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

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.

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.

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 defines coaxiality (C, 𝜃_𝜎=𝜃_𝑑𝜀) and non-coaxiality (NC, 𝜃_𝜎≠𝜃_𝑑𝜀) in terms of 𝜎_𝑖𝑗 and 𝑑𝜀_𝑖𝑗. Proportional (P, 𝜃_𝜎=𝜃_𝑑𝜎) loading corresponds to 𝑑𝜎_𝑖𝑗 proportional to 𝜎_𝑖𝑗, and non-proportional (NP, 𝜃_𝜎≠𝜃_𝑑𝜎) loading refers to changes in principal direction of 𝑑𝜎_𝑖𝑗, when 𝜎_𝑖𝑗 moves along nonlinear stress paths. Incremental coaxiality (IC, 𝜃_𝑑𝜎=𝜃_𝑑𝜀) and incremental non-coaxiality (INC,𝜃_𝑑𝜎≠𝜃_𝑑𝜀) are defined in terms of 𝑑𝜎_𝑖𝑗 and 𝑑𝜀_𝑖𝑗. 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.

(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)

Figure 2 

Non-coaxiality available by three approaches: (a) INC+P; (b) INC+NP; (c) IC+NP.

The incremental non-coaxiality (INC) and non-proportionality (NP) of loading can be defined, respectively, as 𝜃_inc=𝜃_𝑑𝜀−𝜃_𝑑𝜎,𝜃_np=𝜃_𝑑𝜎−𝜃_𝜎. (1.1a)(1.1b)

Therefore, non-coaxiality (NC), incremental non-coaxiality (INC), and non-proportionality (NP) of loading are related by 𝜃_nc=𝜃_inc+𝜃_np, (1.1c) 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.

2. Lame’s Representation of Incremental Non-Coaxiality

Material behaviors are modeled using rate-type linearized solids, that is, 𝑑𝜀_𝑖𝑗=𝐶_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙, (2.1) 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:[𝐶]=𝐶₁₁₁₁𝐶₁₁₂₂𝐶₁₁₁₂𝐶₂₂₁₁𝐶₂₂₂₂𝐶₂₂₁₂𝐶₁₂₁₁𝐶₁₂₂₂𝐶₁₂₁₂. (2.2)

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 𝑑𝜎₁ and 𝑑𝜎₂ (𝑑𝜎₁≥𝑑𝜎₂) and the orientation angle 𝜃_𝑑𝜎 of principal direction. Similarly, the strain increment 𝑑𝜀_𝑖𝑗 is defined using the principal values of strain increment 𝑑𝜀₁ and 𝑑𝜀₂ (𝑑𝜀₁≥𝑑𝜀₂) 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.

(a) Incremental stress ellipse

(b) Incremental strain ellipse

Figure 3 

Lame’s visualization of incremental stress-strain responses.

In Lame’s representation, (2.1) is restated as follows:𝜃_𝑑𝜀=𝑓𝑑𝜎₁,𝑑𝜎₂,𝜃_𝑑𝜎,𝑑𝜀₁=𝑓₂𝑑𝜎₁,𝑑𝜎₂,𝜃_𝑑𝜎,𝑑𝜀₂=𝑓₃𝑑𝜎₁,𝑑𝜎₂,𝜃_𝑑𝜎, (2.3) where the nonlinear functions 𝑓₁, 𝑓₂, and 𝑓₃ depend on 𝐶_{𝑖𝑗𝑘𝑙} and −90^∘≤𝜃_𝑑𝜎≤90^∘.

Without loss of generality, (2.3) may be simplified assuming 𝑑𝜎₂=1 and 𝑑𝜀₂=1 (see the appendix) 𝜃_inc=𝜃_𝑑𝜀−𝜃_𝑑𝜎=𝐹₁𝑅_𝑑𝜎,𝜃_𝑑𝜎,𝑅_𝑑𝜀=𝑑𝜀₁𝑑𝜀₂=𝐹₂𝑅_𝑑𝜎,𝜃_𝑑𝜎, (2.4)(2.5) where functions 𝐹₁ and 𝐹₂ are analogous to 𝑓₁, 𝑓₂ and 𝑓₃, and 𝑅_𝑑𝜎=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, assuming 𝑑𝜎₁>𝑑𝜎₂>0, Table 1 shows five tensors that have the same Lame’s ellipse but different values of 𝜃_𝑑𝜎 and 𝑅_𝑑𝜎.

Table 1 

Orientation angle and ratio of principal stress increments.

3. 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.

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 𝑑𝑒_𝑖𝑗=𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙, (3.1) where 𝑑𝑒_𝑖𝑗=𝑑𝜀_𝑖𝑗−𝛿_𝑖𝑗𝑑𝜀_𝑘𝑘/2, and 𝐶^𝑑_{𝑖𝑗𝑘𝑙}=𝐶_{𝑖𝑗𝑘𝑙}−𝛿_𝑖𝑗𝐶_{𝑎𝑎𝑘𝑙}/2.

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 is𝑑²𝑊^𝑑=𝑑𝜎_𝑖𝑗𝑑𝑒_𝑖𝑗=𝑑𝑠_𝑖𝑗𝑑𝑒_𝑖𝑗=𝑑𝑠_𝑖𝑗𝑑𝑒_𝑖𝑗cos2𝜃_𝑑𝜀−𝜃_𝑑𝜎=𝑑𝑠_𝑖𝑗𝑑𝑒_𝑖𝑗cos2𝜃_inc, (3.2) 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:cos2𝜃_inc=𝑑𝑠_𝑖𝑗𝑑𝑒_𝑖𝑗𝑑𝑠_𝑖𝑗𝑑𝑒_𝑖𝑗=𝑑𝜎_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙𝑑𝑠_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙. (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, cos𝜃_inc is an energy-based parameter that can be extended to higher-dimensional problems.

Assuming 𝑑𝜎₂=1, 𝑑𝜎_𝑖𝑗 can be explicitly expressed as [𝑑𝜎]=𝑑𝜎_𝑎𝑎2[𝛿]+[]=𝑅𝑑𝑠^𝑑𝜎+12+10𝑅01^𝑑𝜎−12cos2𝜃_𝑑𝜎sin2𝜃_𝑑𝜎sin2𝜃_𝑑𝜎−cos2𝜃_𝑑𝜎, (3.4) 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,𝐶₁₂₁₁+𝐶₁₂₂₂=0,𝐶₁₁₁₁−𝐶₂₂₁₁−𝐶₂₂₂₂+𝐶₁₁₂₂=0, (3.5) which corresponds to pressure increments producing no deviatoric strains. With 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0, equation (3.3) becomescos2𝜃_inc=𝑑𝑠_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝑠_𝑘𝑙𝑑𝑠_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝑠_𝑘𝑙=𝑛_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝑛_𝑘𝑙𝑛_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑛_𝑘𝑙, (3.6a) where 𝑛_𝑖𝑗 is[𝑛]=cos2𝜃_𝑑𝜎sin2𝜃_𝑑𝜎sin2𝜃_𝑑𝜎−cos2𝜃_𝑑𝜎. (3.6b)According to (3.6a)-(3.6b),cos2𝜃_inc is independent from 𝑅_𝑑𝜎, and 𝑑²𝑊^𝑑(=𝑑𝜎_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙) and cos2𝜃_inc have identical sign.

Equation (3.1) can be equivalently expressed in matrix form as follows:[𝑑𝑒]=𝐶^𝑑[𝑑𝜎], (3.7a) or 𝑑𝜀₁₁−𝑑𝜀₂₂2𝑑𝜀₂₂−𝑑𝜀₁₁2√2𝑑𝜀̇₁₂=𝐶₁₁₁₁−𝐶₂₂₁₁2𝐶₁₁₂₂−𝐶₂₂₂₂2√2𝐶₁₁₁₂−𝐶₂₂₁₂2𝐶₂₂₁₁−𝐶₁₁₁₁2𝐶₂₂₂₂−𝐶₁₁₂₂2√2𝐶₂₂₁₂−𝐶₁₁₁₂2√2𝐶₁₂₁₁√2𝐶₁₂₂₂2𝐶₁₂₁₂𝑑𝜎₁₁𝑑𝜎₂₂√2𝑑𝜎₁₂. (3.7b)

Complicated spectral analyses for nonsymmetric matrices are avoided by noting that any nonsymmetric matrix A satisfies 𝐱^𝐓𝐀𝐱=𝐱^𝐓((𝐀+𝐀^𝐓)/2)𝐱 for any vector x. Equation (3.3) can be rewrittencos2𝜃_inc=𝑑𝜎_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙𝑑𝑠_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙=𝑑𝜎_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙𝑑𝑠_𝑖𝑗𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙=[𝑑𝜎]^𝑇𝐶^𝑑[𝑑𝜎][𝑑𝑠]^𝑇[𝑑𝑠][𝑑𝑠]^𝑇𝐶^𝑑𝑇𝐶^𝑑[𝑑𝑠], (3.8) where [𝐶^𝑑]=([𝐶^𝑑]+[𝐶^𝑑]^𝑇)/2 (i.e., 𝐶^𝑑_{𝑖𝑗𝑘𝑙}=(𝐶^𝑑_{𝑖𝑗𝑘𝑙}+𝐶^𝑑_{𝑘𝑙𝑖𝑗})/2) is the symmetric part of [𝐶_𝑑], whose equivalent matrix form is 𝐶^𝑑=𝐶₁₁₁₁−𝐶₂₂₁₁2𝐶₂₂₁₁−𝐶₁₁₁₁+𝐶₁₁₂₂−𝐶₂₂₂₂4√22𝐶₁₂₁₁+𝐶₁₁₁₂−𝐶₂₂₁₂4𝐶₂₂₁₁−𝐶₁₁₁₁+𝐶₁₁₂₂−𝐶₂₂₂₂4𝐶₂₂₂₂−𝐶₁₁₂₂2√22𝐶₁₂₂₂+𝐶₂₂₁₂−𝐶₁₁₁₂4√22𝐶₁₂₁₁+𝐶₁₁₁₂−𝐶₂₂₁₂4√22𝐶₁₂₂₂+𝐶₂₂₁₂−𝐶₁₁₁₂42𝐶₁₂₁₂. (3.9)

The eigenvalues of [𝐶^𝑑] are denoted by 𝜆₁, 𝜆₂ and 𝜆₃, and the corresponding orthogonal eigentensors are denoted by 𝛼_𝑖𝑗, 𝛽_𝑖𝑗 and 𝜒_𝑖𝑗. Any stress increments can be expressed as𝑑𝜎_𝑖𝑗=𝑎𝛼_𝑖𝑗+𝑏𝛽_𝑖𝑗+𝑐𝜒_𝑖𝑗, (3.10) where 𝑎, 𝑏, and 𝑐 are the components in the eigendirections.

Equation (3.8) becomes cos2𝜃_inc=𝑎²𝜆₁+𝑏²𝜆₂+𝑐²𝜆₃[𝑑𝑠]^𝑇[𝑑𝑠][𝑑𝑠]^𝑇𝐶^𝑑𝑇𝐶^𝑑[𝑑𝑠], (3.11) which shows that cos2𝜃_inc depends on the signs of 𝜆₁, 𝜆₂ and 𝜆₃.

When 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0, the eigenvalues of [𝐶^𝑑] are 𝜆₁=0,𝜆₂=𝐴^′+𝐵^′2,𝜆₃=𝐴^′−𝐵^′2, (3.12a) where 𝐴^′=𝐶₁₁₁₁−𝐶₂₂₁₁+2𝐶_1212,𝐵^′=𝐶₁₁₁₁−𝐶₂₂₁₁−2𝐶₁₂₁₂²+2𝐶₁₂₁₁+𝐶₁₁₁₂−𝐶₂₂₁₂². (3.12b)(3.12c)Parameters 𝐴^′ and 𝐵^′ solely control the values of eigenvalues. Invoking that 𝜆₂≥𝜆₃, Figure 4 shows four distinct domains:(I)stable; cos2𝜃_inc>0; 𝜆₁=0 and 𝜆₂≥𝜆₃≥0; 𝐴^′≥𝐵^′,(II)unstable; cos2𝜃_inc<0; 𝜆₁=0 and 0≥𝜆₂≥𝜆₃; 𝐴^′<−𝐵^′,(III)partially stable; −1≤cos2𝜃_inc≤1; 𝜆₁=0 and 𝜆₂>0>𝜆₃; −𝐵^′<𝐴^′<𝐵^′.

Figure 4 

Classification 1 (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0).

Point 𝑂^′: Stable-Unstable Transition; cos2𝜃_inc=0; 𝜆₁=0 and 𝜆₂=𝜆₃=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).

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].

𝐶^𝑑_{𝑖𝑗𝑘𝑙} are assumed to have three eigenvalues 𝜆₁, 𝜆₂, and 𝜆₃ and three corresponding eigendirections 𝛼_𝑖𝑗, 𝛽_𝑖𝑗, and 𝜒_𝑖𝑗. The eigenvalues of 𝐶^𝑑_{𝑖𝑗𝑘𝑙} are 𝜆₁≡0,𝜆₂=𝐴+√𝐵2,𝜆₃=𝐴−√𝐵2, (3.13) where 𝐶𝐴=¹¹¹¹−𝐶₂₂₁₁+𝐶₂₂₂₂−𝐶₁₁₂₂+4𝐶₁₂₁₂2,𝐶𝐵=¹¹¹¹−𝐶₂₂₁₁+𝐶₂₂₂₂−𝐶₁₁₂₂−4𝐶₁₂₁₂2²+4𝐶₁₁₁₂−𝐶₂₂₁₂𝐶₁₂₁₁−𝐶₁₂₂₂. (3.14) Consider that a given stress increment𝑑𝜎_𝑖𝑗=𝑎𝛼_𝑖𝑗+𝑏𝛽_𝑖𝑗+𝑐𝜒_𝑖𝑗, (3.15a) where the coefficients a, b, and c are variables comparable to 𝑎, 𝑏 and 𝑐 in (3.10). In the case 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0 (i.e., 𝛼_𝑖𝑗=𝛿_𝑖𝑗)𝑑𝑠_𝑖𝑗=𝑏𝛽_𝑖𝑗+𝑐𝜒_𝑖𝑗. (3.15b)Thus, 𝑑𝑒_𝑖𝑗=𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝜎_𝑘𝑙=𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝑠_𝑘𝑙+12𝑑𝜎_𝑚𝑚𝛿_𝑘𝑙=𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝑑𝑠_𝑘𝑙=𝑏𝜆₂𝛽_𝑘𝑙+𝑐𝜆₃𝜒_𝑘𝑙. (3.16) Equation (3.16) implies that 𝑑𝑒_𝑖𝑗 are coaxial with 𝑑𝜎_𝑖𝑗 when 𝑑𝑠_𝑖𝑗=𝑏𝛽_𝑖𝑗 or 𝑑𝑠_𝑖𝑗=𝑐𝜒_𝑖𝑗.

As shown in (3.13), (3.16) and Figure 5, parameters A and B control 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 (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0).

3.3. Classification 3—Couped Spectral Classification of INC

The third classification builds upon the two previous classifications. Hereafter, we focus on the case 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0 because 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙≠0 creates incremental non-coaxiality (INC) in all directions without possible stable or unstable incremental coaxiality. 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0 implies that 𝐴^′𝐵=𝐴,_′−𝐵=2𝐶₁₂₁₁−𝐶₁₁₁₂+𝐶₂₂₁₂²≥0. (3.17)

Figures 4 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.

Figure 6 

Classification 3 (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0).

When 𝐵^′=𝐵, which results in 𝐶^𝑑_{𝑖𝑗𝑘𝑙}=𝐶^𝑑_{𝑖𝑗𝑘𝑙} (i.e., 𝜆₂=𝜆₂ and 𝜆₂=𝜆₃), 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, 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(a)–3(e) and Figures 7–11 contain 17 examples that apply to anisotropic elasticity as well as elastoplasticity.

Table 3 

(a) 4 SIC

(b) 2 SIC

(c) SIC

(d) No SIC

Figure 7 

𝜃𝑑𝜎−cos2𝜃inc for stable INC (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0).

(a) 4 UIC

(b) 2 UIC

(c) UIC

(d) No UIC

Figure 8 

𝜃𝑑𝜎−cos2𝜃inc for unstable INC (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0).

Figure 9 

𝜃𝑑𝜎−cos2𝜃inc for stable-unstable transition INC (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0, complex 𝜆2,𝜆3).

(a) 4 SIC

(b) 2 SIC

(c) 4 UIC

(d) 2 UIC

(e) 2 SIC and 2 UIC

(f) No SIC/UIC

Figure 10 

𝜃𝑑𝜎−cos2𝜃inc for partially stable INC (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙=0).

(a) No SIC/UIC (real 𝜆2,𝜆3)

(b) No SIC/UIC (complex 𝜆2,𝜆3)

Figure 11 

𝑅𝑑𝜎−𝜃𝑑𝜎−cos2𝜃inc for partially stable INC (𝐶𝑑𝑖𝑗𝑘𝑙𝛿𝑘𝑙≠0).

Figure 7 and Table 3(a) show four cases of stable incremental non-coaxiality when 𝜆₂≥𝜆₃≥0. Figures 7(a) and 7(b) show that there are four and two stable IC (SIC) (i.e., cos2𝜃_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⁺(𝜆₂=𝜆₃=𝜆₂=𝜆₃≥0) in Figure 6. Finally, Figure 7(d) corresponds to complex 𝜆₂ and 𝜆₃ and 𝜆₂≥𝜆₃≥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 𝜆₂≥𝜆₃≥0.

Figure 8 and Table 3(b) show four cases of unstable INC, which corresponds to 0≥𝜆₂≥𝜆₃. 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 cos2𝜃_inc=−1, and fall within Domain II (0≥𝜆₂≥𝜆₃ and 𝜆₂≠𝜆₃<0 and along boundary 𝑎⁻𝐴⁻ (0≥𝜆₂≥𝜆₃ 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=𝜆₂=𝜆₃<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 9 and Table 3(c) shows a particular case when cos2𝜃_inc=0, which corresponds to 𝜆₂=𝜆₃=0 and conjugate complex 𝜆₂ and 𝜆₃. 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 𝜆₂>0>𝜆₃. The case 𝜆₂>0>𝜆₃ 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 Figures 7–10, INC is independent of 𝑅_𝑑𝜎, which results from 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0. On the other hand, Figure 11 presents the variation of INC in terms 𝑅_𝑑𝜎 and 𝜃_𝑑𝜎, when 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙≠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 𝜃_𝑑𝜎 but also on the ratio of principal stress increments, 𝑅_𝑑𝜎. Similar results are founds when 𝜆₂ and 𝜆₃ are real and complex.

In the above examples, the compliance matrices 𝐶_{𝑖𝑗𝑘𝑙} are clearly related to classical anisotropic materials. For instance, isotropic elasticity always creates stable IC (𝜆₂=𝜆₃=𝜆₂=𝜆₃≥0), while orthogonally anisotropy may create stable INC (𝜆₂=𝜆₂≠𝜆₃=𝜆₃>0 and 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0) presented in Figure 7(a), or partially stable INC (𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙≠0) as shown in Figure 11(a). On the other hand, the coefficients of compliance matrix 𝐶_{𝑖𝑗𝑘𝑙} for elastoplastic materials may change and generate several distinct eigenvalue cases with the development of work hardening (softening).

In addition, as shown in Figures 7 and 8, pure stable (or unstable) INC that requires the condition of 𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙=0, may be reproduced by Tresca-type or Mises-type models, which are generally used to simulate incompressible material behaviors. However, pressure sensitive materials (𝐶^𝑑_{𝑖𝑗𝑘𝑙}𝛿_𝑘𝑙≠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.

Appendix

The incremental stress-strain relationship given by (2.1) can be rewritten as follows: 𝑑𝜀_𝑥𝑥=𝐶_{𝑥𝑥𝑥𝑥}𝑑𝜎_𝑥𝑥+𝐶_{𝑥𝑥𝑦𝑦}𝑑𝜎_𝑦𝑦+𝐶_{𝑥𝑥𝑥𝑦}𝑑𝜎_𝑥𝑦𝑑𝜀_𝑦𝑦=𝐶_{𝑦𝑦𝑥𝑥}𝑑𝜎_𝑥𝑥+𝐶_{𝑦𝑦𝑦𝑦}𝑑𝜎_𝑦𝑦+𝐶_{𝑦𝑦𝑥𝑦}𝑑𝜎_𝑥𝑦𝑑𝜀_𝑥𝑦=𝐶_{𝑥𝑦𝑥𝑥}𝑑𝜎_𝑥𝑥+𝐶_{𝑥𝑦𝑦𝑦}𝑑𝜎_𝑦𝑦+𝐶_{𝑥𝑦𝑥𝑦}𝑑𝜎_𝑥𝑦. (A.1)(A.2)(A.3) The principal direction of strain increment can be defined astan2𝜃_𝑑𝜀=2𝑑𝜀_𝑥𝑦𝑑𝜀_𝑥𝑥−𝑑𝜀_𝑦𝑦. (A.4) Substitution of (A.1), (A.2) and (A.3) into (A.4) leads totan2𝜃_𝑑𝜀=2𝐶_{𝑥𝑦𝑥𝑥}𝑑𝜎_𝑥𝑥+2𝐶_{𝑥𝑦𝑦𝑦}𝑑𝜎_𝑦𝑦+2𝐶_{𝑥𝑦𝑥𝑦}𝑑𝜎_𝑥𝑦𝐶_{𝑥𝑥𝑥𝑥}−𝐶_{𝑦𝑦𝑥𝑥}𝑑𝜎_𝑥𝑥+𝐶_{𝑥𝑥𝑦𝑦}−𝐶_{𝑦𝑦𝑦𝑦}𝑑𝜎_𝑦𝑦+𝐶_{𝑥𝑥𝑥𝑦}−𝐶_{𝑦𝑦𝑥𝑦}𝑑𝜎_𝑥𝑦. (A.5) Invoking that stress is related to its principal values by 𝑑𝜎_𝑥𝑥=𝑑𝜎₁+𝑑𝜎₂2+𝑑𝜎₁−𝑑𝜎₂2cos2𝜃_𝑑𝜎𝑑𝜎_𝑦𝑦=𝑑𝜎₁+𝑑𝜎₂2−𝑑𝜎₁−𝑑𝜎₂2cos2𝜃_𝑑𝜎𝑑𝜎_𝑥𝑦=𝑑𝜎₁−𝑑𝜎₂2sin2𝜃_𝑑𝜎. (A.6)(A.7)(A.8) With the introduction of (A6), (A.7) and (A.8) into (A.5), thus (A.5) can be restated astan2𝑞_𝑑𝑒=2𝐶_{𝑥𝑦𝑥𝑥}𝒜+2𝐶_{𝑥𝑦𝑦𝑦}𝒞+2𝐶_{𝑥𝑦𝑥𝑦}𝒟𝐶_{𝑥𝑥𝑥𝑥}−𝐶_{𝑦𝑦𝑥𝑥}𝒜+𝐶_{𝑥𝑥𝑦𝑦}−𝐶_{𝑦𝑦𝑦𝑦}𝒞+𝐶_{𝑥𝑥𝑥𝑦}−𝐶_{𝑦𝑦𝑥𝑦}𝒟, (A.9) where 𝒜, 𝒞, and 𝒟 denote (𝑅_𝑑𝜎+tan²𝜃_𝑑𝜎), (𝑅_𝑑𝜎tan²𝜃_𝑑𝜎+1), and (𝑅_𝑑𝜎tan𝜃_𝑑𝜎−tan𝜃_𝑑𝜎), respectively. Recall that 𝑅_𝑑𝜎=𝑑𝜎₁/𝑑𝜎₂ in (A.9). Based on (A.9), we can concluded (2.4).

Analogous to the above analysis, it can be checked that it is also true for (2.5).

Acknowledgment

The financial support provided by National Natural Science Foundation of China (Grant nos. 10972159 and 50825803) and Natural Science Foundation of Shanghai, China (Grant no. 08ZR1420100).