Erratum to: Foundations of Elastoplasticity: Subloading Surface Model

The original version of this book was revised. The erratum to this book is available at 10.1007/978-3-319-48821-9

Eratum to: K. Hashiguchi, Foundations of Elastoplasticity: Subloading Surface Model, DOI 10.1007/978-3-319-48821-9

The continuum mechanics and the hypoelastic-based plasticity are comprehensively described in this book. In particular, the subloading surface model formulated within the framework of the hypoelastic-based plasticity is described in detail through this book. The pertinence of the subloading surface model will be summarized as the final remarks.

The basic features of the subloading surface model are summarized as follows:

• 1. It is based on the quite natural concept that the plastic deformation is developed as the stress approaches the yield surface and thus it possesses the high generality and the capability of describing accurately irreversible deformations.

• 2. It fulfills the smoothness condition, describing always continuous variation of tangent stiffness modulus and thus depicting the smooth elastic-plastic transition.

• 3. It possesses the automatic controlling functions to attract the stress to the normal-yield surface, so that the stress is pulled-back to the yield surface when it goes over the surface in numerical calculations. In addition, the kinematic hardening variable is pulled-back to the normal-isotropic hardening surface. Further, the back stress is also automatically pulled-back to the stagnation surface.

• 4. It is capable of describing the finite deformation and rotation under an infinitesimal elastic deformation.

By virtue of these rigorous physical backgrounds, the subloading surface model is endowed with the following rigorous descriptions distinguishable from the other elastoplastic constitutive models.

• 1. Plastic strain rate is predicted even for any low stress level. Then, cyclic loading behavior is predicted accurately even for infinitesimal loading amplitude. The other models, e.g. the multi, the two and the superposed kinematic hardening models are incapable of describing the cyclic loading behavior appropriately.

• 2. Deformation of various solids unlimited to metals, e.g. soils, etc. can be described. The other cyclic plasticity models are limited to the description of metal deformation.

• 3. Inelastic strain rate induced by the stress rate tangential to the yield surface, i.e. tangential-inelastic strain rate is described appropriately, fulfilling the continuity condition, which is required to describe the non-proportional loading behavior. All the other models assume the purely-elastic domain so that they predict the tangential-inelastic strain rate induced suddenly at the moment when the stress reaches the yield surface if the tangential-inelastic strain rate is incorporated, violating the continuity condition at the moment.

• 4. Viscoplastic constitutive deformation is described pertinently in a general rate from the quasi-static to the impact loading. The other models are incapable of describing the viscoplastic deformation behavior at high rate, predicting an elastic response for an impact loading.

• 5. Damage phenomenon under cyclic loading is described appropriately, which leads to a softening behavior in general. The other cyclic plasticity models are incapable of describing the cyclic damage phenomenon with a softening pertinently.

• 6. Constitutive equation of fatigue would be described pertinently, which is required to describe plastic strain rate induced in low stress level and small stress amplitude. The other models are incapable of describing the fatigue phenomenon, predicting only an elastic strain rate in low stress level and small stress amplitude.

• 7. Constitutive equation of phase-transformation of metals can be described pertinently. The other models are incapable of describing the phase-transformation phenomenon pertinently.

• 8. Friction phenomenon is described pertinently, including the transition from static to kinetic friction by the sliding, the recovery of static friction with elapse of time and the both of positive and negative rate-sensitivities. The negative and the positive rate sensitivities are relevant to the dry and the lubricated (fluid) friction, respectively. The other model is incapable of describing these friction phenomena.

• 9. Crystal plasticity analysis can be executed pertinently, in which calculation of slips in numerous number of slip systems are required. The yield judgment is unnecessary since the smooth elastic-plastic transition is described and the resolved shear stress is automatically pulled-back to the critical shear stress. The other models are inapplicable to the crystal plasticity analysis rigorously because the yield judgment and the operation to pull back the resolved shear stresses to the critical shear stress are required in numerous number of slip systems. Then, impertinent analysis using the creep model has been performed widely after Pierce et al. (1982, 1983). It is impertinent to adopt the rate-dependent model for the rate-independent deformation phenomenon. In addition, it should be noted that the creep model is impertinent such that it predicts a creep shear strain rate even in unloading process of the resolved shear stress from the critical shear stress. The subloading-overstress model should be adopted in the rate-dependent crystal plasticity analysis.

• 10. The multiplicative hyperelastic-based plasticity can be formulated readily based on the subloading surface model, which is capable of describing the cyclic loading behavior exactly for finite elastoplastic deformation/rotation. The multiplicative hyperelastic-based plasticity cannot be formulated by the other cyclic plasticity models.

Thus, the subloading surface model is endowed inherently with the high generality and flexibility for the description of irreversible mechanical phenomena of solids.

Eventually, it can be concluded that

Subloading surface concept is to be the unified constitutive law which is inevitable to describe irreversible mechanical phenomena in wide classes of solids, ranging from monotonic to cyclic loadings, from quasi-static to impact loadings, from infinitesimal to finite deformations and from micro to macro phenomena.

Then, the elastoplasticity theory will be developed steadily (breaking through the stagnation) by incorporating the exact formulation of the subloading surface model, although it has stagnated during a half century since the study on the unconventional (cyclic) elastoplasticity aiming at description of the plastic strain rate caused by the rate of stress inside the yield surface has started in the 1960s.

The Hashiguchi (subloading surface) model is implemented to the commercial FEM software Marc marketed by MSC Software Corporation, so that Marc users can apply this model to deformation analyses. The implementation was highly supported by Dr. Motohatu Tateishi, MSC Soft. Corp., Japan. The computer program for elastoplastic deformation analysis of metals based on the subloading surface model is provided in Appendix J.

On Various Irrational Models and Formulations Diffused Widely

It is quite regretful for the sound development of applied mechanics and technologies in engineering that various irrational theories have been proposed and are diffused widely. Typical examples are shown below.

Cyclic Kinematic Hardening Models

The cyclic kinematic hardening models, i.e. the multi surface model of Z. Mroz, the two surface model of Y. F. Dafalias (and F. Yoshida) and the superposed-kinematic hardening model of J. L. Chaboche (and N. Ohno) which assume the yield surface enclosing a purely-elastic domain, resulting in the going around in circles endlessly without excluding the serious defect of the conventional elastoplasticity model. Unfortunately, however, they continue to be advertised and have never been withdrawn by these proposers themselves.

Creep Model

The creep model (F. H. Norton, J. W. Hutchinson, etc.) for the rate-dependent irreversible deformation is irrational such that it always predicts the irreversible deformation even in a stress-reducing process in any low stress level. Unfortunately, however, it is applied widely for the prediction of irreversible deformation of not only macroscopic deformation but also crystal plasticity analysis (J. R. Rice, J. W. Hutchinson, R. J. Asaro, A. Needleman, etc.).

Rate-and-State Model

The rate-and-state model (J. H. Dieterich, A. L. Ruina, J. R. Rice, etc.) are used for the prediction of earthquake. Unfortunately, however, it is quite primitive model ignoring the elastoplasticty and thus the disasters due to earthquakes will not be prevented as far as the prediction of earthquake will depend on the rate-and-state model.

Formulation of Plastic Flow Rule by Second Law of Thermodynamics

The formulation for the flow rule of plastic strain rate by exploiting the second law of thermodynamics is fashioned widely (J. A. Lemaitre, S. Murakami, A. Menzel, I. N. Vladimirov, M. Wallin, etc.). However, it is to be the predetermined harmony of the associated flow, so that any valuable result has not been obtained, and thus impedes the serious consideration of the plastic flow rule and the sound development of plasticity theory. Such thermodynamic formalism should be excluded earlier.

Natural scientists are required to take up sincere attitude towards the deep consideration whether existing theories or models flashed into mind are really acceptable. All the defects seen in the above-mentioned irrational models and formulations would have been solved thoroughly by the subloading surface model.