Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

More documents

Recommendations

Info

728 BAHATTIN KILIC AND ERDOGAN MADENCI The average coefficient vector gi jc can be written as in which gi jc = (gci + gcj)/2, (A10) g T c (x) = {gxc, gyc, gzc} = {g∞/gxc, g∞/gyc, g∞/gzc}. (A11) After considering the surface effects, the discrete form of the equations of motion given in (9) is corrected as N� Ne � ρü(xi, t) = b(xi, t) + gi jwj f � u(xi, t), u(x ′ k , t), xi, x ′ k , t� . (A12) e=1 j=1 According to (A10), the vector gi jc is computed by taking the average of the coefficients given in (A11) for two interacting material points. As a result, the surface correction given in (A8) is underestimated for the material point with the larger scaling coefficient and overestimated for the material point with the smaller scaling coefficient. Hence, a single iteration is generally not sufficient to determine the scaling coefficients. Therefore, the scaling coefficients in (A10) are computed iteratively. Each component given in (A10) is set to unity at the beginning of the first iteration step, and then the method outlined above is utilized to determine the coefficients given in (A10). Then, the present method is repeated n times, during which the results of the previous iteration are used as a scaling coefficient. After numerical investigation, it was found that the change in the coefficients in (A10) is insignificant after approximately twenty iterations. Hence, the number of iterations, n, is taken to be twenty in this study. Although this method enables the correction to material behavior due to the surface effects and approximations during the numerical integration, it is still not exact. Appendix B The effects of the peridynamic parameters and discretization are established by considering a rectangular plate (Figure B1). The length and width of the plate are specified as a = 10 in and b = 4 in, respectively. The plate is free of any displacement constraints and is subjected to uniaxial tension in the x-direction. Figure B1. Dimensions and discretization of the plate.
COUPLING OF PERIDYNAMIC THEORY AND THE FINITE ELEMENT METHOD 729 However, point and surface loads cannot be applied directly because the peridynamic equation of motion involves a volume integral that gives rise to zero for point and surface loads. Thus, the uniaxial tension is applied as uniform body forces over the volumes, �b, along the vertical edges having a length, l, of 0.08 in (Figure B1). The longitudinal component of the applied body force is specified as bx = 234375.0 lb/in3 . Hence, the applied tension, F, can be computed in terms of the plate dimensions as F = lbtbx. The material of the plate is isotropic with Young’s modulus and Poisson’s ratio of E = 107 psi and ν = 0.25, respectively. The three-dimensional peridynamic model is constructed by discretization of the plate using cubic subdomains (Figure B1). In each subdomain, eight integration points are utilized to reduce the peridynamic equation of motion to its discrete form. The steady-state solutions are obtained by using the adaptive dynamic relaxation. Within the realm of molecular dynamics, it is well known that the Poisson’s ratio is restricted to 1 4 in three dimensions and 1 3 in two dimensions if atomic interactions are pairwise and do not exhibit any directional dependence. Gerstle et al. [2005] showed that the same restrictions apply to peridynamics equations under the same conditions. Hence, the thickness of the plate affects the observed Poisson’s ratio. Figure B2 shows the variation of the Poisson’s ratio with increasing plate thickness. The Poisson’s ratio is calculated using steady-state displacements at the point located at x0, which can be expressed as and the displacement field at x0 can be written as x T 0 = {x0 = 2.5, y0 = 1.0, z0 = 0.0} (B1) u T 0 = {ux0, u y0, uz0}. (B2) Hence, the Poisson’s ratio is approximated by using initial positions and displacements as ν = − u y0/y0 . (B3) ux0/x0 Figure B2. Effect of internal length on Poisson’s ratio with increasing plate thickness.
Page 1 and 2: Journal of Mechanics of Materials a
Page 3 and 4: JOURNAL OF MECHANICS OF MATERIALS A
Page 5 and 6: R AXIAL COMPRESSION OF HOLLOW ELAST
Page 7 and 8: AXIAL COMPRESSION OF HOLLOW ELASTIC
Page 15: AXIAL COMPRESSION OF HOLLOW ELASTIC
Page 18 and 19: 708 BAHATTIN KILIC AND ERDOGAN MADE
Page 47 and 48: GENETIC MODELING OF COMPRESSIVE STR
Page 57 and 58: epresents 200 150 100 50 -50 -100 G
Page 63: GENETIC MODELING OF COMPRESSIVE STR
Page 66 and 67: 756 XIAO-TING HE, QIANG CHEN, JUN-Y
Page 83 and 84: PLANAR BEAMS: MIXED VARIATIONAL DER
Page 89 and 90:
PLANAR BEAMS: MIXED VARIATIONAL DER
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
JOURNAL OF MECHANICS OF MATERIALS A
Page 107 and 108:
SIFS OF RECTANGULAR TENSILE SHEETS
Page 109 and 110:
Page 111 and 112:
Page 113:
Page 116 and 117:
806 YING LI AND CHANG-JUN CHENG the
Page 118 and 119:
808 YING LI AND CHANG-JUN CHENG 3.
Page 120 and 121:
810 YING LI AND CHANG-JUN CHENG ui(
Page 122 and 123:
812 YING LI AND CHANG-JUN CHENG ( j
Page 124 and 125:
814 YING LI AND CHANG-JUN CHENG Fig
Page 126 and 127:
816 YING LI AND CHANG-JUN CHENG Fig
Page 128 and 129:
818 YING LI AND CHANG-JUN CHENG For
Page 130 and 131:
820 YING LI AND CHANG-JUN CHENG [B
Page 132 and 133:
822 ELIA EFRAIM AND MOSHE EISENBERG
Page 134 and 135:
Page 136 and 137:
Page 138 and 139:
Page 140 and 141:
Page 142 and 143:
Page 144 and 145:
Page 147 and 148:
JOURNAL OF MECHANICS OF MATERIALS A
Page 149 and 150:
MATRIX OPERATOR METHOD FOR THERMOVI
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
��
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
SUBMISSION GUIDELINES ORIGINALITY A
show all

Journal of Mechanics of Materials and Structures vol. 5 (2010 ... - MSP

Create successful ePaper yourself

Delete template?

Save as template?