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Complex Variable Semianalytical Method for Sensitivity Evaluation in Nonlinear Path Dependent Problems: Applications to Periodic Truss Materials

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Computational Modeling, Optimization and Manufacturing Simulation of Advanced Engineering Materials

Part of the book series: Advanced Structured Materials ((STRUCTMAT,volume 49))

Abstract

The evaluation of structural response derivatives with respect to design parameters, usually known as sensitivity analysis, is an issue of paramount importance in gradient-based optimization and reliability analyses in engineering. In the last 20 years, much research has been devoted to develop efficient strategies for the accurate evaluation of sensitivity information. A relatively new and promising procedure combines the semianalytical (SA) approach with the use of complex variables (CVSA). This method allows the use of diminutive perturbations, circumventing the weakness that the traditional SA approach shows when applied to shape design variables. In spite of the great potential of the CVSA, its formulation and application has been restricted to path independent problems. In this chapter we aim to extend the method to handle path dependent problems, emphasizing the treatment of internal variables, such as accumulated plastic strain and damage. In order to make the concept easy to understand, we use the method to evaluate the sensitivity of particular homogenized properties of a 2D periodic truss material (PTM). Optimization of PTMs has encountered great potential in tissue engineering, as well as in automotive and aeronautical applications. Generally PTMs are designed to operate in the linear geometrical and constitutive range. However, using sensitivity analysis we can obtain an insight about how these designed homogenized properties behave when geometrical and/or material nonlinearities are considered.

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Notes

  1. 1.

    The use of the isoparametric approach presented in this work does not lead to a direct extension to continuum elements, but preserves the traditional integral form of the internal force vector and tangent matrix. The matrices and vectors obtained have an analogous in continuum elements and thus, the expressions obtained give a deeper insight into the analogous modifications that should be performed in the continuum elements case.

  2. 2.

    Note that \(X_\ell \) with the “\(\ell \)” subscript stands for the initial value of \(x_\ell \), while X without the “\(\ell \)” subscript describes the global position in the XY system.

  3. 3.

    A virtual displacement field is defined as an infinitesimal displacement that satisfies the boundary conditions of the original configuration of a given body.

  4. 4.

    A given entity is said to be rotated if it is described in the corrotational system.

  5. 5.

    The micro-cracks and voids are small if compared to RVE dimension, but large when compared to the atomic level.

  6. 6.

    In favor of neatness, from this point on, the subscript \((.)_E\) is dropped because it is implicit that only the rotated engineering stress and strain tensors will be used throughout the text.

  7. 7.

    This work focuses on direct analytical sensitivity, since this option offers more advantages than the adjoint approach in path dependent problems even when the number of design variables is smaller than the number of constraints [43].

  8. 8.

    The material parameters adopted do not necessarily correspond to a real material, being chosen for purely academic purposes.

  9. 9.

    The dimensions given to the cell are irrelevant provided the relative density is kept unchanged. This is because the cell size must tend to zero when compared to the macroscopic scale [48].

  10. 10.

    In linear analysis the bulk modulus is constant, which does not occur when considering nonlinear behavior [49].

  11. 11.

    Note that the formulation of bar elements presented in Sect. 2 keeps the area of the cross-sections unchanged, which does not occur strictly. In fact, the effect of the change of the cross-sections along the deformation of the bars can be very important. An extension to account for this effect can be found in the work of Crisfield [50].

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Acknowledgments

The authors wish to express their gratitude to CNPq and CAPES (Brazilian research supporting agencies), and to UDESC for the concession of Master’s scholarships associated to this work.

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  1. Department of Mechanical Engineering, Center for Technological Sciences, Santa Catarina State University - UDESC, Campus Avelino Marcante, Joinville, SC, 89223-100, Brazil

    Geovane A. Haveroth & Pablo A. Muñoz-Rojas

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  1. Geovane A. Haveroth
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Correspondence to Pablo A. Muñoz-Rojas .

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  1. Department of Mechanical Engineering, Santa Catarina State Uni. - UDESC, Joinville, Santa Catarina, Brazil

    Pablo Andrés Muñoz-Rojas

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Haveroth, G.A., Muñoz-Rojas, P.A. (2016). Complex Variable Semianalytical Method for Sensitivity Evaluation in Nonlinear Path Dependent Problems: Applications to Periodic Truss Materials. In: Muñoz-Rojas, P. (eds) Computational Modeling, Optimization and Manufacturing Simulation of Advanced Engineering Materials. Advanced Structured Materials, vol 49. Springer, Cham. https://doi.org/10.1007/978-3-319-04265-7_9

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  • DOI: https://doi.org/10.1007/978-3-319-04265-7_9

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