# Spectral Classification of Non-Coaxiality for Two-Dimensional Incremental Stress-Strain Response

Authors: Jiangu Qian; Jean-Pierre Bardet; Maosong Huang
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

The present study examines the non-coaxial aspects of incremental material behavior, and attempts to classify the incremental non-coaxiality that relates stress and strain increments. In the solid mechanics literature, non-coaxiality (NC) refers usually to incremental strains and stress states having different principal directions. Departing from conventional non-coaxiality, the analysis investigates the incremental non-coaxiality (INC) of linearized rate-type solids. This study uses the concept of deviatoric second-order work for examining the relations between stability and incremental non-coaxiality. Based on a spectral analysis of the constitutive compliance matrix, it proposes three classifications for distinguishing various degrees of incremental non-coaxiality and stability. These classifications determine the conditions for the existence of incremental coaxiality (i.e., colinearity of stress and strain increments), stability, instability, and stable-instable transition (i.e., positive, negative, or zero second-order deviatoric work). The study illustrates these classifications in the cases of generic elastic and elastoplastic constitutive models. The analysis pertains to two-dimensional cases. Additional research is required to extend the analysis from two to three dimensions.


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## Body

## 1. Introduction

The anisotropic and non-coaxial behaviors of geomaterials are challenging to model using the conventional flow theory of plasticity, which assumes that the strain increments and principal stress have identical direction, that is, are coaxial [1–3]. For instance, the associative flow rule of plasticity, which assumes that strain increments are normal to the yield surface, disagrees with many experimental evidences [4, 5] and micromechanical observations [6, 7], which show non-coaxiality, that is, different principal directions for stress states and strain increments.As early as Hill [8], several theories have been proposed to introduce noncoaxiality. For instance, one approach added tangent plasticity to classical coaxial models [1, 9]. Another approach defined strain increment in terms of stress states and material anisotropy, which may lead to an anisotropic hardening law [10–12]. Others have described noncoaxial behaviors using double-shear models [13–15]. Micromechanical studies have related non-coaxiality to the anisotropic fabric resulting from the arrangement of material particles and associated voids [16, 17].Most studies of non-coaxiality focused on the case of non-proportional loading, and assumed coaxiality under proportional loading [18]. However, experiments showed that anisotropic solids, such as geomaterials, may exhibit considerable degrees of noncoaxiality even under proportional loading [5, 19, 20]. This results in two distinct mechanisms relevant to non-coaxiality. According to theoretical [21, 22] and experimental studies [5, 20], non-coaxiality is largely induced by tangent yield effects under non-proportional loading. For proportional loading, experiments showed that non-coaxiality tends to result from induced or inherent material anisotropy [5, 19]. To distinguish the two different mechanisms, this analysis expands the definition of conventional non-coaxiality to incremental non-coaxiality, which is hereafter referred to as the difference between principal directions of strain and stress increments.Figure1 illustrates the relationship among non-coaxiality, incremental non-coaxiality and stress path. It relates these three concepts using (1) stress state σij, (2) stress increment dσij, and (3) strain increment dεij. The principal direction angles of σij,εij, and dεij are defined as θσ,θdσ  and θdε, respectively (Figure 2). Like in the solid mechanics literature [23], Figure 1 defines coaxiality (C, θσ=θdε) and non-coaxiality (NC, θσ≠θdε) in terms of σij and dεij. Proportional (P, θσ=θdσ) loading corresponds to dσij proportional to σij, and non-proportional (NP, θσ≠θdσ) loading refers to changes in principal direction of dσij, when σij moves along nonlinear stress paths. Incremental coaxiality (IC, θdσ=θdε) and incremental non-coaxiality (INC,θdσ≠θdε) are defined in terms of dσij and dεij. Therefore, NC can be generated in three different ways through INC + P (Figure 2(a)), INC + NP (Figure 2(b)) and IC + NP (Figure 2(c)).Figure 1
Illustration of relationship among incremental strain, incremental stress and stress.Non-coaxiality available by three approaches: (a) INC+P; (b) INC+NP; (c) IC+NP.
(a)
Incremental non-coaxiality with proportional loading (θdε≠θdσ=θσ)(b)
Incremental non-coaxiality with non-proportional loading (θdε≠θdσ≠θσ)(c)
Incremental coaxiality with non-proportional loading (θdε=θdσ≠θσ)The incremental non-coaxiality (INC) and non-proportionality (NP) of loading can be defined, respectively, as(1.1a)θinc=θdε-θdσ,(1.1b)θnp=θdσ-θσ.Therefore, non-coaxiality (NC), incremental non-coaxiality (INC), and non-proportionality (NP) of loading are related by(1.1c)θnc=θinc+θnp, where θnp can be determined from the prescribed loading path and θinc is a type-rate material property that needs to be measured from incremental stress-strain response. Clearly, θnc=θinc in the particular case of proportional loading (Figure 2(a)).For stress-controlled problems, the above considerations restrict our attention to incremental non-coaxiality (INC). The present analysis focuses on two aspects: (1) for which materials incremental non-coaxiality (INC) may become negligible or large and (2) in which direction materials may preserve incremental coaxiality (IC). The analysis is presented in two dimensions, but can be extended to three-dimensions with some additional effort.


## 2. Lame’s Representation of Incremental Non-Coaxiality

Material behaviors are modeled using rate-type linearized solids, that is,(2.1)dεij=Cijkldσkl,
where dεij is the incremental strain, dσij is the incremental stress, and Cijkl is the incremental compliance tensor that represents some underlying anisotropy state. The indices are i,j…=1,2 in two-dimensions. Cijkl can be written in matrix form as follows:(2.2)[C]=[C1111C1122C1112C2211C2222C2212C1211C1222C1212].The physical meaning of incremental non-coaxiality can be illustrated using Lame’s ellipse (Figure3), which represents a two-dimensional second-order tensor using an inclined ellipse. dσij is uniquely expressed in terms of principal stress values dσ1 and dσ2 (dσ1≥dσ2) and the orientation angle θdσ of principal direction. Similarly, the strain increment dεij is defined using the principal values of strain increment dε1 and dε2 (dε1≥dε2) and θdε the orientation angle of principal direction. The incremental response is coaxial when θdε=θdσ and non-coaxial when θdε≠θdσ.Lame’s visualization of incremental stress-strain responses.
(a)
Incremental stress ellipse(b)
Incremental strain ellipseIn Lame’s representation, (2.1) is restated as follows:(2.3)θdε=f(dσ1,dσ2,θdσ),dε1=f2(dσ1,dσ2,θdσ),dε2=f3(dσ1,dσ2,θdσ),
where the nonlinear functions f1, f2, and f3 depend on Cijkl and -90∘≤θdσ≤90∘.Without loss of generality, (2.3) may be simplified assuming dσ2=1 and dε2=1 (see the appendix)
(2.4)θinc=θdε-θdσ=F1(Rdσ,θdσ),(2.5)Rdε=dε1dε2=F2(Rdσ,θdσ),
where functions F1 and F2 are analogous to f1, f2 and f3, and Rdσ=dσ1/dσ2 denotes the ratio of incremental principal stresses.Mapping uniquely a stress increment onto a Lame’s ellipse requires assigning a positive sign to principal values. Otherwise, different stress increments would correspond to the same Lame’s ellipses. For instance, assumingdσ1>dσ2>0, Table 1 shows five tensors that have the same Lame’s ellipse but different values of θdσ and Rdσ.Table 1
Orientation angle and ratio of principal stress increments.
Stress increments matrixθdσRdσ[dσ100dσ2]0dσ1dσ2[dσ200dσ1]π/2dσ2dσ1[-dσ100dσ2]π/2-dσ1dσ2[dσ100-dσ2]0-dσ1dσ2[-dσ100-dσ2]π/2dσ1dσ2


## 3. Classifications of Incremental Non-Coaxiality

Three different classifications are proposed for incremental non-coaxiality, based on the values ofCijkl coefficients that indicate some kind of fabric anisotropy. These classifications addresses the following two issues: (1) relation of stability and INC, and (2) number of directions that preserve incremental coaxiality, that is, stress and strain increments have identical or opposite directions.


### 3.1. Classification 1: Energy-Based Classification for INC

The first classification intends to distinguish between materials with strong or weak INC, noting that the degree of INC is intrinsically related to material stability. Since the strain increment has the same principal direction as its deviatoric part, the deviatoric constitutive equation is considered(3.1)deij=Cijklddσkl,
where deij=dεij-δijdεkk/2, and Cijkld=Cijkl-δijCaakl/2.The eigenvalues ofCijkld can be analytically calculated by solving a quadratic equation, which is simpler than solving a cubic equation for the eigenvalues of Cijkl. This is because Cijkld is rank deficient and always possess one zero eigenvalue, leaving only two eigenvalues to determine.The deviatoric second-order work is(3.2)d2Wd=dσijdeij=dsijdeij=|dsij||deij|cos⁡2(θdε-θdσ)=|dsij||deij|cos⁡2θinc,
where dsij is the deviatoric part of dσij, |dsij| and |deij| denote the magnitude of dsij and deij, respectively. One obtains:(3.3)cos⁡2θinc=dsijdeij|dsij||deij|=dσijCijklddσkl|dsij||Cijklddσkl|.The trivial casesdsij=0 and deij=0 can be ignored without loss of generality. In general, there are three independent deviation angles between the principal axes of stress and strain increments in three-dimensional case, for which θinc given by (3.3) does not represent a certain angular orientation of incremental non-coaxiality. However, cosθinc is an energy-based parameter that can be extended to higher-dimensional problems.Assumingdσ2=1, dσij can be explicitly expressed as (3.4)[dσ]=dσaa2[δ]+[ds]=Rdσ+12[1001]+Rdσ-12[cos⁡2θdσsin2θdσsin2θdσ-cos⁡2θdσ],
which implies that dsij and dσij have the same principal directions.In the particular caseCijkldδkl=0, that is,(3.5)C1211+C1222=0,C1111-C2211-C2222+C1122=0,
which corresponds to pressure increments producing no deviatoric strains. With Cijkldδkl=0, equation (3.3) becomes(3.6a)cos⁡2θinc=dsijCijklddskl|dsij||Cijklddskl|=nijCijklddnkl|nij||Cijkldnkl|,
where nij is(3.6b)[n]=[cos⁡2θdσsin2θdσsin2θdσ-cos⁡2θdσ].According to (3.6a)-(3.6b),cos2θinc is independent from Rdσ, and d2Wd(=dσijCijklddσkl) and cos2θinc have identical sign.Equation (3.1) can be equivalently expressed in matrix form as follows:(3.7a)[de]=[Cd][dσ],
or
(3.7b)[dε11-dε222dε22-dε1122dε̇12]=[C1111-C22112C1122-C222222(C1112-C2212)2C2211-C11112C2222-C112222(C2212-C1112)22C12112C12222C1212][dσ11dσ222dσ12].Complicated spectral analyses for nonsymmetric matrices are avoided by noting that any nonsymmetric matrixA satisfies xTAx=xT((A+AT)/2)x for any vector x. Equation (3.3) can be rewritten(3.8)cos⁡2θinc=dσijCijklddσkl|dsij||Cijklddσkl|=dσijC̅ijklddσkl|dsij||Cijklddσkl|=[dσ]T[C̅d][dσ][ds]T[ds][ds]T[Cd]T[Cd][ds],
where [C̅d]=([Cd]+[Cd]T)/2 (i.e., C̅ijkld=(Cijkld+Cklijd)/2) is the symmetric part of [Cd], whose equivalent matrix form is (3.9)[C̅d]=[C1111-C22112C2211-C1111+C1122-C222242(2C1211+C1112-C2212)4C2211-C1111+C1122-C22224C2222-C112222(2C1222+C2212-C1112)42(2C1211+C1112-C2212)42(2C1222+C2212-C1112)42C1212].The eigenvalues of[C̅d] are denoted  by λ̅1, λ̅2 and λ̅3, and the corresponding orthogonal eigentensors are denoted by  α̅ij, β̅ij and χ̅ij. Any stress increments can be expressed as(3.10)dσij=a̅α̅ij+b̅β̅ij+c̅χ̅ij,
where a̅, b̅, and c̅ are the components in the eigendirections.Equation (3.8) becomes
(3.11)cos⁡2θinc=a̅2λ̅1+b̅2λ̅2+c̅2λ̅3[ds]T[ds][ds]T[Cd]T[Cd][ds],
which shows that cos2θinc depends on the signs of λ̅1, λ̅2 and λ̅3.WhenCijkldδkl=0, the eigenvalues of [C̅d] are (3.12a)λ̅1=0,λ̅2=A′+B′2,λ̅3=A′-B′2,
where
(3.12b)A′=C1111-C2211+2C1212,(3.12c)B′=(C1111-C2211-2C1212)2+(2C1211+C1112-C2212)2.Parameters A′ and B′ solely control the values of eigenvalues. Invoking that λ̅2≥λ̅3, Figure 4 shows four distinct domains:(I)
stable;cos2θinc>0; λ̅1=0 and λ̅2≥λ̅3≥0; A′≥B′,(II)
unstable;cos2θinc<0; λ̅1=0 and 0≥λ̅2≥λ̅3; A′<-B′,(III)
partially stable;-1≤cos2θinc≤1; λ̅1=0 and λ̅2>0>λ̅3; -B′<A′<B′.Figure 4
Classification 1 (Cijkldδkl=0).PointO′: Stable-Unstable Transition; cos2θinc=0; λ̅1=0 and λ̅2=λ̅3=0; A′=B′=0.WhenCijkldδkl≠0, the above spectral approach does not fully apply however. This case is similar to partially stable INC (Domain III in Figure 4).


### 3.2. Classification 2—Spectral Classification for IC

The second classification relates to the material directions that preserve stable incremental coaxiality (SIC) or produce unstable incremental coaxiality (UIC). The analysis looks for the principal directions (if any) of anisotropic materials, which may also represent the optimal orientation of anisotropic solids, see, for example, [24].Cijkld are assumed to have three eigenvalues λ1, λ2, and λ3 and three corresponding eigendirections αij, βij, and χij. The eigenvalues of Cijkld are
(3.13)λ1≡0,λ2=A+B2,λ3=A-B2,
where (3.14)A=C1111-C2211+C2222-C1122+4C12122,B=(C1111-C2211+C2222-C1122-4C12122)2+4(C1112-C2212)(C1211-C1222).
Consider that a given stress increment(3.15a)dσij=aαij+bβij+cχij,
where the coefficients a, b, and c are variables comparable to a̅, b̅ and c̅ in (3.10). In the case Cijkldδkl=0 (i.e., αij=δij)(3.15b)dsij=bβij+cχij.Thus,
(3.16)deij=Cijklddσkl=Cijkld(dskl+12dσmmδkl)=Cijklddskl=bλ2βkl+cλ3χkl.
Equation (3.16) implies that deij are coaxial with dσij when dsij=bβij or dsij=cχij.As shown in (3.13), (3.16) and Figure 5, parameters A and Bcontrol the values of eigenvalues λ2 and λ3, which determine the stability and degree of IC. The stability of IC (i.e., SIC or UIC) depends on the signs of λ2 and λ3. The degree of SIC (or UIC) is indicated with the number n of directions along which the incremental stress and deviator strain have identical (or opposite) direction.Figure 5
Classification  2 (Cijkldδkl=0).


### 3.3. Classification 3—Couped Spectral Classification of INC

The third classification builds upon the two previous classifications. Hereafter, we focus on the caseCijkldδkl=0 because Cijkldδkl≠0 creates incremental non-coaxiality (INC) in all directions without possible stable or unstable incremental coaxiality. Cijkldδkl=0 implies that
(3.17)A′=A,B′-B=(2C1211-C1112+C2212)2≥0.Figures4 and 5 were combined into Figure 6 to produce the third classification, which gives information on stability as well as the number of directions for stable or unstable incremental coaxiality. Table 2 is a tabular representation of classifications in Figure 6. This classification will be illustrated using examples in each classification domains.Table 2
Coupled spectral classification of incremental non-coaxiality.
λ2≥λ3≥00>λ2≥λ3λ2>0 &  λ3<0λ2, λ3 complexProperties ofλi(λ1=0) and λ̅iλ2≠λ3λ2=λ3λ2≠λ3λ2=λ3Cijkldδkl=0 (λ̅1=0)λ̅2≥λ̅3 &  λ̅3≥0λ̅2≠λ̅3SIC = 4 stableSIC = 2 stableNo possibilitySIC = UIC = 0 stableλ̅2=λ̅3 & λ̅2≠0SIC = 4 stableSIC stableNo possibilitySIC = UIC = 0 stableλ̅3≤λ̅2& λ̅2≤0λ̅2≠λ̅3No possibilityUIC = 4 unstableUIC = 2 unstableNo possibilitySIC = UIC = 0 unstableλ̅2=λ̅3& λ̅2≠0No possibilityUIC = 4 unstableUIC unstableNo possibilitySIC = UIC = 0 unstableλ̅2≥0 & λ̅3≤0λ̅2≠λ̅3SIC = 4partially stableSIC = 2partially stableUIC = 4partially stableUIC = 2partially stableSIC = UIC = 2partially stableSIC = UIC = 0partially stableλ̅2=λ̅3No possibilitySIC = UIC = 0 Stable-Unstable TransitionCijkldδkl≠0SIC = UIC = 0 partially stableSIC = UIC = 0 partially stableSIC =n: number of directions with identical direction of incremental stress and deviator strain, cos2θinc≡1 (SIC alone means that incremental stress and deviator strain have identical direction).  UIC = n: number of directions with opposite direction of incremental stress and deviator strain, cos2θinc≡-1 (UIC alone means that all incremental stress and deviator strain have opposite direction).Stable-Unstable Transition:cos2θinc≡0; stable: 0≤cos2θinc≤1; unstable: 0>cos2θinc≥-1; partially stable: -1≤cos2θinc≤1.Figure 6
Classification 3 (Cijkldδkl=0).WhenB′=B, which results in Cijkld=C̅ijkld (i.e., λ2=λ̅2 and λ2=λ̅3), points O,O′,a- and a+ coincide, and Domains I, II and III remain only in Figure 6.


## 3.1. Classification 1: Energy-Based Classification for INC

The first classification intends to distinguish between materials with strong or weak INC, noting that the degree of INC is intrinsically related to material stability. Since the strain increment has the same principal direction as its deviatoric part, the deviatoric constitutive equation is considered(3.1)deij=Cijklddσkl,
where deij=dεij-δijdεkk/2, and Cijkld=Cijkl-δijCaakl/2.The eigenvalues ofCijkld can be analytically calculated by solving a quadratic equation, which is simpler than solving a cubic equation for the eigenvalues of Cijkl. This is because Cijkld is rank deficient and always possess one zero eigenvalue, leaving only two eigenvalues to determine.The deviatoric second-order work is(3.2)d2Wd=dσijdeij=dsijdeij=|dsij||deij|cos⁡2(θdε-θdσ)=|dsij||deij|cos⁡2θinc,
where dsij is the deviatoric part of dσij, |dsij| and |deij| denote the magnitude of dsij and deij, respectively. One obtains:(3.3)cos⁡2θinc=dsijdeij|dsij||deij|=dσijCijklddσkl|dsij||Cijklddσkl|.The trivial casesdsij=0 and deij=0 can be ignored without loss of generality. In general, there are three independent deviation angles between the principal axes of stress and strain increments in three-dimensional case, for which θinc given by (3.3) does not represent a certain angular orientation of incremental non-coaxiality. However, cosθinc is an energy-based parameter that can be extended to higher-dimensional problems.Assumingdσ2=1, dσij can be explicitly expressed as (3.4)[dσ]=dσaa2[δ]+[ds]=Rdσ+12[1001]+Rdσ-12[cos⁡2θdσsin2θdσsin2θdσ-cos⁡2θdσ],
which implies that dsij and dσij have the same principal directions.In the particular caseCijkldδkl=0, that is,(3.5)C1211+C1222=0,C1111-C2211-C2222+C1122=0,
which corresponds to pressure increments producing no deviatoric strains. With Cijkldδkl=0, equation (3.3) becomes(3.6a)cos⁡2θinc=dsijCijklddskl|dsij||Cijklddskl|=nijCijklddnkl|nij||Cijkldnkl|,
where nij is(3.6b)[n]=[cos⁡2θdσsin2θdσsin2θdσ-cos⁡2θdσ].According to (3.6a)-(3.6b),cos2θinc is independent from Rdσ, and d2Wd(=dσijCijklddσkl) and cos2θinc have identical sign.Equation (3.1) can be equivalently expressed in matrix form as follows:(3.7a)[de]=[Cd][dσ],
or
(3.7b)[dε11-dε222dε22-dε1122dε̇12]=[C1111-C22112C1122-C222222(C1112-C2212)2C2211-C11112C2222-C112222(C2212-C1112)22C12112C12222C1212][dσ11dσ222dσ12].Complicated spectral analyses for nonsymmetric matrices are avoided by noting that any nonsymmetric matrixA satisfies xTAx=xT((A+AT)/2)x for any vector x. Equation (3.3) can be rewritten(3.8)cos⁡2θinc=dσijCijklddσkl|dsij||Cijklddσkl|=dσijC̅ijklddσkl|dsij||Cijklddσkl|=[dσ]T[C̅d][dσ][ds]T[ds][ds]T[Cd]T[Cd][ds],
where [C̅d]=([Cd]+[Cd]T)/2 (i.e., C̅ijkld=(Cijkld+Cklijd)/2) is the symmetric part of [Cd], whose equivalent matrix form is (3.9)[C̅d]=[C1111-C22112C2211-C1111+C1122-C222242(2C1211+C1112-C2212)4C2211-C1111+C1122-C22224C2222-C112222(2C1222+C2212-C1112)42(2C1211+C1112-C2212)42(2C1222+C2212-C1112)42C1212].The eigenvalues of[C̅d] are denoted  by λ̅1, λ̅2 and λ̅3, and the corresponding orthogonal eigentensors are denoted by  α̅ij, β̅ij and χ̅ij. Any stress increments can be expressed as(3.10)dσij=a̅α̅ij+b̅β̅ij+c̅χ̅ij,
where a̅, b̅, and c̅ are the components in the eigendirections.Equation (3.8) becomes
(3.11)cos⁡2θinc=a̅2λ̅1+b̅2λ̅2+c̅2λ̅3[ds]T[ds][ds]T[Cd]T[Cd][ds],
which shows that cos2θinc depends on the signs of λ̅1, λ̅2 and λ̅3.WhenCijkldδkl=0, the eigenvalues of [C̅d] are (3.12a)λ̅1=0,λ̅2=A′+B′2,λ̅3=A′-B′2,
where
(3.12b)A′=C1111-C2211+2C1212,(3.12c)B′=(C1111-C2211-2C1212)2+(2C1211+C1112-C2212)2.Parameters A′ and B′ solely control the values of eigenvalues. Invoking that λ̅2≥λ̅3, Figure 4 shows four distinct domains:(I)
stable;cos2θinc>0; λ̅1=0 and λ̅2≥λ̅3≥0; A′≥B′,(II)
unstable;cos2θinc<0; λ̅1=0 and 0≥λ̅2≥λ̅3; A′<-B′,(III)
partially stable;-1≤cos2θinc≤1; λ̅1=0 and λ̅2>0>λ̅3; -B′<A′<B′.Figure 4
Classification 1 (Cijkldδkl=0).PointO′: Stable-Unstable Transition; cos2θinc=0; λ̅1=0 and λ̅2=λ̅3=0; A′=B′=0.WhenCijkldδkl≠0, the above spectral approach does not fully apply however. This case is similar to partially stable INC (Domain III in Figure 4).


## 3.2. Classification 2—Spectral Classification for IC

The second classification relates to the material directions that preserve stable incremental coaxiality (SIC) or produce unstable incremental coaxiality (UIC). The analysis looks for the principal directions (if any) of anisotropic materials, which may also represent the optimal orientation of anisotropic solids, see, for example, [24].Cijkld are assumed to have three eigenvalues λ1, λ2, and λ3 and three corresponding eigendirections αij, βij, and χij. The eigenvalues of Cijkld are
(3.13)λ1≡0,λ2=A+B2,λ3=A-B2,
where (3.14)A=C1111-C2211+C2222-C1122+4C12122,B=(C1111-C2211+C2222-C1122-4C12122)2+4(C1112-C2212)(C1211-C1222).
Consider that a given stress increment(3.15a)dσij=aαij+bβij+cχij,
where the coefficients a, b, and c are variables comparable to a̅, b̅ and c̅ in (3.10). In the case Cijkldδkl=0 (i.e., αij=δij)(3.15b)dsij=bβij+cχij.Thus,
(3.16)deij=Cijklddσkl=Cijkld(dskl+12dσmmδkl)=Cijklddskl=bλ2βkl+cλ3χkl.
Equation (3.16) implies that deij are coaxial with dσij when dsij=bβij or dsij=cχij.As shown in (3.13), (3.16) and Figure 5, parameters A and Bcontrol the values of eigenvalues λ2 and λ3, which determine the stability and degree of IC. The stability of IC (i.e., SIC or UIC) depends on the signs of λ2 and λ3. The degree of SIC (or UIC) is indicated with the number n of directions along which the incremental stress and deviator strain have identical (or opposite) direction.Figure 5
Classification  2 (Cijkldδkl=0).


## 3.3. Classification 3—Couped Spectral Classification of INC

The third classification builds upon the two previous classifications. Hereafter, we focus on the caseCijkldδkl=0 because Cijkldδkl≠0 creates incremental non-coaxiality (INC) in all directions without possible stable or unstable incremental coaxiality. Cijkldδkl=0 implies that
(3.17)A′=A,B′-B=(2C1211-C1112+C2212)2≥0.Figures4 and 5 were combined into Figure 6 to produce the third classification, which gives information on stability as well as the number of directions for stable or unstable incremental coaxiality. Table 2 is a tabular representation of classifications in Figure 6. This classification will be illustrated using examples in each classification domains.Table 2
Coupled spectral classification of incremental non-coaxiality.
λ2≥λ3≥00>λ2≥λ3λ2>0 &  λ3<0λ2, λ3 complexProperties ofλi(λ1=0) and λ̅iλ2≠λ3λ2=λ3λ2≠λ3λ2=λ3Cijkldδkl=0 (λ̅1=0)λ̅2≥λ̅3 &  λ̅3≥0λ̅2≠λ̅3SIC = 4 stableSIC = 2 stableNo possibilitySIC = UIC = 0 stableλ̅2=λ̅3 & λ̅2≠0SIC = 4 stableSIC stableNo possibilitySIC = UIC = 0 stableλ̅3≤λ̅2& λ̅2≤0λ̅2≠λ̅3No possibilityUIC = 4 unstableUIC = 2 unstableNo possibilitySIC = UIC = 0 unstableλ̅2=λ̅3& λ̅2≠0No possibilityUIC = 4 unstableUIC unstableNo possibilitySIC = UIC = 0 unstableλ̅2≥0 & λ̅3≤0λ̅2≠λ̅3SIC = 4partially stableSIC = 2partially stableUIC = 4partially stableUIC = 2partially stableSIC = UIC = 2partially stableSIC = UIC = 0partially stableλ̅2=λ̅3No possibilitySIC = UIC = 0 Stable-Unstable TransitionCijkldδkl≠0SIC = UIC = 0 partially stableSIC = UIC = 0 partially stableSIC =n: number of directions with identical direction of incremental stress and deviator strain, cos2θinc≡1 (SIC alone means that incremental stress and deviator strain have identical direction).  UIC = n: number of directions with opposite direction of incremental stress and deviator strain, cos2θinc≡-1 (UIC alone means that all incremental stress and deviator strain have opposite direction).Stable-Unstable Transition:cos2θinc≡0; stable: 0≤cos2θinc≤1; unstable: 0>cos2θinc≥-1; partially stable: -1≤cos2θinc≤1.Figure 6
Classification 3 (Cijkldδkl=0).WhenB′=B, which results in Cijkld=C̅ijkld (i.e., λ2=λ̅2 and λ2=λ̅3), points O,O′,a- and a+ coincide, and Domains I, II and III remain only in Figure 6.


## 4. Examples

Incremental non-coaxiality is modeled for compliance matrices (2.2) that do not necessarily have a major symmetry, that is, Cijkl≠Cklij. As shown in Table 3, all components of Cijkl are dimensionless. With respect to distinct eigenvalue cases, examples are given to demonstrate how the classifications distinguish various degrees of incremental non-coaxiality and stability. Tables 3(a)–3(e) and Figures 7–11 contain 17 examples that apply to anisotropic elasticity as well as elastoplasticity.Table 3(a)
Model parameters for stable INC (Cijkldδkl=0,λ̅1=λ1=0)
Classification Domains in Figure6Cijkl coefficientsSpectral propertiesComputation results in Figure7I[1.5-0.50-0.51.50000.1]λ̅2=2.0,λ̅3=0.2,(a)(B′=B)λ2=2.0,λ3=0.2a+A+[1.0-0.51.5-1.01.5-1.5001.0]λ̅2=3.5,λ̅3=0.5,(b)(B′>B)λ2=λ3=2.0oA+[1.5-0.50-0.51.50001.0]λ̅2=λ̅3=λ2=λ3=2.0(c)(B′=B)VI[1.0-0.51.0-1.01.5-1.0-2.02.01.0]λ̅2=3.0,λ̅3=1.0(d)(B′>B)λ2=2.0+2.8i (complex) λ3=2.0-2.8i (complex)(b)
Model parameters for instable INC (Cijkldδkl=0,λ̅1=λ1=0)
Classification Domains in Figure6Cijkl coefficientsSpectral propertiesComputation results in Figure8II[0.51.501.50.5000-0.1]λ̅2=-0.2,λ̅3=-1.0,(a)(B′=B)λ2=-0.2,λ3=-1.0a-A-[0.52.001.51.000.5-0.5-0.5]λ̅2=-0.5,λ̅3=-1.5,(b)(B′>B)λ2=λ3=-1.0oA-[0.51.501.50.5000-0.5]λ̅2=λ̅3=λ2=(c)(B′=B)λ3=-1.0VII[0.52.01.01.51.0-1.0-1.51.5-0.5]λ̅2=-0.5,λ̅3=-1.5(d)λ2=-1.0+2.45i (complex)(B′>B)λ3=-1.0-2.45i (complex)(c)
Model parameters for stable-instable INC transition  (Cijkldδkl=0,λ̅1=λ1=0)
Classification Domains in Figure6Cijkl coefficientSpectral propertiesComputation results in Figure9Pointo′[1.01.5-0.11.01.50.10.1-0.10.0]λ̅2=λ̅3=0(B′>B)λ2=0.2i (complex) λ3=-0.2i (complex)(d)
Model parameters for partially stable INC (Cijkldδkl=0,λ̅1=λ1=0)
Classification Domains in Figure6Cijkl coefficientsSpectral propertiesComputation results in Figure10IV[1.0-0.50.6-1.01.5-0.61.5-1.51.0]λ̅2=4.1,λ̅3=-0.1(a)(B′>B)λ2=3.9,λ3=0.1oa+[1.0-0.5-4.0-1.01.54.00.00.01.0]λ̅2=6.0,λ̅3=-2.0(b)(B′>B)λ2=λ3=2.0V[0.52.00.251.51.0-0.251.0-1.0-0.6]λ̅2=0.15,λ̅3=-2.35(c)(B′>B)λ2=-0.095,λ3=-2.1oa-[0.52.001.51.002.0-2.0-0.5]λ̅2=1.0,λ̅3=-3.0(d)(B′>B)λ2=λ3=-1.0III[1.5-0.51.0-0.51.5-1.01.0-1.00.6]λ̅2=2.96,λ̅3=-2.16(e)(B′=B)λ2=2.96,λ3=-2.16VIII[1.0-0.50.4-1.01.5-0.4-5.05.00.5]λ̅2=6.13,λ̅3=-3.13(f)(B′>B)λ2=1.5+2.78i (complex) λ3=1.5-2.78i (complex)(e)
Model parameters for partially stable INC (Cijkldδkl≠0,λ1=0)
Cijkl coefficientSpectral propertiesComputation Results in Figure11[1.0-0.50-0.55.00001.0]λ2=3.5,λ3=2.0(a)λ̅1=3.77,λ̅2=2.0,λ̅3=-0.27[1.00.5-1.01.02.01.01.0-1.00.75]λ2=1.125+1.96i (complex)(b)λ3=1.125-1.96i (complex)λ̅1=1.5,λ̅2=0.9,λ̅3=-0.155θdσ-cos2θinc for stable INC (Cijkldδkl=0).
(a)
4 SIC(b)
2 SIC(c)
SIC(d)
No SICθdσ-cos2θinc for unstable INC (Cijkldδkl=0).
(a)
4 UIC(b)
2 UIC(c)
UIC(d)
No UICFigure 9
θdσ-cos2θinc for stable-unstable transition INC (Cijkldδkl=0, complex λ2,λ3).θdσ-cos2θinc for partially stable INC (Cijkldδkl=0).
(a)
4 SIC(b)
2 SIC(c)
4 UIC(d)
2 UIC(e)
2 SIC and 2 UIC(f)
No SIC/UICRdσ-θdσ-cos2θinc for partially stable INC (Cijkldδkl≠0).
(a)
No SIC/UIC (realλ2,λ3)(b)
No SIC/UIC (complexλ2,λ3)Figure7 and Table 3(a) show four cases of stable incremental non-coaxiality when λ̅2≥λ̅3≥0. Figures 7(a) and 7(b) show that there are four and two stable IC (SIC) (i.e., cos2θinc=1). Figure 7(a) falls within Domain I (Figure 6) while Figure 7(b) is on Boundary a+A+ (Figure 6). Figure 7(c) shows a particular case of stable incremental coaxiality in all directions, which is available only if B′=B, corresponding to Boundary OA+(λ2=λ3=λ̅2=λ̅3≥0) in Figure 6. Finally, Figure 7(d) corresponds to complex λ2 and λ3 and λ̅2≥λ̅3≥0, which falls within Domain VI (Figure 6). In this particular case, INC is stable in all directions. As illustrated in Figure 6, λ2>0>λ3 and 0>λ2≥λ3 are two cases that cannot coexist with λ̅2≥λ̅3≥0.Figure8 and Table 3(b) show four cases of unstable INC, which corresponds to 0≥λ̅2≥λ̅3. In contrast to Figure 7, unstable IC is possible along a few directions. Figures 8(a) and 8(b) show four and two unstable IC (UIC), respectively, which correspond to cos2θinc=-1, and fall within Domain II (0≥λ̅2≥λ̅3 and λ2≠λ3<0 and along boundary a-A- (0≥λ̅2≥λ̅3 and λ2=λ3) in Figure 6. Figure 8(c) shows the particular case of unstable IC in all directions, which is available only if B′=B, corresponding to boundary OA- (λ2=λ3=λ̅2=λ̅3<0) in Figure 6. Finally, Figure 8(d) shows that there is unstable INC in all directions, which corresponds to Domain VII in Figure 6.Figure9 and Table 3(c) shows a particular case when cos2θinc=0, which corresponds to λ̅2=λ̅3=0 and conjugate complex λ2 and λ3. Clearly, there is INC in all directions, which corresponds to Point O′ in Figure 6.Figure10 and Table 3(d) show six cases of partially stable INC, which corresponds to λ̅2>0>λ̅3. The case λ̅2>0>λ̅3 can be obtained for any values of λ2 and λ3.  The classifications correspond to Domains III, IV, V, VIII and boundaries Oa+ and Oa-. Stable IC (SIC) can be obtained in 4 directions in Figure 10(a), 2 in Figure 10(b), and 2 in Figure 10(e). Unstable IC (UIC) can be obtained in 4 directions in Figure 10(c), 2 in Figure 10(d) and 2 in Figure 10(e). Figure 10(f) shows that partially stable INC is obtained for all directions, which corresponds to Domain VIII in Figure 6.As shown in Figures7–10, INC is independent of Rdσ, which results from Cijkldδkl=0.  On the other hand, Figure 11 presents the variation of INC in terms Rdσ and θdσ, when Cijkldδkl≠0. The intensity of shaded contours indicates the stability of INC. Table 3(e) lists the corresponding material coefficients. In this case, INC cannot be classified using spectral classification, because it depends not only on the principal direction θdσ but also on the ratio of principal stress increments, Rdσ. Similar results are founds when λ2 and λ3 are real and complex.In the above examples, the compliance matricesCijkl are clearly related to classical anisotropic materials. For instance, isotropic elasticity always creates stable IC (λ2=λ3=λ̅2=λ̅3≥0), while orthogonally anisotropy may create stable INC (λ2=λ̅2≠λ3=λ̅3>0 and Cijkldδkl=0) presented in Figure 7(a), or partially stable INC (Cijkldδkl≠0) as shown in Figure 11(a). On the other hand, the coefficients of compliance matrix Cijkl for elastoplastic materials may change and generate several distinct eigenvalue cases with the development of work hardening (softening).In addition, as shown in Figures7 and 8, pure stable (or unstable) INC that requires the condition of Cijkldδkl=0, may be reproduced by Tresca-type or Mises-type models, which are generally used to simulate incompressible material behaviors. However, pressure sensitive materials (Cijkldδkl≠0) create partially stable INC only, as shown in Figure 11.


## 5. Conclusion

The present study has presented a spectral approach to explore the non-coaxial aspects of incremental material behavior and has classified the non-coaxiality between stress and strain increments. It expands the definition of non-coaxiality (NC), which usually refers to the difference between the principal directions of incremental strain and stress state in the solid mechanics literature. The analysis has investigated the incremental non-coaxiality (INC) induced by incremental stress-strain relations, for example, linearized rate-type solids. Based on the concept of deviatoric second-order work, this study has examined the relations between incremental non-coaxiality and stability. It has proposed three classifications that are based on eigenvalues of the constitutive compliance matrix. These classifications distinguish various degrees of incremental non-coaxiality and stability. They determine the conditions for which stress and strain increments are collinear and result into stability, instability, and stable-instable transition. The classifications have been illustrated using examples of constitutive matrices that are relevant to elastic as well as elastoplastic constitutive modeling. The present classifications are useful for examining the incremental stability of incompressible materials and determining anisotropic state and optimal material orientation.



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