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The elastic theory of shells using geometric algebra

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ABSTRACT

We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.

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Surface geometry.
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RSOS170065F1: Surface geometry.

Mentions: Let B and S be two-dimensional surfaces embedded in three-dimensional Euclidean space . B is the reference configuration of the surface, and S is the spatial configuration, and the two are related by the motionϕt. At time t, the point X∈B is moved to ϕt(X)∈S. Let {Xi} be coordinates over B, and {xi} be coordinates over S. We follow the convention that the indices i,j,k,… run over 1,2, and the indices a,b,c,… run over 1,2,3. We restrict {xi} to be convected coordinates such that , where x∈S. We denote the frame associated with {Xi} by Ei=∂X/∂Xi, and similarly, ei=∂x/∂xi. The reciprocal frames are denoted by {Ei} and {ei}, and are defined to satisfy . The frames on each configuration are illustrated in figure 1.Figure 1


The elastic theory of shells using geometric algebra
Surface geometry.
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC5383861&req=5

RSOS170065F1: Surface geometry.
Mentions: Let B and S be two-dimensional surfaces embedded in three-dimensional Euclidean space . B is the reference configuration of the surface, and S is the spatial configuration, and the two are related by the motionϕt. At time t, the point X∈B is moved to ϕt(X)∈S. Let {Xi} be coordinates over B, and {xi} be coordinates over S. We follow the convention that the indices i,j,k,… run over 1,2, and the indices a,b,c,… run over 1,2,3. We restrict {xi} to be convected coordinates such that , where x∈S. We denote the frame associated with {Xi} by Ei=∂X/∂Xi, and similarly, ei=∂x/∂xi. The reciprocal frames are denoted by {Ei} and {ei}, and are defined to satisfy . The frames on each configuration are illustrated in figure 1.Figure 1

View Article: PubMed Central - PubMed

ABSTRACT

We present a novel derivation of the elastic theory of shells. We use the language of geometric algebra, which allows us to express the fundamental laws in component-free form, thus aiding physical interpretation. It also provides the tools to express equations in an arbitrary coordinate system, which enhances their usefulness. The role of moments and angular velocity, and the apparent use by previous authors of an unphysical angular velocity, has been clarified through the use of a bivector representation. In the linearized theory, clarification of previous coordinate conventions which have been the cause of confusion is provided, and the introduction of prior strain into the linearized theory of shells is made possible.

No MeSH data available.