Shell theory

Bamboo stems

Body as a monocoque

The shell theory deals with the properties and the modeling of shells . Shells are rigid shell structures in which two dimensions compared to the third are large and - in contrast to the flat surface structures disk and disk  may already be unloaded curved in all three dimensions -. The pictures on the right show two typical examples of shell-like structures: the hollow stalks of sweet grass, as shown here with bamboo, and a body . In contrast , limp , thin-walled structures such as flight skins or balloon skins are called membranes .

The origin of the shell theory lies in the 19th century, i.e. in a time when the modern calculation possibilities did not yet exist. Today shell models are designed primarily for the finite element method (FEM). Because of the low bending stiffness of the shells compared to their tensile stiffness in the direction of the thickness, the FEM equation systems are poorly conditioned without shell models and extended, thin-walled structures can only be calculated with the help of shell models. The efficiency of the shell models is essentially a consequence of the introduction of special kinematic assumptions for the cross-sectional deformation of the structure, which allow the three-dimensional continuum to be reduced to a two-dimensional model.

Shells make optimum use of the load-bearing capacity of their material by transferring loads via membrane forces that are constant across the thickness of the shell. This results in the high rigidity of the shells combined with low weight and use of materials, which give them an important role in nature and technology. In nature, shells are found in bird egg shells , the exoskeletons of arthropods (insects, spiders, crabs), bones ( skulls , shoulder blades ) or stalks of grasses . In technology, shells can be found in silos or domed structures , bodies or beams .

Historical summary

The optimal behavior of curved structures was already known to the ancient builders. Numerous preserved viaducts and road bridges, but above all the Pantheon in Rome , are impressive witnesses of this knowledge even today. Attempts to computationally determine the load-bearing behavior of shells were made more than five hundred years ago. Above all, considerations about the flow of forces and the resulting optimal shape of the arching of masonry domes were important. Edoardo Benvenuto and Karl-Eugen Kurrer gave a detailed overview of the historical development of the calculation of arches and shells, which mainly took place in connection with masonry domes .

In the course of the nineteenth century, considerations about the deformation and load-bearing behavior of shells shifted from clear, mechanical considerations and methods of graphic statics to more mathematically oriented, analytical theories. The first attempts to establish a bending theory for the shells were not about the load-bearing behavior of domes, but rather about determining the natural frequencies of bells in order to be able to analytically determine their optimal shape for the desired sound. August EH Love (1888) wrote in his famous essay: “This paper is really an attempt to construct a theory of the vibrations of bells” (in German: This essay is actually an attempt to build up a theory of the vibrations of bells). Refinements to the theory came from Mindlin and Reissner. These theories were still limited to linear elastic media and small deformations.

The brothers Eugène and François Cosserat (1909) introduced the directed continuum by Duhem (1893) into the shell theory, which Naghdi (1972) developed into the forerunner of the so-called geometrically exact shell theory .

With the triumphant advance of FEM since the 1970s, the reduction of the three-dimensional continuum to a two-dimensional one in shell models (by means of integration over the thickness of the shell) is carried out numerically, which allows the consideration of large deformations and any material models.

Markings on the shell

Markings on the shell

The figure on the right shows a section of a bowl (yellow). The shell body is bounded by the shell top (upper reveal surface) upwards and with the shell underside (lower reveal surface) downwards. The middle surface of the bowl (gray) is located in the middle between the top and bottom of the bowl. The distance between the top and bottom of the shell is the shell thickness, which is halved by the middle surface. The thickness direction runs from the shell bottom to the shell top and the tangential direction runs parallel to the shell center surface. Parts of the shell can be cut out with cut surfaces.

Numerics

Typical shell structure

The following problem arises with the numerical solution of initial boundary value problems with the FEM: The tensile stiffness in the thickness direction of a linear elastic, isotropic, square shell with edge length l and thickness h as in the figure is proportional to E l ² / h , if E is the modulus of elasticity of the shell material is. With decreasing shell thickness h, the tensile stiffness in the thickness direction is significantly greater than the bending stiffness, which is proportional to E lh ³ / l ³. The ratio between the tensile stiffness in the direction of the thickness and the flexural stiffness is therefore proportional to ( l / h ) ⁴ , whereby values ​​of l / h > 1000 are not uncommon. When these stiffnesses are added in the computer, the bending stiffness is shifted to the end of the computer word of limited length, which can result in poor conditioning of the resulting system of equations and, with h → 0, even a singular stiffness matrix. By suppressing changes in the shell thickness during the deformation, the tensile stiffness in the tangential direction of the shell, which is proportional to E hl / l , becomes the greatest shell stiffness, which is why the ratio between the greatest and smallest stiffness decreases to ( l / h ) ².

By suppressing distortions in the thickness direction, the maximum permissible time step size for explicit dynamic analyzes according to the Courant-Friedrichs-Lewy condition can be increased from h / c to l / c - often by a factor of ten or more - where c is the Is the speed of wave propagation in the physical structure.

Shell models

Shell kinematics with shell center surface (blue) and a deviation in thickness direction (red) ${\displaystyle {\vec {X}}^{m}}$

The efficiency of the shell theory is essentially a consequence of the introduction of special kinematic assumptions for the cross-sectional deformation of the structure. All shell models have in common that the position vector of a point within the shell is divided into a component that points to the shell center surface, blue in the figure on the right, and a deviation in the thickness direction (red), which is described by directors . Unlike the normal vectors of surfaces , the directors are generally not perpendicular to the shell median surface .

Classification

The shell models can be divided into the following groups:

1. classical shell theory or shell theory based on the degeneration concept,
2. rigid Kirchhoff and flexible shells,
3. Shell models with or without change in thickness during deformation,
4. Higher order shell theories or multi-director theory as well
5. Shell theories of a conventional or a Cosserat continuum.

Classical shell theory and degeneration concept

The two mainstreams of shell theory are classical shell theory and the concept of degeneration. The kinematic assumptions underlying the two concepts are essentially the same. In the classical shell theory, the integration of the stresses over the thickness to the stress resultant is done analytically in advance, which is possible with justifiable effort with linear elastic behavior, small deformations and in flat (disks and plates) and rotationally symmetrical (tubes, spheres and hyperboloids) structures. As a result of the momentum and angular momentum balance, the stress results must satisfy partial differential equations. In the degeneration concept , the integration is carried out numerically via the shell thickness in the specific calculation case. N. Büchter proved in 1992 that shell elements of the two approaches lead to the same results.

Rigid and flexible shells

In the rigid Kirchhoff shell, the directors, which are perpendicular to the shell center surface in the initial state, also remain perpendicular to the shell center surface during the deformation, which corresponds to Bernoulli's assumption for the beams. This assumption is justified in the case of very thin structures, but also there increasingly provides incorrect results for the higher natural frequencies. The extension of the theory to shear-soft shells, where directors can change their angles to the shell median surface, is in good agreement with experimental facts.

Higher order theory and multi-director theory

Approximation of the cross-sectional warping with the p- and h-method

The higher-order shell theories and the multi-director theory differ in the structure of the stress resultant. In the higher-order theory, the deformed shell cross-section represents a kink-free, curved surface. These theories can be derived from a Taylor series expansion of the deformation in the direction of the thickness and lead to hierarchical models that attempt to achieve better solution behavior through higher-value approach functions. In FEM this is called the p-method for improving the approximation, see figure on the right-hand side of the image. The classical shell theories of Kirchhoff-Love and Reissner then represent a theory of the zeroth or first order and arise from neglecting higher order terms in the thickness direction.

The multi-director theories, on the other hand, assume piecewise linear courses of the material lines in the direction of the thickness. This kinematic assumption has often been applied to layered laminate shells. As the pieces are getting shorter and shorter, in principle every function profile can be approximated via the thickness, which corresponds to the h method in FEM, see figure on the right-hand side of the picture.

Thick and thin shells

Shell models that allow the thickness to change during deformation are called “thick shells”, as opposed to “thin shells”, which have an invariable thickness. In the case of thick shells, a careful distinction must be made between the change in orientation and the change in length of material line elements in the direction of the thickness of the structure. The suppression of a change in length of the director during the deformation, as happens in thin shells, is based on the assumption that the normal stresses in the thickness direction of the shell are small and is therefore only permissible if neither concentrated loads - in extreme cases individual forces - nor strongly different opposing forces Surface forces are introduced on the top and bottom of the shell.

If the thickness of the shell is invariable, then the normal strains in the direction of the thickness disappear, which is, however, due to the transverse contraction, in contradiction to the assumption of only small normal stresses in the direction of the thickness. The disappearance of the normal stresses in the direction of the thickness, the degeneration condition , must therefore be taken into account at the material equation level through so-called condensation . Condensed material equations are not available for many material models or are awkward to formulate. In the numerical calculation, e.g. B. in FEM, the condensation can be achieved by numerical methods for any material model. Alternatively, a higher order approach can also be used, which can represent an expansion in thickness.

Conventional or Cosserat Continuum

The higher order shell theory yields equations similar to the shell theory based on Cosserat continua. In a Cosserat continuum, each point in space not only has three degrees of freedom for movement in the three spatial directions, but also additional degrees of freedom of rotation with which the directors of the shell are parameterized in the direct method . The main problem with the direct approach is to relate the internal forces, distortions and curvatures to stresses and distortions of the three-dimensional body. The direct method essentially corresponds to what is now referred to in the literature as the geometrically exact shell theory . The term “exact” here only underlines that the theory does not result from an approximate Taylor series expansion , as does the multi-director theory.

Reduction in dimension

As mentioned at the beginning, the shell theories have been formulated mainly for FEM since the 1970s. A widespread basic equation of FEM is the principle formulated in continuum mechanics by d'Alembert in Lagrange's version, in which the virtual deformation work

${\displaystyle \delta W_{\text{int}}:=\int _{V}\delta w_{\text{int}}\;\mathrm {d} V}$

is to be calculated. Here V is the area of ​​the shell, the virtual work of tension and the volume shape of the shell. The efficiency of the shell models is essentially a consequence of the introduction of a special shell kinematics, which allows the volume integral in the three-dimensional continuum to be reduced to a surface integral of a two-dimensional model by introducing stress results as integral quantities over the thickness. ${\displaystyle \delta w_{\text{int}}}$ ${\displaystyle \mathrm {d} V}$

Shell kinematics

Initial configuration of the shell

The area of ​​the shell is covered with convective coordinates , whereby the first two coordinates parameterize the shell center surface and the thickness direction, see figure on the right. The position vector of a point within the shell (black in the picture) is divided into a component that points to the shell center surface (blue in the picture) and a deviation in the direction of the thickness (red in the picture), which is described by directors : ${\displaystyle {\vec {\Theta }}=(\Theta _{1},\Theta _{2},\Theta _{3})\in \mathbb {R} ^{3}}$ ${\displaystyle \Theta _{3}}$ ${\displaystyle {\vec {X}}^{m}}$ ${\displaystyle {\vec {D}}_{n},n={1,2},\ldots ,N}$

${\displaystyle {\vec {X}}={\vec {\chi }}_{0}({\vec {\Theta }})={\vec {X}}^{m}(\Theta _{1},\Theta _{2})+\sum _{n=1}^{N}f_{n}(\Theta _{3}){\vec {D}}_{n}(\Theta _{1},\Theta _{2})}$.

As usual, sizes in the undeformed starting position are denoted by capital letters at a time . The vector ${\displaystyle t_{0}}$

${\displaystyle {\vec {X}}=\sum _{i=1}^{3}X_{i}{\vec {e}}_{i}}$

contains the material coordinates of the particle with respect to the standard basis . Unlike the normal vectors of surfaces, the directors are generally not perpendicular to the shell median surface. The multi-director theory is based on piecewise linear functions . Instead, potencies of should be used here: ${\displaystyle X_{i}({\vec {\Theta }})}$ ${\displaystyle {\vec {e}}_{1,2,3}}$ ${\displaystyle f_{n}}$ ${\displaystyle \Theta _{3}}$

${\displaystyle {\vec {X}}={\vec {\chi }}_{0}({\vec {\Theta }})={\vec {X}}^{m}(\Theta _{1},\Theta _{2})+\sum _{n=1}^{N}\Theta _{3}^{n}{\vec {D}}_{n}(\Theta _{1},\Theta _{2})=\sum _{n=0}^{N}\Theta _{3}^{n}{\vec {D}}_{n}(\Theta _{1},\Theta _{2})}$,

when is set. In this way, a real shell geometry can be reproduced better and better with the function . ${\displaystyle {\vec {D}}_{0}:={\vec {X}}^{m}}$ ${\displaystyle N\rightarrow \infty }$ ${\displaystyle {\vec {\chi }}_{0}}$

The coordinate surface with then defines the top and bottom side of the shell and indicates the center surface of the shell. Plate and disk structures are included as the special case where the shell median plane and the directors are perpendicular to the shell median plane. ${\displaystyle \Theta _{3}=H(\Theta _{1},\Theta _{2})/2}$ ${\displaystyle \Theta _{3}=-H(\Theta _{1},\Theta _{2})/2}$ ${\displaystyle \Theta _{3}=0}$ ${\displaystyle {\vec {X}}^{m}}$ ${\displaystyle {\vec {D}}_{1,2,\ldots }}$

The movement function

${\displaystyle {\vec {x}}={\vec {\chi }}({\vec {\Theta }},t)=\sum _{n=0}^{N}\Theta _{3}^{n}{\vec {d}}_{n}(\Theta _{1},\Theta _{2},t)}$

depends on the time and supplies the current configuration, the variables of which are designated with lowercase letters. The components of the vector are called "spatial coordinates." ${\displaystyle t}$ ${\displaystyle {\vec {x}}}$

Parameterization of the directors

In the case of non-rigid shells, the directors can change their orientation with respect to the middle surface of the shell over time. The orientation of the directors is described with rotation parameters which - like the directors - only depend on the coordinates . The following rotation parameters are used: ${\displaystyle \Theta _{1},\Theta _{2}}$

1. Euler angles , which indicate angles of rotation about certain axes,
2. Quaternions ,
3. Rotation vectors which determine the axis of rotation through their direction and the angle of rotation is a function of their magnitude, see Orthogonal Tensor , and
4. Differences in the displacements on the top and bottom of the shell.

The Euler angles have the disadvantage that different Euler angles can describe the same rotation. In these cases, a singularity occurs in the derivation of the director according to the Euler angles. This is not the case anywhere when using quaternions. The singularity also does not apply to rotation vectors if their absolute value is the angle of rotation in radians . The parameterization of shell elements based on the differences in the displacements on the upper and lower side of the shell shows properties of continuum elements and shell elements ("continuum shells"). There the advantages of a continuum approach are linked with those of the shell theory.

Basis vectors in shell space

Initial configuration of the shell

The derivation of the location according to the convective coordinates provides the covariant basis vectors (black in the picture on the right) ${\displaystyle \Theta _{1,2,3}}$

${\displaystyle {\begin{array}{rcl}{\vec {G}}_{\alpha }({\vec {\Theta }})&=&\displaystyle {\frac {\partial {\vec {\chi }}_{0}}{\partial \Theta _{\alpha }}}=\sum _{n=0}^{N}\Theta _{3}^{n}{\vec {D}}_{n,\alpha }(\Theta _{1},\Theta _{2}),\;\alpha \in \lbrace {1,2}\rbrace \\{\vec {G}}_{3}({\vec {\Theta }})&=&\displaystyle {\frac {\partial {\vec {\chi }}_{0}}{\partial \Theta _{3}}}=\sum _{n=1}^{N}n\Theta _{3}^{n-1}{\vec {D}}_{n}(\Theta _{1},\Theta _{2})\end{array}}}$.

Here and in the following, Greek indices run from one to two and a comma in the index means a derivative according to the coordinate . The dual basis of the covariant basis is provided by the gradients ${\displaystyle (\cdot )_{,k}}$ ${\displaystyle \Theta _{k}}$

${\displaystyle {\vec {G}}^{\alpha }=\sum _{i=1}^{3}{\frac {\partial \Theta _{\alpha }}{\partial X_{i}}}{\vec {e}}_{i},\quad {\vec {G}}^{3}=\sum _{i=1}^{3}{\frac {\partial \Theta _{3}}{\partial X_{i}}}{\vec {e}}_{i}}$,

which form the contravariant basis vectors (not shown in the picture). The co- and contravariant basis vectors of the median surface at are denoted by a capital A. ${\displaystyle \Theta _{3}=0}$

${\displaystyle {\begin{array}{rclclllcl}{\vec {A}}_{\alpha }(\Theta _{1},\Theta _{2})&:=&{\vec {G}}_{\alpha }(\Theta _{1},\Theta _{2},0)&=&{\vec {D}}_{{0,}\alpha }(\Theta _{1},\Theta _{2}),&{\vec {A}}^{\alpha }(\Theta _{1},\Theta _{2})&:=&{\vec {G}}^{\alpha }(\Theta _{1},\Theta _{2},0)\\{\vec {A}}_{3}(\Theta _{1},\Theta _{2})&:=&{\vec {G}}_{3}(\Theta _{1},\Theta _{2},0)&=&{\vec {D}}_{1}(\Theta _{1},\Theta _{2}),&{\vec {A}}^{3}(\Theta _{1},\Theta _{2})&:=&{\vec {G}}^{3}(\Theta _{1},\Theta _{2},0)\end{array}}}$

and by definition do not depend on the thickness coordinate (blue in the picture). The co- and contravariant base vectors in the current configuration are defined analogously with the spatial coordinates: ${\displaystyle \Theta _{3}}$

${\displaystyle {\begin{array}{rcllrcl}{\vec {g}}_{\alpha }({\vec {\Theta }},t)&=&\displaystyle {\frac {\partial {\vec {\chi }}({\vec {\Theta }},t)}{\partial \Theta _{\alpha }}}=\sum _{n=0}^{N}\Theta _{3}^{n}{\vec {d}}_{n,\alpha }(\Theta _{1},\Theta _{2}),&{\vec {g}}^{\alpha }&=&\displaystyle \sum _{i=1}^{3}{\frac {\partial \Theta _{\alpha }}{\partial x_{i}}}{\vec {e}}_{i}\\{\vec {g}}_{3}({\vec {\Theta }})&=&\displaystyle {\frac {\partial {\vec {\chi }}({\vec {\Theta }},t)}{\partial \Theta _{3}}}=\sum _{n=1}^{N}n\Theta _{3}^{n-1}{\vec {d}}_{n}(\Theta _{1},\Theta _{2}),&{\vec {g}}^{3}&=&\displaystyle {\frac {\partial \Theta _{3}}{\partial x_{i}}}{\vec {e}}_{i}\end{array}}}$.

Shell shifter

The tensor, which transforms the basis vectors of the shell center surface into the shell space, is called the shell shifter or shift tensor : ${\displaystyle {\vec {A}}_{1,2,3}}$

${\displaystyle \mathbf {Z} :=\sum _{i=1}^{3}{\vec {G}}_{i}\otimes {\vec {A}}^{i}}$.

With it, the (covariant) Green-Lagrange strain tensor

${\displaystyle \mathbf {E} =\sum _{i,j=1}^{3}E_{ij}{\vec {G}}^{i}\otimes {\vec {G}}^{j}}$

can be transformed in the shell space to the shell center surface:

${\displaystyle \mathbf {E} ^{m}:=\mathbf {Z} ^{\mathrm {T} }\cdot \mathbf {E} \cdot \mathbf {Z} =\sum _{i,j=1}^{3}E_{ij}{\vec {A}}^{i}\otimes {\vec {A}}^{j}}$.

The stress tensor of the second Piola-Kirchhoff type present in a material point

${\displaystyle {\tilde {\mathbf {T} }}=\sum _{i,j=1}^{3}{\tilde {T}}^{ij}{\vec {G}}_{i}\otimes {\vec {G}}_{j}}$

is contravariant and is made by means of

${\displaystyle \mathbf {Z} ^{-1}\cdot {\tilde {\mathbf {T} }}\cdot \mathbf {Z} ^{\mathrm {T} -1}=\sum _{i,j=1}^{3}{\tilde {T}}^{ij}{\vec {A}}_{i}\otimes {\vec {A}}_{j}=:{\tilde {\mathbf {T} }}^{m}}$

transformed to the shell center surface. The Frobenius scalar product ":" of the tensors remains unaffected by the transformation:

${\displaystyle {\tilde {\mathbf {T} }}:\mathbf {E} ={\tilde {\mathbf {T} }}^{m}:\mathbf {E} ^{m}}$.

Cutting sizes

The components of the Green-Lagrange strain tensor

${\displaystyle {\begin{array}{rclrcl}E_{\alpha \beta }=E_{\beta \alpha }&=&\displaystyle {\frac {1}{2}}({\vec {g}}_{\alpha }\cdot {\vec {g}}_{\beta }-{\vec {G}}_{\alpha }\cdot {\vec {G}}_{\beta })&=&\displaystyle {\frac {1}{2}}\sum _{k=0}^{N}\sum _{l=0}^{N}\Theta _{3}^{k+l}({\vec {d}}_{k,\alpha }\cdot {\vec {d}}_{l,\beta }-{\vec {D}}_{k,\alpha }\cdot {\vec {D}}_{l,\beta })\\E_{\alpha 3}=E_{3\alpha }&=&\displaystyle {\frac {1}{2}}({\vec {g}}_{\alpha }\cdot {\vec {g}}_{3}-{\vec {G}}_{\alpha }\cdot {\vec {G}}_{3})&=&\displaystyle {\frac {1}{2}}\sum _{k=0}^{N}\sum _{l=1}^{N}\Theta _{3}^{k+l-1}l({\vec {d}}_{k,\alpha }\cdot {\vec {d}}_{l}-{\vec {D}}_{k,\alpha }\cdot {\vec {D}}_{l})\\E_{33}&=&\displaystyle {\frac {1}{2}}({\vec {g}}_{3}\cdot {\vec {g}}_{3}-{\vec {G}}_{3}\cdot {\vec {G}}_{3})&=&\displaystyle {\frac {1}{2}}\sum _{k=1}^{N}\sum _{l=1}^{N}\Theta _{3}^{k+l-2}kl({\vec {d}}_{k}\cdot {\vec {d}}_{l}-{\vec {D}}_{k}\cdot {\vec {D}}_{l})\end{array}}}$

are divided into components that only depend on the coordinates of the shell center surface: ${\displaystyle E_{nij}^{m}}$

${\displaystyle E_{ij}({\vec {\Theta }})=\sum _{n=0}^{2N}\Theta _{3}^{n}E_{nij}^{m}(\Theta _{1},\Theta _{2})}$.

Multiplication with the dyads and summation over all indices provide the decomposition: ${\displaystyle {\vec {A}}^{i}\otimes {\vec {A}}^{j}}$

${\displaystyle \mathbf {E} ^{m}=\sum _{n=0}^{2N}\Theta _{3}^{n}\mathbf {E} _{n}^{m}(\Theta _{1},\Theta _{2})}$.

The virtual stress work is the Frobenius scalar product of the stresses with virtual strains which, like the components of the strain tensor, are developed into the thickness coordinate:

${\displaystyle \delta w_{\text{int}}={\tilde {\mathbf {T} }}:\delta \mathbf {E} ={\tilde {\mathbf {T} }}^{m}:\delta \mathbf {E} ^{m}=\sum _{n=0}^{2N}\Theta _{3}^{n}{\tilde {\mathbf {T} }}^{m}:\delta \mathbf {E} _{n}^{m}}$.

For the volume integral, the volume element of the shell space is multiplicatively divided into the surface element of the middle surface and a remainder:

${\displaystyle \mathrm {d} V=\det({\vec {G}}_{1},{\vec {G}}_{2},{\vec {G}}_{3})\,\mathrm {d} \Theta _{1}\mathrm {d} \Theta _{2}\mathrm {d} \Theta _{3}=\underbrace {\frac {\mathrm {det} ({\vec {G}}_{1},{\vec {G}}_{2},{\vec {G}}_{3})}{|{\vec {A}}_{1}\times {\vec {A}}_{2}|}} _{:=Z}\;\mathrm {d} \Theta _{3}\,\underbrace {|{\vec {A}}_{1}\times {\vec {A}}_{2}|\mathrm {d} \Theta _{1}\mathrm {d} \Theta _{2}} _{:=\mathrm {d} A^{m}}=Z\,\mathrm {d} \Theta _{3}\mathrm {d} A^{m}}$.

Integration of the virtual stress work over the shell volume delivers with the definitions provided

${\displaystyle {\begin{array}{rcl}\delta W_{\text{int}}&=&\displaystyle \int _{V}\delta w_{\text{int}}\,\mathrm {d} V=\int _{A^{m}}\int _{-H/2}^{H/2}\delta w_{\text{int}}Z\,\mathrm {d} \Theta _{3}\,\mathrm {d} A^{m}\\&=&\displaystyle \int _{A^{m}}\sum _{n=0}^{2N}\underbrace {\int _{-H/2}^{H/2}\Theta _{3}^{n}{\tilde {\mathbf {T} }}^{m}Z\,\mathrm {d} \Theta _{3}} _{:={\hat {\mathbf {N} }}_{n}}:\delta \mathbf {E} _{n}^{m}\mathrm {d} A^{m}=\int _{A^{m}}\sum _{n=0}^{2N}{\hat {\mathbf {N} }}_{n}:\delta \mathbf {E} _{n}^{m}\mathrm {d} A^{m},\end{array}}}$

thus an area integral over the work of the effective internal force tensor on virtual distortions in the shell center surface . The components ${\displaystyle {\hat {\mathbf {N} }}_{n}}$ ${\displaystyle \delta \mathbf {E} _{n}^{m}}$

${\displaystyle N_{nij}=\int _{-H/2}^{H/2}\Theta _{3}^{n}{\tilde {T}}^{ij}Z\,\mathrm {d} \Theta _{3}}$

are called equivalent internal forces or symmetrical internal forces, because they do not arise through the integration of internal forces over a cutting surface but are symmetrical ( ). ${\displaystyle N_{nij}=N_{nji}}$

With linear elasticity

${\displaystyle {\tilde {\mathbf {T} }}^{m}={\stackrel {4}{\mathbf {C} }}{}^{m}:\mathbf {E} ^{m}}$

With a constant fourth-order elasticity tensor, the expansion of the strain tensor in the thickness coordinate can be used advantageously: ${\displaystyle {\stackrel {4}{\mathbf {C} }}{}^{m}}$

${\displaystyle \mathbf {N} _{n}=\int _{-H/2}^{H/2}\Theta _{3}^{n}{\tilde {\mathbf {T} }}^{m}Z\,\mathrm {d} \Theta _{3}=\sum _{l=0}^{2N}\int _{-H/2}^{H/2}\Theta _{3}^{n+l}{\stackrel {4}{\mathbf {C} }}{}^{m}Z\,\mathrm {d} \Theta _{3}:\mathbf {E} _{l}^{m}.=:\sum _{l=0}^{2N}{\stackrel {4}{\mathbf {D} }}_{l}:\mathbf {E} _{l}^{m}.}$

A material equation can be specified for the symmetrical internal size.

The internal size tensor, which arises from the integration of internal stresses over the cut surfaces of the shell, is generally asymmetrical. This is due to the fact that the intersection stresses are related to the intersection normals via the generally asymmetrical first Piola-Kirchoff stress tensor. The effective internal size tensor is only the same as the internal size tensor in flat surface structures (panes and plates).

See also

• Classic laminate theory

literature

• Évariste Sanchez-Palencia , Olivier Millet, Fabien Bechet: Singular Problems in Shell Theory: Computing and Asymptotics . Springer, 2010, ISBN 978-3-642-13814-0 .
• H. Altenbach, J. Altenbach, R. Rikards: Introduction to the mechanics of laminate and sandwich structures . German publishing house for basic industry, 1996.
• PM Naghdi: The theory of shells and plates . In: S. Flügge (Ed.): Handbook of Physics , Volume 6a 2. Solid Mechanics . Springer-Verlag, 1972, ISBN 3-540-05535-5 , ISBN 0-387-05535-5 .

Web links

• J. Matheas: Main focus of work on shell theory

Individual evidence

1. ^ E. Benvenuto: An Introduction to the History of Structural Mechanics, Part II: Vaulted Structures and Elastic Systems . Springer, 1991.
2. ↑ K.-E. Kurrer: The History of the Theory of Structures. Searching for Equilibrium . Ernst & Son, 2018.
3. ↑ ^{a b} A.EH Love: On the small vibrations and deformations of thin elastic shells . In: Philosophical Transactions of the Royal Society , 179, 1888, pp. 491 ff.
4. ↑ ^{a b c} J. Irslinger: Mechanical basics and numerics of three-dimensional shell elements . 2013, ISBN 978-3-00-044707-5 (Report No. 61, Institute for Structural Analysis, University of Stuttgart).
5. ↑ ^{a b} E. Reissner: The effect of transverse shear deformation on the bending of elastic plates . In: Journal of applied mechanics , Trans. ASME, Vol. 67, No. 2, 1945, pp. 69-77
6. ^ ^{A b} E. Cosserat, F. Cosserat: Théorie des corps déformables. Hermann & Cie, Paris, 1909
7. ↑ P. Duhem: Le potentiel thermodynamique et la pression hydrostatique. In: Annales scientifiques de l'École Normale Supérieure. 10 (1893), pp. 183-230.
8. ↑ ^{a b} P. M. Naghdi (1972)
9. ↑ ^{a b} M. Bischoff: Theory and numerics of a three-dimensional shell formulation . 1999 ( ibb.uni-stuttgart.de [PDF] report no. 30, Institute for Structural Analysis, University of Stuttgart).
10. ↑ ^{a b} N. Büchter: Merging of the degeneration concept and the shell theory for finite rotations . 1992 ( ibb.uni-stuttgart.de [PDF] report no. 14 of the Institute for Structural Analysis at the University of Stuttgart).
11. ↑ Altenbach [1996, p. 290ff]
12. ↑ F. Gruttmann: Theory and numerics of thin-walled fiber composite structures. Habilitation thesis at the Faculty of Civil Engineering and Surveying at the University of Hanover, report no. F 96/1, University of Hanover, 1996 ( uni-hannover.de ).
13. ^ RD Mindlin: Influence of Rotatory Inertia and Shear on Flexural Motions of Isotropic Elastic Plates . In: Journal of Applied Mechanics . No. 18 , 1951, ISSN  0021-8936 , p. 31-38 .
14. ↑ C. Huettel, A. Miller Matze: Consistent discretization of thickness strains in thin shells Including 3D material models . In: Communications in applied numerical methods . tape 15 , 1999, p. 283-293 .