Trans RINA, Vol 153, Part B2, Intl J Small Craft Tech, 2011 Jul-Dec T
c² s²
2cs
s² c²
2cs ; s 2cs ² c cos sin . 2cs c s²
These observations lead to the mixed stress – strain wrinkling criterion
1 1
2
0
0 0
slack taut
and2 wrinkled 0
and following modification of the membrane formulation assuming uniaxial tension in direction :
σ H ε εw m m11 Figure 3: Element and material coordinate systems
A more detailed description of the structural method is given in [2] and [3].
4. ANISOTROPIC WRINKLING
A significant shortcoming of the basic membrane stress – strain formulation is its behaviour under compressive in- plane loads. “Real” sailcloth has a negligible bending stiffness and therefore negligible buckling strength, with compressive in-plane loads causing the cloth to wrinkle. Unfortunately, the basic membrane formulation has the same stress-strain gradient under compression as well as under tension.
This shortcoming is corrected by using a wrinkling model. Some wrinkling models ([4], [5]) calculate the wrinkling angle directly
stiffness matrix in case of wrinkling. Yet, until now this approach has only been described for isotropic materials and, in fact, doesn’t replicate the real behaviour of materials. Other wrinkling models ([6], [7], [9]) modify the deformation vector under following observations:
A wrinkled membrane is in a state of uniaxial tension.
The wrinkles are aligned with this uniaxial tension.
Material stresses are invariant to strain changes perpendicular to the wrinkles as long as the membrane is not coming under tension in this direction.
In anisotropic materials principal stresses and strains are not aligned.
If we assume the taut state as a starting point
and reduce principle stress in direction two (2), the basic membrane formulation holds up to -
and including - the point where 2 is exactly zero but the material not yet wrinkled. From this point on material stress remains unchanged while element strain changes further
(this
assuming principle stress equal than principle stress 2)
being greater or 22 H H32 H H33
H H33 H H31 23
21
©2011: The Royal Institution of Naval Architects
23 22
11 from strains and alter the
Figure 4: material under natural uniaxial tension (AB’C’D) and uniaxial tension (wrinkled, ABCD) [5]
ABCD are the corners of a wrinkled membrane element under uniaxial tension in direction t. Material stress in direction w is zero, yet strain in direction w is negative finite. If we extend the membrane direction w exactly up to the point where the wrinkles vanish but material stress is still zero (AB’C’D), we find the state of natural uniaxial tension. Up to this state material stress is invariant, yet from this state on the regular membrane formulation holds true.
In the state of natural uniaxial tension we can write:
σ H ε Defining σ
m
m11;0;0 m T we can rewrite: and the wrinkling strain εw w22
0 0
with the material stress σm
0 0
measure for the amount of wrinkling.
Geometrically this modification can be described as shown in Figure 4:
, where 22w is a
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