Elastic potentials as yield surfaces for isotropic materials

Jorge Castro

doi:10.1371/journal.pone.0275968

Abstract

This paper proposes that elastic potentials, which may be rigorously formulated using the negative Gibbs free energy or the complementary strain energy density, may be used as the yield surface of elasto-plastic constitutive models. Thus, the yield surface may be assumed in some materials as an elastic potential surface for a specific level of critical complementary strain energy density. Traditional approaches, such as the total strain energy criterion, only consider second order terms, i.e., the elastic potential is centred at the origin of the current stress state. Here, first order terms are considered, and consequently, the elastic potential may be translated, which allows to reproduce the desired level of tension-compression asymmetry. The proposed approach only adds two additional parameters, e.g., uniaxial compressive and tensile yield limits, to the elastic ones. For linear elasticity, the proposed approach provides elliptical yield surfaces and shows a correlation between the shape of the ellipse and the Poisson’s ratio, which agree with published experimental data for soils and metallic glasses. This elliptical yield surface also fits well experimental values of amorphous polymers and some rocks. Besides, the proposed approach automatically considers the influence of the intermediate stress. For non-linear elasticity, a wider range of elastic potentials, i.e., yield surfaces, are possible, such as distorted ellipsoids. For the case of incompressible non-linear materials, the yield surfaces are between von Mises and Tresca ones.

Citation: Castro J (2022) Elastic potentials as yield surfaces for isotropic materials. PLoS ONE 17(10): e0275968. https://doi.org/10.1371/journal.pone.0275968

Editor: Kevin M. Moerman, National University of Ireland Galway, Galway, Ireland, IRELAND

Received: March 28, 2022; Accepted: September 27, 2022; Published: October 26, 2022

Copyright: © 2022 Jorge Castro. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: This paper is part of the R&D project “Strain energy density in rock strength at different temperatures (EnergyRock)” (Ref.: PID2020-116138GB-I00) funded by MCIN/ AEI /10.13039/501100011033 and its conceptualization started as part of a previous project entitled “The Critical Distance in Rock Fracture” (Ref.: BIA2015-67479-R) funded by the Spanish Ministry of Economy and Competitiveness and the European Regional Development Fund (ERDF). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

List of symbols: a,b,c,d, Parameters of the complementary strain energy function; C_ij, Compliance matrix; E, Young’s modulus; f, Yield surface; g, Plastic potential; G, Shear modulus; K, Bulk modulus; M, Stress ratio at critical state; n, Non-linearity material parameter; p, Mean stress; p_c, Hydrostatic compressive yield stress; p_t, Hydrostatic tensile yield stress; q, Deviatoric stress; U₀, Strain energy density; U_c0, Complementary strain energy density; α, Shape factor in the Ellipse failure criterion; ε, Strain; ε_ij, Strain tensor component; ν, Poisson’s ratio; σ, Normal stress; σ*, Positive “model” stress; σ_b, “Back” stress; σ_c, Uniaxial compressive yield stress; σ_i, Principal stress; σ_t, Uniaxial tensile yield stress; σ₀, Initial or shifting stress; σ_ij, Stress tensor component; τ, Shear stress; ϕ_cr, Critical state friction angle; Subscripts/superscripts 0, initial; 1,2,3, principal stress directions; d,v, deviatoric, volumetric; e,p, elastic, plastic; i,j,k, principal stress directions in contracted notation; oct, octahedral; ref, reference; y, yield

1. Introduction

Although the concepts of work and energy are essential in continuum solid mechanics, they are not so commonly or easily integrated in yield criteria, which limit elastic and plastic states. Notable attempts, such as the total strain energy criterion [1, 2], are not currently used. In the related field of fracture mechanics, energy is generally accepted as a criterion for crack initiation; in fact, its origin is due to Griffith [3], who originally applied the first law of thermodynamics to solve the failure problem of a cracked glass and proposed a critical energy criterion. Nowadays, the Theory of Critical Distances (e.g., [4]) allows to apply stress-based criteria, avoiding the singularity of stresses at the crack front, and, on the other hand, the average strain energy density criterion [5] is another successful method for fracture assessment. Fracture mechanics is one of the fields that the author has worked in and motives this work [6, 7], and the other is constitutive modelling of soft soils using elasto-plastic models [8, 9].

Energy concepts are also helpful in providing additional techniques to solve elasticity problems (e.g., [10]). Also, a restrictive form of elasticity that is usually called Green elasticity or hyperelasticity (e.g., [11]) requires the existence of strain energy potential functions (Fig 1): (1)

Download:

Fig 1. Strain energy for uniaxial stress.

https://doi.org/10.1371/journal.pone.0275968.g001

Cauchy stresses and small strains are considered in this paper.

Hyperelastic formulations guarantee that the first law of thermodynamics is satisfied. For isothermal conditions, the strain energy per unit volume, U₀, also called strain energy density, and the complementary strain energy per unit volume, U_c0, are equivalent to the Helmholtz free energy and Gibbs free energy with negative sign, respectively (e.g., [12]).

The search for more accurate yield surfaces has led to the development of complex surfaces with an increasing number of fitting parameters. Besides, the yield criteria are material-specific. In rocks, for example, some recent advanced models (e.g., [13]) require calibration of fitting functions, while the empirical Hoek-Brown [14] criterion is still the most popular for rocks due to its simplicity. In this manner, yield criteria should preferably be based on solid theoretical principles, cover an ample range of materials and have a few parameters that are easily calibrated, e.g., the uniaxial tension and compressive yield limits (e.g., [15]).

This paper tries to open a path for theoretically based yield surfaces using elastic potentials, which may be rigorously formulated using the negative Gibbs free energy or the complementary strain energy density (e.g., [16]). Thus, the yield surface may be assumed in some materials as an elastic potential surface for a specific level of critical complementary strain energy density. Here, rate-independent continuous materials under isothermal conditions are considered. Contrary to the total strain energy criterion, the elastic potential is not necessary centred at the origin of the current stress state because first order terms are also considered here, and consequently, the elastic potential may be translated. The proposed approach only adds two additional parameters, e.g., uniaxial compressive and tensile yield limits, to the elastic ones. This allows to correlate, for example, the shape of the yield surface and the Poisson’s ratio, which controls the shape of the elastic potential.

The basis and capabilities of the proposed approach are presented and some of its main features are validated using data available in the literature. In this manner, Section 2 presents the case of linear isotropic materials, both incompressible and compressible materials, where elastic potentials lead to von Mises and elliptical yield surfaces, respectively. Sections 3, 4, 5 and 6 present application to soils, rocks, metallic glasses and polymers, respectively. Section 7 further examines non-linear isotropic elasticity, which provides distorted elliptical yield surfaces and, for the case of an incompressible non-linear material, could lead to Tresca criterion. Finally, some conclusions are drawn.

2. Linear isotropic elasticity

2.1 Elastic potential

Linear elasticity may be easily formulated within the hyperelastic framework (e.g., [10]); it is enough to assume that the elastic potential is a quadratic form: (2)

Here, contracted notation is used for the sake of brevity and elastic potentials are presented in terms of unordered principal stresses, σ_i, for the sake of visualization in the principal stress space. As the material is isotropic, the behaviour for each principal direction should be identical. It is quickly demonstrated that: (3)

Thus, using the more common elastic parameters of Young’s modulus (E) and Poisson’s ratio (ν), the elastic potential for linear elasticity is: (4)

The shape of the elastic potential is an ellipsoid in the principal stress space and an ellipse using the octahedral normal and shear stresses (Fig 2). For the particular case of an incompressible material (ν = 0.5), the elastic potential degenerates into a cylinder (Von Mises); for the case of ν = 0, it is a sphere, and for the strange case of ν = −1, it degenerates into two planar surfaces of maximum mean stress. The positive definite property of the strain energy (U_c0≥0) gives the limit values of the Poisson’s ratio (−1≤ν≤1/2). From a geometrical point of view, this means that the elastic potentials should be convex surfaces (e.g., [17]).

Download:

Fig 2. Linear elastic potentials for different Poisson’s ratios in octahedral stress plot.

https://doi.org/10.1371/journal.pone.0275968.g002

Using Eqs (1) and (4), it may be shown that ν controls the strain path (shape of the elastic potential), while 1/2E acts like a kind of elastic multiplier: (5)

2.2 Yield surface

The proposed approach here assumes that the yield surface should correspond to an elastic potential surface for a critical value of the complementary strain energy density (U_c0 = U_c0,y). The subscript “y” is used here to denote the critical specific value of the strain energy that limits the elastic domain (yield surface). For linear elasticity, U₀ and U_c0 are the same and it is not necessary to distinguish. However, the critical value should be defined in terms of U_c0 and not U₀ because it is the one that constitutes a local minimum according to the principle of minimum complementary energy (e.g., [10]).

(6)

This implies a yield surface that is an ellipsoid in the principal stress space. For the particular case of an incompressible material (ν = 0.5), the yield surface is the von Mises cylinder (Fig 2). In this manner, the elastic potential is proposed here to be used both to establish the stress-strain relationship (Eq 4) and to define the yield surface (Eq 6).

2.3 Tension-compression yield asymmetry

Most materials show a tension-compression yield (and strength) asymmetry, i.e., the yield stress at compression is usually higher than that at tension. In Eq (6), no distinction between compressive and tensile stresses was made because the predicted behaviour is symmetric at tension and compression. Besides, the quadratic form proposed in Eq (2) for linear elasticity is not fully general because the linear term in stresses and the constant term, which are irrelevant for the linear elastic stress-strain behaviour, are missing. If those terms are added to Eq (2), the following general form of the complementary strain energy density is obtained: (7)

Introducing the linear and constant terms implies that the current (initial) state does not necessary correspond to a zero-stress and zero-strain state. This seems reasonable because the material may have previously suffered whatever strains and stresses and, from a practical point of view, only requires an incremental calculation of the elastic deformations from the current (initial) state.

In Eq (7), the strain energy density has units of energy per volume, i.e., pressure. Thus, coefficients a and b have units of the inverse of pressure (Eq 3), c is dimensionless and d has units of pressure and it may be combined with the strain energy density for the sake of simplicity. Hence, Eq (7) may be simplified by reorganizing the parameters as follows: (8) where b/a = −2ν (Eq 3). As for the Poisson’s ratio, the values of the other parameters have certain limits to ensure that U_c0≥0.

The yield surface may be obtained by imposing a limit value of the complementary strain energy density, U_c0,y in Eq (8). Once the Poisson’s ratio is determined, the two remaining parameters may be obtained from the yield stresses for two different stress paths. For example, using the uniaxial tensile and compressive yield stresses (−σ_t and σ_c), the yield surface is (9) or using the hydrostatic tensile and compressive yield stresses (−p_t and p_c), the yield surface is (10)

Thus, the yield surface has 3 parameters (e.g., ν, σ_c and σ_t, Eq 9). Please, note that compressive stresses are assumed as positive and the parameters σ_t and p_t are defined as positive (absolute) values.

The third term in Eqs (7–10) causes a translation of the elastic potential, i.e., the yield surface, which may be interpreted as a shifted origin or an initial hydrostatic stress state, σ₀, so that the elastic potential ellipsoid (Fig 2) is shifted and its origin is at σ₀. Please, note that σ₀ is not an “apparent” or measurable initial stress and may be considered simply as a broad idealization of internal forces, stress history, atmospheric pressure… Using σ₀, the elastic potential (Eq 7) or the yield surface (U_c0,y) may be alternatively expressed as: (11)

The relationship between σ₀ and coefficients c and d is given by Eqs (7) and (11).

(12)

Using Eqs (9–11), the initial or shifting stress (σ₀) may be expressed as a function of the yield stresses (13) (14)

3. Application of the linear case to soils

Soil response is clearly non-linear, but the application of the linear formulation (Section 2) gives a first approximation, as will be shown. As commonly done for soils, compressive stresses are assumed to be positive and the stress invariants p and q are used: (15)

Thus, using those stress invariants (p and q) and K and G, the yield surface based on the elastic potential (Eq 6) may be expressed as (16)

It is necessary to apply the translation to account for the tension-compression asymmetry (Eq 11). Besides, in the case of soils, tensile stresses are usually null (p_t≈0) and the hydrostatic compressive yielding stress (p_c) is usually called the mean preconsolidation pressure. Hence, the translation given by Eq (14) is p_c/2 and Eq (16) becomes (17)

The shape of the elastic potential and yield surface given by Eq (17) is completely analogous to the yield surface and plastic potential of the Modified Cam Clay (MCC) model [18] (18) where M is the stress ratio at critical state and may be correlated with the critical state friction angle for triaxial compression (σ₂ = σ₃, ordered principal stresses) as follows (19)

Thus, the analogy between the proposed yield surface (Eq 17) and that of the MCC model (Eq 18) gives the following equivalences: (20) (21)

Eq (20) implies a direct relationship between the Poisson’s ratio (an elastic parameter of the soil) and the stress ratio at critical state (a plastic and failure parameter of the soil). Interestingly, neither of those two parameters depend on the confining pressure. The relationship is plotted in Fig 3 and generally agrees with published values (S1 Appendix). The published values correspond to parameters for specific constitutive models calibrated from laboratory experiments, mainly drained triaxial compression tests. Some scatter in the data may arise from soil anisotropy, soil nonlinearity and Poisson’s ratio determination, calibration or specific meaning within the used constitutive model. On the other hand, Federico and Elia [19] explored several empirical correlations between the Poisson’s ratio and the friction angle of soils. For comparison with experimental data, they interpreted values provided by Wroth [20]. These data are also included in Fig 3, showing a good agreement with the proposed relationship.

Download:

Fig 3. Relationship between Poisson’s ratio and critical state friction angle for soils.

https://doi.org/10.1371/journal.pone.0275968.g003

4. Application to rocks

The proposed yield surface may also be applied to some rocks (Fig 4). Since most rocks behave as quasi-brittle materials, the yield surface becomes a failure envelope. The experimental data in Fig 4 correspond to Solnhofen limestone (after Mogi [21]). The main capabilities of the yield surface (Eq 9) are:

It is able to reproduce different compression/tension strength asymmetry ratios. In fact, the ratio may directly be an input value. In rocks, which are natural materials, this is particularly useful because the uniaxial compression/tension strength ratio is usually known and it varies in the range 5–40.
It is able to automatically consider the influence of the intermediate stress (σ₂) in a logical and natural manner. Fig 4 shows experimental results of true triaxial tests [21] and the influence of σ₂ is properly captured by the model.
The shape of the failure surface (i.e., elliptical, Eq 9) is similar to a parabolic curve in the area of interest (please, refer to the black solid line, σ₂ = σ₃, in Fig 4). Due to the large compression/tension strength asymmetry ratio, experimental values are only available for relatively low confining stresses (area of interest). Besides, the most common failure criterion for rocks is parabolic and empirical [14].

Download:

Fig 4. Elastic potentials as yield surfaces for Solnhofen limestone.

Lines: Eq 9. Symbols: experimental data after Mogi [21]. Compressive stresses positive.

https://doi.org/10.1371/journal.pone.0275968.g004

For rock types other than Solnhofen limestone, the proposed linear model (Eq 9) usually underestimates the influence of the confining stress. Non-linear models within the proposed framework (Section 7) could be used to better reproduce the influence of the confining stress.

5. Application to metallic glasses

Metallic glasses are macroscopically isotropic, exhibit nearly zero tensile ductility and very limited compressive plasticity, and cannot be work-hardened. Thus, the yield surface may be assumed to be the failure surface, as previously done for rocks (Section 4). Here, the data gathered by Liu et al. [22] are reinterpreted within the proposed framework for linear elastic isotropic materials (Section 2).

For the sake of comparison with Liu et al. [22], the yield surface (Eq 11) is represented as a shifted ellipse in the Mohr’s diagram (normal and shear stresses on the failure plane (σ, τ)): (22) where σ₀ is the initial (or shifting) stress and τ_y and σ_y are the vertical and horizontal semi-axes of the ellipse, respectively. Their ratio is the parameter that controls the shape of the ellipse, α = τ_y/σ_y, and may be related to the Poisson’s ratio, ν. For the sake of consistency with other parts of this paper, compressive stresses are assumed to be positive.

From Eq (11), using the Mohr’s circle and assuming triaxial (σ₂ = σ₃) or plane strain (σ₂ = ν(σ₁+σ₃)) conditions, the following relationships between α and ν may be found: (23) (24)

Fig 5 shows the relationship between α and ν. The correlation is analogous to that shown in Fig 3 for soils. Although some uncertainties arise in the comparison because the stress triaxiality of the data is not clear and experimental α values are influenced by their calculation process, the correlation between the shape of the yield surface and the Poisson’s ratio is clear.

Download:

Fig 5. Relationship between α and ν for metallic glasses.

https://doi.org/10.1371/journal.pone.0275968.g005

Similarly to Eq (13), the shifting stress σ₀ in Eq (22) may be determined based on the uniaxial compression and tensile strengths (σ_c and −σ_t, respectively): (25)

As an example of the matching properties of the proposed yield surface (Eq 22), the experimental data by Qu et al. [23] are fitted in Fig 6. The fitting is based on the tensile and compressive strengths (σ_c = 1.84 GPa and σ_t = 1.66 GPa) and the α value given by Qu et al. [23] (α = 0.41), which is in the range provided by triaxial (Eq 23, α = 0.36) and plane strain (Eq 24, α = 0.45) conditions using the reported value [22] of the Poisson’s ratio (ν = 0.37).

Download:

Fig 6. Fitted yield surface of a metallic glass tested by Qu et al. [23].

https://doi.org/10.1371/journal.pone.0275968.g006

6. Application to amorphous polymers

Synthetic polymers are highly popular nowadays and their mechanical behavior is a subject under intense study. Raghava et al. [24] presented an interesting study on the macroscopic yield behavior of amorphous synthetic polymers. Their experimental results on polycarbonate (PC) and polyvinyl chloride (PVC) are compared with the proposed linear model (Eq 9) in Fig 7. The results are normalized by the uniaxial tensile yield stress (σ_t). Raghava et al. [24] measured Poisson’s ratios of ν = 0.42 and 0.38 and yield asymmetry ratios of σ_c/σ_t = 1.2 and 1.33 for PC and PVC, respectively.

Download:

Fig 7. Fitted yield surfaces of amorphous polymers tested by Raghava et al. [24].

Polycarbonate (PC) (σ_c/σ_t = 1.2; ν = 0.42) [24]; Polyvinyl Chloride (PVC) (σ_c/σ_t = 1.33; ν = 0.38) [24]; Polystyrene (PS) [25].

https://doi.org/10.1371/journal.pone.0275968.g007

The agreement between the experimental data and the proposed yield surface in Fig 7 is good and the slight differences between PC and PVC are also captured. To populate Fig 7 in the compressive side, the experimental data by Whitney and Andrews [25] on Polystyrene (PS) are also included. Here, the goal is just to envisage some of the potential capabilities of the proposed framework, but further and more detailed validations are required in polymers and other materials.

7. Non-linear isotropic elasticity

7.1 A general example

Some materials, such as granular materials, show a non-linear response (Fig 1), even for the elastic range. In these stress-dependent materials, the stiffness is assumed to vary with the stress state.

For isotropic materials, the complementary strain energy density may be expressed just as a function of stress invariants. The principal stresses will be here used for visualization of the elastic potential. Many different types of non-linear elasticity may be formulated within the hyperelastic framework (e.g., [12, 26]); here, for demonstration, the following complementary strain energy density function is assumed as an example: (26) where n is a material parameter that controls the material non-linearity. This formulation has the advantage that the stiffness is stress-dependent, not just mean pressure-dependent, and it may be reduced to the linear case (Eq 7) by assuming n = 1. Besides, the stiffness roughly follows a power law (approximately ). Thus, the common range is between n = 1 (constant modulus) and n = 0.5 (roughly linear stress-dependency of the stiffness). It is worth noting that non-linear hyperelastic models always introduce a “stress-induced” anisotropy (e.g., [27]). The analysis of the non-linear elastic behaviour of this hyperelastic model is detailed in S2 Appendix.

It is convenient to introduce two mathematical tweaks in Eq (26). Firstly, negative values of the stress are not possible in Eq (26) when n≠1. Introducing a “back” stress, σ_b, (pressure) is useful to avoid negative values. Hence, positive “model” stress values, σ*, are: (27)

This type of translation of the stress axes is quite common, for example, with the atmospheric pressure.

Secondly, it is useful to introduce a reference stress (pressure), , so that the dimensions of constants a and b do not depend on n and may be expressed as a function of a reference Young’s modulus and a reference Poisson’s ratio, E_ref and ν_ref, for that reference stress.

may be arbitrarily chosen, but σ_b is a fitting parameter that determines the stress for which the stiffness is null. Thus, Eq (26) may be expressed as: (28)

Similarly to the linear elastic case, the yield surface may be defined as the elastic potential (Eq 28) for a limit value of the complementary strain energy density (U_c0,y). Consequently, once the non-linear elastic constants (E_ref, ν_ref, n, σ_b) have been determined, the additional two constants of the yield surface (c, U_c0,y−d) may be obtained from the yield stresses for two different stress paths, for example, the uniaxial or hydrostatic yield stresses at tension and compression.

For the non-linear case, the shape of the elastic potentials (i.e., yield surfaces) in the principal stress space are distorted ellipsoids (Figs 8–11). In Figs 8–11, simple values have been chosen for the constants, namely p_t = 0, σ_b = 0.1p_c, ν_ref = 0.3 and = 1 (arbitrary units). The non-linear elastic potentials reflect the asymmetries caused by the non-linear elastic behaviour, such as larger elastic regions for higher compressive stresses. This kind of distorted elliptical yield surfaces have experimentally been measured, for example, for clays (e.g., [28]). For a detailed validation of a particular model within the proposed approach, a specific laboratory campaign is required.

Download:

Fig 8.

Non-linear elastic potential in 3D principal stress space (n≈0.5, ν_ref = 0.3, p_t = 0, σ_b = 0.1p_c): (a) compressive side view; (b) tensile side view.

https://doi.org/10.1371/journal.pone.0275968.g008

Download:

Fig 9. Biaxial non-linear elastic potentials (σ₃ = 0).

https://doi.org/10.1371/journal.pone.0275968.g009

Download:

Fig 10. Non-linear elastic potentials in octahedral stress plot for the triaxial plane (σ₂ = σ₃).

https://doi.org/10.1371/journal.pone.0275968.g010

Download:

Fig 11. Non-linear elastic potentials in deviatoric plane.

https://doi.org/10.1371/journal.pone.0275968.g011

In the deviatoric plane (Fig 11), the stress-dependency (non-linearity) distorts the circular section for linear elastic materials (n = 1) towards rounded triangles, similar to Lade-Duncan [29] or Matsuoka-Nakai [30] surfaces, which in turn may be viewed as a kind of rounded Mohr-Coulomb.

7.2 Incompressible non-linear isotropic materials

A simple way of formulating a non-linear incompressible hyperelastic model is by using just the deviatoric component of the complementary strain energy density: (29)

Similarly to Eq (26), the quadratic power of 2 is replaced by 2n in Eq (29). Besides, ordered principal stresses are considered in Eq (29) to avoid negative values in the base of the 2n exponent. For full visualization in the principal stress space, they may be alternated. Assuming σ_ref = 1 (arbitrary units) for the sake of simplicity, the yield surface is: (30)

Eq (30) is equivalent to the Hosford yield criterion [31]. The results for several n values are plotted in the deviatoric or π–plane (Fig 12) and they do not vary with the mean pressure. Interestingly, the yield surface reduces to the von Mises criterion for the linear case (n = 1) (as already mentioned) and to the Tresca criterion for a linear stress-dependency (n = 0.5). The information available in the literature [32] confirms that the yielding of non-linear incompressible materials, such as soft soils under undrained conditions, is better captured by Tresca than by von Mises criterion.

Download:

Fig 12. Influence of non-linearity for incompressible materials: Deviatoric plane section of the yield surface.

https://doi.org/10.1371/journal.pone.0275968.g012

8. Conclusions

This paper proposes that yield surfaces may be assumed to be elastic potential surfaces for specific levels of critical complementary strain energy density in some materials. Traditional approaches, such as the total strain energy criterion, only consider second order terms, i.e., the initial strain energy is null and the elastic potential is centred at the origin of the current stress state. Here, first order terms are considered, and consequently, the elastic potential may be translated, which allows to reproduce the desired level of tension-compression asymmetry.

The proposed approach shows a correlation between the shape of the yield surface and the Poisson’s ratio, which control the shape of the elastic potential. This correlation agrees well with published values in the literature for soils and metallic glasses. Besides, the yield surface is elliptical as experimentally measured. Published experimental data of amorphous synthetic polymers also agree well with this elliptical yield surface. For rocks, the tension-compression strength asymmetry is large and the elliptical yield surface in the area of interest (i.e., low confining stresses) fits well experimental data points, which have traditionally been fitted using parabolic curves. Besides, the proposed approach automatically considers, in a natural and logical manner, the influence of the intermediate stress (σ₂), yet the linear case usually underestimates the influence of the confining stress and non-linear models are required.

Introducing non-linear elasticity in the proposed approach gives a wide range of elastic potentials, such as distorted ellipsoids, which are similar to those experimentally measured, for example, for clays. In the deviatoric plane, the shapes of the yield surfaces are similar to that of Matsuoka-Nakai, for example. Further experimental investigation is needed for detailed validations of specific models. For the case of incompressible non-linear materials, the yield surfaces are between von Mises and Tresca ones.

As hyperelasticity or associated plasticity, the proposed framework to derive yield surfaces using elastic potentials may be considered just as a classifying criterion and a possible approach to formulate yield surfaces in some specific materials.

Compressive stresses and strains are assumed to be positive

Supporting information

S1 Appendix. Soil data from the literature.

https://doi.org/10.1371/journal.pone.0275968.s001

(PDF)

S2 Appendix. Example of non-linear hyperelastic model.

https://doi.org/10.1371/journal.pone.0275968.s002

(PDF)

References

1. Beltrami E. 1885. Sulle condizioni di resistenza dei corpi elastici. Il Nuovo Cimento, 18(1): 145–155.
- View Article
- Google Scholar
2. Haigh B.P. 1920. Strain-energy Function and the Elastic Limit. Report of the 87th Meeting of the British Association for the Advancement of Science, p. 486–495. John Murray, London.
- View Article
- Google Scholar
3. Griffith A.A. 1921. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, 221, 163–198.
- View Article
- Google Scholar
4. Taylor D. 2007. The Theory of Critical Distances: a new perspective in fracture mechanics. Elsevier, London, UK.
5. Lazzarin P., Zambardi R. 2001. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int. J. Fract. 112, 275–298.
- View Article
- Google Scholar
6. Justo J., Castro J., Cicero S., Sánchez-Carro M.A., Husillos R. 2017. Notch effect on the fracture of several rocks: Application of the Theory of Critical Distances. Theoretical and Applied Fracture Mechanics 90, 251–258.
- View Article
- Google Scholar
7. Justo J., Castro J. 2021. Mechanical properties of 4 rocks at different temperatures and fracture assessment using the strain energy density criterion. Geomechanics for Energy and the Environment 25: 100212.
- View Article
- Google Scholar
8. Castro J., Sivasithamparam N. 2017. A Constitutive Model for Soft Clays Incorporating Elastic and Plastic Cross-Anisotropy. Materials 10 (6), 584. pmid:28772938
- View Article
- PubMed/NCBI
- Google Scholar
9. Sivasithamparam N., Castro J. 2016. An anisotropic elastoplastic model for soft clays based on logarithmic contractancy. International Journal for Numerical and Analytical Methods in Geomechanics 40: 596–621.
- View Article
- Google Scholar
10. Sadd M. H. 2014. Elasticity Theory, Applications and Numerics. 3rd Ed. Elsevier, Oxford, UK.
11. Fung Y.C. 1965. Foundations of solid mechanics. Prentice-Hall, New Jersey.
- View Article
- Google Scholar
12. Houlsby G.T., Puzrin A.M. 2006. Principles of hyperplasticity: an approach to plasticity theory based on thermodynamics principles. Springer-Verlag, London.
13. Ma X., Rudnicki J.W., Haimson B.C. 2017. The application of a Matsuoka-Nakai-Lade-Duncan failure criterion to two porous sandstones. International Journal of Rock Mechanics and Mining Sciences 92, 9–18.
- View Article
- Google Scholar
14. Hoek E., Brown E.T. 1980. Empirical Strength Criterion for Rock Masses. Journal of Geotechnical Engineering, ASCE 106(GT9), 1013–35.
- View Article
- Google Scholar
15. Christensen R.M. 2013. The theory of materials failure. Oxford University Press, Oxford.
16. Hill R., Rice R. 1973. Elastic potentials and the structure of inelastic constitutive laws. SIAM Journal of Applied Mathematics 25, 448–461.
- View Article
- Google Scholar
17. Callen H.B. 1985. Thermodynamics and an introduction to thermosstatistics. Wiley, New York.
18. Roscoe K.H., Burland J.B. 1968. On the generalised stress-strain behaviour of ‘wet’ clay. In Engineering Plasticity (Heyman J, Leckie FA, eds), p. 563–609. Cambridge University Press, Cambridge.
- View Article
- Google Scholar
19. Federico A., Elia G. 2009. At-rest earth pressure coefficient and Poisson’s ratio in normally consolidated soils. In Proc. 17th International Conference on Soil Mechanics and Geotechnical Engineering (Hamza M et al., eds), IOS Press.
20. Wroth C.P. 1975. In-situ measurement of initial stress and deformation characteristics. Proc., Specialty Conf. on In-Situ Measurement of Soil Properties, Raleigh, North Carolina (New York: ASCE), vol. 2, 181–230.
- View Article
- Google Scholar
21. Mogi K. 2007. Experimental rock mechanics. CRC Press.
22. Liu Z.Q., Qu R.T., Zhang Z.F. 2015. Elasticity dominates strength and failure in metallic glasses. Journal of Applied Physics 117, 014901.
- View Article
- Google Scholar
23. Qu R.T., Eckert J., Zhang Z.F. 2011. Tensile fracture criterion of metallic glass. Journal of Applied Physics 109, 083544.
- View Article
- Google Scholar
24. Raghava R., Caddell R.M., Yeh G.S.Y. 1973. The macroscopic yield behaviour of polymers. Journal of Materials Science 8, 225–232.
- View Article
- Google Scholar
25. Whitney W., Andrews R.D. 1967. Yielding of glassy polymers: volume effects. Journal of Polymer Science: Part C 16, 2981–2990.
- View Article
- Google Scholar
26. Humrickhouse P.W., Sharpe J.P., Corradini M.L. 2010. Comparison of hyperelastic models for granular materials. Physical Review E 81, 011303. pmid:20365364
- View Article
- PubMed/NCBI
- Google Scholar
27. Puzrin A.M. 2012. Constitutive modelling in geomechanics: Introduction. Springer Verlag, Berlin.
28. Graham J., Noonan M.L., Lew K.V. 1983. Yield states and stress-strain relationships in a natural plastic clay. Canadian Geotechnical Journal 20: 502–516.
- View Article
- Google Scholar
29. Lade P.V., Duncan J.M. 1974. Elasto plastic stress-strain theory for cohesionless soil. Journal of the Geotechnical Engineering Division, ASCE 101, 1037–1053.
- View Article
- Google Scholar
30. Matsuoka H., Nakai T. 1974. Stress-deformation and strength characteristics of soil under three different principal stresses. Proc. JSCE 232, 59–70.
- View Article
- Google Scholar
31. Hosford W.F. 1972. A generalized isotropic yield criterion. Journal of Applied Mechanics 39, 607–609.
- View Article
- Google Scholar
32. Lade P.V. 1988. Effects of voids and volume changes on the behaviour of frictional Materials. International Journal for Numerical and Analytical Methods in Geomechanics 12, 351–370.
- View Article
- Google Scholar

[ref1] 1. Beltrami E. 1885. Sulle condizioni di resistenza dei corpi elastici. Il Nuovo Cimento, 18(1): 145–155.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Haigh B.P. 1920. Strain-energy Function and the Elastic Limit. Report of the 87th Meeting of the British Association for the Advancement of Science, p. 486–495. John Murray, London.
View Article
Google Scholar

[5] View Article

[6] Google Scholar

[ref3] 3. Griffith A.A. 1921. The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, 221, 163–198.
View Article
Google Scholar

[8] View Article

[9] Google Scholar

[ref4] 4. Taylor D. 2007. The Theory of Critical Distances: a new perspective in fracture mechanics. Elsevier, London, UK.

[ref5] 5. Lazzarin P., Zambardi R. 2001. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int. J. Fract. 112, 275–298.
View Article
Google Scholar

[12] View Article

[13] Google Scholar

[ref6] 6. Justo J., Castro J., Cicero S., Sánchez-Carro M.A., Husillos R. 2017. Notch effect on the fracture of several rocks: Application of the Theory of Critical Distances. Theoretical and Applied Fracture Mechanics 90, 251–258.
View Article
Google Scholar

[15] View Article

[16] Google Scholar

[ref7] 7. Justo J., Castro J. 2021. Mechanical properties of 4 rocks at different temperatures and fracture assessment using the strain energy density criterion. Geomechanics for Energy and the Environment 25: 100212.
View Article
Google Scholar

[18] View Article

[19] Google Scholar

[ref8] 8. Castro J., Sivasithamparam N. 2017. A Constitutive Model for Soft Clays Incorporating Elastic and Plastic Cross-Anisotropy. Materials 10 (6), 584. pmid:28772938
View Article
PubMed/NCBI
Google Scholar

[21] View Article

[22] PubMed/NCBI

[23] Google Scholar

[ref9] 9. Sivasithamparam N., Castro J. 2016. An anisotropic elastoplastic model for soft clays based on logarithmic contractancy. International Journal for Numerical and Analytical Methods in Geomechanics 40: 596–621.
View Article
Google Scholar

[25] View Article

[26] Google Scholar

[ref10] 10. Sadd M. H. 2014. Elasticity Theory, Applications and Numerics. 3rd Ed. Elsevier, Oxford, UK.

[ref11] 11. Fung Y.C. 1965. Foundations of solid mechanics. Prentice-Hall, New Jersey.
View Article
Google Scholar

[29] View Article

[30] Google Scholar

[ref12] 12. Houlsby G.T., Puzrin A.M. 2006. Principles of hyperplasticity: an approach to plasticity theory based on thermodynamics principles. Springer-Verlag, London.

[ref13] 13. Ma X., Rudnicki J.W., Haimson B.C. 2017. The application of a Matsuoka-Nakai-Lade-Duncan failure criterion to two porous sandstones. International Journal of Rock Mechanics and Mining Sciences 92, 9–18.
View Article
Google Scholar

[33] View Article

[34] Google Scholar

[ref14] 14. Hoek E., Brown E.T. 1980. Empirical Strength Criterion for Rock Masses. Journal of Geotechnical Engineering, ASCE 106(GT9), 1013–35.
View Article
Google Scholar

[36] View Article

[37] Google Scholar

[ref15] 15. Christensen R.M. 2013. The theory of materials failure. Oxford University Press, Oxford.

[ref16] 16. Hill R., Rice R. 1973. Elastic potentials and the structure of inelastic constitutive laws. SIAM Journal of Applied Mathematics 25, 448–461.
View Article
Google Scholar

[40] View Article

[41] Google Scholar

[ref17] 17. Callen H.B. 1985. Thermodynamics and an introduction to thermosstatistics. Wiley, New York.

[ref18] 18. Roscoe K.H., Burland J.B. 1968. On the generalised stress-strain behaviour of ‘wet’ clay. In Engineering Plasticity (Heyman J, Leckie FA, eds), p. 563–609. Cambridge University Press, Cambridge.
View Article
Google Scholar

[44] View Article

[45] Google Scholar

[ref19] 19. Federico A., Elia G. 2009. At-rest earth pressure coefficient and Poisson’s ratio in normally consolidated soils. In Proc. 17th International Conference on Soil Mechanics and Geotechnical Engineering (Hamza M et al., eds), IOS Press.

[ref20] 20. Wroth C.P. 1975. In-situ measurement of initial stress and deformation characteristics. Proc., Specialty Conf. on In-Situ Measurement of Soil Properties, Raleigh, North Carolina (New York: ASCE), vol. 2, 181–230.
View Article
Google Scholar

[48] View Article

[49] Google Scholar

[ref21] 21. Mogi K. 2007. Experimental rock mechanics. CRC Press.

[ref22] 22. Liu Z.Q., Qu R.T., Zhang Z.F. 2015. Elasticity dominates strength and failure in metallic glasses. Journal of Applied Physics 117, 014901.
View Article
Google Scholar

[52] View Article

[53] Google Scholar

[ref23] 23. Qu R.T., Eckert J., Zhang Z.F. 2011. Tensile fracture criterion of metallic glass. Journal of Applied Physics 109, 083544.
View Article
Google Scholar

[55] View Article

[56] Google Scholar

[ref24] 24. Raghava R., Caddell R.M., Yeh G.S.Y. 1973. The macroscopic yield behaviour of polymers. Journal of Materials Science 8, 225–232.
View Article
Google Scholar

[58] View Article

[59] Google Scholar

[ref25] 25. Whitney W., Andrews R.D. 1967. Yielding of glassy polymers: volume effects. Journal of Polymer Science: Part C 16, 2981–2990.
View Article
Google Scholar

[61] View Article

[62] Google Scholar

[ref26] 26. Humrickhouse P.W., Sharpe J.P., Corradini M.L. 2010. Comparison of hyperelastic models for granular materials. Physical Review E 81, 011303. pmid:20365364
View Article
PubMed/NCBI
Google Scholar

[64] View Article

[65] PubMed/NCBI

[66] Google Scholar

[ref27] 27. Puzrin A.M. 2012. Constitutive modelling in geomechanics: Introduction. Springer Verlag, Berlin.

[ref28] 28. Graham J., Noonan M.L., Lew K.V. 1983. Yield states and stress-strain relationships in a natural plastic clay. Canadian Geotechnical Journal 20: 502–516.
View Article
Google Scholar

[69] View Article

[70] Google Scholar

[ref29] 29. Lade P.V., Duncan J.M. 1974. Elasto plastic stress-strain theory for cohesionless soil. Journal of the Geotechnical Engineering Division, ASCE 101, 1037–1053.
View Article
Google Scholar

[72] View Article

[73] Google Scholar

[ref30] 30. Matsuoka H., Nakai T. 1974. Stress-deformation and strength characteristics of soil under three different principal stresses. Proc. JSCE 232, 59–70.
View Article
Google Scholar

[75] View Article

[76] Google Scholar

[ref31] 31. Hosford W.F. 1972. A generalized isotropic yield criterion. Journal of Applied Mechanics 39, 607–609.
View Article
Google Scholar

[78] View Article

[79] Google Scholar

[ref32] 32. Lade P.V. 1988. Effects of voids and volume changes on the behaviour of frictional Materials. International Journal for Numerical and Analytical Methods in Geomechanics 12, 351–370.
View Article
Google Scholar

[81] View Article

[82] Google Scholar

Figures

Abstract

1. Introduction

2. Linear isotropic elasticity

2.1 Elastic potential

2.2 Yield surface

2.3 Tension-compression yield asymmetry

3. Application of the linear case to soils

4. Application to rocks

5. Application to metallic glasses

6. Application to amorphous polymers

7. Non-linear isotropic elasticity

7.1 A general example

7.2 Incompressible non-linear isotropic materials

8. Conclusions

Supporting information

S1 Appendix. Soil data from the literature.

S2 Appendix. Example of non-linear hyperelastic model.

References