modulus and Poissons ratio compare favorably with those based on existing models and experimental data.

Finite element simulation of the orthogonal metal cutting process for qualitative understanding of the effects of crater wear on the chip formation process

Abstract The orthogonal cutting of oil hardening tool steel O1 is simulated using a fully coupled thermomechanical finite element model. ABAQUS program is the computational tool in this model. Emphasis is placed on the effects of the geometric variations of the tool rake face, resulting from crater wear, on the cutting process. Johnson–Cook’s model is employed as the constitutive equation for the workpiece material. The sticking and sliding tool–chip frictional behavior is described by Coulomb’s law, and the chip separation by a critical stress criterion. Four cases are simulated, which are associated with three distinct tool face geometries, i.e., flat tool and two types of cratered tools. Representative results, including deformed meshes, contours of the von Mises equivalent plastic strain, the von Mises equivalent stress and cutting temperature, distributions of the contact stresses on the tool–chip interface, and cutting forces, are provided for each case and are compared with each other. The cutting forces produced with the flat tool agree well with the experimental data.

Elasto-plastic analysis of an internally pressurized thick-walled cylinder using a strain gradient plasticity theory

Abstract An analytical solution for the stress, strain and displacement fields in an internally pressurized thick-walled cylinder of an elastic strain-hardening plastic material in the plane strain state is presented. A strain gradient plasticity theory is used to describe the constitutive behavior of the material undergoing plastic deformations, whereas the generalized Hooke’s law is invoked to represent the material response in the elastic region. The solution gives explicit expressions for the stress, strain and displacement components. The inner radius of the cylinder enters these expressions not only in non-dimensional forms but also with its own dimensional identity, unlike classical plasticity-based solutions. As a result, the current solution can capture the size (strengthening) effect at the micron scale. The classical plasticity-based solution of the same problem is shown to be a special case of the present solution. Numerical results for the maximum effective stress in the cylinder wall are also provided to illustrate applications of the newly derived solution.

Micromechanical Modeling of Viscoelastic Properties of Carbon Nanotube-Reinforced Polymer Composites

A micromechanics model is developed for predicting the linearly viscoelastic properties of carbon nanotube-reinforced polymer composites. By employing the Correspondence Principle in viscoelasticity, the Mori-Tanaka method is extended to the Carson domain. The inversion of the creep compliances from the Carson (transformed) domain to the time (physical) domain is accomplished numerically by using a recently developed multi-precision algorithm. The new micromechanics model is validated by comparing with existing experimental data. By applying the presently developed model, a parametric study for the creep behavior of carbon nanotube-reinforced polymer composites is conducted, with testing temperature, nanotube aspect ratio, nanotube volume fraction and nanotube orientation as the controlling parameters. For composites having unidirectionally aligned nanotubes, numerical results indicate that the increase of the nanotube aspect ratio significantly enhances their axial creep resistance but has insignificant influences on their transverse, shear and plane strain bulk creep compliances. Also, the random orientation of nanotubes provides more effective plane strain bulk creep resistance but less effective axial creep resistance than the aligned orientation does. In addition, the effect of the nanotube orientation on the shear compliances is negligibly small. Furthermore, for composites with aligned or randomly oriented nanotubes, all the compliances are found to decrease monotonically with the increase of the nanotube volume fraction. Finally, the influences of testing temperature on the composite creep compliances (except for the bulk strain compliance) are similar to those on the compliance of the matrix.

A general solution of an infinite elastic plate with an elliptic hole under biaxial loading

Abstract A general analytical solution is obtained in this paper for an infinite elastic plate with a traction-free elliptic hole subjected to arbitrary biaxial loading. The boundary-value problem is solved by using the complex potential method, but the usual two-fold conformal transformations are avoided by employing the elliptic-hyperbolic coordinate system, which is physical and natural. All expressions for stress and displacement fields are derived in explicit form to provide a complete analysis and to make the solution ready for engineering use. With two adjustable parameters—the biaxial loading factor λ and the orientation angle β—contained in these expressions, the present solution furnishes a most general account of the elliptic hole problem. It is shown that all existing solutions, including the solution of a cracked plate under biaxial loading, can be obtained from this general solution. In addition, the solution for an important biaxial loading case characterizing thin-walled cylindrical pressure vessels, which has not been reported before, is also derived in the paper as a specific case of the general solution.

An exact elasto-plastic solution for an open-ended thick-walled cylinder of a strain-hardening material

Abstract In this paper, a closed-form analytical solution for the elasto-plastic stress, strain, and displacement components of an internally pressurized open-ended thick-walled cylinder made of a strain-hardening material is obtained by using the deformation theory of Hencky and the yield criterion of Von Mises and choosing stress components as basic unknowns. On the basis of the elastic power-law plastic material model and a modified Nadais auxiliary-variable method, this solution provides a new analytical pattern for the elasto-plastic analysis and the strength design of a strain-hardening-material open-ended thick-walled cylinder. It is shown that the solution is a general one with on the one hand Nadais known solution for stress components and on the other P.C.T. Chens solution for strain components of an open-ended thick-walled cylinder made of elastic-perfectly-plastic material as its two specific cases. The calculation formula for the plastic-limit pressure of the cylinder given in the paper, which is able to account for the strain-hardening effect of material, furnishes a new and more reasonable theoretical basis for the strength design by plastic-limit analysis and the autofrettage design of an open-ended thick-walled cylinder made of strain-hardening material.

Preparation, characterization, and modeling of carbon nanofiber/epoxy nanocomposites

There is a lack of systematic investigations on both mechanical and electrical properties of carbon nanofiber (CNF)-reinforced epoxy matrix nanocomposites. In this paper, an in-depth study of both static and dynamic mechanical behaviors and electrical properties of CNF/epoxy nanocomposites with various contents of CNFs is provided. A modified Halpin-Tsai equation is used to evaluate the Youngs modulus and storage modulus of the nanocomposites. The values of Youngs modulus predicted using this method account for the effect of the CNF agglomeration and fit well with those obtained experimentally. The results show that the highest tensile strength is found in the epoxy nanocomposite with a 1.0 wt% CNFs. The alternate-current (AC) electrical properties of the CNF/epoxy nanocomposites exhibit a typical insulator-conductor transition. The conductivity increases by four orders of magnitude with the addition of 0.1 wt% (0.058 vol%) CNFs and by ten orders of magnitude for nanocomposites with CNF volume fractions higher than 1.0 wt% (0.578 vol%). The percolation threshold (i.e., the critical CNF volume fraction) is found to be at 0.057 vol%.

Variational solution for a cracked mosaic model of woven fabric composites

Abstract A variational solution for a cracked mosaic laminate model of woven fabric composites is presented using the principle of minimum complementary energy. The solution is derived for the woven laminate in either the plane strain or the plane stress state, with the warp/fill yarn materials being either orthotropic or transversely isotropic, unlike other existing solutions in the literature of laminate elasticity. The stress components are given in closed-form expressions in terms of a perturbation function, which is governed by two (uncoupled) fourth-order inhomogeneous ordinary differential equations (i.e., Euler–Lagrange equations) when the thermal effects are included. All possible expressions of this perturbation function are obtained in closed forms. Young’s modulus of the cracked laminate is calculated using the determined minimum complementary energy. The present closed-form solution can account for different yarn materials, applied loads (crack densities), geometrical dimensions, or their combinations. To demonstrate the solution, a total of 60 sample cases are analyzed using three different composite systems (i.e., glass fiber/epoxy, graphite fiber/epoxy and ceramic fiber/ceramic) and ten different crack densities. The obtained numerical results are also compared to two existing elasticity solutions for cross-ply laminates.

Strain gradient solution for Eshelby’s ellipsoidal inclusion problem

Eshelby’s problem of an ellipsoidal inclusion embedded in an infinite homogeneous isotropic elastic material and prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is analytically solved. The solution is based on a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter in addition to two classical elastic constants. The fourth-order Eshelby tensor is obtained in analytical expressions for both the regions inside and outside the inclusion in terms of two line integrals and two surface integrals. This non-classical Eshelby tensor consists of a classical part and a gradient part. The former involves Poisson’s ratio only, while the latter includes the length scale parameter additionally, which enables the newly obtained Eshelby tensor to capture the inclusion size effect, unlike its counterpart based on classical elasticity. The accompanying fifth-order Eshelby-like tensor relates the prescribed eigenstrain gradient to the disturbed strain and has only a gradient part. When the strain gradient effect is not considered, the new Eshelby tensor reduces to the classical Eshelby tensor, and the Eshelby-like tensor vanishes. In addition, the current Eshelby tensor for the ellipsoidal inclusion problem includes those for the spherical and cylindrical inclusion problems based on the SSGET as two limiting cases. The non-classical Eshelby tensor depends on the position and is non-uniform even inside the inclusion, which differ from its classical counterpart. For homogenization applications, the volume average of the new Eshelby tensor over the ellipsoidal inclusion is analytically obtained. The numerical results quantitatively show that the components of the newly derived Eshelby tensor vary with both the position and the inclusion size, unlike their classical counterparts. When the inclusion size is small, it is found that the contribution of the gradient part is significantly large. It is also seen that the components of the averaged Eshelby tensor change with the inclusion size: the smaller the inclusion, the smaller the components. Moreover, these components are observed to approach the values of their classical counterparts from below when the inclusion size becomes sufficiently large.