Damageability and Integrity of Materials: New Concepts of the Damage and Healing Fields

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.

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:

$$ \overline{\sigma}=\frac{\sigma }{1-{\phi}^{\ast }}=\frac{\sigma }{1-\sqrt{2\phi -{\phi}^2}} $$

(117)

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:

$$ \overline{\sigma}=\frac{\sigma\;\left(1+\sqrt{2\phi -{\phi}^2}\right)}{{\left(1-\phi \right)}^2} $$

(118)

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:

$$ 2{\phi}^2-2\phi =0 $$

(119)

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:

$$ \phi \circ \psi \circ \phi \equiv \phi \left(\psi \left(\phi \right)\right)=\phi \left(\psi \left(1-\frac{\overline{A}}{A}\right)\right)=\phi \left({e}^{-\left(1-\frac{\overline{A}}{A}\right)}\right)=1-{e}^{\frac{\overline{A}}{A}-1} $$

(120)

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:

$$ \psi \circ \phi \circ \psi \equiv \psi \left(\phi \left(\psi \right)\right)=\psi \left(\phi \left({e}^{-\frac{\overline{A}}{A}}\right)\right)=\psi \left(1-{e}^{-\frac{\overline{A}}{A}}\right)={e}^{-\left(1-{e}^{-\frac{\overline{A}}{A}}\right)}={e}^{\left({e}^{-\frac{\overline{A}}{A}}-1\right)} $$

(121)

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:

$$ \phi \circ L\circ \phi \equiv \phi \left(L\left(\phi \right)\right)=\phi \left(L\left(1-\frac{\overline{A}}{A}\right)\right)=\phi \left(-\ln \left(1-\frac{\overline{A}}{A}\right)\right)=1+\ln \left(1-\frac{\overline{A}}{A}\right) $$

(122)

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:

$$ L\circ \phi \circ L\equiv L\left(\phi (L)\right)=L\left(\phi \left(-\ln \frac{\overline{A}}{A}\right)\right)=\psi \left(1+\ln \frac{\overline{A}}{A}\right)=-\ln \left(1+\ln \frac{\overline{A}}{A}\right) $$

(123)

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.