By C. Truesdell
Reprinted from the magazine of rational mechanics and research, volumes 1-3, 1952-54 with additional corrections and footnotes further in 1962.
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The symbolic summation over the elements defines the global tangent stiffness matrix ˜ K = K. 60) elements The assembly procedure for the global tangent stiffness matrix K and for the total internal forces q is illustrated in the following example. 2. Element formulation of two-element truss. 1 the equilibrium equations and the tangent stiffness of the twoelement truss shown in Fig. 6 were derived directly from the elastic energy. This example shows how the current internal forces and the incremental equilibrium equations of the truss are obtained by assembling the element contributions.
Malvern (1969) or Holzapfel (2000). It is clear from the definition of the angle of rotation ϕ that it is independent of any choice of coordinate system, while the components [n1 , n2 , n3 ] of the direction of the rotation axis refer to the chosen coordinate system. It is therefore to be expected that the angle of rotation can be extracted from the trace of the rotation tensor. 10), whereby the trace of the rotation tensor is tr(R) = Rii = 1 + 2 cos ϕ. 10). The principles used to extract the rotation angle ϕ and the components [n1 , n2 , n3 ] of the axis of rotation constitute a simple example of the use of invariance and symmetry.
These general features are encountered repeatedly in various settings in the following chapters. 1 Deformation and strain In most finite element formulations for structures and solids the displacements are the primary variables of the problem. The displacements may lead to deformation of the elements, and this in turn to internal forces. It is important to use deformation measures that vanish identically for rigid body displacements. In a theory with finite displacements this requirement can be satisfied by several different strain definitions.
The Mechanical foundations of elasticity and fluid dynamics by C. Truesdell
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