A rectangular membrane is remotely loaded by a uniform traction acting along two opposite edges. The material is linear elastic and anisotropic. A geometrical discontinuity, in the shape of a void or of a rigid inclusion, is placed centrally. Four kinds of discontinuities are analyzed in detail. First, we consider two typical stress concentration problems where the discontinuity is a circular hole or a circular rigid inclusion. Then, we analyze two stress intensification problems, corresponding to a small slit or a rigid segment with their axis placed perpendicularly to the stress trajectories. These problems have been widely considered in the past by assuming several material laws. In all cases, the membrane was treated as a thin plate so that plane stress conditions were assumed (standard membrane theory). However, these solutions cannot be representative of the compression-free state of stress generated within a real membrane if this is susceptible of wrinkling. In fact, because of the complicated pattern of ridges, cusps and wrinkles which emerges at equilibrium within a real membrane, it is nearly impossible to forecast its actual displacement field. Therefore, the assessment of the membrane equilibrium state remains up to date a highly challenging task, difficult to solve, either analytically or numerically. In many cases the analysis is aimed to assess the stress distribution only, so that some approximated solutions can be obtained by recourse to a relaxed energy approach whereby the most significant geometrical non-linearities are turned into a material non-linearity. However, the same approach cannot be used to deal with anisotropic materials since an explicit expression for the relaxed energy is still missing. In this paper, the state of stress around geometrical discontinuities is determined by a previous model developed by the authors and following as close as possible the main ideas of the relaxed energy approach, but properly adapted to include cases of material anisotropy. The influence of both anisotropy and wrinkling is emphasized by comparison with the corresponding isotropic cases.
Geometrical Discontinuities in Partly Wrinkled Anisotropic Membranes
BARSOTTI, RICCARDO;LIGARO', SALVATORE SERGIO
2004-01-01
Abstract
A rectangular membrane is remotely loaded by a uniform traction acting along two opposite edges. The material is linear elastic and anisotropic. A geometrical discontinuity, in the shape of a void or of a rigid inclusion, is placed centrally. Four kinds of discontinuities are analyzed in detail. First, we consider two typical stress concentration problems where the discontinuity is a circular hole or a circular rigid inclusion. Then, we analyze two stress intensification problems, corresponding to a small slit or a rigid segment with their axis placed perpendicularly to the stress trajectories. These problems have been widely considered in the past by assuming several material laws. In all cases, the membrane was treated as a thin plate so that plane stress conditions were assumed (standard membrane theory). However, these solutions cannot be representative of the compression-free state of stress generated within a real membrane if this is susceptible of wrinkling. In fact, because of the complicated pattern of ridges, cusps and wrinkles which emerges at equilibrium within a real membrane, it is nearly impossible to forecast its actual displacement field. Therefore, the assessment of the membrane equilibrium state remains up to date a highly challenging task, difficult to solve, either analytically or numerically. In many cases the analysis is aimed to assess the stress distribution only, so that some approximated solutions can be obtained by recourse to a relaxed energy approach whereby the most significant geometrical non-linearities are turned into a material non-linearity. However, the same approach cannot be used to deal with anisotropic materials since an explicit expression for the relaxed energy is still missing. In this paper, the state of stress around geometrical discontinuities is determined by a previous model developed by the authors and following as close as possible the main ideas of the relaxed energy approach, but properly adapted to include cases of material anisotropy. The influence of both anisotropy and wrinkling is emphasized by comparison with the corresponding isotropic cases.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
