# Comments on the Paper "Some Thoughts Concerning the Existence or Otherwise of an Isotropic Bimodulus Material" by Z. Rigbi, Journal of Engineering Materials and Technology, Transactions of the ASME 102(4) (1980) 383, DOI: 10.1115/1.3224829

Zolochevsky Alexander

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Zolochevsky Alexander. Comments on the Paper "Some Thoughts Concerning the Existence or Otherwise of an Isotropic Bimodulus Material" by Z. Rigbi, Journal of Engineering Materials and Technology, Transactions of the ASME 102(4) (1980) 383, DOI: 10.1115/1.3224829. Research Center "Polytech", Kharkiv, Ukraine. 2025, pp.1-17. hal-05149414

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# Comments on the Paper "Some Thoughts Concerning the Existence or Otherwise of an Isotropic Bimodulus Material" by Z. Rigbi, Journal of Engineering Materials and Technology, Transactions of the ASME 102(4) (1980) 383, DOI: 10.1115/1.3224829

Zolochevsky Alexander

Research Center «Polytech», Kharkiv, Ukraine

e-mail: zolochevsky55@ukr.net

**ABSTRACT.** Many initially isotropic materials (both brittle and ductile) have different stress-strain diagrams under tensile and compressive loading types. A bimodulus elastic model is the simplest bilinear approximation of the strongly nonlinear material behavior in the case with the tension-compression asymmetry. Then we have two elastic moduli (one in tension and the other in compression). It is shown that the two assumptions accepted by Rigbi (1980) for the initially isotropic bimodulus material are wrong. First, the absolute values of the principal strains in the shear plane under the conditions of pure shear are not the same for the isotropic bimodulus material. Second, the superposition principle is not applicable to the bimodulus material. Such statements were made using the constitutive model developed by Zolochevsky in the early 1980s.

**KEY WORDS:** nonlinear elasticity, constitutive modeling, isotropy, tension-compression asymmetry, bimodulus material, pure shear, superposition principle

**For citation:**

Zolochevsky A. Comments on the Paper "Some Thoughts Concerning the Existence or Otherwise of an Isotropic Bimodulus Material" by Z. Rigbi, Journal of Engineering Materials and Technology, Transactions of the ASME 102(4) (1980) 383, DOI: 10.1115/1.3224829. Technical Report. Kharkiv: Research Center «Polytech»; 2025. 17p.

## INTRODUCTION

Tension-compression asymmetry is the characteristic feature commonly found in many artificial and natural materials [1-5]. This material feature occurs when comparing the stress-strain diagrams obtained in basic experiments under uniaxial tension and uniaxial compression. A bimodulus elastic model is the simplest bilinear approximation of the strongly nonlinear material behavior in the case with the tension-compression asymmetry. Then we have two elastic moduli (one in tension and the other in compression).

For example, the stress-strain diagrams of elastomeric foam in tension and compression [6] are given in Fig. 1a. Here stresses have been normalized with respect to the tangent modulus *E*, *E*=40kPa. Experimental data in Fig. 1a are shown by solid line. The results of calculations are represented in Fig. 1a by dashed lines: the thick dashed line corresponds to the strongly nonlinear approximation by Medri [6], and the thin dashed line corresponds to the bilinear approximation of the stress-strain diagrams. Finally, we arrive at the idealization of elastomeric foam in the form of a bimodulus material (Fig. 1b). Furthermore, we obtain for elastomeric foam under study that the elastic modulus in tension is 2.5 times higher than the elastic modulus in compression [6].

©Alexander Zolochevsky, 2025

<figure><img src="image_2.png" /></figure>

It is licensed under a Creative Commons Attribution 4.0 International License.
<figure><img src="image_3.png" /><figcaption>Fig. 1. Uniaxial stress-strain diagrams for elastomeric foam [6] (a) and for bimodulus material (b)</figcaption></figure>

In 1980, Rigbi [7] analyzed the stress-strain state in an initially isotropic bimodulus material under the conditions of pure shear (Fig. 2), when only the tangential stress $\tau$ acts on the four faces of the element *abcd*. In this case the principal stresses are

$$ \sigma_1 = \tau, \sigma_3 = -\tau, \sigma_2 = 0 \quad (1) $$

The principal directions 1 and 3 are the directions along the diagonals *ac* and *bd* (Fig. 2) while the principal direction 2 is perpendicular to the element *abcd*.

<figure><img src="image_4.png" /><figcaption>Fig. 2. Schematic representation of pure shear by Rigbi for isotropic bimodulus material</figcaption></figure>

The shear strain can be defined as

$$ \gamma = \frac{\tau}{G} \quad (2) $$

where *G* is the shear modulus. The diagonal *ac* takes longer after deformation while the diagonal *bd* takes shorter. Thus, the square *abcd* transforms to the rhombus *a'**b'**c'**d'* under the conditions of pure shear.

**Assumption 1 by Rigbi [7].** The principal strains in the shear plane are defined as
$$ \varepsilon_1 = \frac{\gamma}{2}, \varepsilon_3 = -\frac{\gamma}{2} \qquad (3) $$

In other words, it was assumed that $aa' = bb' = cc' = dd'$ in Fig. 2.

**Assumption 2 by Rigbi [7].** It was accepted the applicability of the superposition principle to the bimodulus material.

Using these two assumptions Rigbi made a conclusion [7] that an initially isotropic bimodulus material does not exist. Paper [7] was referenced in [6, 8-13], however, there is no evidence in [6, 8-13] to support this speculation by Rigbi.

From general point of view, it is necessary to note that the relation between elastic strains $\varepsilon_1, \varepsilon_3$ and stresses given by Eq. (1) can be defined using the constitutive model developed for the isotropic bimodulus materials. Started since Ambartsumian and Khachatrian [14], a number of the theories of elasticity have been proposed in the literature to describe the nonlinear elastic deformation of the initially isotropic and anisotropic bimodulus materials. Review of these constitutive models can be found in [1, 2, 15].

In the present report it will be shown using the bimodulus elastic model by Zolochevsky [16] developed in the early 1980s that the two assumptions by Rigbi are wrong.

## CONSTITUTIVE FRAMEWORK BY ZOLOCHEVSKY

The elastic models under consideration at small strains are based on the hypothesis of the existence of the potential function

$$ f = \frac{1}{2} \sigma_e^2 \qquad (4) $$

and law

$$ \varepsilon_{ij} = \frac{\partial f}{\partial \sigma_{ij}} \qquad (5) $$

Here $\sigma_e$ is the so-called equivalent stress which allows us to generalize the data obtained in the basic experiments to the case of a multiaxial stress state, $\sigma_e \ge 0$; $\varepsilon_{kl}$ is the tensor of elastic strains; $\sigma_{kl}$ is the stress tensor.

The equivalent stress for the initially isotropic bimodulus materials was introduced by Zolochevsky in [16] as follows

$$ \sigma_e = \sigma_0 + \sigma \qquad (6) $$

where

$\sigma_0 = \sqrt{AI_1^2 + CI_2}$, $\sigma = BI_1$; $I_1$ and $I_2$ are the first and the second invariants of the stress tensor; $I_1 = \sigma_{kl}\delta_{kl}$, $I_2 = \sigma_{kl}\sigma_{kl}$; the third invariant of the stress tensor is not taken into account; $\delta_{kl}$ is the Kronecker symbol; A, B, C are some material parameters.

Using Eqs. (4)-(6), we obtain the constitutive equation of the isotropic bimodulus elasticity, such as [16]

$$ \varepsilon_{kl} = \sigma_e \left( \frac{AI_1\delta_{kl} + C\sigma_{kl}}{\sigma_0} + B\delta_{kl} \right) \qquad (7) $$

Now we consider a method of determining the parameters A, B and C in Eq. (7). For this purpose, we need the results of the basic experiments under uniaxial tension, uniaxial compression and pure torsion taken on the standard specimens in which uniform stress states in the isotropic bimodulus material under study have been realized.

We assume that considering uniaxial tension, we have the following relation in the direction of loading

$$ \varepsilon_{11} = \frac{\sigma_{11}}{E_{+}} \qquad (8) $$

By analogy, the relation in the case of uniaxial compression is

$$ \varepsilon_{11} = -\frac{|\sigma_{11}|}{E_{-}} \qquad (9) $$
Also, we assume that the relation in the case of pure torsion coincides with Eq. (2). Here $E_+$ and $E_-$ are the elastic modulus in tension and elastic modulus in compression, respectively.

On the other hand, we can use Eq. (7) to write relations describing the elastic deformation in the basic experiments under discussion.

In this regard, we obtain in the case of uniaxial tension that

$$ \varepsilon_{11} = (\sqrt{A+C}+B)^2 \sigma_{11} \quad (10) $$

Similarly, we have for uniaxial compression that

$$ \varepsilon_{11} = -(\sqrt{A+C}-B)^2 |\sigma_{11}| \quad (11) $$

In the case of pure torsion with the tangential stress $\sigma_{12} = \tau$ we obtain from Eq. (7) that

$$ \gamma = 2\varepsilon_{12} = 2C\tau \quad (12) $$

Comparing then Eqs. (8) and (10), (9) and (11), (2) and (12), respectively, we have

$$ C = \frac{1}{2G}, \quad A = \frac{1}{4} \left( \frac{1}{\sqrt{E_+}} + \frac{1}{\sqrt{E_-}} \right)^2 - C, \quad B = \frac{1}{2} \left( \frac{1}{\sqrt{E_+}} - \frac{1}{\sqrt{E_-}} \right) \quad (13) $$

Thus, the material parameters in the theory of the isotropic bimodulus elasticity can be determined by means of Eq. (13).

If it is found from the basic experiments that

$$ E_+ = E_- \quad (14) $$

then we obtain from the third equality in Eq. (13) that $B=0$. In this case, the constitutive equation (7) transforms into relation

$$ \varepsilon_{kl} = AI_1\delta_{kl} + C\sigma_{kl} \quad (15) $$

Thus, the classical theory of elasticity defined by Eq. (15) can be used under condition (14).

## DISCUSSION

A number of comments need to be made in reference to the assumptions by Rigbi and constitutive framework by Zolochevsky.

**Comment 1.** We will determine the elastic strains in an initially isotropic bimodulus material with the biaxial stress given by Eq. (1) under the conditions of pure shear using the constitutive equation (7). Then we have

$$ \varepsilon_1 = (C + B\sqrt{2C})\tau, \quad \varepsilon_3 = (-C + B\sqrt{2C})\tau \quad (16) $$

Using additionally the first equality in Eq. (13) we obtain that

$$ \varepsilon_1 = \frac{\gamma}{2} + B\sqrt{2C}\tau, \quad \varepsilon_3 = -\frac{\gamma}{2} + B\sqrt{2C}\tau \quad (17) $$

Thus, the absolute values of the principal strains in the shear plane under the conditions of pure shear are not the same for the isotropic bimodulus material, and the first assumption by Rigbi [7] is wrong. Also, it is clear that Eq. (17) coincides with Eq. (3) only if $B=0$ under condition (14), i.e. for traditional materials with the same elastic moduli in tension and compression.

**Comment 2.** Considering biaxial stress state defined by Eq. (1) as a combined action of uniaxial tension in the principal direction 1 and uniaxial compression in the principal direction 3 (Fig. 2), we will show that the superposition principle is not applicable to the bimodulus material.

First, in the particular case of uniaxial tension in the principal direction 1 with stress $\sigma_{11} = \tau$ we obtain from Eq. (7) strains in the direction of loading and transverse direction, such as
$$ \epsilon_{11}^{(1)} = (\sqrt{A+C}+B)^2 \tau, \quad \epsilon_{33}^{(1)} = (\sqrt{A+C}+B)\tau \left( \frac{A}{\sqrt{A+C}} + B \right) \qquad (18) $$

Second, in the case of uniaxial compression in the principal direction 3 with stress $\sigma_{33} = -\tau$ we have from Eq. (7) the following strains in the direction of loading and transverse direction

$$ \epsilon_{33}^{(2)} = -(\sqrt{A+C}-B)^2 \tau, \quad \epsilon_{11}^{(2)} = (\sqrt{A+C}-B)\tau \left( -\frac{A}{\sqrt{A+C}} + B \right) \qquad (19) $$

Finally, using the superposition principle in a form $\epsilon_1 = \epsilon_{11}^{(1)} + \epsilon_{11}^{(2)}$, $\epsilon_3 = \epsilon_{33}^{(1)} + \epsilon_{33}^{(2)}$, as well as, Eqs. (18) and (19), we arrive at the following relations

$$ \epsilon_1 = \tau \left[ (\sqrt{A+C}+B)^2 + (\sqrt{A+C}-B) \left( B - \frac{A}{\sqrt{A+C}} \right) \right] \\ \epsilon_3 = -\tau \left[ (\sqrt{A+C}-B)^2 - (\sqrt{A+C}+B) \left( B + \frac{A}{\sqrt{A+C}} \right) \right] \qquad (20) $$

Comparing Eqs. (16) and (20), we can conclude that Eq. (20) transforms into Eq. (16) only if $B=0$ under condition (14), i.e. for traditional materials with the same elastic moduli in tension and compression. In other words, the superposition principle is not applicable to the bimodulus material, and **the second assumption by Rigbi** [7] is **wrong**. Also, it is necessary to note that, from the general point of view, the bimodulus model belongs to the nonlinear models, and the superposition principle cannot be used in the nonlinear analysis.

**Comment 3.** It is known [17, 18] that a sufficient, but not necessary, condition of the uniqueness of the solution of the elastic boundary value problems is the satisfaction of the Drucker's postulate in the formulation

$$ \delta\sigma_{ij}\delta\epsilon_{ij} \ge 0 \qquad (21) $$

Since nonsatisfaction of the Drucker's postulate sometimes means that solution is not unique, let us determine the limitations for the parameters in Eq. (7) under condition (21). In this regard, we have from Eq. (7) that

$$ \delta\sigma_{ij}\delta\epsilon_{ij} = (\delta\sigma_e)^2 + \sigma_e\delta^2\sigma_e \qquad (22) $$

where

$$ \delta^2\sigma_e = \frac{A(\delta I_1)^2 + C\delta\sigma_{kl}\delta\sigma_{kl} - (\delta\sigma_0)^2}{\sigma_0} \qquad (23) $$

Using Eq. (22) and inequality $(\delta\sigma_e)^2 \ge 0$ we obtain that the condition (21) comes down to the requirement of nonnegativity of the equivalent stress, i.e. $\sigma_e \ge 0$, natural in all constitutive equations, as well as, to the inequality

$$ \delta^2\sigma_e \ge 0 \qquad (24) $$

The last condition means that the equivalent stress should be the convex function of the components of the stress tensor.

Now we consider the requirement of positive determinacy of the quadratic form $\sigma_0^2 = AI_1^2 + CI_2$. In this regard, we can write

$$ A(S_{nn}^2) + CS_{kl}S_{kl} > 0 \qquad (25) $$

where

$$ S_{kl} = \sigma_{kl}\delta\sigma_0 - \delta\sigma_{kl}\sigma_0 \qquad (26) $$

Taking into account condition (25) and Eq. (26) we have

$$ A(\delta I_1)^2 + C\delta\sigma_{kl}\delta\sigma_{kl} - (\delta\sigma_0)^2 > 0 \qquad (27) $$
Considering then Eqs. (22) and (23), as well as, conditions (21) and (27), we obtain the satisfaction of the Drucker's postulate. Thus, despite the fact that the verification of the convexity for the potential function should be regarded as a certain limitation on the parameters in a number of the constitutive models [19-25], it can be argued that the Drucker's postulate does not impose severe requirements on the use of the bimodulus model by Zolochevsky.

**Comment 4.** The equivalent stress given by Eq. (6) was used in the theories of plasticity [26] and creep [27] for the initially isotropic materials with tension-compression asymmetry. Also, the modified version of the equivalent stress (6) was used in the fatigue limit criterion and theory of low-cycle fatigue damage [28].

**Comment 5.** The third invariant $I_3 = \sigma_{ij}\sigma_{jk}\sigma_{ki}$ of the stress tensor was included into the expression of the equivalent stress, i.e.

$$ \sigma_e = \sqrt{AI_1^2 + CI_2} + BI_1 + \alpha\sqrt{DI_1^3 + KI_1I_2 + LI_3} \quad (28) $$

based on the three invariants $I_1, I_2, I_3$ and six material parameters $A, B, C, D, K, L$. Assuming in Eq. (28) that $\alpha=0$, we arrive from Eq. (28) at Eq. (6). The equivalent stress given by Eq. (28) was used to describe nonlinear elasticity [29], plasticity [30], and creep [31, 32] of the initially isotropic materials with different behavior in tension and compression.

In 2023 [33], the Hosford-type equivalent stress was formulated by Zolochevsky, such as

$$ \sigma_e = \left[ \left( \sqrt{AI_1^2 + CI_2} \right)^r + (BI_1)^r + \alpha \left( \sqrt[3]{DI_1^3 + KI_1I_2 + LI_3} \right)^r \right]^{-\frac{1}{r}} \quad (29) $$

Here the integer exponent $r$ takes odd values chosen from 1, 3, 5... It is not difficult to see that the Hosford-type equivalent stress given by Eq. (29) coincides with the equivalent stress given by Eq. (28) under condition $r=1$.

**Comment 6.** The equivalent stress given by Eq. (6) is a particular case of the equivalent stress

$$ \sigma_e = \sqrt{a_{ijkl}\sigma_{ij}\sigma_{kl}} + b_{ij}\sigma_{ij} \quad (30) $$

developed for the initially anisotropic materials with tension-compression asymmetry. Here $b_{ij}$ and $a_{ijkl}$ are the material tensors of the second rank and fourth rank, respectively. Actually, considering the following equations to be valid

$$ b_{ij} = B\delta_{ij}, \quad a_{ijkl} = A\delta_{ij}\delta_{kl} + \frac{C}{2}(\delta_{ik}\delta_{jl} + \delta_{li}\delta_{jk}) \quad (31) $$

we arrive from Eq. (30) at Eq. (6).

The equivalent stress given by Eq. (30) was used to describe nonlinear elasticity [34-37], plasticity [38], and creep [39-42] of the initially anisotropic materials with different behavior in tension and compression. Such constitutive framework by Zolochevsky was applied to steels [43-49], light alloys [50-59], nickel-base superalloys [60], composite materials [61-76], rocks [77], woods [78], bone tissues [79-89] and other biological objects [90-92].

For orthotropic materials under assumption that the coordinate axes coincide with the principal directions of anisotropy 1, 2 and 3, the following relations in Eq. (30) are valid

$$ \begin{aligned} a_{1211} &= a_{1222} = a_{1233} = a_{1311} = a_{1322} = a_{1333} = a_{2311} = 0 \\ a_{2322} &= a_{2333} = a_{1213} = a_{1223} = a_{1323} = 0 \\ b_{12} &= b_{23} = b_{13} = 0 \end{aligned} \quad (32) $$

Thus, the symmetric tensors $b_{ij}$ and $a_{ijkl}$ include three and nine independent components, i.e. $b_{11}, b_{22}, b_{33}$, as well as $a_{1111}, a_{2222}, a_{3333}, a_{1122}, a_{1133}, a_{2233}, a_{1212}, a_{1313}$ and $a_{2323}$, respectively. Additionally, we can accept the condition by Hill [93]

$$ \begin{aligned} \frac{1}{2}(a_{1111} + a_{2222} - a_{3333}) &= H_0 = -a_{1122}, & \frac{1}{2}(a_{2222} + a_{3333} - a_{1111}) &= F_0 = -a_{2233} \\ \frac{1}{2}(a_{1111} + a_{3333} - a_{2222}) &= G_0 = -a_{1133}, & N_0 &= 2a_{1212}, L_0 &= 2a_{2323}, M_0 &= 2a_{1313} \end{aligned} \quad (33) $$
Then taking into account Eqs. (32) and (33), we arrive from Eq. (30) at the equivalent stress of simpler structure, such as

$$ \sigma_e = b_{11}\sigma_{11} + b_{22}\sigma_{22} + b_{33}\sigma_{33} + \sqrt{H_0(\sigma_{11} - \sigma_{22})^2 + F_0(\sigma_{22} - \sigma_{33})^2 + G_0(\sigma_{33} - \sigma_{11})^2 + 2N_0\sigma_{12}^2 + 2L_0\sigma_{23}^2 + 2M_0\sigma_{13}^2} \quad (34) $$

Practical recommendations on how and when to use the equivalent stress given by Eq. (34) for the orthotropic materials can be found in [54, 69, 72].

**Comment 7.** An extended version of the equivalent stress for the initially anisotropic materials with tension-compression asymmetry was developed [94, 95], such as

$$ \sigma_e = \sqrt{a_{ijkl}\sigma_{ij}\sigma_{kl}} + b_{ij}\sigma_{ij} + \alpha \sqrt{c_{ijklmn}\sigma_{ij}\sigma_{kl}\sigma_{mn}} \quad (35) $$

Here $c_{ijklmn}$ is the material tensor of the sixth rank. Assuming in Eq. (35) that $\alpha=0$, we arrive from Eq. (35) at Eq. (30). We note also that considering Eq. (31) and, additionally,

$$ c_{ijklmn} = D\delta_{ij}\delta_{kl}\delta_{mn} + \frac{K}{6}(\delta_{ij}\delta_{km}\delta_{ln} + \delta_{ij}\delta_{kn}\delta_{lm} + \delta_{kl}\delta_{im}\delta_{jn} + \delta_{kl}\delta_{in}\delta_{jm} + \delta_{mn}\delta_{ik}\delta_{jl} + \delta_{mn}\delta_{il}\delta_{jk}) + \frac{L}{8}(\delta_{ik}\delta_{jm}\delta_{ln} + \delta_{ik}\delta_{jn}\delta_{lm} + \delta_{il}\delta_{km}\delta_{jn} + \delta_{il}\delta_{kn}\delta_{jm} + \delta_{im}\delta_{kj}\delta_{ln} + \delta_{im}\delta_{kn}\delta_{lj} + \delta_{in}\delta_{kj}\delta_{lm} + \delta_{in}\delta_{km}\delta_{lj}) $$

to be valid, we arrive from the equivalent stress given by Eq. (35) at the equivalent stress given by Eq. (28) for isotropic materials.

**Comment 8.** In 2023 [33], the Hosford-type equivalent stress for the initially anisotropic materials with tension-compression asymmetry was formulated by Zolochevsky, such as

$$ \sigma_e = (\sigma_0^r + \sigma^r + \alpha\sigma_1^r)^{\frac{1}{r}} \quad (36) $$

Here $\sigma_0^2 = a_{ijkl}\sigma_{ij}\sigma_{kl}$, $\sigma = b_{ij}\sigma_{ij}$, $\sigma_1^3 = c_{ijklmn}\sigma_{ij}\sigma_{kl}\sigma_{mn}$, the integer exponent $r$ takes odd values chosen from 1, 3, 5... It is not difficult to see that under condition $r=1$ we arrive from Eq. (36) at Eq. (35).

Using Eq. (36), hypothesis of the existence of the potential

$$ F = \sigma_e $$

and associated law

$$ e_{ij} = \lambda \frac{\partial F}{\partial \sigma_{ij}} $$

we obtain the following constitutive relation [33] between the kinematic tensor $e_{ij}$ and stress tensor

$$ e_{ij} = \lambda(\sigma_0^r + \sigma^r + \alpha\sigma_1^r)^{\frac{1}{r-1}}(\sigma_0^{r-2}a_{ijkl}\sigma_{kl} + \sigma^{r-1}b_{ij} + \alpha\sigma_1^{r-3}c_{ijklmn}\sigma_{kl}\sigma_{mn}) \quad (37) $$

Here $\lambda$ is the certain scalar factor to be determined when specifying the theory of elasticity, plasticity or creep.

In the case of the nonlinear elasticity, the kinematic tensor in Eq. (37) is the tensor of the elastic strain; the scalar $\lambda$ can be defined as some function of the equivalent stress $\sigma_e$ in the absence of elastic damage, i.e. $\lambda = \zeta(\sigma_e)$. In the simplest case we can accept, for example, the power function $\zeta(\sigma_e) = \sigma_e^m$, and, in particular, $m=1$ under consideration of the bimodulus model.

The condition of plasticity for anisotropic materials under study with the strain hardening measure $q$ in the absence of plastic damage can be written as

$$ \zeta(\sigma_e) = q \quad (38) $$

Considering plastic deformation, the kinematic tensor $e_{ij}$ in Eq. (37) can be accepted as the tensor of the rates of the plastic strain; the scalar $\lambda$ in Eq. (37) can be defined as $\lambda \equiv \dot{q}$, and, therefore

$$ \lambda = \chi\zeta'(\sigma_e)\dot{\sigma}_e \quad (39) $$
Here the dot above the symbol denotes a derivative with respect to the loading parameter; the prime denotes a derivative with respect to $\sigma_e$, and $\chi=1$ for the monotonic loading processes when condition (38) is valid simultaneously with inequality $\dot{\sigma}_e > 0$. Also, it is necessary to take in Eq. (39) that $\chi=0$ when $v(\sigma_e) < q$ under elastic deformation, and either in the case of unloading or neutral loading when Eq. (38) takes place together with condition $\dot{\sigma}_e \le 0$. Additionally, we can accept, for example, the power approximation $\zeta(\sigma_e) = \sigma_e^k$ in Eqs. (38) and (39).

In the case of creep, the kinematic tensor $e_{ij}$ in Eq. (37) is the tensor of the rates of the creep strain. Two structural parameters, $Q$ and $\omega$, are taken into account to describe the creep hardening and creep softening, respectively. The scalar $\lambda$ in Eq. (37) can be introduced for the creep deformation as

$$ \lambda = \frac{v(\sigma_e)h(Q)\omega^\beta}{(1-\omega)^p} \qquad (40) $$

Here function $v(\sigma_e)$ can be given in one of the following approximations: the exponential law $v(\sigma_e) = \exp(\sigma_e/d)$ or $v(\sigma_e) = 1 - \exp(-\sigma_e/d)$, the law of hyperbolic sine $v(\sigma_e) = \sh(\sigma_e/g)$ and the Norton-type power function

$$ v(\sigma_e) = \sigma_e^n \qquad (41) $$

The exponents $n$ within the range of 3 to 8 in the power function (41) are related to the dislocation creep [27, 32, 39-42, 96, 97]. The value $n \approx 1$ corresponds to the diffusion creep [98-100]. Also, the fractional power function [1, 42, 101-104] $v(\sigma_e) = [(\sigma_e - \Sigma_1)/(\Sigma_2 - \sigma_e)]^\eta$ can be used in the creep modeling. The Kachanov-Rabotnov damage parameter $\omega$ is increasing with time from the initial value $\omega=0$ at the reference instant of time to the final value $\omega=1$ at the instant of creep rupture. The structural parameters $Q$ and $\omega$ in Eq. (40) will be determined when specifying the physical mechanisms of creep.

For metallic materials with the three creep stages (primary, secondary and tertiary), we can assume the strain hardening with $h(Q) = 1$ in Eq. (40) and introduce the Kachanov-Rabotnov damage parameter, such as [94, 96, 97, 105-108]

$$ \omega = \frac{\varphi}{\varphi_*} \qquad (42) $$

and

$$ \dot{\varphi} = \sigma_{kl} e_{kl} \qquad (43) $$

Here $\varphi$ is the specific energy dissipated at time $t$, and $\varphi_*$ is the specific energy dissipated at the instant of creep rupture. Thus, we can conclude that the condition $\omega \in [0,1]$ is valid taking into account Eqs. (42) and (43). In the case of the time hardening the parameter $Q$ in Eq. (40) is time, $\beta=0$, and such representations are available as $h(Q) = t^{-m}$ or $h(Q) = \exp(-ft)$. Using Eqs. (36), (37), (40) and (43) we obtain for metallic materials the following creep damage evolution equation

$$ \dot{\varphi} = \lambda \sigma_e $$

The characteristic feature of carbon-based cathode materials is the intensification of creep at high temperatures due to migration of metallic sodium into cathode structure during aluminum electrolysis [101, 109-114]. Furthermore, the creep curves of carbon-based cathode materials show the first and second stages without the third creep stage. In this case metallic sodium in the carbon structure can be considered as damage due to the foreign atoms in cathode. Therefore, the Kachanov-Rabotnov damage parameter for carbon-based cathode materials can be introduced as the sodium concentration $C$ which can be found as a function of time and coordinates considering the initial-boundary value problem for sodium transport [101, 109, 115]. Thus, in order to reproduce the creep
deformation of carbon-based cathode materials it is necessary to accept in Eqs. (37) and (40) the following representations

$$Q = \varphi, h(Q) = \varphi^{\gamma}, \omega = C$$

It is not difficult to see that damage parameter $\omega$ is increasing with time from the initial value $\omega = C_0 = 0$ at the reference instant of time to the hypothetical value $\omega = C_* = 1$ at the instant of creep rupture. Therefore, the necessary condition $\omega \in [0, 1]$ takes place in the present study.

The creep constitutive equations (37) and (40) were also used for perovskite-based materials XYO$_{3-\delta}$. Here X is a cation (cations) larger than the cation (cations) Y, O is an oxygen anion, $\delta$ is the oxygen nonstoichiometry which can be found as a function of time and coordinates considering the Fick's second law, as well as, the initial and boundary conditions for oxygen vacancies [116, 117]. Note that the creep curves of perovskite-based materials demonstrate only the steady state stage without primary and tertiary creep stages [98, 99, 118-122]. In this case it is necessary to accept in Eqs. (37), (40) and (41) the following requirements [98, 99]

$$h(Q) = 1, \omega = \frac{\delta}{3}, p = \beta, n = 1$$

Also, it is interesting to note that ignoring the coupling between creep deformation and oxygen vacancies transport in perovskite-based materials may lead to the unrealistically large deformation in structures [123].

**Comment 9.** The model predictions with the equivalent stress (6) within the constitutive framework by Zolochevsky taking into account the strongly nonlinear stress-strain diagrams of gray cast iron under tension and compression (Fig. 3) were compared with the predictions based on the bimodulus model (Fig. 1b) with $E_+ = 50$ GPa and $E_- = 100$ GPa. In this way, numerical calculations based on the other elastic models under assumption of the initial isotropy are also performed. For this purpose, we consider the thin cylindrical shell of length 0.2 m, radius of the middle surface 0.2 m and thickness 0.01 m, subjected to internal pressure of 4 MPa. One edge of the shell is fixed, and the other edge is loaded by the axial compressive force of 1602 kN/m.

<figure><img src="image_5.png" /><figcaption>**Fig. 3. Uniaxial stress-strain diagrams for gray cast iron under tension (1) and compression (2) [1, 36, 37]**</figcaption></figure>

Comparison of predictions for different elastic models is given in Fig. 4a, b. Numerical results obtained taking into account for the strongly nonlinear stress-strain diagrams of gray cast iron under tension and compression are shown by solid line. It is seen in Fig. 4a, b that such results are different significantly from the analogous results obtained under assumption of the linear elasticity (— — —), nonlinear deformation with identical properties under tension and compression (— — —) and bimodulus material (— — —). More details can be found in [1].
<figure><img src="image_6.png" /><figcaption>Fig. 4. Distribution of normal displacement (a) and axial displacement (b) along the shell</figcaption></figure>

The above mentioned comparison show that at present the bimodulus model can be considered rather as a kind of the numerical simulation, but not a realistic attempt to reproduce the real material behavior in structures with tension-compression asymmetry. In this regard, right now the practical significance of research [11, 13, 124-137] based on the bimodulus model is doubtful.

**Comment 10.** The equivalent stresses by Zolochevsky given by Eq. (6), (28)-(30), (35) and (36) are the homogeneous functions of the components of the stress tensor, i.e. $\sigma_e(\kappa\sigma_{kl}) = \kappa\sigma_e(\sigma_{kl}) \forall \kappa \neq 0$. On the other hand, a number of the nonhomogeneous equivalent stresses can be found, for example, in [133-135]. However, the use of the homogeneous equivalent stresses in practical applications is more appropriate in comparison with the nonhomogeneous equivalent stresses [136]. In particular, in the case of the homogeneous equivalent stress we have

$$ \frac{\partial \sigma_e}{\partial \sigma_{kl}} \sigma_{kl} = \sigma_e, \quad \lambda = \frac{\sigma_{kl} e_{kl}}{\sigma_e}, \text{ and, finally, we arrive at the constitutive equation (37) of simpler structure and simpler determination of the material parameters in comparison with the physical relations [137-140] based on the nonhomogeneous equivalent stresses.} $$

**Comment 11.** Basic experiments show that the transverse strain coefficient of the modern metamaterials changes from negative to positive during the conversion of tension and compression [141,142]. It is possible to reproduce such experimental phenomenon using the constitutive framework by Zolochevsky and numerical modeling tools [143,144] for numerical analysis.

## CONCLUSIONS

The state of the art in solid mechanics is characterized by advancements in the study of the tension-compression material asymmetry in structures.

It is given a response to speculation by Rigbi (1980) for the initially isotropic bimodulus material.

It is shown that the bimodulus model can be considered rather as a kind of the numerical simulation, but not a realistic attempt to reproduce the real material behavior in structures with tension-compression asymmetry.

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