Realizable (Average Stress, Average Strain) Pairs in a Plate with Holes

Here a complete characterization is given of the set of all possible (average stress, average strain) pairs that can exist in a plate containing a fixed volume fraction f of holes. Specifically, for a given average stress, the range of values the average strain takes as the microgeometry is varied (while keeping f fixed) is determined. It is shown that multiple rank laminate materials suffice to generate all possible values of the average strain. When the microgeometry is restricted to be a periodic array of holes, with only one hole per unit cell, the average strain takes a smaller range of values as the hole shape is varied. A certain necessary condition for optimality must be satisfied if the hole is such that the average strain is on the boundary of this range. Numerical results are obtained for the range in the limit where the holes are well separated and occupy a small volume fraction. Optimal hole shapes, associated with average strains on the boundary of the range of admissible values, are identif...

[1]  William H. Press,et al.  Numerical recipes , 1990 .

[2]  Alexander Movchan,et al.  The Pólya–Szegö matrices in asymptotic models of dilute composites , 1997, European Journal of Applied Mathematics.

[3]  Andrej Cherkaev,et al.  Microstructures of Composites of Extremal Rigidity and Exact Bounds on the Associated Energy Density , 1997 .

[4]  Robert V. Kohn,et al.  Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions , 1993 .

[5]  S. Vigdergauz.  Two-Dimensional Grained Composites of Extreme Rigidity , 1994 .

[6]  Mark Kachanov,et al.  Elastic Solids with Many Cracks and Related Problems , 1993 .

[7]  E. Garboczi,et al.  The elastic moduli of a sheet containing circular holes , 1992 .

[8]  K. Hwang,et al.  Reduced dependence of defect compliance on matrix and inclusion elastic properties in two-dimensional elasticity , 1996, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[9]  Grégoire Allaire,et al.  On optimal microstructures for a plane shape optimization problem , 1999 .

[10]  Yury Grabovsky,et al.  The cavity of the optimal shape under the shear stresses , 1998 .

[11]  G. Milton,et al.  Bounding the current in nonlinear conducting composites , 2000 .

[12]  G. Milton,et al.  Which Elasticity Tensors are Realizable , 1995 .

[13]  Luc Tartar,et al.  Remarks on the homogenization method in optimal design problems , 1995 .

[14]  K. Hwang,et al.  Two–dimensional elastic compliances of materials with holes and microcracks , 1997, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[15]  Robert V. Kohn,et al.  Microstructures minimizing the energy of a two phase elastic composite in two space dimensions. , 1995 .

[16]  G. Allaire,et al.  Optimal design for minimum weight and compliance in plane stress using extremal microstructures , 1993 .

[17]  S. Vigdergauz.  RHOMBIC LATTICE OF EQUI-STRESS INCLUSIONS IN AN ELASTIC PLATE , 1996 .

[18]  Andrej Cherkaev,et al.  Design of Composite Plates of Extremal Rigidity , 1997 .

[19]  W. Press,et al.  Numerical Recipes: The Art of Scientific Computing , 1987 .

[20]  Graeme W. Milton,et al.  Invariant properties of the stress in plane elasticity and equivalence classes of composites , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[21]  A. Cherkaev.  Variational Methods for Structural Optimization , 2000 .

[22]  Mark Kachanov,et al.  Solids with non-spherical cavities : simplifiedrepresentations of cavity compliance tensors and theoverall anisotropy , 1999 .

[23]  S. Vigdergauz,et al.  A hole in a plate, optimal for its biaxial extension - compression , 1986 .

[24]  Gilles A. Francfort,et al.  Homogenization and optimal bounds in linear elasticity , 1986 .

[25]  S. Vigdergauz,et al.  Energy-minimizing inclusions in a planar elastic structure with macroisotropy , 1999 .

[26]  G. Milton.  The Theory of Composites , 2002 .

[27]  G. Hu,et al.  A new derivative on the shift property of effective elastic compliances for planar and three–dimensional composites , 2001, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.