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The State of Stress and Strain Adjacent to Notches in a New Class of Nonlinear Elastic Bodies

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Abstract

In this paper we study the deformation of a body with a notch subject to an anti-plane state of stress within the context of a new class of elastic models. These models stem as approximations of constitutive response functions for an elastic body that is defined within the context of an implicit constitutive relation between the stress and the deformation gradient. Gum metal and many metallic alloys are described well by such constitutive relations. We consider the state of anti-plane stress of a body with a smoothened V-notch within the context of constitutive relations for the linearized strain in terms of a power-law for the stretch. The problem is solved numerically and the convergence and the stability of the solution is studied.

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Notes

  1. Noll [42] later generalized the concept of his definition of a Simple Material but not to include implicit relationships. We shall not get into a discussion of the same here.

  2. Rajagopal [55] classified the material symmetry possessed by the sub-classes of bodies whose histories of the stress, deformation gradient, density, etc., are given in terms of an implicit constitutive relations.

  3. Bulíček et al. [7] study several mathematical aspects concerning strain limiting bodies.

  4. Since we consider constitutive relations of the type (4b), the anti-plane stress state is equivalent to the classical definition of anti-plane strain.

  5. We are not using the same model and obtaining two different sets of values for the material parameters under two loading conditions. The NLB and NLS models are two different models that provide equally good fits. We used different experiments to obtain the material parameters for the two different models. These different sets of values fit the data equally well for the both the experiments. Which of these two models better explains the response of the body can only be determined by having other experiments against which these models can be corroborated. Such experiments are not available at the present time.

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Acknowledgements

Josef Málek thanks the Czech Science Foundation for support through the project 18-12719S. K.R. Rajagopal thanks the Office of Naval Research for support of this work.

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Authors and Affiliations

  1. Charles University, Faculty of Mathematics and Physics, Mathematical Institute, Sokolovská 83, 186 75, Prague, Czech Republic

    Vojtěch Kulvait & Josef Málek

  2. Department of Mechanical Engineering, Texas A&M University, College Station, TX, 77843, USA

    K. R. Rajagopal

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  1. Vojtěch Kulvait
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  2. Josef Málek
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  3. K. R. Rajagopal
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Correspondence to K. R. Rajagopal.

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Dedicated to the memory of Professor Walter Noll

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Kulvait, V., Málek, J. & Rajagopal, K.R. The State of Stress and Strain Adjacent to Notches in a New Class of Nonlinear Elastic Bodies. J Elast 135, 375–397 (2019). https://doi.org/10.1007/s10659-019-09724-0

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