New Tensors for Anisotropic Damage in Continuum Damage Mechanics

2012 ◽  
Vol 134 (2) ◽  
Author(s):  
George Z. Voyiadjis ◽  
Mohammed A. Yousef ◽  
Peter I. Kattan
Keyword(s):  
Mechanical Behavior ◽  
Elastic Energy ◽  
Elastic Strain ◽  
Damage Mechanics ◽  
Continuum Damage Mechanics ◽  
Damage Effect ◽  
Anisotropic Damage ◽  
Continuum Damage ◽  
Energy Equivalence ◽  
Scalar Invariant

In this work, new proposed damage tensors are studied in order to investigate the damage effect variables in the mechanical behavior of materials. All cases studied in this work are defined in terms of the elasticity of the material and based on the hypotheses of both elastic strain equivalence and elastic energy equivalence. Moreover, the new proposed damage tensors are anisotropically expressed in terms of the well-known damage effect tensor M. The principal-valued damage effect tensor is used to obtain the first scalar invariant of that tensor and its inverse, which are employed in expressing and verifying the new proposed damage tensors. The study demonstrates that most of the new proposed damage tensors are verified within the framework of continuum damage mechanics. In addition, new hybrid damage tensors are proposed which are defined in terms of the damage effect tensor and the new proposed damage tensors. The new hybrid damage tensors are eventually expressed in terms of the damage effect tensor.

Decomposition of Elastic Stiffness Degradation in Continuum Damage Mechanics

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
George Z. Voyiadjis ◽  
Peter I. Kattan
Keyword(s):  
Elastic Energy ◽  
Damage Mechanics ◽  
Continuum Damage Mechanics ◽  
Elastic Stiffness ◽  
Stiffness Degradation ◽  
Continuum Damage ◽  
Energy Equivalence ◽  
The One ◽  
Special Case ◽  
Decomposition System

The degradation of elastic stiffness is investigated systematically within the framework of continuum damage mechanics. Consistent equations are obtained showing how the degradation of elastic stiffness can be decomposed into a part due to cracks and another part due to voids. For this purpose, the hypothesis of elastic energy equivalence of order n is utilized. In addition, it is shown that the hypothesis of elastic strain equivalence is obtained as a special case of the hypothesis of elastic energy equivalence of order n. In the first part of this work, the formulation is scalar and applies to the one-dimensional case. The tensorial formulation for the decomposition is also presented that is applicable to general states of deformation and damage. In this general case, one cannot obtain a single explicit tensorial decomposition equation for elastic stiffness degradation. Instead, one obtains an implicit system of three tensorial decomposition equations (called the tensorial decomposition system). Finally, solution of the tensorial decomposition system is illustrated in detail for the special case of plane stress.

Shakedown-Ratcheting Analysis of a Spherical Pressure Vessel by Anisotropic Continuum Damage Mechanics

2018 ◽  
Author(s):  
Ali Nayebi ◽  
Azam Surmiri ◽  
Hojjatollah Rokhgireh
Keyword(s):  
Pressure Vessel ◽  
Damage Mechanics ◽  
Kinematic Hardening ◽  
Continuum Damage Mechanics ◽  
Mapping Method ◽  
Anisotropic Damage ◽  
Continuum Damage ◽  
Interaction Diagram ◽  
Plastic Shakedown ◽  
Spherical Pressure

In cyclic loading and when plastic flow occurs, discontinuities grow. In this research, interaction diagram of Bree has been developed when the spherical pressure vessel contains discontinuities such as voids and microcracks. Bree’s diagram is used for ratcheting assessment of pressurized equipment in ASME III NH. Nature of these defects leads to an anisotropic damage. Anisotropic Continuum Damage Mechanics (CDM) is considered to account effects of these discontinuities on the behavior of the structure. Shakedown – ratcheting response of a hollow sphere under constant internal pressure and cyclic thermal loadings are studied by using anisotropic CDM theory coupled with nonlinear kinematic hardening of Armstrong-Frederick m’s model (A-F). Return mapping method is used to solve numerically the developed relations. Elastic, elastic shakedown, plastic shakedown and ratcheting regions are illustrated in the modified Bree’s diagram. Influence of anisotropic damage due to the plastic deformation is studied and it was shown that the plastic shakedown region is diminished because of the developed damage.

scholarly journals On continuum damage mechanics

2019 ◽  
Vol 52 (3) ◽  
pp. 125-147
Author(s):  
Kari Juhani Santaoja
Keyword(s):  
Stress Tensor ◽  
Effective Stress ◽  
Elastic Strain ◽  
Damage Mechanics ◽  
Volume Fraction ◽  
Strain Tensor ◽  
Continuum Damage Mechanics ◽  
Continuum Damage ◽  
The Matrix ◽  
Elastic Strain Tensor

A material containing spherical microvoids with a Hookean matrix response was shown to take the appearance usually applied in continuum damage mechanics. However, the commonly used variable damage D was replaced with the void volume fraction f , which has a clear physical meaning, and the elastic strain tensor \Bold {ε}^e with the damage-elastic strain tensor \Bold {ε}^{de}. The postulate of strain equivalence with the effective stress concept was reformulated and applied to a case where the response of the matrix obeys Hooke’s law. In contrast to many other studies, in the derived relation between the effective stress tensor \Bold {\Tilde{σ}} and the stress tensor \Bold {σ}, the tensor \Bold {\Tilde{σ}} is symmetric. A uniaxial bar model was introduce for clarifying the derived results. Other candidates for damage were demonstrated by studying the effect of carbide coarsening on creep rate.

An Anisotropic Creep Damage Model for Anisotropic Weld Metal

2007 ◽  
Author(s):  
S. Peravali ◽  
T. H. Hyde ◽  
K. A. Cliffe ◽  
S. B. Leen
Keyword(s):  
Weld Metal ◽  
Test Data ◽  
Damage Evolution ◽  
Damage Mechanics ◽  
Continuum Damage Mechanics ◽  
Isotropic Case ◽  
Anisotropic Damage ◽  
Secondary Creep ◽  
Continuum Damage ◽  
Anisotropic Creep

Past studies from creep tests on uniaxial specimens and Bridgman notch specimens, for a P91 weld metal, showed that anisotropic behaviour (more specifically transverse isotropy) occurs in the weld metal, both in terms of creep (steady-state) strain rate behaviour and rupture times (viz. damage evolution). This paper describes the development of a finite element (FE) continuum damage mechanics methodology to deal with anisotropic creep and anisotropic damage for weld metal. The method employs a second order damage tensor following the work of Murakami and Ohno [1] along with a novel rupture stress approach to define the evolution of this tensor, taking advantage of the transverse isotropic nature of the weld metal, to achieve a reduction in the number of material constants required from test data (and hence tests) to define the damage evolution. Hill’s anisotropy potential theory is employed to model the secondary creep. The theoretical model is implemented in a material behaviour subroutine within the general-purpose, non-linear FE code ABAQUS [2]. The validation of the implementation against established isotropic continuum damage mechanics solutions for the isotropic case is described. A procedure for calibrating the multiaxial damage constants from notched bar test data is described for multiaxial implementations. Also described is a study on the effect of uniaxial specimen orientation on anisotropic damage evolution.

Kinematic Description of Damage

1998 ◽  
Vol 65 (1) ◽  
pp. 93-98 ◽  
Author(s):  
Taehyo Park ◽  
G. Z. Voyiadjis
Keyword(s):  
Fourth Order ◽  
Damage Mechanics ◽  
Effective Strain ◽  
Continuum Damage Mechanics ◽  
Second Order ◽  
Finite Strains ◽  
Damage Effect ◽  
Energy Equivalence ◽  
Simple Mapping ◽  
Damage Tensor

In this paper the kinematics of damage for finite elastic deformations is introduced using the fourth-order damage effect tensor through the concept of the effective stress within the framework of continuum damage mechanics. However, the absence of the kinematic description of damage deformation leads one to adopt one of the following two different hypotheses. One uses either the hypothesis of strain equivalence or the hypothesis of energy equivalence in order to characterize the damage of the material. The proposed approach in this work provides a relation between the effective strain and the damage elastic strain that is also applicable to finite strains. This is accomplished in this work by directly considering the kinematics of the deformation field and furthermore it is not confined to small strains as in the case of the strain equivalence or the strain energy equivalence approaches. The proposed approach shows that it is equivalent to the hypothesis of energy equivalence for finite strains. In this work, the damage is described kinematically in the elastic domain using the fourth-order damage effect tensor which is a function of the second-order damage tensor. The damage effect tensor is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. The constitutive equations of the elastic-damage behavior are derived through the kinematics of damage using the simple mapping instead of the other two hypotheses.

Anisotropic Damage Effect Tensors for the Symmetrization of the Effective Stress Tensor

1997 ◽  
Vol 64 (1) ◽  
pp. 106-110 ◽  
Author(s):  
G. Z. Voyiadjis ◽  
T. Park
Keyword(s):  
Stress Tensor ◽  
Effective Stress ◽  
Fourth Order ◽  
Damage Mechanics ◽  
Continuum Theory ◽  
Continuum Damage Mechanics ◽  
Damage Effect ◽  
Anisotropic Damage ◽  
Tensor Algebra ◽  
Physical Description

Based on the concept of the effective stress and on the description of anisotropic damage deformation within the framework of continuum damage mechanics, a fourth order damage effective tensor is properly defined. For a general state of deformation and damage, it is seen that the effective stress tensor is usually asymmetric. Its symmetrization is necessary for a continuum theory to be valid in the classical sense. In order to transform the current stress tensor to a symmetric effective stress tensor, a fourth order damage effect tensor should be defined such that it follows the rules of tensor algebra and maintains a physical description of damage. Moreover, an explicit expression of the damage effect tensor is of particular importance in order to obtain the constitutive relation in the damaged material. The damage effect tensor in this work is explicitly characterized in terms of a kinematic measure of damage through a second-order damage tensor. In this work, tensorial forms are used for the derivation of such a linear transformation tensor which is then converted to a matrix form.

Anisotropic (Continuum Damage Mechanics)-based multi-mechanism model for semi-crystalline polymer

2016 ◽  
Vol 27 (3) ◽  
pp. 357-386 ◽  
Author(s):  
Walid Ayadi ◽  
Lucien Laiarinandrasana ◽  
Kacem Saï
Keyword(s):  
Experimental Data ◽  
Shape Factor ◽  
Damage Mechanics ◽  
Crystalline Polymer ◽  
Continuum Damage Mechanics ◽  
Degree Of Crystallinity ◽  
Continuum Damage ◽  
Mechanism Model ◽  
Energy Equivalence ◽  
Proposed Model

In this work, the anisotropic damage of semi-crystalline polymers is investigated. The model, developed within a thermodynamic framework, includes the following features: (i) the degree of crystallinity; (ii) the hydrostatic pressure effect; and (iii) the damage anisotropy. The adopted tensorial damage variable is based on the Continuum Damage Mechanics approach under the energy equivalence assumption. For the quantification of the anisotropy, a parameter called “shape factor” is defined as the ratio between the void mean diameter and the void mean height. This parameter is linked to the main axial and the main radial damage components. Experimental data taken from the recent literature using the tomography technique were selected to assess the model capability. Finite element simulations of notched round bar specimens subjected to tensile test stopped at three key loading stages are systematically compared with experimental data. The proposed model was able not only to accurately simulate the macroscopic response of the material, but more interestingly, to reproduce the spatial distribution of the shape factor. This demonstrates the anisotropy effects of the material under study induced at different stages of the deformation.

Numerical Analysis of Growing the Ductile Damage in Structures Reinforced by SMA Using Continuum Damage Mechanics Approach

2018 ◽  
Vol 10 (07) ◽  
pp. 1850070 ◽  
Author(s):  
M. Baghani ◽  
M. Ganjiani ◽  
M. Rezaei
Keyword(s):  
Mechanical Behavior ◽  
Damage Mechanics ◽  
Numerical Study ◽  
Continuum Damage Mechanics ◽  
Continuum Damage ◽  
Nonlinear Finite Element Method ◽  
Damage Growth ◽  
Simulation Results ◽  
Tension Loading ◽  
Aluminum Material

In this paper, the numerical study on mechanical behavior of materials reinforced with shape memory alloys (SMAs) in framework of continuum damage mechanics has been investigated. The investigated structure is an aluminum notched piece reinforced with SMA under tension loading. Simulation of the structure has been conducted with nonlinear finite element method. In numerical simulations, the SMA is embedded in the aluminum material and it is assumed that there is no slip between the aluminum and SMAs. To properly account for the mechanical behavior with damage effects, Lemaitre constitutive model via UMAT code have been developed. Simulation results were verified by experimental results. The Brinson model is used to consider the thermomechanical behavior of SMA. The simulation results showed that the mechanical behavior of aluminum with reinforced aluminum is rather different. Also, the presence of SMAs in the notched piece leads to increasing the energy absorption, piece fracture in higher loadings and a decreasing in the damage growth rate.

Complex Damage Variables in Continuum Damage Mechanics

2015 ◽  
Vol 784 ◽  
pp. 3-10
Author(s):  
George Z. Voyiadjis ◽  
Peter I. Kattan
Keyword(s):  
Damage Mechanics ◽  
Continuum Damage Mechanics ◽  
Elastic Stiffness ◽  
Continuum Damage ◽  
Cross Sectional ◽  
Practical Applications ◽  
Area Reduction ◽  
Variable Approach ◽  
Energy Equivalence ◽  
Mathematical Equations

The concept of complex damage variables is introduced in this work. These damage variables have both real and imaginary parts. They are introduced not to use them in practical applications but to try to derive a direct relationship between the damage due to cross-sectional area reduction and the damage due to elastic stiffness degradation. In addition this concept can provide an insight in addressing the concept of healing that the authors have extensively published as well as the concept of undamageable materials. Toward this goal some success is achieved in the sense that some complicated relationships between the two damage processes are formulated. These relationships and the complex variable approach are novel ideas in Continuum Damage Mechanics. Throughout the formulation the hypothesis of strain equivalence is used in order to simplify the mathematical equations. This work can be extended and generalized by substituting the hypothesis of energy equivalence but this will complicate the equations unnecessarily.

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