Herbert A. Mang

A new method for the coupling of finite element and boundary element discretized subdomains of elastic bodies

Abstract A new and efficient approach for the coupling of subregions of elastic solids discretized by means of finite elements (FE) and boundary elements (BE), respectively, is presented. The method is characterized by so-called ‘bi-condensation’ of nodal degrees of freedom followed by the transformation of the resulting BEM-related traction-displacement equations for the interface(s) of the BE subregion(s) and the FE subdomain(s) to ‘FEM-like’ force-displacement relations which are assembled with the FEM-related force-displacement equations for the interface(s). The presented ‘local FE coupling approach’ is computationally more economic than a global coupling approach since it only requires the inversion of BEM-related coefficient matrices referred to the interfaces of BE subregions and FE subdomains. Depending on whether the principle of virtual displacements or the principle of minimum of potential energy is used for the generation of force-displacement equations for the coupling interface(s), unsymmetric or symmetric coefficient matrices are obtained. Since the two principles are mechanically equivalent, identical results would be achieved in the limit of finite discretizations. The numerical investigation has shown that, depending on the problem and the discretization, the results obtained on the basis of symmetric coefficient matrices may be poor. This applies to ‘edge problems’ characterized by discontinuous tractions along the edges. On the basis of unsymmetric coefficient matrices, however, satisfactory results are obtained even for relatively coarse discretizations.

A Finite Element Shell Analysis of Guard Cell Deformations

ABSTRACT IN this paper the width of the stomatal aperture, as postulated by von Mohl in 1856, is shown to be a function of the hydrostatic (turgor) pressure in the guard cells, Pg, and the pressure, Ps, of the immediately surrounding epidermal cells, which will be referred to as the subsidiary cells in this paper. The aperture does not depend solely upon the pressure difference (Pg-Ps) as believed by Ursprung and Blum (1924) and StSlfelt (1966). Instead, aperture width is shown to be a simple multi-linear relationship (i.e., a linear com-bination) of Pg and Ps. The recent re-search by Glinka (1971) and Edwards, Meidner and Sheriff (1976), showing the relative contributions of the oppos-ing pressures Pg and Ps, is, thus, given a simple and lucid interpreta-tion. The analysis of a guard cell as an elliptical torus shows that a stomate could function without either of the two conditions classically believed to be essential (Meyer et al. 1960, p. 84, Meidner and Mansfield 1968, pp. 14-17, Bidwell 1974, p. 298). The thick-ened wall (ventral wall) of the guard cell facing the aperture need not neces-sarily be stiffer than the dorsal wall common to the adjacent epidermal cell for the proper functioning of a stomate. The radially oriented cellu-lose microfibrils in the guard cell wall are not vital but are important for quite different reasons than claimed by Aylor et al. (1973). Consideration of the radial stiffening by means of the in-troduction of a mechanically equiva-lent orthogonally anisotropic (i.e., or-thotropic) material causes the aperture width to be more sensitive to a unit increment in Ps than to a unit incre-ment in Pg (for parameters of physi-cal interest3). The guard cell volume, however, is smaller than the adja-cent cell volume and Pg is believed to be larger than Ps, in general. We conjecture that this increased sensi-tivity for the subsidiary cell (i.e., clos-ing) pressure is important for the functioning of the feedback control loops regulating the aperture width. We define an antagonism ratio to characterize this property. The pore length in the model is shown to be surprisingly constant during opening, as is reported for many species (Meidner and Mansfield 1968, p. 12). The guard cell is generally believed to bulge into the neighboring epider-mal cell upon opening (Meidner and Mansfield 1968, p. 15). However, the shell model suggests that the outer-most portion of the guard cell at the widest point (and not visible in an in vivo situation) actually moves away from the neighboring cell. The ap-proximate point at which the exposed surface of the epidermal cell joins the guard cell exhibits only limited mo-tion. Note that there are especially thin regions here in the epidermal cell thought to behave as hinges (Hautgelenke). Even when the ex-treme, unphysiological case of a fixed aperture length is imposed, the outermost perimeter moves away from the adjacent cell. If the epidermis is opaque, the view from outside the leaf suggests that the guard cell bulges into the neighboring cell, as claimed in the classical hypothesis, provided the stiffening effect of the micellae is sufficiently prominent and provided the guard cell pressure is significantly larger than the epidermal cell pres-sure. The opposing influence of the turgor pressure in the guard cells and in the adjacent epidermal cells is shown to be an inherent part of the stomatal mechanism (von Mohl 1856). Pressure influence coefficients for the guard cell are defined and related to parameter changes, e. g., material and thickness. The multilinear relationship of aperture width to the opposing turgor pressures was found and revealed that pore width does not depend solely on the pressure difference between the guard cell and the adjacent epidermal cell. Finally, the theory developed is shown to embrace and to clarify the experimental results of Glinkas plas-molytic study (1971) and the direct method of Edwards et al. (1976).

Wind-loaded reinforced-concrete cooling towers: buckling or ultimate load?

Abstract Wind loadings govern the design of most cooling towers. Until now, proof of suffiecient safety against buckling under wind load has been a major concern for the designers of such shells. In this paper it is demonstrated that a typical cooling tower made of reinforced concrete would not buckle—at least not in the classical sense of the word. Failure would rather be initiated by rapid propagation of cracks in tensile zones followed by temporary stiffening and, finally, by yielding of the reinforcement. The theoretical part of this paper is restricted to a presentation of the constitutive model, discussion of the equation for incremental-iterative ultimate-load analysis and of the condition for instability. The numerical part contains a detailed study of a built hyperbolic cooling tower. It is shown that: (a), buckling loads resulting from linear and geometrically nonlinear prebuckling analyses are considerably larger than the ultimate load; and (b), results based on a certain form of ‘equivalent axisymmetric pressure’ are on the unsafe side of corresponding results from the ‘actual’ wind load. It is also demonstrated that the ‘crack load’, representing a lower bound to the ultimate load, can be estimated by means of a linear-elastic nonaxisymmetric analysis of the cooling tower.

Chemoplastic material model for the simulation of early-age cracking: From the constitutive law to numerical analyses of massive concrete structures

This paper deals with the development and application of a three-dimensional material model for the simulation of early-age cracking of concrete. The starting point is the determination of the intrinsic material function for the fracture energy of early-age concrete. For this purpose, results of beam bending tests reported in [Proceedings of the SEM/RILEM International Conference on Fracture of Concrete and Rock. Houston, Texas, USA: 1987. p. 409] are employed. The intrinsic material function serves as input for the calibration of the Rankine fracture criterion formulated in the framework of chemoplasticity. Finally, the developed 3D material model is employed for a chemomechanical analysis of a roller-compacted-concrete dam. The temperature fields and the field of the degree of hydration required for this analysis are obtained from a preceding thermochemical analysis of the dam.

Hybrid method for quantification of stress states in shotcrete tunnel shells: combination of 3D in situ displacement measurements and thermochemoplastic material law

Abstract A hybrid method for quantification of the loading of shotcrete tunnel shells, combining thermochemomechanical material modeling of shotcrete with 3D in situ displacement measurements in the framework of the non-linear finite element method, is presented. Histories of displacement fields, determined from in situ measurements by suitable interpolation functions, are prescribed on the outer boundary of a part of the tunnel, discretized in 2D or 3D. The main goal is the determination of fields of safety degrees, amounting to 0% for the unloaded shell and to 100% for the material loaded up (locally) to the compressive strength. A comprehensive investigation of the significance of the individual material characteristics of shotcrete as well as that of the third dimension in space on the structural behavior is performed. The Sieberg tunnel in Lower Austria, constituting a part of the high capacity railway line between Vienna and Salzburg, is used as the vehicle for this investigation. This tunnel was very recently completed.

A new 3-D finite element model for cord-reinforced rubber composites: application to analysis of automobile tires

Abstract A 3-D finite element model for cord-reinforced rubber composites is proposed. It is characterized by superimposing so-called “rebar” elements, consisting of one or more reinforcing cord layers with arbitrary orientation, on corresponding 3-D rubber elements. The rubber material and the different cord materials are represented independently. A Lagrange multiplier method is employed for the large strain analysis of rubber which is modelled by the incompressible Mooney material law. The compressible Neo-Hookean material law is adopted for the cords (rebar elements). It is modified for the special case of uniaxial stress states. The proposed method provides a realistic representation of cord-reinforced rubber composites while minimizing the necessary discretization effort. It is applied to 3-D finite element analysis of an automobile tire, involving determination of the pressure distribution in the contact zone and of the radial load-displacement curve. Very good agreement between analysis predictions and experimental results has been obtained.

A Galerkin-type BE-FE formulation for elasto-acoustic coupling

The interactions between the vibrations of an elastic structure and the sound field in the surrounding fluid are taken into account by coupling a symmetric Galerkin formulation of the Boundary Element Method for acoustic radiation and scattering with a standard finite element formulation for the dynamic behavior of elastic structures.

3D finite element analysis of rubber‐like materials at finite strains

The paper starts with a review of constitutive equations for rubber‐like materials, formulated in the invariants of the right Cauchy—Green deformation tensor. A general framework for the derivation of the stress tensor and the tangent moduli for invariant‐based models, for both the reference and the current configuration, is presented. The free energy of incompressible rubber‐like materials is extended to a compressible formulation by adding the volumetric part of the free energy. In order to overcome numerical problems encountered with displacement‐based finite element formulations for nearly incompressible materials, three‐dimensional finite elements, based on a penalty‐type formulation, are proposed. They are characterized by applying reduced integration to the volumetric parts of the tangent stiffness matrix and the pressure‐related parts of the internal force vector only. Moreover, hybrid finite elements are proposed. They are based on a three‐field variational principle, characterized by treating th...

Meshless LBIE formulations for simply supported and clamped plates under dynamic load

Simply supported and clamped thin elastic plates under dynamic loads are analyzed. Both harmonic and impact loads are considered. Viscous damping is taken into account. The governing partial differential equation (PDE) of fourth order is decomposed into two coupled PDEs of second order for the deflection and its Laplacian. Both equations contain time-dependent variables. The Laplace transform is used to eliminate the time dependence of the variables. Unknown Laplace transforms are computed from the local boundary integral equations. The meshless approximation based on the moving least square method is employed for the implementation. Time-dependent values are obtained by the Durbin inversion technique.

Meshless local boundary integral equation method for simply supported and clamped plates resting on elastic foundation

Abstract Simply supported and clamped thin elastic plates resting on a two-parameter foundation are analyzed in the paper. The governing partial differential equation of fourth order for a plate is decomposed into two coupled partial differential equations of second order. One of them is Poisson’s equation whereas the other one is Helmholtz’s equation. The local boundary integral equation method is used with meshless approximation for both the Poisson and the Helmholtz equation. The moving least square method is employed as the meshless approximation. Independent of the boundary conditions fictitious nodal unknowns used for the approximation of bending moments and deflections are always coupled in the resulting system of algebraic equations. The Winkler foundation model follows from the Pasternak model if the second parameter is equal to zero. Numerical results for a square plate with simply and/or clamped edges are presented to prove the efficiency of the proposed formulation.