Pseudo-kinematic invariants - gems in FE structural analyses

Invariants are real treasures -- not just of the sciences! Knowledge about them may considerably simplify their computation. The first invariants, students of mechanics are usually confronted with, are the three invariants of the stress tensor with respect to the coordinate systems used for their computation. Different from such invariances are the ones of the acceleration and the initial speed of a fictitious particle, moving non-uniformly on an originally unknown twisted curve on the unit sphere. These invariances are the hypothetical basis of the present research project. They refer to an arbitrary real symmetric matrix as one of the two coefficient matrices of a class of linear eigenvalue problems in the framework of the Finite Element Method (FEM). The other one is the tangent stiffness matrix. The suspected invariant quantities are termed as pseudo kinematic because they are computed by a pseudo time that represents an arc length. Contrary to the epistemological significance of the suspected invariances, their practical significance is, at first sight, rather limited. However, a mathematical expression, containing only the two invariant quantities, was recently shown by the applicant to be equal to the non-membrane percentage of the strain energy of shells, folded plates, arches, and frames. This theoretically remarkable and practically relevant result was obtained by solving the so-called consistently linearized eigenvalue problem, with the tangent stiffness matrix and its derivative with respect to a dimensionless load parameter as the two coefficient matrices. If the hypothetically assumed invariance of the aforementioned pseudo kinematic quantities can be verified, the second coefficient matrix, which generally must be approximated by a finite-difference expression, could be replaced by a simpler matrix -- ideally by the unit matrix. The quoted mathematical expression would then remain valid, even if the tangent stiffness matrix was not differentiable, as may be the case for inelastic deformations. To prove these invariances numerically, one special N-dimensional eigenvector each of two different linear eigenvalue problems will be used as a tool for determination of curves on the unit sphere, along which fictitious particles are moving. According to the underlying hypotheses, the acceleration and the initial speed of the two vectors, the vertices of which describe the two different surface curves, should be equal. If the proof is successful, topical program modules will be developed and subsequently implemented in a multi-purpose FE program system to allow for a more efficient computation of the non-membrane percentage of the strain energy of a wide class of civil and mechanical engineering structures with, as well as without, stability limits in the elastic or the elasto-plastic material regime. Keeping this quantity as small as possible is generally an important goal of their design.

Approximately one year after the start of the present research project it turned out that a change of the original research goal was necessary. Determination of mathematical conditions for extreme values of the stiffness of proportionally loaded structures became the new research goal. Sufficient strength and stiffness are crucial requirements on buildings. Hence, lack of such conditions represents a significant gap of the literature in the area of structural mechanics. The reason for the change of the original research goal were unforeseeable deficiencies of the mathematical tool in the form of the Consistently Linearized Eigenvalue Problem in the framework of the Finite Element Method. In hindsight, the aforementioned change of the original research goal turned out to be a scientific stroke of luck! Rather incidentally, a novel linear eigenvalue problem with two real, symmetric, indefinite coefficient matrices was discovered. One of the two matrices refers to the actual load intensity. The other one follows from specialization of the first one for the onset of loading of the investigated structure. The tangent stiffness matrix is a submatrix of the two coefficient matrices. The reason for the indefiniteness of the two matrices is a hybrid finite element in the library of a commercial computer program. It is an extension of a displacement element. However, the rationale behind this extension has nothing to do with the present research project. The existence of two indefinite coefficient matrices in a linear eigenvalue problem is a necessary condition for complex eigenvalues and eigenvectors. Hence, it needs a mechanical reason for the originally real fundamental eigenvalue to become indeed one of two conjugate complex eigenvalues in a finite region of the proportionally increased load. This reason is the occurrence of a minimum value of the stiffness of the structure concerned, at a load level inside the said region. The mathematical condition, found for this extreme value, is the one of a point of inflection of the real part of the complex fundamental eigenvalue function. The complex region of this function signals the aversion from the structure's original tendency towards a stability limit. This explains why the mathematical condition for a maximum value of the stiffness of a proportionally loaded structure, which was found later, is the one of a point of inflection of a real, in contrast to a complex, fundamental eigenvalue function. In this case, a trend change of the structural behavior occurs in the opposite direction, namely, in the structure's tendency towards a stability limit. The respective buckling condition is characterized by a zero position of the real fundamental eigenvalue function. The numerically verified conditions for extreme values of the stiffness of proportionally loaded structures represent a significant novelty in the history of structural mechanics.