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The methodology for determining the upper bounds on the homogenized linear elastic properties of cellular solids, described for the two-dimensional case in Dimitrovová and Faria (1999), is extended to three-dimensional open-cell foams. Besides the upper bounds, the methodology provides necessary and sufficient conditions on optimal media. These con...
Contexts in source publication
Context 1
... order to justify the definition stated above, it is necessary to verify that a curved beam cannot from part of the optimal low-density media. Let us suppose that the i-beam of a micro-frame basic cell is curved. Then a local coordinate system (z 1 , z 2 ) can be introduced so that z 1 connects the centers of the joints (Fig. 2). The middle axis of the beam is given by z 2 =a(z 1 ) and r designates the curved coordinate. Let us separate the beam of active length from the joints by the cuts shown in Fig. 2. It is assumed that there exists a plane containing the i-beam middle curve and that the macroload acts in such a way that the generalized internal forces in ...
Context 2
... that the i-beam of a micro-frame basic cell is curved. Then a local coordinate system (z 1 , z 2 ) can be introduced so that z 1 connects the centers of the joints (Fig. 2). The middle axis of the beam is given by z 2 =a(z 1 ) and r designates the curved coordinate. Let us separate the beam of active length from the joints by the cuts shown in Fig. 2. It is assumed that there exists a plane containing the i-beam middle curve and that the macroload acts in such a way that the generalized internal forces in the beam cuts are also contained in this plane. The geometrical parameters a(r), a 0 k , a 0 m , h k , h m , v k , v m , l, p, the generalized internal forces in the beam cuts F, ...
Context 3
... acts in such a way that the generalized internal forces in the beam cuts are also contained in this plane. The geometrical parameters a(r), a 0 k , a 0 m , h k , h m , v k , v m , l, p, the generalized internal forces in the beam cuts F, B and D and other local auxiliary coordinate systems ( ~ , ~ z z 1 2 ) and ($ , $ z z 1 2 ) are specified in Fig. 2. l and p are projections of the theoretical and the active lengths on z 1 and the bending moment along the beam is separated into its (average) constant (D) and "antisymmetric" ...
Context 4
... must be uniquely defined in a way applicable to any beam from the basic cell. Coordinate systems ( ~ , ~ z z 1 2 ) and(z z 1 2 $ , $ ) are introduced as specified in Fig. 2. With respect to (z 1 , z 2 ) this yields from (6): ...
Context 5
... order to justify the definition stated above, it is necessary to verify that a curved beam cannot from part of the optimal low-density media. Let us suppose that the i-beam of a micro-frame basic cell is curved. Then a local coordinate system (z 1 , z 2 ) can be introduced so that z 1 connects the centers of the joints (Fig. 2). The middle axis of the beam is given by z 2 =a(z 1 ) and r designates the curved coordinate. Let us separate the beam of active length from the joints by the cuts shown in Fig. 2. It is assumed that there exists a plane containing the i-beam middle curve and that the macroload acts in such a way that the generalized internal forces in ...
Context 6
... that the i-beam of a micro-frame basic cell is curved. Then a local coordinate system (z 1 , z 2 ) can be introduced so that z 1 connects the centers of the joints (Fig. 2). The middle axis of the beam is given by z 2 =a(z 1 ) and r designates the curved coordinate. Let us separate the beam of active length from the joints by the cuts shown in Fig. 2. It is assumed that there exists a plane containing the i-beam middle curve and that the macroload acts in such a way that the generalized internal forces in the beam cuts are also contained in this plane. The geometrical parameters a(r), a 0 k , a 0 m , h k , h m , v k , v m , l, p, the generalized internal forces in the beam cuts F, ...
Context 7
... acts in such a way that the generalized internal forces in the beam cuts are also contained in this plane. The geometrical parameters a(r), a 0 k , a 0 m , h k , h m , v k , v m , l, p, the generalized internal forces in the beam cuts F, B and D and other local auxiliary coordinate systems ( ~ , ~ z z 1 2 ) and ($ , $ z z 1 2 ) are specified in Fig. 2. l and p are projections of the theoretical and the active lengths on z 1 and the bending moment along the beam is separated into its (average) constant (D) and "antisymmetric" ...
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