Print version ISSN 0100-7386
J. Braz. Soc. Mech. Sci. vol.22 no.4 Rio de Janeiro 2000
Instability of a Square Sheet under Symmetric Biaxial Loading
An early experiment found that a square rubber sheet under symmetric biaxial loading may not remain square. This curious result has been one of the most instructive examples in finite elasticity. Here thermodynamic considerations are used to analyze this instability.
Keywords: Thermodynamic stability criterious, Mooney-Rivilin Material, bifurcation, availability function
An early experiment found that a square rubber sheet under symmetric biaxial loading may not remain square. This curious result has been one of the most instructive examples in finite elasticity. The instability of square rubber sheet under symmetric biaxial loading was first reported by Treloar in 1948 and since then it has been discussed and analyzed quite often in the literature (Kearsley (1986), Chen (1987), Ericksen (1991), Müller (1996)).
Although the subject is a mechanical problem in nature, unlike the previous works, we shall rely on thermodynamic considerations to establish a stability criterion with proper boundary conditions of a biaxially loaded square sheet for the analysis of instability.
We consider an incompressible isotropic elastic body of Mooney-Rivlin materials, whose free (strain) energy function y is given by
where I and II are the first and the second invariants of the Cauchy-Green strain tensor, and r is the mass density. Both a and b are material constants, which according to experimental results for rubber satisfy the following inequalities:
If b = 0, the material is called Neo-Hookean.
We shall consider biaxial loading of a rubber sheet laying in the x-y plane, given by a time-dependent homogeneous deformation which takes a material point at X = (X,Y,Z) to a point at x = (x,y,z) with
For this deformation, the invariants of the Cauchy-Green strain tensors are given by
We shall first establish a criterion for thermodynamic stability. For a body in a region n at a uniform constant temperature q and free of external supplies, we have the energy equation,
and the entropy inequality,
where T is the Cauchy stress tensor, e is the internal energy and h is the entropy density. By eliminating the heat flux q between (5) and (6), we obtain
where y = e - qh is the free energy density. Let the region occupied by the body in the reference state be denoted by nk then the above condition can be written in the reference state as
where Tk is the Piola-Kirchhoff stress tensor.
Let the region nk occupied by the square sheet in the reference state be given by 0 £ X £ 1, 0 £ Y £ 1, and 0 £ Z £ D. The sheet is uniformly loaded on the lateral surfaces by the forces per unit area F1 and F2 in the X and Y directions respectively and is stress-free on the top and the bottom surfaces, i.e.,
where ex and ey are the unit base vectors of the coordinate system. Moreover from (3), we have
Therefore, by the boundary conditions (8) and (9), it follows that
where A1 e A2 are the lateral surfaces of the region at X = 1 and Y =1 respectively.
Now, considering the deformation process under prescribed biaxial loading, and assuming the process is quasi-static (with negligible acceleration), we have from (7) that
which upon integration gives
since the process is homogenous. Therefore if we define
the relation (10) becomes
We call A(t) the availability function of the square sheet.
We call a deformed state under a prescribed biaxial loading characterized by the stretches (l1, l2) a stable equilibrium state if any small perturbation from this state will eventually return to this state as time tends to infinity. Suppose that such a perturbation is represented by a process (l1 (t), l2 (t)), then is stable if
Since the availability A(t) is a decreasing function of time by the condition (12), it must have a local minimum at . Therefore, by regarding A as a function of (l1, l2), this criterion is equivalent to the following conditions:
and the Hessian matrix of A is positive semi-definite, or equivalently,
where E denotes the evaluation at the stable equilibrium state .
By the use of the free energy given in (1) and (4), the expression (11) leads to the following equilibrium conditions from (13):
where the equations are evaluated at the equilibrium state and the overhead bars are suppressed for simplicity.
In the case of symmetric loading, F1 = F2, from (15) we obtain
where h = b / a is a non-negative material constant from the empirical inequalities (2). This immediately gives the symmetric solution, l1 = l2. Since l1, l2.and h are positive quantities, no other solution exists if hl1l 2, < 1, which rules out the possibility of an asymmetric solution for Neo-Hookean materials (h=0).
The asymmetric solution may exist and can be found from the equation,
For such a solution, l1 and l2 are different in general, and the square sheet becomes rectangular after stretching.
Furthermore, the conditions (14) lead to
which is identically satisfied, and
Let the left hand side of the relation (19) be denoted by f (l1,l2), then we have
which is the condition for an equilibrium state( l1,l2 )to be stable.
We have plotted the function f (l , l) against l for the symmetric solution in Fig. 1 for h=0.1. It shows that for l £ lB = 3.1685. the function f (l,l ) is non-negative and therefore according to the condition (20) the symmetric solution is stable. However, for l > lB the function f (l,l) becomes negative and hence the square sheet is not longer stable under symmetric loading.
For asymmetric solution l1 ¹ l2 under symmetric loading, from the condition (17) one can solve for l2 in terms of l1 so that l2 = g(l1) and hence f (l1, g (l1 )) becomes a function of l1 only. Doing this numerically, we can easily verify the condition (20) by plotting the function f (l1, l2) against l1 as shown in Fig (2), from which we conclude that f (l1l 2) is non-negative and hence the asymmetric solution is always stable.
The author (ISL) acknowledges the financial support from the research fellowship of National council of Scientific and Technological Development -- CNPq of Brazil.
Chen, Y. C., 1987, "Stability of homogeneous deformations of an incompressible elastic body under dead-load surface tractions", J. Elasticity, 17, 223-248. [ Links ]
Ericksen, J. L., 1991, Introduction to Thermodynamics of Solids, Chapman & Hall, London. [ Links ]
Müller, I., 1996, "Two instructive instabilities in non-linear elasticity: Biaxially loaded membrane and rubber balloons", Mecanica, 31, 386-395. [ Links ]
Kearsley, E.A., 1986, "Asymmetric stretching of a symmetrically loaded elastic sheet", Int. J. Solids Structures, 22, 111-119. [ Links ]
Treloar, L.R.G., 1948, "Stresses and birefringence in rubber subjected to general homogeneous strain". Proc. Phys. Soc., 60, 135-144. [ Links ]
Manuscript received: May 1999. Technical Editor: Angela Ourívio Nieckele.