Axial deformation - Hollow spheres

Publications on the mechanics of hollow thin spherical shells date back to the second half of the nineteenth century.^43,44 The first comprehensive treatment of the deformation of thin elastic shells was presented by Love in 1888 and has become a "classic" paper since.⁴⁵ For his shell theory he adopted another "classic" - Kirchhoff’s assumptions for thin plates.⁴⁶ The "Kirchhoff-Love assumptions" provide the framework for thin shell theory which will be the basic concept for the description of the physics of capsule deformation in this section (cf. fig. 2.3).

Indeed, the restriction to thin shells is sensible as many shell structures - be it in the microscopic world or in architecture - meet the criterion of thinness, i.e. a ratio of shell thickness h to radius R equal or less than 1/20.⁴⁷ Actually, herein lies the strength of shell constructions: mechanical stability and protection of the interior with a minimum of materials consumption. Beyond the requirement of thin shells the Kirchhoff-Love assumptions add the following constraints: the shell material is considered as homogeneous, isotropic and linear elastic (Hooke’s law); only small deformations (with regard to shell thickness) are allowed; compression of the shell as well as stretching or in-plane shear is neglected. In other words, the shell deforms in pure bending upon normal axial loading.

This simplification can be illustrated by the energy contributions of stretching and bending.² For a given deformation d of a shell with elastic modulus E the energy per unit area for bending (E_B) and stretching (E_S) can be expressed as:

E_B ∝Eh³( d

R²)² (2.12)

E_S ∝Eh(d

R)² (2.13)

For the ratio of these two follows:

E_S E_B ∝(R

h)² ≥_thinshell 400 (2.14)

This means that for a thin shell the energy required to stretch it is orders of magnitude higher than the energy necessary to bend it. In other words, upon external loading the shell will primarily deform in bending.

Figure 2.3. Schematic representation of basic requirements in thin shell theory.

Eric Reissner was one of the first scientists to present an analytical solution to the problem of shallow shell deformation.^48–52 His approach is based on the approximations

for thin shell theory as outlined above. For an axisymmetric shell under point load he finds a linear relation between force F and resulting deformation d:

F = h²

R · 4E

p3(1−ν²) ·d (2.15)

Here,ν is Poisson’s ratio. Despite its simplicity equation 2.15 has been shown to apply to a range of capsule systems, also for non-point-like loading,e.g. with an AFM colloidal probe (for details see 2.2).^53–57

However, the behavior of many relevant capsule systems deviates from the aforementioned simplifications. Additional contributions to the shell mechanics may originate from membrane pre-tensions. One of the first experimental and theoretical studies treating this issue examined Pickering emulsion droplets.⁵⁸ Here, a model is presented that includes both the contribution of surface tension and the mechanical tension of the shell. It is successfully applied to data from compression tests and comparison to results from continuum mechanics shows clearly the prominent role of the particle stabilized water-oil interface. A later work on a comparable system provided further evidence for the importance of surface tensions in shell mechanics.⁵⁹ In our paper on pH-responsive hydrogel capsules we also encountered a mechanical shell stiffening effect (as compared to substrate supported films) that may be explained by the occurrence of surface or membrane tension.⁶⁰

Another feature that is observed already for moderate deformations is the buckling instability of the shell. At a certain point the shell may respond to external pressure by an inward folding, thus minimizing its energy. Pogorelov proposed a force-deformation relation with a square root scaling to describe this phenomenon.⁶¹ Other researchers deduced from the onset of buckling information about the shell’s elastic properties,e.g.

with the help of osmotic pressure effects.^62–69

Finally, "large" deformations are generally expected to impose further deformation modes (such as stretching or shearing of the shell) in addition to bending, even in the absence of buckling. In the introduction, one of the Kirchhoff-Love assumptions was "small"

deformations with respect to the shell thickness. Typically, deformations are considered

"small" when they are on the order of one to two times the shell thickness. At this point it is helpful to consider again how an applied force scales with deformation. In the case of pure bending the force scales with relative deformation ε (deformation divided by capsule radius) like⁷⁰

F ∝ 4

3Eh²ε (2.16)

which is equivalent to Reissner’s equation 2.15. However, pure bending is only an approximation; in fact, there will always be (at least) one more deformation mode, namely the stretching of the shell. For the limiting case of an impermeable shell with volume conservation this additional term scales like⁷¹

F ∝ 16π

3 EhRε³. (2.17)

Figure 2.4 shows these two scaling laws calculated exemplarily for two microcapsules of 10 µm radius, an elastic modulus of 1 MPa and a shell thickness of 50 nm (500 nm, the upper limit for thin shells!). We see that up to a relative deformation of ca. 5‰(20‰) shell permeability or stretching is negligible. Theε value corresponds to a deformation of approximately (half) the thickness of the shell. So, within this range thin shell theory can be expected to hold. However, for larger deformations volume effects become increasingly important (cubic scaling!) and need to be incorporated in physical modeling.

Up to now there are only few works that systematically investigated the burst of microcapsules in consequence of mechanical compression. Being an industrial relevant system melamine-formaldehyde (MF) capsule were studied by Zhang and co-workers.^72,73 They found a linear correlation between capsule diameter and the force or deformation necessary to burst the capsules. Moreover, capsule burst was observed to occur for relative deformations of about 70%, independent of capsule size. A theoretical description of such

Figure 2.4. Bending and stretching contributions to the deformation of a microcapsule, 20 micron in diameter, shell thickness 500 nm and Young’s modulus 1 MPa;

inset shows the calculations for the same capsule with reduced shell thickness (50 nm).

large deformations leading to capsule burst has been proposed by Mercadé-Prieto et al.

with the help of finite element modeling (FEM).⁷⁴ Good agreement with experimental data on MF capsules is found treating the shell material as elastic-perfectly plastic with strain hardening. In a later work the deformation of thick-shelled PMMA capsules was studied experimentally and by FEM.⁷⁵

Further aspects concerning the theory of capsule deformation can be found in [76].