Abstract
Within the framework of continuum damage mechanics, several novel and basic notions are proposed in this chapter. These ideas focus on the nature of the two damage and healing processes, as well as providing a consistent and systematic description for the concepts of damageability and material integrity. To that goal, the following four sections are presented: The logarithmic and exponential damage variables are introduced in section “A New Damage Variable” along with comparisons to the classical damage variable. Section “Integrity and Damageability of Materials” introduces a novel damage mechanics formulation that includes the two aspects of damage-integrity and healing-damageability. The damage and integrity variables can all be derived from the damage-integrity angle, while the healing variable and damageability variable can be calculated from the healing-damageability angle. Section “The Integrity Field” introduces the new integrity field concept, whereas section “The Healing Field” introduces the new healing field concept. These two domains are offered as a broadening of the traditional damage and integrity ideas.
Similar content being viewed by others
References
C. Basaran, S. Nie, An irreversible thermodynamic theory for damage mechanics of solids. Int. J. Damag. Mech. 13(3), 205–224 (2004)
C. Basaran, H. Tang, Implementation of a thermodynamic framework for damage mechanics of solder interconnects in microelectronic packaging. Int. J. Damag. Mech. 11(1), 87–108 (2002)
C. Basaran, C.Y. Yan, A thermodynamic framework for damage mechanics of solder joints. Trans. ASME J. Electron. Packag. 120, 379–384 (1998)
C. Basaran, M. Lin, H. Ye, A thermodynamic model for electrical current induced damage. Int. J. Solids Struct. 40(26), 7315–7327 (2003)
C. Basaran, H. Tang, S. Nie, Experimental damage mechanics of microelectronics solder joints under fatigue loading. Mech. Mater. 36, 1111–1121 (2004)
I. Carol, E. Rizzi, K. William, An ‘Extended’ Volumetric/Deviatoric Formulation of Anisotropic Damage Based on a Pseudo-Log Rate. Technical Report No. GT-023, ETSECCPB-UPC, E-08034. (Barcelona, 2002)
D.J. Celentano, P.E. Tapia, J.-L. Chaboche, Experimental and numerical characterization of damage evolution in steels, in Mecanica Computacional, ed. by G. Buscaglia, E. Dari, O. Zamonsky, vol. XXIII, (Bariloche, 2004)
M.K. Darabi, R.K. Abu Al-Rub, D.N. Little, A continuum damage mechanics framework for modeling micro-damage healing. Int. J. Solids Struct. 49, 492–513 (2012)
I. Doghri, Mechanics of Deformable Solids: Linear and Nonlinear, Analytical and Computational Aspects (Springer-Verlag, Berlin, 2000)
E.M. Gunel, C. Basaran, Damage characterization in non-isothermal stretching of acrylics: Part I theory. Mech. Mater. 43(12), 979–991 (2011a)
E.M. Gunel, C. Basaran, Damage characterization in non-isothermal stretching of acrylics: Part II experimental validation. Mech. Mater. 43(12), 992–1012 (2011b)
N.R. Hansen, H.L. Schreyer, A thermodynamically consistent framework for theories of elastoplasticity coupled with damage. Int. J. Solids Struct. 31(3), 359–389 (1994)
L. Kachanov, On the creep fracture time. Izv Akad, Nauk USSR Otd Tech 8, 26–31 (1958) (in Russian)
P.I. Kattan, G.Z. Voyiadjis, A coupled theory of damage mechanics and finite strain Elasto-plasticity – Part I: damage and elastic deformations. Int. J. Eng. Sci. 28(5), 421–435 (1990)
P.I. Kattan, G.Z. Voyiadjis, A plasticity-damage theory for large deformation of solids – Part II: Applications to finite simple shear. Int. J. Eng. Sci. 31(1), 183–199 (1993)
P.I. Kattan, G.Z. Voyiadjis, Decomposition of damage tensor in continuum damage mechanics. J. Eng. Mech. ASCE 127(9), 940–944 (2001a)
P.I. Kattan, G.Z. Voyiadjis, Damage Mechanics with Finite Elements: Practical Applications with Computer Tools (Springer-Verlag, Berlin, 2001b)
D. Krajcinovic, Damage Mechanics (North Holland, 1996, 776 page)
P. Ladeveze, J. Lemaitre, Damage effective stress in quasi-unilateral conditions, in The 16th International Congress of Theoretical and Applied Mechanics, (Lyngby, Denmark, 1984)
P. Ladeveze, M. Poss, L. Proslier, Damage and fracture of tridirectional composites, in Progress in Science and Engineering of Composites. Proceedings of the Fourth International Conference on Composite Materials, Japan Society for Composite Materials, vol. 1, (1982), pp. 649–658
H. Lee, K. Peng, J. Wang, An anisotropic damage criterion for deformation instability and its application to forming limit analysis of metal plates. Eng. Fract. Mech. 21, 1031–1054 (1985)
G. Lubineau, A pyramidal Modeling scheme for laminates – Identification of transverse cracking. Int. J. Damag. Mech. 19(4), 499–518 (2010)
G. Lubineau, P. Ladeveze, Construction of a micromechanics-based intralaminar mesomodel, and illustrations in ABAQUS/standard. Comput. Mater. Sci. 43(1), 137–145 (2008)
B. Luccioni, S. Oller, A directional damage model. Comput. Methods Appl. Mech. Eng. 192, 1119–1145 (2003)
Y. Rabotnov, Creep rupture, in Proceedings, Twelfth International Congress of Applied Mechanics, Stanford, 1968, ed. by M. Hetenyi, W. G. Vincenti, (Springer-Verlag, Berlin, 1969), pp. 342–349
J.R. Rice, Inelastic constitutive relations for solids: An internal variable theory and its application to metal plasticity. J. Mech. Phys. Solids 19, 433–455 (1971)
F. Sidoroff, Description of anisotropic damage application in elasticity, in IUTAM Colloqium on Physical Nonlinearities in Structural Analysis, (Springer-Verlag, Berlin, 1981), pp. 237–244
G.Z. Voyiadjis, Degradation of elastic modulus in elastoplastic coupling with finite strains. Int. J. Plast. 4, 335–353 (1988)
G.Z. Voyiadjis, P.I. Kattan, A coupled theory of damage mechanics and finite strain elasto-plasticity – Part II: Damage and finite strain plasticity. Int. J. Eng. Sci. 28(6), 505–524 (1990)
G.Z. Voyiadjis, P.I. Kattan, A plasticity-damage theory for large deformation of solids – Part I: Theoretical formulation. Int. J. Eng. Sci. 30(9), 1089–1108 (1992)
G.Z. Voyiadjis, P.I. Kattan, Damage Mechanics (Taylor and Francis (CRC Press), 2005)
G.Z. Voyiadjis, P.I. Kattan, Advances in damage mechanics: Metals and metal matrix composites with an introduction to fabric tensors, 2nd edn. (Elsevier, 2006)
G.Z. Voyiadjis, P.I. Kattan, A comparative study of damage variables in continuum damage mechanics. Int. J. Damag. Mech. 18(4), 315–340 (2009)
G.Z. Voyiadjis, P.I. Kattan, Mechanics of damage processes in series and in parallel: A conceptual framework. Acta Mech. 223(9), 1863–1878 (2012)
G.Z. Voyiadjis, P.I. Kattan, Healing and super healing in continuum damage mechanics. Int. J. Damag. Mech. 23(2), 245–260 (2014)
G.Z. Voyiadjis, A. Shojaei, G. Li, A thermodynamic consistent damage and healing model for self-healing materials. Int. J. Plast. 27(7), 1025–1044 (2011)
G.Z. Voyiadjis, A. Shojaei, G. Li, P.I. Kattan, A theory of anisotropic healing and damage mechanics of materials. Proc. R. Soc. A Math. Phys. Eng. Sci. 468(2137), 163–183 (2012a)
G.Z. Voyiadjis, A. Shojaei, G. Li, P.I. Kattan, Continuum damage-healing mechanics with introduction to new healing variables. Int. J. Damag. Mech. 21, 391–414 (2012b)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Section Editor information
Appendix
Appendix
Appendix I: Are There Any Limits to the Damage Variable
The restrictions on the values of the damage classical damage variable are derived in this Appendix using a mathematical formulation that is completely consistent. The derivation is dependent on mathematical manipulations rather than the obvious physical components of the problem.
Consider the Eq. (48) as the form of an alternate damage variable ϕ∗. In this example, the effective stress can be expressed as follows:
When the numerator and denominator of Eq. (117) are multiplied by the quantity \( 1+\sqrt{2\phi -{\phi}^2} \) and the result is simplified, one gets:
When one compares Eq. (118) for alternative damage variables to Eq. (2) for classical damage variables, equating these two equations, canceling the stress, and simplifying, one gets the quadratic equation in ϕ:
Appendix II: How to Compose Damage Variables
Damage variable composition is provided as a new way for creating more complex and adaptable damage variables. This method is based on the calculus concept of function composition. Consider the following two functions f(x) andg(x), where x is an independent variable. The function (f ∘ g)(x) = f(g(x)) is described as the combination of the functions f ∘ g.
Damage variables will be defined using the preceding definition of function composition. Consider the following two damage variables: \( {\phi}_1\left(\frac{\overline{A}}{A}\right) \) and \( {\phi}_2\left(\frac{\overline{A}}{A}\right) \), with \( x\equiv \frac{\overline{A}}{A} \) as the independent variable. When considering composition ϕ1 ∘ ϕ2, one will notice that it does not create a consistent damage variable. However, further research reveals that the triple composition ϕ1 ∘ ϕ2 ∘ ϕ1 will generate a new and consistent damage variable. One also discovers that the other triple composition ϕ2 ∘ ϕ1 ∘ ϕ2 may be used to generate a valid damage variable. A variety of basic examples are provided here to demonstrate these observations. Consider the classical damage variable ϕ in Eq. (1) and the exponential damage variable ψ in Eq. (39). The composition will then create a new damage variable as follows:
The new damage variable ϕ ∘ ψ ∘ ϕ starts at 0 for virgin (undamaged) material (at \( \frac{\overline{A}}{A}=1 \)) and develops monotonically to its maximum value of 1 − 1/e = 0.632 upon rupture (at \( \frac{\overline{A}}{A}=0 \)), as shown in Eq. (120). This new damage variable would be beneficial in cases where the maximum amount of damage is constrained and cannot exceed 1, causing the effective stress to burst at infinity (like the classical damage variable).
The next step is to explore a different combination of the two damage factors, ϕ and ψ. As follows, the composition ψ ∘ ϕ ∘ ψ will create a new damage variable as follows:
The new damage variable ψ ∘ ϕ ∘ ψ starts at the value \( {e}^{\frac{1}{e}-1}=0.531 \) for virgin (undamaged) material (at \( \frac{\overline{A}}{A}=1 \)) and grows monotonically to its maximum value of 1 at rupture (at \( \frac{\overline{A}}{A}=0 \)), as shown in Eq. (121) above. This additional damage variable would be beneficial in instances when the minimum damage value is nonzero and may be increased to 1. This differs from the conventional damage variable, which starts at 0. The classical damage variable ϕof Eq. (1) and the logarithmic damage variable L of Eq. (7) will be used to show two more damage variable compositions. The composition ϕ ∘ L ∘ ϕ will then create a new damage variable as follows:
The new damage variable ϕ ∘ L ∘ ϕ starts at negative infinity for virgin (undamaged) material (at \( \frac{\overline{A}}{A}=1 \)) and grows linearly to its maximum value of 1 at rupture, as shown in Eq. (122) above (at \( \frac{\overline{A}}{A}=0 \)). This new damage variable would be beneficial in instances when the damage variable should have a nonzero initial value.
The next step is to explore a different combination of the two damage variables ϕ and L. As follows, the composition L ∘ ϕ ∘ L will create a new damage variable:
The new damage variable L ∘ ϕ ∘ L starts at 0 for virgin (undamaged) material (at \( \frac{\overline{A}}{A}=1 \)) and grows linearly to infinity at rupture (at \( \frac{\overline{A}}{A}=0 \)), as seen in Eq. (123) above. The logarithmic damage variable ψ and this new damage variable are quite comparable.
Other compositions are conceivable, but the four examples above will suffice for this work. Compositions such as ψ ∘ L ∘ ψ, L ∘ ψ ∘ L, ϕ ∘ ψ ∘ L, ψ ∘ ϕ ∘ L, L ∘ ϕ ∘ ψ, L ∘ ψ ∘ ϕ, and others may be of interest to the reader. One can develop (or construct) a customized damage variable that meets his or her demands using the defined process of damage variable composition as provided below. For a damage variable to be legitimate, the following requirements must be met:
-
1.
For the given range of acceptable \( \frac{\overline{A}}{A} \) values, the damage variable must have positive values.
-
2.
The damage variables must increase in a monotonic manner.
-
3.
The third criterion is optional, although it is desired. The values of the damage variables must be in the range of 0–1. This is not required because certain damage variables fall outside of this range. The logarithmic damage variable, for example, can reach infinity.
Each damage variable created via the process of damage variable composition must meet the first two requirements stated above.
Rights and permissions
Copyright information
© 2021 Springer Science+Business Media, LLC, part of Springer Nature
About this entry
Cite this entry
Voyiadjis, G.Z., Kattan, P.I., Jeong, J. (2021). Damageability and Integrity of Materials: New Concepts of the Damage and Healing Fields. In: Voyiadjis, G.Z. (eds) Handbook of Damage Mechanics. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-8968-9_83-1
Download citation
DOI: https://doi.org/10.1007/978-1-4614-8968-9_83-1
Received:
Accepted:
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-8968-9
Online ISBN: 978-1-4614-8968-9
eBook Packages: Springer Reference EngineeringReference Module Computer Science and Engineering