Higher order strain energy forms are studied and introduced in this section. These new forms are usually linked to non-linear stress-strain relations and are studied in detail in this work These new proposed types of materials are called here Voyiadjis-Kattan materials (Voyiadjis and Kattan, 2013a, 2013b).

One first starts with the linear relation. The linear stress-strain relation σ = E   ε corresponds to the classsical strain energy form U = 1 2 σ   ε . Now, suppose higher powers of the stain are suggested in the expressions like the following 1 2 σ   ε 2 , 1 2 σ   ε 3 , What happens to the stress-strain relations in these cases? This issue is studies in the sequel.

Use will be made of the terminology by Voyiadjis-Kattan material of order n to designate any non-linear elastic material that has a higher-order strain energy of the form 1 2 σ   ε n .

One first starts with the most general from of the stress-strain equation: σ = E   f ( ε ) , where f ( ε ) is an unknown function of the strain that is to be determined. Ther strain energy U in this case is obtained using the following equation: U = ∫ σ   d ε (1)

One now illustrates the general case using the higher-order strain energy form U = 1 2 σ   ε n . Substituting this expression for U into Eq. 1, one obtains: 1 2 σ   ε n =   ∫ σ   d ε (2)

Next, one substitutes the general stress-strain relation σ = E   f ( ε ) into Eq. 2 to obtain: 1 2 E   f ( ε )     ε n =   E ∫ f ( ε ) d ε (3)

Simplifying the above relation, one obtains:   f ( ε )     ε n = 2 ∫ f ( ε ) d ε (4)

Differentiating both sides of the above equations lead to the following: f / ( ε )   ε n + n   f ( ε )   ε n − 1 = 2 f ( ε ) (5)

The above is the governing differential equation of the system and is solved using the MATLAB Symbolic Math Toolbox. The solution is obtained as follows: f ( ε ) =   1 ε n e − 2 / [ ( n − 1 ) ε ( n − 1 ) ] (6)

Substituting the above expression into the general constitutive relation σ = E   f ( ε ) , one obtains: σ = E     1 ε n e − 2 / [ ( n − 1 ) ε ( n − 1 ) ] (7)

The above solution is obtained after applying the initial condition that the stress is zero when the strain is zero. The above equation is a non-linear stress-strain relationship that governs the behavior of the Voyiadjis-Kattan material of order n .

For certain selected values of n, the results are shown in Table 1. Other existing materials similar to the Voyiajdis-Kattan material are shown in Table 2. Figure 1 showns a graph of the various stress-strain relations of Table 1. Figure 1 is generated based on Eq. 7 and proper units appear on the figure.

One considers a linear elastic material with modulus of elasticity E . Another confiuguration of the material is considered that is fictitious with no damage with the modulus Ε ¯ . In order to compute the effective elastic modulus Ε ¯ in this case, one may use the hypothesis of elastic energy equivalence where the elastic strain energy is assumed to be equal in both configurations (Sidoroff, 1981). Figure 2 is obtained based on Eq. 9 and Eq. 11 below. Proper units and a proper legend now appear on the figure.

The scalar damage variable ℓ is defined in terms of the reduction in the elastic modulus as follows: ℓ = E ¯ − E E (8) where E is the elastic modulus in the damaged state while E ¯ is the effective elastic modulus (in the fictitious state) with E ¯ > E (see Figure 2). Other researchers used the new damage variable in their wrok—Celentano et al. (2004) and Voyiadjis (1988) and Voyiadjis and Kattan (2009). The expression in Eq. 8 can be re-written as follows: E ¯ = E ( 1 + ℓ ) (9)

Using the hypothesis of elastic energy equivalence one assumes the complementary elastic strain energy ( σ 2 2 E ) to be equal in both configurations, i.e., σ 2 2 E = σ ¯ 2 2 E ¯ (10)

Using the hypothesis of elastic energy equivalence and using Eq. 10, one obtains σ ¯ = E ¯ E   σ . In this case, it can be easily shown that the damage variable ℓ = E ¯ − E E will yield the relation σ ¯ = σ   1 + ℓ .

Postulating a new hypothesis of higher-order energy equivalence in the form 1 2 σ 2   ε = 1 2 σ ¯ 2   ε ¯ (11a)

one consequently obtains: σ 3 2 E = σ ¯ 3 2 E ¯ (11b) Finally, one obtains the relation σ ¯ = E ¯ E 3 (11c)

In this case, it is easily shown that using ℓ = E ¯ − E E will yield the relation σ ¯ = σ     1 + ℓ 3 (11d)

For the general case and for a general value of n, one obtains: σ ¯ = E ¯ E n (12a)

In this case, it is easily shown that using ℓ = E ¯ − E E will yield the general relation σ ¯ = σ     1 + ℓ n (12b)

Several curves are plotted on the same graph paper to show the relations between the ratio of the stresses σ ¯ σ and ℓ using Eqs 11b, 12b (see Figure 3). It is clear that for the limiting case when n → ∞ , the curve has a constant value at 1. Figure 3 is generated based on Eq. 12b and proper units appear on the figure.

One now explains the above results using the formulas derived for ℓ . Starting with the formula σ ¯ = σ     1 + ℓ n of Eq. 12b which was derived in the previous paragraphs one now studies the case when n → ∞ . In this case, the following is obtained: σ ¯ = σ   1 + ℓ n = σ   ( 1 + ℓ ) 1 n = σ   ( 1 + ℓ ) 1 ∞ = σ   ( 1 + ℓ ) 0 = σ ⋅ 1 = σ (13)

Therefore one obtains σ ¯ = σ irrespective of the value of the damage variable ℓ . The following is a summary of the main concepts and results in this section:

1. The Voyiadjis-Kattan material of order n is a non-linear elastic material which has strain energy of the form 1 2 σ   ε n , where n is greater than 1.

In this section, the precise equations governing the elastic stiffness transformation for Voyiadjis-Kattan materials are derived (Voyiadjis and Kattan, 2013a; 2012b; 2012c; 2013b; 2014). For this derivation, use is made of the classical damage variable that is defined in terms of area reduction. In this regard, the effective stress is given by: σ ¯ = σ 1 − φ (14) where σ is the Cauchy stress and ϕ is the classical damage variable.

Utilizing a hypothesis of higher-order energy equivalence in the following form: 1 2   σ ¯   ε ¯ n = 1 2   σ   ε n (15) and substituting Eq. 14 into Eq. 15, and simplifying, one obtains the following expression for the effective strain: ε ¯ n = ( 1 − φ )   ε n (16)

It should be noted that the stress-strain relationship for the Voyiadjis-Kattan material of order n is given by Eq. 7. Rewriting Eq. 7 in the effective fictitious configuration, one obtains: σ ¯ = E ¯     1 ε ¯ n e − 2 / [ ( n − 1 ) ε ¯ ( n − 1 ) ] (17)

Substituting for the effective stress from Eq. 14 into Eq. 17, one obtains: σ = ( 1 − φ   ) E ¯     1 ε ¯ n e − 2 / [ ( n − 1 ) ε ¯ ( n − 1 ) ] (18)

Next, substituting for the stress from Eq. 7 into Eq. 18 and simplifying the resulting equation, one obtains: E E ¯   ε ¯ n ε n = ( 1 − φ )   e − 2 / ( ( n − 1 ) ε n − 1 ) e − 2 / ( ( n − 1 ) ε ¯ n − 1 ) (19) and furthermore substituting Eq. 16 into Eq. 19 and simplifying one obtains: E E ¯ = e 2 ε ¯ n − 1 − 2 ε n − 1 ( n − 1 )   ε ¯ n − 1 ε n − 1 (20)

The above relation can be re-written in the following form: ln E E ¯ = 2 ε ¯ n − 1 − 2 ε n − 1 ( n − 1 )   ε ¯ n − 1 ε n − 1 (21)

Again, substituting Eq. 16 into Eq. 21 and simplifying, one obtains: ln E E ¯ = 2   [ ( 1 − φ ) 1 − 1 / n − 1 ] ( n − 1 )   ( 1 − φ ) 1 − 1 / n ε n − 1 (22)

It should be noted that as one approaches infinity, a dimension is crossed and evolves into of four dimensions. Thus undamageable materials are a type of four-dimensional material. Their realization in the manufacturing technology will require work in the fourth dimension. Another type of four-dimensional materials will be the bimodal material of Section 3.3 below where infinity is approached and a dimension is crossed again.

Based on the recent work of the authors (Voyiadjis and Kattan, 2017a; 2017b), one utilizes a new scalar bur non-linear damage variable φ ∗ defined as follows: φ ∗ = 2   φ − φ 2 (23)

It is noted from Figure 4 that both damage variables satisfy the same boundary condition but while φ is linear, the new damage variable φ ∗ is non-linear. Figure 4 is generated based on Eq. 23 and it now appears with units and a proper legend.

Based on the above equations, one can then write the following relation: σ ¯ = σ 1 − φ ∗ = σ 1 − 2 φ − φ 2 (24)

The values of the effective stress are real when the damage variable φ has values in the range 0 < φ < 2 . A plot of the expression of the effective stress of Eq. 24 is shown in Figure 5 for the range of values 0 < φ < 2 . The value of two for the damage variable is twice the rupture value of 1 for the damage variable in classical damage mechanics. However, the authors have no physical interpretation for the value of two for the damage variable.

The following observations are made regarding Figure 5 and the associated Eq. 7: Note that Figure 5 is generated based on Eq. 24 and appears with proper units.

1. The simple expression shown in Eq. 24, along with Figure 5, clearly describes a damage stage that is followed by a healing stage.

Using the hypothesis of elastic strain equivalence, substituting the elastic constitutive relations ε = σ / E and ε ¯ = σ ¯ / E ¯ , along with using Eq. 24, and simplifying, one obtains the following expression for the elastic stiffness transformation: E = E ¯   ( 1 − 2   φ − φ 2 ) (25)

Alternatively, using the hypothesis of elastic energy equivalence, substituting Eq. 24 into Eq. 10, and simplifying (while assuming n = 1 ), one obtains the following expression for the elastic strain transformation: ε ¯ = ε   ( 1 − 2   φ − φ 2 ) (26)

Finally one obtains the following expression for the elastic stiffness in this case: E = E ¯   ( 1 − 2   φ − φ 2 ) 2 (27)

The relations of Eqs 27, 25 are plotted in Figure 6. The results of this section are summarized in Table 3.

Further developments of the theory derived in Section 4 for self-regenerating materials are shown in this section especially with the loss of stiffness and its further recovery. These results can be extended to the hypothetical case when n → ∞ . In this case interesting results are obtained, and a new type of material emerges that can be constructed mathematically. This new limit material is termed a bimodal material.

Both Eqs 25, 27 for the elastic stiffness transformation due to damage can be generalized using the following expression: E = E ¯   ( 1 − 2   φ − φ 2 ) n (28) where n is an integer exponent with n = 1 ,   2 ,   3 ,   4 ,     ..... . It is noted that Eq. 25 of the hypothesis of elastic strain equivalence is recovered using n = 1 , while Eq. 27 of the hypothesis of elastic energy equivalence is recovered using n = 2 . The material behavior described by the generalized Eq. 11 is called a self-regenerating material of order n (see Figure 6). Note that Figure 6 is generated based on Eq. 28 and appears with proper units.

The expression of Eq. 28 is plotted in Figure 7 for several values of the integer exponent n . The special case when n → ∞ is illustrated separately in Figure 6. The curve obtained in Figure 8 represents the limit of the sequence of curves appearing in Figure 7 as n → ∞ . This limiting case is very interesting as it gives rise to a new type of material that has some curious and strange characteristics. It should be noted that both the self-regenerating materials of order n of Figure 7, as well as the hypothetical limit material of Figure 8 do not currently exist except as biological tissue which is explained in Section 4. Figures 7, 8 are generated based on Eq. 28 and appear with proper units.

It is very interesting and paramount to observe in this section the strange behavior of the limit material of Figure 8 which naturally exists as biological tissue with its capability to fully heal itself (see Section 4) As shown in Figure 8, the elastic stiffness of this material is zero everywhere except at the two end points, i.e., at φ = 0 and φ = 2 . This means that the stiffness of the material vanishes as soon as the loading starts and remains vanished until the final load is applied at the end of the deformation and damage process. At the final point of loading, it seems that the elastic stiffness appears suddenly, behaving in a bimodal way, to its full extent. Thus, this material exhibits vital behavior in the sense that the elastic stiffness disappears due to excessive damage at the start of loading, and biologically mends itself through tissue regeneration at the end of loading. The elastic stiffness vanishes throughout the loading process between the start point and the end point. Therefore, the material exhibiting the characteristics shown in Figure 8 is termed a bimodal material. It is emphasized that the bimodal material is the limit of the self-regenerating material of order n as n → ∞ . The bimodal material does not exist currently but the basic equations governing its behavior are formulated in this work.

The main characteristics of the postulated bimodal material are summarized below based on Eq. 28 and Figure 8:

1. The bimodal material suffers a sudden drop of its elastic stiffness from E ¯ to zero at the starting point of loading.

It should be noted that as one reaches infinity, a dimension is crossed and enters the mathematics of four dimensions. Thus this bimodal material is another type of four-dimensional materials. The first type of four-dimensional material was the undamageable material of Section 2.