The modeling and numerical analysis of wrinkled membranes

Abstract

SUMMARYIn this paper three fundamental issues regarding modeling and analysis of wrinkled membranes are addressed. First, a new membrane model with viable Young's modulus and Poisson's ratio is proposed, which physically characterizes stress relaxation phenomena in membrane wrinkling, and expresses taut, wrinkled and slack states of a membrane in a systematic manner. Second, a parametric variational principle is developed for the new membrane model. Third, by the variational principle, the original membrane problem is converted to a non-linear complementarity problem in mathematical programming. A parametric ÿnite element discretization and a smoothing Newton method are then used for numerical solution. The proposed membrane model and numerical method are capable of delivering convergent results for membranes with a mixture of wrinkled and slack regions, without iteration of membrane stresses. Three numerical examples are provided.

Simulation of light‐weight membrane structures by wrinkling model

“…The maximum tensile stress occurs in the corners with a value of about 0.11 Pa. Stress direction figures and values agree well with the result published in Reference [28]. At the end of the simulation elements state become wrinkled from first to fifth element, slack for the sixth and taut for seventh and eighth elements.…”

“…In particular, the test is performed using the scheme reported in Reference [28] for the example 1 and shown in Figure 7(a). A rectangular membrane of size 8 × 4 m is fixed at its four sides; at the corners it is subjected to four tension loads (P = 0.001 N with = 45 • ).…”

Analysis of wrinkled membrane structures based on a wrinkle-wave model

“…To measure partially wrinkled membranes, Stein and Hedgepeth introduced the concept of a viable Poisson's ratio, 4 which was extended by Mikulas 5 in a numerical and experimental study. On the basis of these works, Ding and Yang proposed a new membrane model with viable Young's modulus and Poisson's ratio, 6 which was later studied by Rossi et al 7 and Jarasjarungkiat et al 8 However, although tension field theory can describe certain characteristics of wrinkling, it cannot predict wrinkle amplitude, number, or wavelength. 9 To overcome the difficulty of predicting the wrinkling behavior, a stability theory was developed.…”

“…10 However, the finite element method cannot express taut, slack, and wrinkled states in a systematic and continuous fashion, which can lead to divergent results. 6 Diaby et al used another method, bifurcation analysis, to analyze the buckling and wrinkling phenomena of pre-stressed membranes. 11 The bifurcation theorem was introduced to track the bifurcation path, which requires considerable computational effort.…”

Wrinkling reduction of membrane structure by trimming edges

“…The square membrane was loaded uniformly and incrementally in steps of 1N. 15 Circular and positive contours of the second principal stresses were formed in the center to comply with an equal biaxial stress state. Their magnitude decreased as they moved outwards towards the corners, where they became compressive.…”