Theory manual

This pages is the theory manual of SMART+. Its scope is:

  • to demonstrate the formalism in which the tensors are represented in SMART+. Also to explain important aspects for the tensors, like their representation in local and global coordinates,
  • to introduce briefly the theoretical framework in which the analyses are based on.

Mathematics for mechanical problems

  1. Preliminaries
  2. Tensors conventions: Fourth order tensors in SMART+, Examples of SMART+ formalism
  3. Tensors rotators

Constitutive laws

  1. Elasticity: Isotropic materials, Transversely isotropic materials
  2. Elastoplasticity
  3. Viscoelasticity
  4. Shape memory alloys
  5. Return mapping algorithms: Convex cutting plane, Closest point projection

Micromechanics and homogenization

  1. General micromechanics concepts: Definition of homogenization, Average theorems, Hill’s lemma and Hill-Mandel theorem, Concentration tensors, Eshelby problem
  2. Mori Tanaka
  3. Self consistent
  4. Periodic homogenization: Composite laminates



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Eshelby, J.D., 1957. The Determination of the Elastic Field of an Ellipsoidal Inclusion, and Related Problems. Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 241 (1226), 376–396.

Gavazzi, A.C., Lagoudas, D.C., 1990. On the numerical evaluation of Eshelby’s tensor and its application to elastoplastic fibrous composites. Computational Mechanics 7, 13–19.

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Lai, W.M., Rubin, D., Kremple, E., 2010. Introduction to continuum mechanics, 4th Edition. Elsevier.

Lagoudas, D. (Ed.), 2008. Shape Memory Alloys: Modeling and Engineering Applications. Springer.

Lagoudas, D., Hartl, D., Chemisky, Y., Machado, L., Popov, P., 2012. Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys. International Journal of Plasticity 32-33, 155–183.

Lemaitre, J., Chaboche, J.-L., 2002. Mechanics of Solid Materials. Cambridge University Press.

Mura, T., 1987. Micromechanics of Defects in Solids. In: Mechanics of elastic and inelastic solids, Second, Revised Edition. Kluwer Academic Publishers.

Nemat-Nasser, S., Hori, M., 1999. Micromechanics: overall properties of heterogeneous materials, 2nd Edition. North-Holland.

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Ortiz, M., Simo, J.C., 1986. An analysis of a new class of integration algorithms for elastoplastic constitutive relations. International Journal for Numerical Methods in Engineering 23, 353–366.

Qidwai, M.A., Lagoudas, D.C., 2000. Numerical implementation of a shape memory alloy thermomechanical constitutive model using return mapping algorithms. International Journal for Numerical Methods in Engineering 47, 1123–1168.

Qu, J., Cherkaoui, M., 2006. Fundamentals of Micromechanics of Solids. Wiley.

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Suquet, P.M., 1987. Elements of homogenization for inelastic solid mechanics. In: Lecture Notes in Physics. Vol. 272. Springer, 193–278.

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