# Matlab R2009b Crack Free 138

Overall, we developed a scalable NPC film-assisted transfer method to fabricate crack-free and tear-free, millimeter-scale suspended single-layer CVD graphene films, allowing us to observe and understand the temperature-dependent single-component and mixture gas transport through the intrinsic defects in graphene. Graphene films with a minuscule porosity of 0.025% displayed attractive H2 permeance and H2/CH4 selectivities approaching the performance of 1-Âµm-thick state-of-the-art polymer membranes. Improvements in the H2 permeance and/or H2/CH4 selectivity were demonstrated by ozone functionalization. Overall, the methods developed here bring deployment of the single-layer nanoporous graphene membranes for gas separation a step closer to reality.

## Matlab R2009b Crack Free 138

Based on Euler-Bernoulli beam theory and a continuous stiffness beam model, the free vibration of rectangular-section beams made of functionally graded materials (FGMs) containing open edge cracks is studied. Assuming the material gradients follow exponential distribution along beam thickness direction, the conversion relation between the vibration governing equations of a FGM beam and that of an isotropic homogenous beam is deduced. A continuous function is used to characterize the bending stiffness of an edge cracked FGM beam. Thus, the cracked FGM beam is treated as an intact beam with continuously varying bending stiffness along its longitudinal direction. The characteristic equations of beams with different boundary conditions are obtained by transfer matrix method. To verify the validity of the proposed method, natural frequencies for intact and cracked FGM beams are calculated and compared with those obtained by three-dimensional finite element method (3D FEM) and available data in the literature. After that, further discussions are carried out to analyze the influences of crack depth, crack location, material property, and slenderness ratio on the natural frequencies of the cracked FGM beams.

While numerous studies have been performed to investigate the influences of the cracks on the dynamic behaviors of isotropic homogenous beams, there are also some research works focusing on the edge cracked FGM beams which are concerned in this paper. The following presents a literature review related to FGM beams with open edge cracks. Yang and Chen [27] studied the free vibration of cracked FGM beams based on Euler-Bernoulli beam theory and the rotational spring model. The influences of crack location, material properties, and slenderness ratio on the natural frequency of cracked FGM beams were discussed. Ke et al. [28] improved the method described in [27] for cracked Timoshenko FGM beams. Yu and Chu [29] proposed a p-version finite element method to determine the natural frequencies of cracked FGM beams and dealt with the crack identification of the slender cantilever FGM beams containing open edge cracks. Based on Euler-Bernoulli beam theory and the rotational spring model, Aydin [30] used a third-order determinant to analyze the natural frequencies of damaged FGM beams containing arbitrary numbers of cracks.

As is known, natural frequencies are the most commonly used parameters employed in the crack identification of a damaged structure [30]. However, the literature review above shows that a limited amount of research works has been carried out to discuss the vibration problems of edge cracked FGM beams. Moreover, in most reports, rotational springs are used to simulate the cracks in FGM beams. Therefore, the present paper investigates the free vibration of FGM beams containing open edge cracks and proposes a new method to determine the natural frequencies of these beams. More specifically, based on Euler-Bernoulli beam theory, the conversion relation between vibration governing equation of a FGM beam and that of an isotropic homogenous beam is deduced, and a cracked FGM beam is simulated by a continuous beam model. Moreover, the cracked FGM beam is treated as an intact beam with its bending stiffness varying continuously along beam longitudinal direction. Dividing the beam into sufficient number of segments and employing the transfer matrix method, the characteristic equation of the beam is obtained. The validity of the proposed method is verified by the three-dimensional finite element method (3D FEM) and the data in references. Furthermore, the influences of crack depth, crack location, material gradient, and slenderness ratio on the first three natural frequencies of the cracked FGM beams are discussed. The method and the results in this paper can be used for nondestructive testing of beam structures made of FGMs.

According to the continuous bending stiffness characterized by (15), a cracked FGM beam can be treated as an intact beam with continuously varying bending stiffness along its longitudinal direction. The governing equation of free vibration of the cracked FGM beam takes the form of a nonuniform beam aswhere , , and have been, respectively, defined by (14) and (15). Being different from (13), the analytical solutions of (18) cannot be solved by separation variable method. However, the numerical solutions of (18) can be obtained by means of transfer matrix method. For a beam with varying section parameters along its longitudinal direction, we can divide the beam into segments. If the length of each segment is short enough, the bending stiffness and the linear density for each segment can be treated as constants. Then, the governing equation of each segment can also take the following form of a uniform beam: In (19), the bending stiffness and the linear density of (i)th segment can be described aswhere , and denote the length and the coordinates of left end and right end of the (i)th segment.