Coursework1 - CEGEM090

8/11/2019 Coursework1 - CEGEM090

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Coursework CEGEM090/CEGEG090 Advanced Structural Analysis

(due-in date: Mon 10 November 2014, noon)

Wrinkling of a stamped elastic plate

Wrinkling patterns are generally observed when thin plates are put under strong in-plane edgetension leading to compressive hoop stresses. Here we consider a different kind of wrinklingstudied recently by Hure et al. [1]. In this problem the plate boundary is free from tension butthe amplitude of the wrinkles is highly constrained. This situation occurs in various industrialprocesses where metal plates are plastically embossed in a mould, the mismatch in Gaussiancurvature generally leading to regular wrinkles. In the elastic case crumpling singularities firstappear as the mould is progressively closed down (see Fig. 1c) and evolve into a pattern ofapparently smooth radial wrinkles under strong confinement (Fig. 1d,e).

We analyse a simplified one-dimensional version of the problem (see Fig. 2) for a beam (ora plate of unit width) of length L and thickness h. In-plane displacements /2 are imposed at

both ends. In addition, the out-of-plane displacement of the plate (beam) is constrained to a

Figure 1: Experimental setup. A circular plate of radiusR is compressed between two rigidtransparent spherical dies of radius . Image taken from [1].

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Figure 2: Notation for a single wrinkle in the simplified 1D model.

maximum value . The amplitude A of the midplane of the plate is thus A = h. For largevalues ofa single wrinkle (as in Fig. 2) forms when the axial force exceeds Eulers criticalload.

(a) We will use an energy approach to analyse this problem. The total strain energy can bewritten as the following integral over the volume Vof the plate:

U=12

V

T dV =12

h/2

h/2

(xx+ y y+ z z+ xyxy+ xzxz+ yz yz) dxdydz.

Here, the barred quantities x, etc. stand for the strains and stresses at an arbitrary pointthrough the thickness, while the unbarred quantities denote the corresponding strains andstresses at the midsurface (i.e., at z= 0).

By employing the Kirchhoff hypothesis for thin plates (implying xz = 0, yz = 0, z = 0)and Eqs. (1), (2) and (3) in the lecture notes, and performing the z integration in theabove integral, show that the strain energy of the plate can be written as

U=Um+ Ub,

where

Um =C

2

2x+

2

y+ 2xy+1

2 2xy

dxdy

is the membrane strain energy and

Ub=D

2

2x+

2

y+ 2xy+ 2(1 )2

xy

dxdy

is the bending strain energy, CandD being the axial and bending rigidities of the plate.

(b) In our present case of a plate of unit width, with bending only in the x direction, theenergy expressions reduce to

Um=C

2

L/2L/2

2x dx, U b=D

2

L/2L/2

2w

x2

2dx, (1)

where we have written the relevant curvature as x = 2w/x2. By in-plane equilibrium

we have Nx/x = 0 and hence x/x = 0 (Eqs. (6) and (5) in the lecture notes), so xis constant along the plate. It can therefore be computed as

x= 1

L

L/2L/2

x dx= 1

L

L/2L/2

u

x+

1

2

w

x

2dx (2)

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(i.e., the average over the length of the plate), where we have used the nonlinear strain-displacement relations to allow for deflections w of the order of the thickness of the plate.

We approximate the deflection w of a (small-amplitude) single wrinkle by taking

w(x) =A

2[1 + cos(2x/)] ,

where is the wavelength (see Fig. 2). By using Eqs. (1) and (2) above, with boundaryconditions u(L/2) = one/2 and u(L/2) = one/2, show that for this single wrinklethe membrane and bending strain energies are given by

Um =C

2

2A2

42

one

2, Ub=

D4A2

3 ,

where one is the in-plane end displacement of a single wrinkle.

(c) We now consider a series ofn successive sinusoidal wrinkles as in (b) along the length Lof the plate. Deduce from the result under (b) that the total strain energy of a pattern

ofnwrinkles (so thatn= L and none= ) is given by

U=

n4

D4A2

L4 +

C

2

n22A2

4L2

L

2L.

The actual pattern formed will be such as to minimise this total energy. Thus, by min-imisingUfor an imposed amplitudeA with respect to the number of wrinkles, derive theestimate

n2 = 12L

32A2 + 82h2

and conclude that the number of wrinkles tends to a finite value

nmax 0.39L

h

L

as the amplitude A vanishes. Note that this result does not depend on the materialproperties of the plate.

(d) As an example we take data from experiments with polypropylene films reported in [1](see Fig.1). A strip of this film of length 250 mm and thickness 250 m is clamped ona rigid plate with an imposed displacement of 0.2 %. Compute the maximum number ofwrinkles formed as the resulting blister is stamped.

We can also get an estimate for the circular disc in Fig.1d by takingL = 2R. Computethe maximum number of radial wrinkles in this case for a disc radius R of 25 mm, athickness of 58 m and again an imposed displacement of 0.2 %.

References

[1] J. Hure, B. Roman, J. Bico, Stamping and wrinkling of elastic plates, Physical ReviewLetters109, 054302 (2012).

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Coursework1 - CEGEM090