Applications of tensor functions to the formulation of constitutive equations involving damage and initial anisotropy

doi:10.1016/0013-7944(86)90023-8

Engineering Fracture Mechanics

Volume 25, Issues 5–6, 1986, Pages 573-584

https://doi.org/10.1016/0013-7944(86)90023-8 Get rights and content

Abstract

In this paper some results of the tensor function theory are applied to the formulation of constitutive equations of isotropic and anisotropic materials in the secondary and tertiary creep stage. The creep process, in its tertiary phase, is characterized by a damage tensor. Because of its microscopic nature, damage has, in general, an anisotropic character even in cases where the material was originally isotropic, i.e. isotropic in its virgin state. Fissure orientation and length cause anisotropic macroscopic behaviour. In the first part of the paper some possible ways of representing constitutive equations involving (initial) anisotropy of the material (e.g. from rolling) and involving anisotropic creep-damage are dealt with. The formulations of such equations are based upon theorems concerning tensor-valued functions. Furthermore, some simplified constitutive equations for more practical use are discussed. The main problem of this part is: to find an irreducible set of tensor generators. Besides the problem of finding such tensor generators it is very important to determine the scalar coefficients in constitutive equations as functions of the invariants and experimental data. The second part of the paper is concerned with the determination of the scalar functions. This can be done by using tensorial interpolation methods as pointed out in detail.

References (15)

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Interpolation methods for tensor functions
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Zur Aufstellung von Stoffgleichungen in der Kriechmechanik anisotroper Körper
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J. Betten
Constitutive equations of isotropic and anisotropic materials in the secondary and tertiary creep stage
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Material gleichungen zur Beschreibung des sekundären und tertiären Kriechverhaltens anisotroper Stoffe
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On the time to failure under creep conditions (in Russian)
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(1958)

There are more references available in the full text version of this article.

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The effective stress tensor of anisotropic damage was introduced by definition of different tensorial form of damage parameter to explain damage induced anisotropy. For example, the 2nd rank symmetric damage tensor was proposed by some authors [16,10,17,14]. Transformation of real stress tensor to the effective stress tensor was also carried out by the definition of the fourth rank damage tensor [18,19].
In this study, the effect of developed anisotropic damage on the ratcheting is studied. Damage occurs during extra loading in an anisotropic manner. A modified anisotropic continuum damage model of Murakami is developed in this paper and combined with nonlinear kinematic hardening model of Chaboche. The constitutive relations for anisotropic damage of elastoplastic materials are developed based on irreversible thermodynamics. 2nd rank symmetric damage tensor, the kinematic hardening tensor and tensor of damage potential surface movement are specified according to the internal state manifold which is consistent to the thermodynamic state of solids. The symmetrization of effective stress tensor is carried out by the proposed method of Cordebois and Sidoroff (1982). The proposed yield surface distortion model consists of equivalent effective stress and it distorts in the stress space by evolution of damage parameter. Thus, the induced anisotropy is due to plastic deformation and damage evolution. Combination of the nonlinear kinematic hardening model of Chaboche and the model of anisotropic continuum damage is applied to cyclic loading modeling and the ratcheting strains are predicted under various loading paths.
Ductile damage prediction in sheet and bulk metal forming
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This paper is dedicated to the presentation of an advanced 3D numerical methodology for virtual sheet and/or bulk metal forming simulation to predict the anisotropic ductile defects occurrence. First, the detailed formulation of thermodynamically-consistent fully coupled and fully anisotropic constitutive equations is given. The proposed constitutive equations account for the main material nonlinearities as the anisotropic plastic flow, the mixed isotropic and kinematic hardening and the anisotropic ductile damage under large inelastic strains. Second, the related numerical aspects required to solve the initial and boundary value problem (IBVP) are very briefly presented in the framework of the 3D finite element method. The global resolution schemes as well as the local integration schemes of the fully coupled constitutive equations are briefly discussed. Finally, some typical examples of sheet and bulk metal forming processes are numerically simulated using the proposed numerical methodology.
Damage anisotropy and its effect on the plastic anisotropy evolution under finite strains
2015, International Journal of Solids and Structures
Citation Excerpt :
This approach finds its limitation when the damage develops in plans of different orientations. Damage representation by second-rank tensors (Vakulenko and Kachanov, 1971; Dragon and Mroz, 1979; Cordebois and Sidoroff, 1979, 1980; Kachanov, 1980; Murakami and Ohno, 1981; Oda, 1983; Betten, 1986; Murakami, 1987; Suaris, 1987; Murakami, 1988, 2012; Chaboche, 1992, 1993; Ladevèze, 1993; Hansen and Schreyer, 1994; Voyiadjis and Kattan, 1996; Desmorat, 2000; Lemaitre et al., 2000; Brünig, 2002, 2003; Lemaitre and Desmorat, 2005; Desmorat and Cantournet, 2007; among others.) : The geometrical properties of the microcracks may be described, in a continuum sense, by a second-rank tensor.
In this work, non-associative finite strain anisotropic elastoplastic model fully coupled with anisotropic ductile damage is considered. First, the kinematics of large deformation based on multiplicative decomposition of the total transformation gradient is briefly recalled. Using a thermodynamically-consistent framework, the effective state variables are defined to introduce the effect of the anisotropic damage through the total energy equivalence assumption. For an accurate plastic anisotropy description, non-associative plasticity framework is considered for which equivalent stresses in yield function and in plastic potential are separately defined. The evolution equations governing the overall dissipative phenomena are deduced from the generalized normality rule applied to the plastic potential, while the consistency condition is still deduced from the yield function. Applications are made to a generic material defined by selected values of material parameters and considering different typical strain controlled loading paths as uniaxial tension (UT), plane tension (PT) and equibiaxial tension (ET). For each loading path, the effect of the anisotropic damage evolution on the plastic anisotropy and vice-versa is carefully studied in the context of finite plasticity assuming small elastic strains.
A thermodynamically consistent framework for saturated viscoplastic rock-materials subject to damage
2012, Mechanics Research Communications
The aim of this study is to build a thermodynamically consistent theoretical framework to model viscoplasticity and damage in saturated geomaterials. The induced anisotropic damage is represented by a second-order tensor. The key point of the model formulation is the definition of a “double effective stress”, stemming from the concept of “damaged effective stress” (used in Continuum Damage Mechanics to couple damage and viscoplasticity in solids), and from the concept of “hydro-mechanical effective stress” (used in poromechanics to account for solid–fluid interactions). The dissipation potential is expressed as a function of the “double effective stress”, which makes it possible to couple creep, fluid flow and damage in a consistent framework.
A temperature dependent creep damage model for polycrystalline ice
2012, Mechanics of Materials
We present a continuum damage model for the temperature dependent creep response of polycrystalline ice under a multiaxial state of stress, suited for ice in polar regions. The proposed model is based on a thermo-viscoelastic constitutive law for ice creep and a local orthotropic damage accumulation law for tension, compression and shear loadings. Orthotropic damage is represented by a symmetric second-order damage tensor and its effect on creep is incorporated through the effective stress concept. The unknown model parameters are first calibrated using published experimental data from constant uniaxial stress tests and then predictions are made for constant strain rate and multiaxial loadings. The predicted results are in good agreement with both experimental and numerical results in the literature illustrating the viability of the proposed model. The model is mainly intended for studying the failure mechanisms of polar ice at low deformation rates with depth varying temperature profiles.
On kinematic, thermodynamic, and kinetic coupling of a damage theory for polycrystalline material
2010, International Journal of Plasticity
The paper proposes a new consistent formulation of polycrystalline finite-strain elasto-plasticity coupling kinematics and thermodynamics with damage using an extended multiplicative decomposition of the deformation gradient that accounts for temperature effects. The macroscopic deformation gradient comprises four terms: thermal deformation associated with the thermal expansion, the deviatoric plastic deformation attributed to the history of dislocation glide/movement, the volumetric deformation gradient associated with dissipative volume change of the material, and the elastic or recoverable deformation associated with the lattice rotation/stretch. Such a macroscopic decomposition of the deformation gradient is physically motivated by the mechanisms underlying lattice deformation, plastic flow, and evolution of damage in polycrystalline materials. It is shown that prescribing plasticity and damage evolution equations in their physical intermediate configurations leads to physically justified evolution equations in the current configuration. In the past, these equations have been modified in order to represent experimentally observed behavior with regard to damage evolution, whereas in this paper, these modifications appear naturally through mappings by the multiplicative decomposition of the deformation gradient. The prescribed kinematics captures precisely the damage deformation (of any rank) and does not require introducing a fictitious undamaged configuration or mechanically equivalent of the real damaged configuration as used in the past.

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Applications of tensor functions to the formulation of constitutive equations involving damage and initial anisotropy

Abstract

Zur Aufstellung von Stoffgleichungen in der Kriechmechanik anisotroper Körper

Rheol. Acta

Constitutive equations of isotropic and anisotropic materials in the secondary and tertiary creep stage

Material gleichungen zur Beschreibung des sekundären und tertiären Kriechverhaltens anisotroper Stoffe

Z. angew. Math. Mech.

Elastizitäts- und Plastizitätslehre

Tensor functions involving second-order and fourth-order argument tensors

On the time to failure under creep conditions (in Russian)

Izv. Akad. Nauk USSR Otd. Tekh. Nauk