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

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.


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 noncoaxiality 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 , noncoaxiality 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.
Figure 1 illustrates the relationship among non-coaxiality, incremental non-coaxiality and stress path.It relates these three concepts using 1 stress state σ ij , 2 stress increment dσ ij , and 3 strain increment dε ij .The principal direction angles of σ ij , ε ij , and dε ij are defined as θ σ , θ dσ and θ dε , respectively Figure 2 .Like in the solid mechanics literature 23 , Figure 1 Therefore, non-coaxiality NC , incremental non-coaxiality INC , and non-proportionality NP of loading are related by where θ np can be determined from the prescribed loading path and θ inc is a type-rate material property that needs to be measured from incremental stress-strain response.Clearly, θ nc θ inc 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.

Lame's Representation of Incremental Non-Coaxiality
Material behaviors are modeled using rate-type linearized solids, that is, where dε ij is the incremental strain, dσ ij is the incremental stress, and C ijkl is the incremental compliance tensor that represents some underlying anisotropy state.The indices are i, j . . . 1, 2 in two-dimensions.C ijkl can be written in matrix form as follows: The physical meaning of incremental non-coaxiality can be illustrated using Lame's ellipse Figure 3 , which represents a two-dimensional second-order tensor using an inclined ellipse.dσ ij is uniquely expressed in terms of principal stress values dσ 1 and dσ 2 dσ 1 ≥ dσ 2 and the orientation angle θ dσ of principal direction.Similarly, the strain increment dε ij is defined using the principal values of strain increment dε 1 and dε 2 dε 1 ≥ dε 2 and θ dε the orientation angle of principal direction.The incremental response is coaxial when θ dε θ dσ and non-coaxial when θ dε / θ dσ .
In Lame's representation, 2.1 is restated as follows: where the nonlinear functions f 1 , f 2 , and f 3 depend on C ijkl and −90 • ≤ θ dσ ≤ 90 • .Without loss of generality, 2.3 may be simplified assuming dσ 2 1 and dε 2 1 see the appendix where functions F 1 and F 2 are analogous to f 1 , f 2 and f 3 , and R dσ dσ 1 /dσ 2 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, assuming dσ 1 > dσ 2 > 0, Table 1 shows five tensors that have the same Lame's ellipse but different values of θ dσ and R dσ .

Classifications of Incremental Non-Coaxiality
Three different classifications are proposed for incremental non-coaxiality, based on the values of C ijkl 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.

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 where The eigenvalues of C d ijkl can be analytically calculated by solving a quadratic equation, which is simpler than solving a cubic equation for the eigenvalues of C ijkl .This is because C d ijkl is rank deficient and always possess one zero eigenvalue, leaving only two eigenvalues to determine.
The deviatoric second-order work is where ds ij is the deviatoric part of dσ ij , |ds ij | and |de ij | denote the magnitude of ds ij and de ij , respectively.One obtains:

3.3
The trivial cases ds ij 0 and de ij 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 θ inc given by 3.3 does not represent a certain angular orientation of incremental non-coaxiality.However, cos θ inc is an energybased parameter that can be extended to higher-dimensional problems.
Assuming dσ 2 1, dσ ij can be explicitly expressed as which implies that ds ij and dσ ij have the same principal directions.
In the particular case C d ijkl δ kl 0, that is,

3.7b
Complicated spectral analyses for nonsymmetric matrices are avoided by noting that any nonsymmetric matrix A satisfies x T Ax x T A A T /2 x for any vector x.Equation The eigenvalues of C d are denoted by λ 1 , λ 2 and λ 3 , and the corresponding orthogonal eigentensors are denoted by α ij , β ij and χ ij .Any stress increments can be expressed as When C d ijkl δ kl 0, the eigenvalues of where

3.12c
Parameters A and B solely control the values of eigenvalues.Invoking that λ 2 ≥ λ 3 , Figure 4 shows four distinct domains: Mathematical Problems in Engineering 9 Point O : Stable-Unstable Transition; cos 2θ inc 0; λ 1 0 and λ 2 λ 3 0; A B 0. When C d ijkl δ kl / 0, the above spectral approach does not fully apply however.This case is similar to partially stable INC Domain III in Figure 4 .

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 .
C d ijkl are assumed to have three eigenvalues λ 1 , λ 2 , and λ 3 and three corresponding eigendirections α ij , β ij , and χ ij .The eigenvalues of C d ijkl are

3.14
Consider that a given stress increment where the coefficients a, b, and c are variables comparable to a, b and c in 3.10 .In the case  i.e., SIC or UIC depends on the signs of λ 2 and λ 3 .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.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.

Examples
Incremental non-coaxiality is modeled for compliance matrices 2.2 that do not necessarily have a major symmetry, that is, C ijkl / C klij .As shown in Table 3, all components of C ijkl 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  Figure 7 and Table 3 a show four cases of stable incremental non-coaxiality when λ 2 ≥ λ 3 ≥ 0. Figures 7 a and 7 b show that there are four and two stable IC SIC i.e., cos 2θ inc 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 λ 2 λ 3 λ 2 λ 3 ≥ 0 in Figure 6.Finally, Figure 7 d corresponds to complex λ 2 and λ 3 and λ 2 ≥ λ 3 ≥ 0, which falls within Domain VI Figure 6 .In this particular case, INC is stable in all directions.As illustrated in Figure 6, λ 2 > 0 > λ 3 and 0 > λ 2 ≥ λ 3 are two cases that cannot coexist with λ 2 ≥ λ 3 ≥ 0.
Classification Domains in Figure 6 C ijkl coefficients Spectral properties Computation results in Figure 8 II 0.5 1.5 0 1.5 0.5 0 0 0 −0.1    Figure 9 and Table 3 c shows a particular case when cos 2θ inc 0, which corresponds to λ 2 λ 3 0 and conjugate complex λ 2 and λ 3 .Clearly, there is INC in all directions, which corresponds to Point O in Figure 6.
Figure 10 and Table 3 d show six cases of partially stable INC, which corresponds to λ 2 > 0 > λ 3 .The case λ 2 > 0 > λ 3 can be obtained for any values of λ 2 and λ 3 .The classifications correspond to Domains III, IV, V, VIII and boundaries Oa and Oa − .Stable IC  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 θ dσ but also on the ratio of principal stress increments, R dσ .Similar results are founds when λ 2 and λ 3 are real and complex.

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 stableinstable 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.

Figure 1 :
Figure 1: Illustration of relationship among incremental strain, incremental stress and stress.

Figure 2 :
Figure 2: Non-coaxiality available by three approaches: a INC P; b INC NP; c IC NP.

λ 1 b 2 λ 2 c 2 λ 3
where a, b, and c are the components in the eigendirections.Equation 3.8 becomes cos 2θ inc a 2 ds T ds ds T C d T C d ds , 3.11 which shows that cos 2θ inc depends on the signs of λ 1 , λ 2 and λ 3 .
a -3 e and Figures 7-11 contain 17 examples that apply to anisotropic elasticity as well as elastoplasticity.

Figure 8 and
Figure 8 and Table 3 b show four cases of unstable INC, which corresponds to 0 ≥ λ 2 ≥ λ 3 .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 cos 2θ inc −1, and fall within Domain II 0 ≥ λ 2 ≥ λ 3 and λ 2 / λ 3 < 0 and along boundary a − A − 0 ≥ λ 2 ≥ λ 3 and λ 2 λ 3 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 − λ 2 λ 3 λ 2 λ 3 < 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.

Figure 8 c
Figure 8 and Table 3 b show four cases of unstable INC, which corresponds to 0 ≥ λ 2 ≥ λ 3 .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 cos 2θ inc −1, and fall within Domain II 0 ≥ λ 2 ≥ λ 3 and λ 2 / λ 3 < 0 and along boundary a − A − 0 ≥ λ 2 ≥ λ 3 and λ 2 λ 3 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 − λ 2 λ 3 λ 2 λ 3 < 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.

Table 1 :
Orientation angle and ratio of principal stress increments.
.6b , cos 2θ inc is independent from R dσ , and d 2 W d dσ ij C d ijkl dσ kl and cos 2θ inc have identical sign.
Equation 3.1 can be equivalently expressed in matrix form as follows: