Search

Critical drawback of poor stiffness in traditional auxetic materials limits them to be applied where negative Poisson’s ratio behavior and excellent load-bearing capacity are simultaneously desired. In this work, a general prescription for designing auxetic lattice structures with high stiffness and controllable isotropic/anisotropic elastic properties is proposed. Based on it, we rationally design a class of novel auxetic lattice structures whose deformation is mainly governed by the stretching mechanisms and conducted an investigation on a tree-like re-entrant structure. A combination of theoretical predictions, numerical simulations and tensile test experiments have been carried out to gain a comprehensive understanding of the structural in-plane elastic mechanical properties along both the principal axes as well as in the off-axis directions. Besides, theoretical model for designing tree-like re-entrant structure with isotropic elastic properties is established and experimentally verified with specimens fabricated with CFRP composite. The results of this research has proven the tree-like re-entrant structure exhibits excellent effective Young’s modulus, shear modulus and obvious negative Poisson’s ratio effect along omni-direction. Investigation on the directional dependence of structural effective elastic properties has further indicated the proposed structure presents a great potential for designing auxetic material with controllable anisotropic elastic properties. In particular, the proposed structure is capable in achieving isotropic design with any negative Poisson’s ratio between −1 and 0. The design philosophy presented in this paper provides a general prescription for solving the challenge of poor stiffness in traditional auxetic materials and paves a new way for isotropic auxetic design.

This paper introduces a new methodology for generating three-dimensional (3D) negative Poisson’s ratio behavior with lattice representations of the rotating rigid mechanism as a starting point. Based on it, a class of new 3D auxetic lattice structures is proposed. The elastic properties of a representative 3D auxetic lattice structure, including the homogenized Young’s modulus and Poisson’s ratio along the three principal axes, are systematically investigated in a combination of analytical predictions, numerical simulations and experimental tests. Moreover, effects of the structural geometrical parameters and specimen size on the elastic properties as well as structural nonlinear mechanical responses along the principal axes are carefully discussed. Different from most traditional 3D auxetic materials that are only capable of achieving negative or positive Poisson's ratio, our results suggest that the homogenized Poisson’s ratio effect of the proposed 3D lattice structure along all principal axes can be tuned from positive to negative in a wide range. The excellent design flexibility would help to expand and hasten the adoption of the new 3D auxetic lattice structures in engineering applications.

The non-local theory solution to a rectangular crack in a 3D infinite orthotropic elastic medium is investigated by using the generalized Almansi’s theorem and the Schmidt method in the present paper. The problem is formulated through the double Fourier transform into three pairs of dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. For solving the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. Numerical examples are provided to illustrate the effects of the geometric shape of the rectangular crack and the lattice parameter on the stress fields near the crack edges. Different from the classical solutions, the present solutions exhibit no stress singularity along the rectangular crack edges in an orthotropic elastic medium. Thus, this allows us to use the maximum stress as a fracture criterion.

In this paper, the basic solution of a Mode-I limited-permeable crack in functionally graded piezoelectric/piezomagnetic materials subjected to a stress loading was investigated using the generalized Almansi’s theorem. In the analysis, the electric permittivity and the magnetic permeability of the air inside the crack were considered. The problem was formulated through Fourier transform into two pairs of dual integral equations about the unknown jumps of displacements across the crack surfaces. To solve the dual integral equations, the jumps of displacements across the crack surfaces were directly expanded as a series of Jacobi polynomials. The solution of the present paper shows that the singular stresses, the singular electric displacements and the singular magnetic flux at crack tips in functionally graded piezoelectric/piezomagnetic materials carry the same forms as those in homogeneous piezoelectric/piezomagnetic materials; however, the magnitudes of the magnetoelectric intensity factors depend on the gradient of functionally graded piezoelectric/piezomagnetic materials. It is also revealed that the effects of the electric and magnetic boundary conditions on the electric displacement and magnetic flux fields near crack tips can not be ignored.

In this paper, the basic solutions of two parallel mode-I cracks or four parallel mode-I cracks in the piezoelectric materials were investigated by means of the Schmidt method for the limited-permeable electric boundary conditions. The electric permittivity of air in the crack was considered. Through the Fourier transform, the problems can be solved with the help of two pairs of dual integral equations, in which the unknown variables were the jumps of the displacements across the crack surfaces, not the dislocation density functions. To solve the dual integral equations, the jumps of the displacements across the crack surfaces were directly expanded in a series of Jacobi polynomials. Finally, the effects of the distance between two parallel cracks, the distance between two collinear cracks and the electric boundary conditions on the stress and the electric intensity factors in the piezoelectric materials are analyzed. These results can be used for the strength evaluation of the piezoelectric materials with multi-cracks. The crack shielding effect is also present in the piezoelectric materials.

In this paper, the behavior of a Mode-I crack in the piezoelectric/piezomagnetic materials subjected to a uniform tension loading is investigated by the generalized Almansi’s theorem. Through the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations, in which the unknown variables are the jumps of the displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps are directly expanded in a series of Jacobi polynomials. Then the closed form solution of this problem can be obtained.

In this paper, the interaction of two collinear cracks in functionally graded materials subjected to a uniform anti-plane shear loading is investigated by means of nonlocal theory. The traditional concepts of the nonlocal theory are extended to solve the fracture problem of functionally graded materials. To make the analysis tractable, it is assumed that the shear modulus varies exponentially with the coordinate vertical to the crack. By use of the Fourier transform, the problem can be solved with the help of a pair of triple integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the triple integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularity is present near the crack tips. The nonlocal elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials. The magnitude of the finite stress field depends on the crack length, the distance between two cracks, the parameter describing the functionally graded materials and the lattice parameter of the materials.

In this paper, the dynamic behavior of two parallel symmetric cracks in piezoelectric materials under harmonic anti-plane shear waves is investigated by use of the non-local theory for permeable crack surface conditions. To overcome the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the problem to obtain the stress occurs near the crack tips. By means of the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations that the unknown variables are the jumps of the displacement along the crack surfaces. These equations are solved using the Schmidt method. Numerical examples are provided. Contrary to the previous results, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the frequency of the incident wave, the distance between two cracks and the lattice parameter of the materials, respectively. Contrary to the impermeable crack surface condition solution, it is found that the dynamic electric displacement for the permeable crack surface conditions is much smaller than the results for the impermeable crack surface conditions. The results show that the dynamic field will impede or enhance crack propagation in the piezoelectric materials at different stages of the dynamic load.

In this paper, the behavior of two parallel symmetry permeable interface cracks in a piezoelectric layer bonded to two half piezoelectric materials planes subjected to an anti-plane shear loading is investigated by using Schmidt method. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. This process is quite different from that adopted previously. The normalized stress and electrical displacement intensity factors are determined for different geometric and property parameters for permeable crack surface conditions. Numerical examples are provided to show the effect of the geometry of the interacting cracks, the thickness and the materials constants of the piezoelectric layer upon the stress and the electric displacement intensity factor of the cracks. Contrary to the impermeable crack surface condition solution, it is found that the electric displacement intensity factors for the permeable crack surface conditions are much smaller than the results for the impermeable crack surface conditions.

In this paper, the interaction between two collinear cracks in the piezoelectric materials under anti-plane shear loading was investigated by using the non-local theory for impermeable crack-face conditions. By using the Fourier transform, the problem can be solved with the help of two pairs of triple integral equations. The solutions are obtained by using the Schmidt method. Numerical examples are provided to show the effect of the geometry of the interacting cracks. Contrary to the previous results, it is found that no stress and electric displacement singularity is present near the crack tip.

The problem of determining the stress field in an elastic strip of finite width when the uniform tension is applied to the faces of two collinear symmetrical cracks situated within it is considered. By using the Fourier transform, the problem can be solved with a set of triple integral equations. These equations are solved using Schmidts method. This method is suitable for solving the strips problem of arbitrary width.

Field equations of the non-local elasticity are solved to determine the state of stress in a plate with a Griffith crack subject to the anti-plane shear. Then a set of dual-integral equations is solved using Schmidts method. Contrary to the classical elasticity solution, it is found that no stress singularity is present at the crack tip. The significance of this result is that the fracture criteria are unified at both the macroscopic and the microscopic scales.