Numerical solution methods for viscoelastic orthotropic materials

Numerical solution methods for viscoelastic orthotropic materials, specifically fiber reinforced composite materials, are examined. The methods include classical lamination theory using time increments, direction solution of the Volterra Integral, Zienkiewicz's linear Prony series method, and a new method called Nonlinear Differential Equation Method (NDEM) which uses a nonlinear Prony series. The criteria used for comparison of the various methods include the stability of the solution technique, time step size stability, computer solution time length, and computer memory storage. The Volterra Integral allowed the implementation of higher order solution techniques but had difficulties solving singular and weakly singular compliance function. The Zienkiewicz solution technique, which requires the viscoelastic response to be modeled by a Prony series, works well for linear viscoelastic isotropic materials and small time steps. The new method, NDEM, uses a modified Prony series which allows nonlinear stress effects to be included and can be used with orthotropic nonlinear viscoelastic materials. The NDEM technique is shown to be accurate and stable for both linear and nonlinear conditions with minimal computer time

Stability analysis of second- and fourth-order finite-difference modelling of wave propagation in orthotropic media

The stability of the finite-difference approximation of elastic wave propagation in orthotropic homogeneous media in the three-dimensional case is discussed. The model applies second- and fourth-order finite-difference approaches with staggered grid and stress-free boundary conditions in the space domain and second-order finite-difference approach in the time domain. The numerical integration of the wave equation by central differences is conditionally stable and the corresponding stability criterion for the time domain discretisation has been deduced as a function of the material properties and the geometrical discretization. The problem is discussed by applying the method of VonNeumann. Solutions and the calculation of the critical time steps is presented for orthotropic material in both the second- and fourth-order case. The criterion is verified for the special case of isotropy and results in the well-known formula from the literature. In the case of orthotropy the method was verified by long time simulations and by calculating the total energy of the system

Collapse of orthotropic spherical shells

We report on the buckling and subsequent collapse of orthotropic elastic spherical shells under volume and pressure control. Going far beyond what is known for isotropic shells, a rich morphological phase space with three distinct regimes emerges upon variation of shell slenderness and degree of orthotropy. Our extensive numerical simulations are in agreement with experiments using fabricated polymer shells. The shell buckling pathways and corresponding strain energy evolution are shown to depend strongly on material orthotropy. We find surprisingly robust orthotropic structures with strong similarities to stomatocytes and tricolpate pollen grains, suggesting that the shape of several of Nature's collapsed shells could be understood from the viewpoint of material orthotropy.Comment: 7 pages, 5 figure

Extension-twist coupled laminates for aero-elastic compliant blade design

A definite list of laminate configurations with extension-twisting (and shearing-bending) coupling is derived for up to 21 plies of identical thickness. The list comprises individual stacking sequences, containing standard angle-ply and cross-ply sub-sequences; combinations which are contrary to the previously assumed form for this class of laminate. The list also contains dimensionless parameters from which the extensional, coupling and bending stiffness terms are readily calculated for any fiber/matrix system. Lamination parameters are shown graphically to illustrate the extent of the design space with up to 21 plies. A special sub-group from this class of coupled laminate is identified that can be manufactured flat under a standard elevated temperature curing process; this sub-group possesses hygro-thermally curvature-stable behavior. Finally, bounds on the compression buckling strength are assessed using a closed form solution for all the laminate groups presented

Disappearance of stretch-induced wrinkles of thin sheets: a study of orthotropic films

A recent paper (Healey et al., J. Nonlin. Sci., 2013, 23:777-805.) predicted the disappearance of the stretch-induced wrinkled pattern of thin, clamped, elastic sheets by numerical simulation of the F\"oppl-von K\'arm\'an equations extended to the finite in-plane strain regime. It has also been revealed that for some aspect ratios of the rectangular domain wrinkles do not occur at all regardless of the applied extension. To verify these predictions we carried out experiments on thin 20 micrometer thick adhesive covered), previously prestressed elastomer sheets with different aspect ratios under displacement controlled pull tests. On one hand the the adjustment of the material properties during prestressing is highly advantageous as in targeted strain regime the film becomes substantially linearly elastic (which is far not the case without prestress). On the other hand a significant, non-ignorable orthotropy develops during this first extension. To enable quantitative comparisons we abandoned the assumption about material isotropy inherent in the original model and derived the governing equations for an orthotropic medium. In this way we found good agreement between numerical simulations and experimental data. Analysis of the negativity of the second Piola-Kirchhoff stress tensor revealed that the critical stretch for a bifurcation point at which the wrinkles disappear must be finite for any aspect ratio. On the contrary there is no such a bound for the aspect ratio as a bifurcation parameter. Physically this manifests as complicated wrinkled patterns with more than one highly wrinkled zones on the surface in case of elongated rectangles. These arrangements have been found both numerically and experimentally. These findings also support the new, finite strain model, since the F\"oppl-von K\'arm\'an equations based on infinitesimal strains do not exhibit such a behavior.Comment: 16 pages, 5 figure

Fracture parameters formulation for single edge notched AS4 stitched warp-knit fabric composite plate

The three-dimensional problem of the fracture for the single edge notched tension plate (SENT) of orthotropic material is considered in this paper. The finite element solution is used to evaluate the singular and non-singular terms of the William series, i.e. Stress intensity factor (SIF) and T-stresses namely T11, T13 and T33. Based on the obtained numerical results, a fitting procedure is performed in order to propose analytical formulations giving the fracture parameters near the crack tip. The obtained results are in good agreement with the finite elements calculation and other literature results

Fracture parameters formulation for single edge notched AS4 stitched warp-knit fabric composite plate

The three-dimensional problem of the fracture for the single edge notched tension plate (SENT) of orthotropic material is considered in this paper. The finite element solution is used to evaluate the singular and non-singular terms of the William series, i.e. Stress intensity factor (SIF) and T-stresses namely T11, T13 and T33. Based on the obtained numerical results, a fitting procedure is performed in order to propose analytical formulations giving the fracture parameters near the crack tip. The obtained results are in good agreement with the finite elements calculation and other literature results

Fracture parameters formulation for single edge notched AS4 stitched warp-knit fabric composite plate

The three-dimensional problem of the fracture for the single edge notched tension plate (SENT) of orthotropic material is considered in this paper. The finite element solution is used to evaluate the singular and non-singular terms of the William series, i.e. Stress intensity factor (SIF) and T-stresses namely T11, T13 and T33. Based on the obtained numerical results, a fitting procedure is performed in order to propose analytical formulations giving the fracture parameters near the crack tip. The obtained results are in good agreement with the finite elements calculation and other literature results

Multistability of free spontaneously-curved anisotropic strips

Multistable structures are objects with more than one stable conformation, exemplified by the simple switch. Continuum versions are often elastic composite plates or shells, such as the common measuring tape or the slap bracelet, both of which exhibit two stable configurations: rolled and unrolled. Here we consider the energy landscape of a general class of multistable anisotropic strips with spontaneous Gaussian curvature. We show that while strips with non-zero Gaussian curvature can be bistable, strips with positive spontaneous curvature are always bistable, independent of the elastic moduli, strips of spontaneous negative curvature are bistable only in the presence of spontaneous twist and when certain conditions on the relative stiffness of the strip in tension and shear are satisfied. Furthermore, anisotropic strips can become tristable when their bending rigidity is small. Our study complements and extends the theory of multistability in anisotropic shells and suggests new design criteria for these structures.Comment: 20 pages, 10 figure

Multiparameter actuation of a neutrally-stable shell: a flexible gear-less motor

We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multiparameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disk, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic deformation, in a perfectly isotropic system, the shell is theoretically neutrally stable, owning a continuous manifold of stable cylindrical shapes corresponding to the rotation of the axis of maximal curvature. Small imperfections render the actual structure bistable, giving preferred orientations. A three-parameter piezoelectric actuation, exerted through micro-fiber-composite actuators, allows us to add a small perturbation to the plastic inelastic curvature and to control the direction of maximal curvature. This actuation law is designed through a geometrical analogy based on a fully non-linear inextensible uniform-curvature shell model. We report on the fabrication, identification, and experimental testing of a prototype and demonstrate the effectiveness of the piezoelectric actuators in controlling its shape. The resulting motion is an apparent rotation of the shell, controlled by the voltages as in a "gear-less motor", which is, in reality, a precession of the axis of principal curvature.Comment: 20 pages, 9 figure