Jörg Hohe

A nonlinear theory for doubly curved anisotropic sandwich shells with transversely compressible core

Abstract In the present paper, an advanced geometrically nonlinear shell theory of doubly curved structural sandwich panels with transversely compressible core is presented. The model is based on the adoption of the Kirchhoff theory for the face sheets and a second/third order power series expansion for the core displacements. The theory accounts for dynamic effects as well as for initial geometric imperfections. In the v. Karman sense, large displacement theory is employed with respect to the transverse direction while the displacement gradients with respect to the tangential directions are assumed to be small. The equations of motion are derived by means of Hamilton’s principle and hold valid for all types of elastic and elastic–plastic material models. The theory is illustrated by an analysis of the elastic buckling and postbuckling behavior of flat and curved sandwich panels using an extended Galerkin scheme. Owing to the assumed transverse flexibility of the core, both the global and the local (face wrinkling) instability modes can be addressed.

Advances in the Structural Modeling of Elastic Sandwich Panels

The area of sandwich construction has been an active field of research for more than five decades. During this period, a large number of mathematical models for structural sandwich panels has been provided, ranging from the early sandwich membrane models to more sophisticated recent approaches. Especially during the past decade, the increasing demand for high-performance, lightweight structures has stimulated a strong trend toward the develoment of refined models for sandwich plates and shells. The main focus of the present paper is a survey of recent developments and contemporary trends in the modeling of the deformation and buckling behavior of sandwich shells. The development of refined sandwich models is illustrated by an approach recently proposed by the authors. The basic ideas and features of this model are outlined and compared to alternative recent approaches presented by other authors.

A refined analysis of the effective elasticity tensor for general cellular sandwich cores

Abstract The aim of the present study is the determination of the components of the effective elasticity tensor for two-dimensional cellular sandwich cores in consideration of core face sheet constraints. The microstructure is homogenized by means of a strain-energy based RVE concept assuming that strain states, which are equivalent on the macroscopic level, lead to equal strain energy in a representative volume element whether the real microstructure or the quasi-homogeneous “effective” medium is considered. The strain energy can be evaluated analytically if the cellular structure is decomposed into the individual cell wall elements, and assumptions are made for the displacement field of each cell wall. The displacement field of the core is approximated by a weighted superposition of the displacement field of the unconstrained core and an extension of the displacements of the face sheets into the core region. Since the approach is based on a kinematically admissible strain field in conjunction with the principle of minimum strain energy, the results provide rigorous Voigt type bounds for the effective normal and shear moduli. In general, a good agreement of the analytical results and the results of a finite element analysis is observed.

Effective elastic properties of triangular grid structures

An energetic homogenization procedure is proposed for the determination of effective elastic properties of grid structures such as sandwich cores. The method is based on the assumption of the same average distribution of specific strain energy in the real microstructure and the effective homogenized medium. For a representative volume element, the strain energy is evaluated comparatively by an analytical approach and by the finite element method. The effective elastic grid properties are derived by differentiation of the specific strain energy with respect to the macroscopic strain components. The procedure is applied successfully to the analysis of the effective elastic properties of triangular grid structures. The dependence of the effective properties on both, the grid geometry and the cell wall material is discussed in detail.

Effective elastic properties of hexagonal and quadrilateral grid structures

The energetic homogenization procedure for the determination of the effective elasticity tensor of grid structures as proposed in an earlier paper is extended to the analysis of general hexagonal and quadrilateral grid structures. The method is based on the assumption that for macroscopically equal deformation the same strain energy has to be stored in a representative volume element whether the given microstructure or the effective homogenized medium is considered. The components of the effective elasticity tensor are determined by partial derivation of the strain energy density with respect to the macroscopic strain components. The analysis is performed comparatively by an analytical approach and by means of the finite element method. Standard commercial sandwich core geometries are considered as well as highly non-orthotropic grid structures. The influence of the cell geometry on the effective properties is discussed in detail. Special interest is directed to negative Poissons ratios as well as to effectively non-orthotropic elastic behaviour.

An energetic homogenisation procedure for the elastic properties of general cellular sandwich cores

Abstract The present study provides a general procedure for the determination of the effective elastic properties of two-dimensional cellular sandwich cores with arbitrary cell topology and geometry. The scheme uses a strain energy-based representative volume element procedure assuming that macroscopically equivalent strain states have to cause the same strain energy in a representative volume element whether the real microstructure or the “effective” homogenised medium is considered. The strain energy can be evaluated either by analytical or pure numerical methods. Both approaches agree well in a number of examples considering different sandwich core geometries.

A direct homogenisation approach for determination of the stiffness matrix for microheterogeneous plates with application to sandwich panels

The present study is concerned with the numerical determination of the effective plate stiffness matrix for composite shells with microheterogeneous layers. Contrary to the standard two-step procedure, where the effective plate or shell stiffness matrix is derived by a projection of the effective elasticity tensor onto the plate or shell reference surface, a direct method is proposed. This method uses a strain energy based RVE-procedure which assumes mechanical equivalence of a representative volume element for the given microstructure and a corresponding plate element if the strain energy in both elements is equal, provided that the effective deformation is equal in an average sense. In parametric studies concerning sandwich plates with hexagonal honeycomb cores, it is observed that the direct method and the two-step procedure are equivalent for the determination of the in-plane and the transverse shear properties while a significant deviation of the results obtained by both methods is found in case of the effective bending properties.

A mechanical model for two-dimensional cellular sandwich cores with general geometry

The aim of the present study is the determination of the effective elastic properties of two-dimensional cellular materials using cellular sandwich cores as a material example. A strain energy-based homogenisation procedure is proposed which assumes equivalence of microstructure and effective medium if the strain energy stored in a representative volume element is equal whether the real microstructure or the effective medium is considered provided that the strain state in both cases is equal in the average sense. Within this scheme, the strain energy can be evaluated either analytically or numerically. Both approaches are found to agree well in two examples considering commercial sandwich core geometries.

Assessment of the delamination hazard of the core face sheet bond in structural sandwich panels

Delamination of the adhesive bond between face sheets and cellular core of structural sandwich panels is a major problem in sandwich construction. Due to incompatibilities in the modes of deformation associated with the face sheets and the cellular core, stress concentrations and singularities can occur even in absence of cracks. These stress concentrations are assumed to govern the onset of delamination. In the present study, a mesoscale concept for a first-order assessment of the delamination hazard induced by the incompatibility in the modes of deformation at the interface between core and face sheets is presented. The approach is based on a fourth order tensor which can easily be derived from the effective elasticity tensor for the cellular core. Due to the general formulation, the concept is applicable to all types of two dimensional cellular sandwich cores irrespectively of cell geometry and loading conditions. The approach is illustrated by an analysis of three examples concerning commercial sandwich core geometries as well as a more general non-orthotropic cellular structure.

Singular stress fields in cellular cores for structural sandwich panels

The present study is concerned with the singularity of the stress fields in the cell walls of hexagonal honeycomb cores used in sandwich construction. The stress singularity under consideration occurs at the intersections of the cell walls at the core face sheet interface due to incompatibilities in the modes of deformation associated with the unconstrained cellular core and the face sheets. The stress fields are investigated by means of a closed-form asymptotic analysis and numerically by means of the finite element method. The stress singularity at the intersection is found to be of the pure power-law type. The dependence of order and intensity of the singularity on core and face sheet geometry and material as well as on the loading conditions is investigated in detail. A brief discussion of possible fracture concepts closes the study.