This paper extends the approach for determining the three-dimensional global displaced shape of slender structures from a limited set of scalar surface strain measurements. It is an exhaustive approach that captures the effect of curvature, shear, torsion, and elongation. The theory developed provides both a determination of the uniaxial strain (in a given direction) anywhere in the structure and the deformed shape, given a set of strain values. The approach utilizes Cosserat rod theory and exploits a localized linearization approach that helps to obtain a local basis function set for the displacement solution in the Cosserat frame. For the assumed deformed shape (both the midcurve and the crosssectional orientation), the uniaxial value of strain in any given direction is obtained analytically, and this strain model is the basis used to predict the shape via an approximate local linearized solution strategy. Error analysis due to noise in measured strain values and in uncertainty in the proximal boundary condition is performed showing uniform convergence with increased sensor count.

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This paper investigates the variational formulation and numerical solution of a higher-order, geometrically exact Cosserat type beam with a deforming cross-section, instigated from generalized kinematics presented in earlier works. The generalizations include the effects of a fully-coupled Poisson's and warping deformations in addition to other deformation modes in Simo-Reissner beam kinematics.The kinematics at hand renders the deformation map to be a function of not only the configuration of the beam but also on the elements of the tangent space of the beam's configuration (axial strain vector, curvature, warping amplitude, and their derivatives). This complicates the process of deriving the balance laws and exploring the variational formulation of the beam, at the same time, make it worthwhile. The weak and strong form is derived for the dynamic case considering a general boundary.We restrict ourselves to linear small-strain elastic constitutive law and the static case for numerical implementation. The finite element modeling of this beam has higher regularity requirements. The matrix (discretized) form of the equation of motion is derived. Finally, numerical simulations comparing various beam models are presented.

In this paper, we discuss about reconstructing the global deformed shape of slender structures such as pipelines, tethers, or cables from a limited set of scalar surface strain measurements. We present a comprehensive approach that captures the effect of curvature, shear, torsion, and axial deformation. Our primary focus is to demonstrate the applicability of the approach to aid in damage detection algorithms. This theory utilizes Cosserat rod theory and exploit localized linearization approach that helps to obtain local basis function set for the displacement solution in director frame. The uniaxial strain vector and the surface strain for the Cosserat beam incorporating the abovementioned effects are obtained and used to develop the reverse algorithm to reconstruct global shape of the structure. Error analysis due to noise in measured strain values is performed and results are discussed.

This is the extended version of paper submitted to Applied Mathematics Letters Journal, Elsevier with the title "On the derivatives of curvature of framed space curve and their time-updating scheme".Abstract. This paper deals with the concept of curvature of framed space curves, their higher-order derivatives, variations, and co-rotational derivatives. We realize that parametrizing rotation tensor using the Gibbs vector is effective in deriving a closedform formula to obtain any order derivative of the curvature tensor as the summation of functions of the parametrizing quantity and its derivatives. We use these results for formulating a linearized updating algorithm for curvature and its derivatives when the configuration of the curve acquires a small increment. Finally, the MATLAB code to obtain updated curvature (spatial and material) and its derivatives is presented.

In this paper, we present an improved methodology for the global shape reconstruction of rod-like structures that capture the effect of curvature, shear, torsion, axial deformation, and Poisson's transformation. The inclusion of Poisson's effect relaxes Euler-Bernoulli's rigid cross-section assumption such that the cross-section could now undergo planar deformation (shrinking or expansion in the same plane). This scenario is particularly useful for the inflatable structures and pipelines subjected to large radial pressure.The theory of shape sensing utilizes the concept of curve framing using Cosserat director triad also called as Cosserat kinematics. The idea is to develop an algorithm to reconstruct the global shape of the rods using the local differential geometry parameters (finite strain parameters) of the midcurve. The deformed configuration of the object lies in ℝ ×𝑆𝑂(3)× ℝ space. The presented theory exploits localized linearization approach that helps to obtain local basis set for the approximation of the midcurve position vector and the director triad, whereas moving least square approximation is utilized to estimate the axial strain field. The uniaxial surface strain incorporating all the effects mentioned above is derived and used to develop the shape-sensing algorithm. A simulation describing the idea is presented at the end.

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