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Journal ArticleDOI

A new thermodynamically consistent continuum model for hardening plasticity coupled with damage

01 Dec 2002-International Journal of Solids and Structures (Pergamon)-Vol. 39, Iss: 25, pp 6241-6271
TL;DR: In this paper, a phenomenological model for hardening-softening elasto-plasticity coupled with damage is presented, where specific kinematic internal variables are used to describe the mechanical state of the system.
About: This article is published in International Journal of Solids and Structures.The article was published on 2002-12-01 and is currently open access. It has received 68 citations till now. The article focuses on the topics: Elastic energy & Internal energy.

Summary (3 min read)

Introduction

  • The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
  • The main characteristics of the proposed model are summarised in the following.
  • First, an unified format for the evolution laws of any internal variable is obtained, and they all derive from a single potential of the driving forces.

2.1. State variables and ambient spaces

  • A simple material is considered and the constitutive relations are developed within the framework of the Generalised Standard Material Model (Germain, 1973; Halphen and Nguyen, 1975).
  • The assumption of local state is kept valid, i.e. the equilibrium state of a material point is assumed independent of the state of the neighbouring elements.
  • The state of the system is phenomenologically described assigning a set of internal variables and the related mechanisms for energy exchange, distinguishing the reversible phenomena that modify the stored energy and the irreversible ones that cause energy dissipation.
  • The last term in (2) represents the heat exchange while the other terms correspond to the mechanical power.
  • Since for any closed system the power of the internal variables a, x must be zero, their total value vanishes, i.e. their elastic and plastic parts are opposite (Kluitenberg, 1962; Ziegler, 1977).

2.2. Internal energy functional

  • The internal energy, dependent only on the reversible part of the kinematic variables, is postulated as the sum of a deformation energy /ðee;xe; seÞ and of a hardening energy wðae;xe; seÞ coupled with damage: eðee; ae;xe; seÞ ¼ /ðee;xe; seÞ þ wðae;xe; seÞ þ InXðxeÞ ð6Þ.
  • In the sequel, as isothermal processes are considered, the dependence on entropy or temperature will be dropped.
  • Furthermore, the function EðxeÞ can be assumed convex in xe.
  • The form (8) has been often used in mechanics, and is based on the idea of the ‘‘effective stress’’ r̂, first proposed by Kachanov (1958).
  • The operators M1, M2 are in general fourth order tensor (Hansen and Schreyer, 1994).

2.3. The dissipation functional

  • The irreversible behaviour is ruled by the dissipation potential d, that has to comply with the second principle of thermodynamics, stating the irreversibility of entropy production (Chaboche, 1999): q_sþ div q T r T P 0 ð22Þ where q is the material density, r the density of the internal heat production and q the heat flux for unit area.
  • Eq. (34) implies that the dissipation functional is equal to the support function of the convex domain K : dð _gpÞ ¼ suppK ð35Þ.
  • Furthermore, forms of the internal energy or of the dissipation potential different from those used in the paper can be applied, subjected only to the thermodynamic restrictions (28).
  • Additional degrees of freedom can be obtained if the exponent n of the damage law is assumed to be different for the elastic and the plastic moduli.

3. The admissible domain

  • In paragraph 2 it has been shown that the maximum dissipation principle (34) implies the existence of an elastic domain K of the generalised stresses that can be described by means of the yield functional (43).
  • Aim of this paragraph is to examine and compare some of them with the present model.
  • All the convex domains having their borders between the solid and the dashed lines in Fig. 4, corresponding to different choices of the dissipation d, are admissible.
  • Two distinct Lagrangian multipliers are thus introduced.
  • The admissible domain (53) in the uniaxial case is examined in Fig. 7 where the undamaged elastic path for uniaxial stress is represented.

4.1. The case of isotropic damage

  • The functions g1 and g2 are the classical expressions of the Mises and Drucker-Prager criteria with the addition of two variables: the isotropic hardening variable v, that rules the homothetic expansion of the plastic surface and the isotropic damage energy f, dual of the damage variable x, that describes the contraction of the domain when damage is active.
  • A functional dependency on the stresses can also be introduced to reproduce coupling phenomena of plasticity and fracture.
  • It is observed that the considered yield functions are all positively homogeneous, contrarily to the expressions obtained in the case an effective stress plastic criterion is used (see Eq. (53)).
  • The model is completed by the elastic relations Eq. (17).

4.2. Predictions of the model

  • The damage response of the model presented in Sections 2 and 4.1 is analysed considering some simple uniaxial processes.
  • The loading unloading behaviour is considered in Fig. 9b.
  • Due to the presence of damage the application of this model to structural problems will cause strain localisation with the consequent mesh-dependency of the numerical results.
  • The parameters c and n particularly influence the degradation of the elastic modulus.

4.3. Multiaxial compression processes: comparison with experimental results

  • The predictions of the model examined in Section 4.2 are compared to experimental data obtained from tests on confined concrete (Van Mier, 1984).
  • In order to avoid the occurrence of localisation, only the data in the hardening phase, before the peak, are used.
  • (Van Mier, 1984), from which the experimental relation elim1 rlim1 has been derived for comparison with the prediction of the model given by Eq. (65).
  • Fig. 18. Envelope of limit uniaxial elastic states for hardening isotropically damaged material.
  • Constant n. Fig. 21. Experimental degradation of the elastic modulus and model prediction for the test of Fig. 20.

5. Numerical algorithm

  • As it has been previously observed, the numerical implementation of the presented coupled plastic model is a generalisation of the standard algorithm for elastoplasticity with internal variables.
  • De, it can be found the algorithmic consistent tangent operator (Cuomo, submitted for publication) UepðeÞ ¼ sup s ½hg; si ecðsÞ dcðsÞ hg0p; si ð66Þ where the vector g0p collects the values of accumulated irreversible kinematic variables at the beginning of the step.
  • The second form of (67) derives from an augmented Lagrangian Regularisation (Bertsekas, 1982) that turns the inequality constraint gi6 0 into the equality constraint gl ¼ 0 and has several advantages.
  • This form is particularly useful in the case of corner points.
  • The coupled Mises criterion (56) has been used for the material with different values of the material constant c. Fig. 25 shows the total reaction versus the imposed vertical displacement for three different values of the damage parameter c (c ¼ 0 is equivalent to absence of damage) in plane strain.

6. Conclusions

  • An internal variable model of plasticity coupled with damage has been formulated in the framework of GSMM.
  • Various damaging material behaviours have been modelled, as plastic-hardening or cohesive fracturelike behaviour, depending on the choice of the dissipation functional, and therefore of the generalised elastic domain.
  • Consequently the evolution laws of any internal variable are obtained in a unified way.
  • These can be easily identified by means of usual experimental tests.
  • Several mechanical behaviours have been modelled, both in monotonic and in cyclic processes.

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Citations
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Journal ArticleDOI
TL;DR: In this article, a triaxial damage-plastic model for the failure of reinforced concrete columns is presented, where the plasticity part is based on the effective stress and the damage model is driven by the plastic strain.

477 citations

Journal ArticleDOI
TL;DR: Gabrio Piola's scientific papers have been underestimated in mathematical physics literature as mentioned in this paper, but a careful reading of them proves that they are original, deep and far-reaching, and even even...
Abstract: Gabrio Piola’s scientific papers have been underestimated in mathematical physics literature. Indeed, a careful reading of them proves that they are original, deep and far-reaching. Actually, even ...

362 citations

Posted Content
TL;DR: In this paper, the authors show that non-local and higher gradient continuum mechanics was conceived already in Piola's works and explain the reasons of the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola contribution.
Abstract: Gabrio Piola's scientific papers have been underestimated in the mathematical-physics literature. Indeed a careful reading of them proves that they are original, deep and far reaching. Actually -even if his contribution to mechanical sciences is not completely ignored- one can undoubtedly say that the greatest part of his novel contributions to mechanics, although having provided a great impetus and substantial influence on the work of many preminent mechanicians, is in fact generally ignored. It has to be remarked that authors [10] dedicated many efforts to the aim of unveiling the true value of Gabrio Piola as a scientist; however, some deep parts of his scientific results remain not yet sufficiently illustrated. Our aim is to prove that non-local and higher gradient continuum mechanics was conceived already in Piola's works and to try to explain the reasons of the unfortunate circumstance which caused the erasure of the memory of this aspect of Piola's contribution. Some relevant differential relationships obtained in Piola [Piola, 1845-6] are carefully discussed, as they are still nowadays too often ignored in the continuum mechanics literature and can be considered the starting point of Levi-Civita's theory of Connection for Riemannian manifolds.

335 citations

Journal ArticleDOI
TL;DR: In this article, a Lagrangian action is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments.
Abstract: In this paper a stationary action principle is proved to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments. We remark that these fluids are sometimes also called Korteweg–de Vries or Cahn–Allen fluids. In general, continua whose deformation energy depends on the second gradient of placement are called second gradient (or Piola–Toupin, Mindlin, Green–Rivlin, Germain or second grade) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both the material and spatial descriptions and the corresponding Euler–Lagrange equations and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and ∇C or on C−1 and ∇C−1, where C is the Cauchy–Green deformation tensor. When particularized to energies which characterize fluid materia...

222 citations

Journal ArticleDOI
TL;DR: In this article, the authors focus on the design of wave-guides aimed to control wave propagation in micro-structured continua, with particular attention to piezoelectromechanical structures, having a strong coupling between macroscopic motion and some internal degrees of freedom.

199 citations

References
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TL;DR: In this article, the authors present Mathematical Preliminaries for Kinematics and Kinetics, and Thermodynamic Orthogonality, and Viscoelastic Bodies.
Abstract: Preface. Chapters: 1. Mathematical Preliminaries. 2. Kinematics. 3. Kinetics. 4. Thermodynamics. 5. Material Properties. 6. Ideal Liquids. 7. Linear Elasticity. 8. Inviscid Gases. 9. Viscous Fluids. 10. Plastic Bodies. 11. Viscoelasticity. 12. General Tensors. 13. Large Displacements. 14. Thermodynamic Orthogonality. 17. Plasticity. 18. Viscoelastic Bodies. Bibliography. Subject Index.

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382 citations


"A new thermodynamically consistent ..." refers background in this paper

  • ...Coupling between damage and plasticity is usually introduced redefining the plastic relations for the so called effective stresses (Kachanov, 1958; Cordebois and Sidoroff, 1982; Krajcinovic, 1984; Lemaitre and Chaboche, 1985; Ju, 1989)....

    [...]

Journal ArticleDOI
TL;DR: In this article, a unified framework for coupled elastoplastic and damage theories is developed and a rigorous thermodynamic procedure is followed that is sufficiently general to include anisotropic plasticity and damage formulations, and the concept of effective stress is the critical mechanism for coupling these theories.

381 citations


"A new thermodynamically consistent ..." refers background or result in this paper

  • ...On the contrary, many authors account for damage coupling defining the yield function in the effective stress space (Lemaitre and Chaboche, 1985; Hansen and Schreyer, 1994; Marotti de Sciarra, 1997): ĝ ¼ ĝðr̂; v̂Þ r̂ ¼ Mr v̂ ¼ Mv ð49Þ where M ¼ MðxeÞ is the effective stress operator introduced in…...

    [...]

  • ...Note that the tangent operator is symmetric and it is a generalisation of the standard plasticity tangent operator, as opposite to the one found by other models (e.g. Hansen and Schreyer, 1994)....

    [...]

  • ...The nature of these variables is generally tensorial (Hansen and Schreyer, 1994; Zhu and Cescotto, 1995), although they can be reduced to scalar quantities if only isotropic damage is considered....

    [...]

  • ...…E0M T1 HðxeÞ ¼ M 12 H0M T2 ð10Þ The constitutive equations (7) yield: r ¼ M 11 E0M T1 ee v ¼ M 12 H0M T2 ae ð11Þ and in the effective spaces it holds: r̂ ¼ M1r ¼ E0M T1 ee ¼ E0êe v̂ ¼ M2v ¼ H0M T2 ae ¼ H0âe ð12Þ The operators M1, M2 are in general fourth order tensor (Hansen and Schreyer, 1994)....

    [...]

  • ...…the plastic strain rates are consequently obtained including the damage variable as a parameter, which, in turn, is determined through a specific evolution law independent of the plastic potential (Hansen and Schreyer, 1994; Simo and Ju, 1987; Klisinski and Mr oz, 1988; Marotti de Sciarra, 1997)....

    [...]

DOI
01 Jan 1984
TL;DR: A submitted manuscript is the author's version of the article upon submission and before peer-review as mentioned in this paper, and the final published version features the final layout of the paper including the volume, issue and page numbers.
Abstract: • A submitted manuscript is the author's version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers.

358 citations


"A new thermodynamically consistent ..." refers methods or result in this paper

  • ...The predictions of the model examined in Section 4.2 are compared to experimental data obtained from tests on confined concrete (Van Mier, 1984)....

    [...]

  • ...The results, presented in Table 1, are in reasonable agreement with the experimental data reported by Van Mier (1984)....

    [...]

  • ...Under the classical hypothesis that f is constant w.r.t. the plastic components (Kluitenberg, 1962; Besseling and Van der Giessen, 1994), and observing that _f ¼ of oee _ee þ of oae _ae þ of oxe _xe þ of oT _T ¼ r _ee þ v _ae þ f _xe qs _T ð25Þ the entropy production (24) becomes, using (4): c ¼ r _ep þ v _ap þ f _xp q T gradT P 0 ð26Þ In the case the mechanical dissipation is not coupled with the thermal dissipation, the following inequalities are separately satisfied: s _gp ¼ r _ep þ v _ap þ f _xp P 0 q T gradT P 0 ð27Þ In the sequel, as isothermal processes will be considered, thermal dissipation will be neglected....

    [...]

  • ...(Van Mier, 1984), from which the experimental relation elim1 rlim1 has been derived for comparison with the prediction of the model given by Eq....

    [...]