Hyperelastic Membrane Model

A rubberlike material can be used to cover the rigid skeleton of a MAV design to obtain flexibility of the wing (see Fig. 4.6). The large deformations observed for this kind of material in the Reynolds number range of operation indicate that the linear

elasticity assumption may not be valid. To address this issue, a hyperelastic model to describe the 3D membrane material behavior is employed [162]. The stress-strain curve of a hyperelastic material is non-linear, but follows the same path in loading and unloading (below the plastic limit, which is significantly higher than in metals). Compared with the previously discussed 2D linear model, a 3D membrane model introduces several complicated factors. First, for three-dimensional membranes, the tension is defined as a biaxial tension along the lines of principal stress [436]. Second, the geometric and material properties may vary along the spanwise direction and need to be described in detail. A third factor is membrane compression, leading to wrinkles when one of the principal tensions vanishes. In addition, it is desirable to account for the membrane mass when solving for the dynamic equations of the membrane movement.

A finite element analysis of the static equilibrium of an inflated membrane undergoing large deformations is presented by Oden and Sato [437]. A review of the earlier work on the dynamic analysis of membranes can be found in Jenkins and Leonard [438]. An update of the state-of-the-art models in membrane dynamics is presented by Jenkins [439]. Verron et al. [440] studied, both numerically and experimentally, the dynamic inflation of a rubberlike membrane. Ding et al. [441] numerically studied partially wrinkled membranes.

In a recent effort, Stanford et al. [442] proposed an accurate linear model for 3D membranes used in MAV design. Their experimental measurements showed that the maximum strain value is quite small; therefore they constructed a linear approx­imation of the stress-strain curve, centered about the membrane wing’s prestrain value. The linear constitutive equation used for membrane modeling is Poisson’s equation:

dW d 2w P(X’У) (4—21)

dx2 + dy2 t, ()

where Wis the out-of-plane membrane displacement, p(x, y) is the applied pressure (wing loading, in this case), and T is the membrane tension per unit length. The aerodynamic loads are computed on a rigid wing and fed into the structural model, assuming that the change in shape of the membrane wing did not overtly redistribute the pressure field. They obtained good agreement between the experimental data and computations.

A 3D membrane model was developed by Lian et al. [163]. The model gives good results for membrane dynamics with large deformations, but has limited capability to handle the wrinkle phenomenon that occurs when the membrane is compressed. The membrane material considered obeys the hyperelastic Mooney-Rivlin model [443]. A brief review of their membrane model is given next.

The Mooney-Rivlin model is one of the most frequently employed hyperelas­tic models because of its mathematical simplicity and relatively good accuracy for reasonably large strains (less than 150 percent) [443]. For an initially isotropic mem­brane, Green and Adkins [444] defined a strain energy function, W, as

where Ib I2, and I3 are the first, second, and third invariants of the Green deformation tensor, respectively. More details about the model, its validation, and its numerical implementation can be found in the literature (e. g., [154] [160] [440]).

4.2.1 Flat Plate and Shell Models

A flat 3D wing can be modeled as a plate that allows for spanwise and chordwise bending and twist. For a thin isotropic plate, the small transverse displacement is governed by the classical plate equation,

(4-23)

where W is transverse displacement, ps the density, hs the thickness, E the modulus of elasticity, v the Poisson’s ratio of the wing, and /ext the distributed external force per unit length acting in the vertical direction.

The most general structural class, which includes the previously introduced model, is the shell model. A thin shell is modeled as a curved 2D structure with the in-plane and out-of-plane displacements coupled via its curvature. A shell finite element (FE) can carry bending and membrane forces. With a thin shell FE model, Nakata and Liu [445] investigated the non-linear dynamic response of anisotropic flexible hawkmoth wings flapping in air, and Chimakurthi, Cesnik, and Stanford [446] simulated flapping plate/shell-like wing structures undergoing small strains and large displacements/rotations.

More research is needed to further refine the structural model for the anisotropic, batten-enforced mechanical properties of the highly flexible structures such as wings and to better understand the dynamics of the resulting fluid-structure interactions.