# Clarification of the notion of associated flow for pressure-dependent plasticity

Baptiste Reyne, Frédéric Barlat

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Baptiste Reyne, Frédéric Barlat. Clarification of the notion of associated flow for pressure-dependent plasticity. 2023. hal-04167192

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# Clarification of the notion of associated flow for pressure-dependent plasticity

Baptiste Reyne<sup>a,*</sup>, Frédéric Barlat<sup>b</sup>

<sup>a</sup>Department of Materials Science and Engineering, NTNU, N-7491 Trondheim, Norway

<sup>b</sup>Graduate Institute of Ferrous and Energy Materials Technology, POSTECH, Gyeongbuk 37673, Republic of Korea

## Abstract

It is often found in the literature that the plastic strains may not associate with a pressure-dependent yield function, unless plasticity can bring volumetric changes. This paper shows that, on the contrary, pressure dependence and the absence of volume change can coexist in a fully associated, thermodynamically consistent model. To do so in this paper, the thermodynamics of continua are followed in the general case of plasticity. A main aspect differs from the classical presentation in that the hydrostatic and deviatoric contributions to the free energy are explicitly considered independent. The implications of this assumption are analyzed at each step. It is demonstrated that a standard generalized model can use a hydrostatic stress dependent yield condition under the condition of isochoric plastic deformation without any lack of consistency. This conclusion leads to the clarification of several definitions and the distinction of facts from intuitions about plastic models.

**Keywords:** pressure dependent plasticity, thermodynamics, solid mechanics, soils, metals, Drucker-Prager, Spitzig-Richmond,

## 1. Introduction

Pressure-dependent plasticity designates the tendency of the yield stress to vary with respect to the hydrostatic component of the applied load. As shown further, this phenomenon concerns many materials used in modern industries. But in spite of a possible pressure dependence, several materials (metals in particular) deform plastically in an isochoric (or incompressible) manner, that

<footnote>*Corresponding author.</footnote>
<figure><img src="image_2.png" /><figcaption>Figure 1: Stress differential effect as measured in martensitic steel MART1500 using the methods of Lee et al. [25].</figcaption></figure>

is without volume change. This translates into a kinematic restriction that forces the plastic strain rate to be purely deviatoric.

On the other hand, in the *classical* [26, 6, 5] thermodynamic description of plasticity, the associated framework uses the yield function as plastic potential. It is consistent with the postulate of maximum plastic work [5, page 328]. But it requires that the plastic strain rate inherits a hydrostatic component from any pressure dependence of the yield function.

One must therefore abandon associativity in order to guaranty isochoricity, and use different functions to describe the plastic yield and flow. Such a contradiction is a rare exception among basic constitutive models, which leads to expect possible improvements and clarifications of the theory. In other cases such as porous materials, there exist plastic volume changes that an associated flow rule often fails to accurately represent [24, page 144].

The issue is vastly cross-cutting as shown by the wide variety of materials it impacts. For example, *metals* are usually presented as pressure-independent [26, page 16] but evidences gathered by Spitzig and Richmond [36] show that their yield limit follows a linear dependency to pressure. It causes, for example, tension-compression asymmetry as clearly shown for a martensitic steel in figure 1. It is nonetheless experimentally shown that the plastic volume change remains negligible and that the associated prediction is not satisfactory [33] for these metals. Several models have
been proposed to recreate this phenomenon. They use a linear dependency to the hydrostatic stress;
they are non-associated, with [9] or without [37, 2, 3, 11, 4] plastic volume changes. With the rise of
advanced high strength steels for their applications in the automotive industry, yield limits exceed a
gigapascal and new challenges appear. For instance, the tension–compression asymmetry becomes
significant and alters the quality of springback predictions in forming simulations [43]. These
concerns are not limited to metals. Similar experimental conclusions are drawn for *polycarbonates*
and *polyethylene* [35, 33] as well.

But often, associated plasticity simply does not suffice to recreate an accurate deviatoric to
volumetric ratio of strains [27]. This can be the case for *cellular materials* and *foams* [8, 18, 40],
*polymers* or *fiber composites* [13, 39] that exhibit a yield pressure dependency or tension–compression
asymmetry. The behavior of *granular materials* and *aggregate composites* such as soils and snows,
rocks or concrete, is conditioned by their internal cohesion that increases with higher pressure
[16, 45, 44, 46]. To that end, experimental evidences motivated the development of non-associated
or non-normal models in the second half of the last century [34, 1]. The models cited above all
require a yield function different from the flow rule: the principle of associativity is not respected
because too restrictive to account for the complexity of their deformation. Indeed, associated
models are applicable to a subclass of simple plastic behaviors. This paper shows how this subclass
can be extended when several criteria are met.

The organisation of the paper is illustrated in figure 2.

* The guiding thread is the classical outline of the thermodynamics of irreversible processes (filled boxes in figure 2).

* It is particularized following classical developments and definitions: the spine of this paper is a review.

* The added contribution is to explicitly account for an isochoric assumption within the thermodynamic formulation (framed input in figure 2). The novelty is not in the assumption itself, but in the analysis of its consequences.

* We show that an entire class of plastic–isochoric, pressure–dependent materials still satisfy the requirements of associativity (framed output in figure 2).
<figure><img src="image_3.png" /><figcaption>Figure 2: Organization of the paper.</figcaption></figure>
The assumption (hypothesis 1) is stated in section 2 where the general implications of the Clausius-Duhem inequality are considered. Rate-independent isochoric plasticity is treated throughout section 3 and leads to clarifications concerning associativity and normality for such a class of models. After some examples of applications (section 4), the implications for rate-dependence and plasticity with volume changes are analyzed in section 5, where the connection with the classical description of thermodynamics is recovered. This paper brings a first set of solutions to the problem of compatibility between the associated framework and pressure dependent materials, as well as clarifications about maximum work principles in plasticity.

## 2. General behavior

The present section is not limited to plasticity or pressure dependence, it concerns any material continuum [29, 17, 26]. For the sake of simplicity and the use of the Cauchy stress and linearized Green-Lagrange measures, the kinematics are considered to respect the small strains assumption. The Clausius-Duhem inequality is obtained after the second principle of thermodynamics (SPT) and expresses the total volumic dissipation as

$$ \mathcal{D}_{\text{tot}} = \mathcal{D} + \mathcal{D}_{\text{th}} \geq 0, \qquad (1) $$

$$ \text{where} \quad \mathcal{D} = \sigma : \dot{\epsilon} - \rho(\dot{\psi} + \eta \dot{\theta}), \qquad (2) $$

$$ \text{and} \quad \mathcal{D}_{\text{th}} = -\frac{q}{\theta} \text{grad} \theta. \qquad (3) $$

$\mathcal{D}$ represents the mechanical contribution and includes the power of internal forces ($\sigma : \dot{\epsilon}$), the specific free energy $\psi$, the entropy $\eta$ and the absolute temperature $\theta$. The dot notation refers to time derivation and $\rho$ is the mass density. $\mathcal{D}_{\text{th}}$ refers to the thermal contribution, with the heat flux $q$. The present study only focuses on the mechanical dissipation that is expected positive independently of the thermal contribution. In this context, building a constitutive model consists in giving an expression to the free energy $\psi$ and connecting the state variables to one another.

### 2.1. Internal variables and admissible states

In the classical thermodynamic description of elasto-plastic systems [29], a general form for the specific free energy would exhibit dependencies with the total strain $\epsilon$, the temperature $\theta$ and any number of internal variables. Before unfolding the rational theory of thermodynamics applied to elasto-plasticity, let us make the following assumption.
**Hypothesis 1.** The hydrostatic and deviatoric part of the strains contribute individually and independently to the specific free energy of the system (formalized in equation (7)).

Let the hydrostatic and deviatoric parts of the stress tensor be independently taken into account.

$$ \sigma = s + hI \qquad (4) $$

where $s = \text{dev } \sigma$ and $h = \text{tr } \sigma/3$ are the stress deviator and hydrostatic stress, respectively. The corresponding strain naturally splits into its deviator $e = \text{dev } \varepsilon$ and the volumetric ratio $v = \text{tr } \varepsilon$

$$ \varepsilon = e + (v/3)I \qquad (5) $$

It follows that the internal power of forces can be decomposed as:

$$ \sigma : \dot{\varepsilon} = s : \dot{\varepsilon} + h\dot{v}. \qquad (6) $$

While the classical theory is merely reformulated the assumed independence of the deviatoric and hydrostatic contributions allows a further differentiation of the plastic evolution from its hydrostatic stress dependence.

The specific free energy takes for dependencies:

$$ \psi(e, v, \theta, V_k), \quad 1 \le k \le n \qquad (7) $$

where $V_k$ is the short notation for {$V_1, V_2, \dots, V_n$} that designates the set of added internal variables. The above expression is a small point of divergence with the classical theory: the independence of $s$ and $h$ is explicitly stated here, in their contribution to the specific free energy.

**Remark 1.** Hypothesis 1 is costless since it is implicit in elastic models; for example (neo-)Hookean elasticity where compressible and incompressible contributions are distinct. The authors could not find any elastic model to serve as counterexample.

**Remark 2.** The uncoupling of hypothesis 1 is not new and already part of all tensorial descriptions of plasticity from the literature. A most early and fundamental example in the von Mises equivalent stress [41] that measures the deviatoric part of the stress tensor, independently of its spheric part. The hydrostatic– deviatoric split is found explicitly in the formulation of yield criteria [16, 34, 2], in strain energy relations [30] or in the power of internal forces [5]. The light brought by this work is in the consequences of this assumption for pressure-dependent plasticity, that goes against a simplification usually made.
**Definition 1 (internal variables).** An internal variable is a quantity that (i) imports in the behavior of the material, (ii) does not appear explicitly in the Clausius–Duhem inequality (1) and (iii) appears in the dependencies of the state potential, here the specific free energy $\psi$.

## 2.2. Dissipation

Based on the above assumptions, the mechanical dissipation can be extended

$$ \mathcal{D} = s : \dot{e} + h\dot{v} - \rho(\dot{\psi} + \eta\dot{\theta}) \quad (8) $$

$$ = \left( s - \rho \frac{\partial \psi}{\partial e} \right) : \dot{e} + \left( h - \rho \frac{\partial \psi}{\partial v} \right) \dot{v} + \rho \left( \eta + \frac{\partial \psi}{\partial \theta} \right) \dot{\theta} - \rho \frac{\partial \psi}{\partial V_k} \dot{V}_k. \quad (9) $$

Postulating (i) the existence of elastic-only transformations and (ii) an arbitrary thermal evolution at $\dot{V}_k=0$, leads to the equations of state:

$$ s = \rho \frac{\partial \psi}{\partial e}, \quad h = \rho \frac{\partial \psi}{\partial v}, \quad \eta = - \frac{\partial \psi}{\partial \theta}. \quad (10) $$

As expected, the hydrostatic and deviatoric stresses independently conjugate with their respective strains counterparts. The remaining terms allow the definition of the remaining state equations

$$ A_k = \rho \frac{\partial \psi}{\partial V_k}. \quad (11) $$

**Definition 2 (forces and fluxes).** In this outline like in the literature [10], thermodynamic fluxes are all the variables that the free energy depends upon. Thermodynamic forces are then obtained by derivation. The terms “generalized forces” and “generalized velocities” are respectively synonyms. It is said that forces conjugate to fluxes, and conversely [10, 20].

**Definition 3 (equations of state).** Equations of state [17], also called Coleman relations [29], connect thermodynamic fluxes to forces and derive from a state potential.

## 2.3. External variables evolution

After taking the four equations of state (10) and (11) into account, the mechanical dissipation becomes

$$ \mathcal{D} = -A_k \dot{V}_k. \quad (12) $$
In order to close the model, the time-rate of change of fluxes must be given by complementary equations. They take the general form

$$ \dot{V}_k = -M_k(s, h, A_k) \qquad (13) $$

where each equation $M_k$ of the model may depend upon any combination of thermodynamic forces.

**Definition 4 (equations of evolution).** Evolution (or complementary [17]) equations connect the rate of change of internal variables to their conjugate. Their are the last set of equations needed to close a model's formulation.

**Remark 3 (laws and models).** Equations of state and complementary equations are often called laws [17, 6]. In the paper, the words "function", "equation" or "model" are preferred since they reflect better their empirical nature. This is in opposition to balance laws or thermodynamic principles that are more absolute and unbreakable.

The model is then thermodynamically admissible if the mechanical dissipation (12) is positive, i.e. if $\sum_k A_k M_k \ge 0$. It is sufficient to build a function $\Omega(s, h, A_k)$ named dissipation pseudo-potential<fnref n="1" /> such that

$$ \dot{V}_k = -\frac{\partial \Omega}{\partial A_k}, \qquad (14) $$

which makes the model generalized standard and comes with several advantages:

* the positivity of dissipation is automatically verified if $\Omega$ is positive, convex and nil at its origin;

* the constitutive model is entirely defined by only two scalar-valued functions, $\psi$ and $\Omega$;

* its numerical implementation can fully profit from the power of modern optimization methods.

<footnote n="1">¹The prefix "pseudo" is often added to signify that the dissipation potential exists only by construction, as opposed to the free energy that can appear explicitly in the second principle of thermodynamics.</footnote>
**Definition 5 (generalized standard materials).** A model is called generalized standard when all evolution equations (definition 4) derive from a unique scalar function [20, 26], then called dissipation pseudo-potential.

This general property leads to sub categories detailed further (see associativity, definition 6).

### 3. Rate-independent isochoric plasticity

This section covers the case of plasticity with isochoric (no volume change) internal bonds, and without rate dependence. It remains relevant with or without a yield dependence to hydrostatic stress. The possible exchanges of energies involving temperature are ignored.

**Hypothesis 2 (isochoric plasticity).** Plastic strains take place without volume change.

The plastic phenomenon is then modeled considering that strains evolve with an irreversible component, traceless and denoted

$$V_1 = \epsilon^p, \quad \text{tr} \, \epsilon^p = 0; \qquad (15)$$

that does not contribute to the free energy. The nil trace condition enforces the interdiction of plastic volume changes. Other internal variables are now denoted $V_q$ and may correspond to, for example, isotropic or distortional hardening. Disregarding the temperature-entropy influence, the specific free energy becomes

$$\psi(e - \epsilon^p, v, V_q), \quad 2 \le q \le n \qquad (16)$$

Since the rate of change of energy varies in equal and opposed manner with respect to the strain deviator and the plastic strains, a new state equation is inferred:

$$-A_1 = s = -\rho \frac{\partial \psi}{\partial \epsilon^p}. \qquad (17)$$

#### 3.1. Admissibility of internal variables: the yield surface

The evolution of internal variables is classically conditioned by a scalar function $f$ that may depend upon any combination of thermodynamic forces. Let the yield function $f(s, h, A_q)$ define the elastic domain ($f < 0$) and the plastic domain ($f = 0$). This section deals in details with the rate-independent case. The visco-plastic case ($f \ge 0$) is treated in section 5.
## 3.2. Evolution of internal variables: complementary equations

The last equation defining admissible states, the rate of change of internal variables is to be defined by complementary constitutive equations. In the *rate-independent* general case, they are written as functions of the thermodynamic forces:

$$ \begin{bmatrix} \dot{\boldsymbol{\epsilon}}^p = \dot{\lambda}M_1(\boldsymbol{s}, h, A_q) \\ \dot{V}_q = -\dot{\lambda}M_q(\boldsymbol{s}, h, A_q) \end{bmatrix} \quad \text{(general case).} \qquad (18) $$

This is a particularized form<fnref n="2" /> of (13) where $\dot{\lambda}$ is called the plastic multiplier. It is used to regularize the problem in order to ensure that the elastic-plastic conditions of section 3.1 are respected. To do so, the plastic multiplier is governed by the Kuhn-Tucker conditions:

$$ \dot{\lambda}f = 0, \quad \dot{\lambda} \ge 0, \quad f \le 0; \qquad (19) $$

which formalize the assertion: $f = 0$ in plasticity and $f < 0$ in elasticity.

It is common and useful to build a scalar-valued function $F(\boldsymbol{s}, h, A_q)$ called *plastic potential* [22], such that the fluxes are obtained by derivation with respect to their corresponding thermodynamic force:

$$ \begin{bmatrix} \dot{\boldsymbol{\epsilon}}^p = -\dot{\lambda} \frac{\partial F}{\partial(-\boldsymbol{s})} = \dot{\lambda} \frac{\partial F}{\partial \boldsymbol{s}} \\ \dot{V}_q = -\dot{\lambda} \frac{\partial F}{\partial A_q} \end{bmatrix} \quad \text{(normal case).} \qquad (20) $$

Here the model is not generalized standard (definition 5) since it is defined by more than two functions (a free energy, the yield surface, and the plastic potential $F$). It is often practical and simpler to choose the yield function $f$ as potential for some internal variables, which leads to the concept of associativity.

**Definition 6 (associativity).** A thermodynamic flux is called associated when it derives from the yield surface with respect to its conjugate [32]. When at least the plastic strain derives from the yield surface, the model is called simply associated.

The key of this work is to notice that associativity depends on how variables conjugate together, which itself depends solely on the chosen state potential dependencies.

<footnote n="2">²The minus sign is chosen to simplify the expression of the mechanical dissipation.</footnote>
**Remark 4 (range of use).** The concept of associativity only makes sense when a yield function is used to describe the material. The more general notion of normality can be used whenever a variable derives from a potential. The attribute "standard" can be found as a synonym of "simply associated" [26, page 63]. In the same logic, one could use "fully associated" interchangeably with "generalized standard"—although such expression seems absent from the literature.

**Remark 5 (associativity in the literature).** Part of the literature on plasticity [24, 42, 5] does not discuss its thermodynamic foundations and defines associativity directly as the proportionality between the plastic strain rate $\dot{\varepsilon}^p$ and the yield surface normal in the entire space of stresses. A number of reference works [29, 26, 6] do specify the notions of work-conjugacy and associativity, but still conclude on the previous definition since hypothesis 1 is not made.

This is legitimate in their general context since the application of hypothesis 1 is conceptual and important to the special cases presented in this paper. But it also justifies the need for these remarks to exist in the literature.

The model is generalized standard (definition 5) if every internal variable is associated, i.e. all complementary equations derive from the yield function:

$$ \left. \begin{aligned} \dot{\varepsilon}^p &= \dot{\lambda} \frac{\partial f}{\partial s} \\ \dot{V}_q &= -\dot{\lambda} \frac{\partial f}{\partial A_q} \end{aligned} \right\} \quad (\text{standard generalized case}). \qquad (21) $$

As can be seen from (20) or (21),

* the possible hydrostatic stress dependence of the yield function conserves the simply associated character of the model. In other words, any plastic model can be written as in (20) or (21). This is because $\dot{\varepsilon}^p$ conjugates with the stress deviator alone.

* $\dot{\varepsilon}^p$ remains deviatoric regardless of any hydrostatic stress dependence in $\frac{\partial f}{\partial s}$, since the derivative of a function with respect to a deviator is deviatoric.

Therefore isochoric plasticity is associated regardless of the chosen model, provided the free energy separates the deviatoric and hydrostatic strains contributions (hypothesis 1).

### 3.3. Plastic multiplier derivation

Finally, the plastic multiplier is computed using the consistency condition:

$$ \dot{\lambda} f = 0 \qquad (22) $$
which is implied by the Kuhn-Tucker conditions (19). In the plastic case, $\dot{\lambda} > 0$ and

$$ \dot{f} = 0 = \frac{\partial f}{\partial \mathbf{s}} : \dot{\mathbf{s}} + \frac{\partial f}{\partial \dot{\mathbf{h}}} \dot{\mathbf{h}} + \frac{\partial f}{\partial A_q} \dot{A}_q \quad (23) $$

The following two substitutions can be made using (i) the chain rule and equation (11); then (ii) the constitutive models (18).

$$ \dot{A}_q = \frac{\partial^2 \rho \psi}{\partial V_q^2} \dot{V}_q = - \frac{\partial^2 \rho \psi}{\partial V_q^2} \dot{\lambda} M_q. \quad (24) $$

The second one is only possible (and needed) for the forces that conjugate with dissipative variables. The consistency condition becomes:

$$ \dot{\lambda} = \left( \frac{\partial f}{\partial \mathbf{s}} : \dot{\mathbf{s}} + \frac{\partial f}{\partial \dot{\mathbf{h}}} \dot{\mathbf{h}} \right) \left( \frac{\partial f}{\partial A_q} \frac{\partial^2 \rho \psi}{\partial V_q^2} M_q \right). \quad (25) $$

This expression resembles the classical derivation that yet exhibits no hydrostatic-deviatoric independence.

### 3.4. Dissipation

Considering a simply associated model, the mechanical dissipation (12) becomes

$$ \mathcal{D} = -A_k \dot{V}_k = \mathbf{s} : \dot{\mathbf{\epsilon}}^p - A_q \dot{V}_q = \dot{\lambda} \left( \mathbf{s} : \frac{\partial f}{\partial \mathbf{s}} + A_q M_q \right). \quad (26) $$

The model is thermodynamically admissible if the dissipation is positive or zero. This results in the condition $\mathbf{s} : \frac{\partial f}{\partial \mathbf{s}} + A_q M_q \ge 0$ to impose on the model's formulation.

When the model is standard generalized, all of the internal variables are associated and

$$ \mathcal{D} = \dot{\lambda} \left( \mathbf{s} : \frac{\partial f}{\partial \mathbf{s}} + A_q \frac{\partial f}{\partial A_q} \right). \quad (27) $$

In this context, certain properties of the yield function can be used to automatically verify the SPT. If $f(\mathbf{s}, \mathbf{h}, A_q)$ is positive, convex with respect to $\mathbf{s}$ and $A_q$ for any $\mathbf{h}$; and if $f(0, 0, 0) = 0$ then

$$ \mathbf{s} : \frac{\partial f}{\partial \mathbf{s}} \ge f, \quad A_q \frac{\partial f}{\partial A_q} \ge f. \quad (28) $$

After substitution, it follows that

$$ \mathcal{D} \ge \dot{\lambda} f = 0 \quad (29) $$
since the Kuhn–Tucker conditions (19) are enforced. Note that f was introduced with a potential dependency upon every thermodynamic force (s, h and A_q) but the convexity with respect to hydrostatic stress is not required. This goes against the usual formulation that requires convexity with respect to every variable, and is due to the fact that the hydrostatic stress does not conjugate with any dissipative variable.

## 4. Applications

The considered examples focus on a pressure-dependent, elastic, isochoric-plastic material with isotropic hardening. The internal variables are the plastic strain ε^p = V_1 and the accumulated plastic strain ε̄ = V_2. The specific free energy takes the form

$$ \psi(e, v, \epsilon^p, \bar{\epsilon}) = \psi_e(e - \epsilon^p, v) + \psi_p(\bar{\epsilon}) \quad (30) $$

where ψ_e represents recoverable work and ψ_p the energy that is neither dissipated nor recoverable. For isotropic linear elasticity, the elastic potential takes the quadratic form

$$ \rho\psi_e = \frac{1}{2}(Kv^2 + 2\mu(e - \epsilon^p) : (e - \epsilon^p)) \quad (31) $$

with K the bulk modulus and μ the shear modulus. The state potential yields the state equations:

$$ s = \rho \frac{\partial \psi_e}{\partial e} = -\rho \frac{\partial \psi_e}{\partial \epsilon^p}, \quad h = \rho \frac{\partial \psi_e}{\partial v} \quad \text{and} \quad r = \rho \frac{\partial \psi_p}{\partial \bar{\epsilon}}. \quad (32) $$

The new hardening variable r = A_2 is the thermodynamic force that conjugates with the accumulated plastic strain. Its constitutive expression derives from ψ_p, which is calibrated from experimental measurements in order to ensure the evolution conditions. Two variants of these conditions are analysed in the following lines.

### 4.1. Drucker–Prager plasticity

The Drucker–Prager [16] yield function describes a linear sensitivity to pressure, expressed as

$$ f(s, h, r) = \bar{\sigma}(s) - (1 - \alpha h)\sigma_y - r \quad (33) $$

where α is the pressure coefficient; σ_y is the initial yield stress, and $\bar{\sigma}(s)$ is a first-degree homogeneous equivalent stress. The equation f = 0 describes a cone in stress space (s, h). The initial Drucker–Prager criterion [16] corresponds to such function with an equivalent stress of the von Mises form
<figure><img src="image_4.png" /><figcaption>Figure 3: Comparison of the two considered models. The lines represent the yield condition $f = 0$ in the stress-space and allow the comparison of their hardening behaviors. The acronym SPT stands for: second principle of thermodynamics.</figcaption></figure>

$\bar{\sigma} = \sqrt{s : s/2}$. These authors intended to generalize the Mohr–Coulomb criterion that is obtained with the equivalent stress of Tresca. The form (33) is widely used for granular materials but also for metals by Lou et al. [28].

Let the model be standard generalized: the formulation is closed by the associated complementary equations

$$ \dot{\epsilon}^p = \dot{\lambda} \frac{\partial f}{\partial s} = \dot{\lambda} \frac{\partial \bar{\sigma}}{\partial s} \quad \text{and} \quad \dot{\epsilon} = -\dot{\lambda} \frac{\partial f}{\partial r} = \dot{\lambda}. \qquad (34) $$

As expected in the present theory, the plastic strain rate remains associated while deviatoric. After including the Kuhn–Tucker conditions (19), the model is entirely formulated and only remains the Clausius–Duhem inequality verification.

The positivity of dissipation is not automatically verified since the yield function does not meet the requirements of positivity or nullity at the origin. But its expression given by (27) can be rewritten using Euler's theorem ($\bar{\sigma}$ is homogeneous):

$$ \mathcal{D} = \dot{\lambda} (\bar{\sigma}(s) - r) \geq 0. \qquad (35) $$

Now enforcing the Kuhn–Tucker conditions (19), two possible cases are to be treated. Elasticity (i): when $\dot{\lambda} = 0$ the dissipation is zero and the Clausius–Duhem inequality trivially satisfied. Plasticity
(ii): when $\dot{\lambda} \ge 0$, $f = 0$ implies $\bar{\sigma}(s) - r = \sigma_y(1 - \alpha h)$ which leads to rewriting the condition as

$$ \alpha h \le 1. \qquad (36) $$

This condition defines the vertex of the initial Drucker-Prager cone as a limit state as shown in figure 3. Note that the limit value of the hydrostatic stress remains fixed regardless of the widening of the cone.

## 4.2. Spitzig-Richmond plasticity

The Spitzig-Richmond [35] yield function accounts for a bi-linear dependency to pressure and hardening, that is:

$$ f(s, h, r) = \bar{\sigma}(s) - (1 - \alpha h)(\sigma_y + r) \qquad (37) $$

where the quantities have the same definition as in the Drucker-Prager yield function (33). This type of function proposed after experimental investigations in metals is found in a number of related works [33, 2, 4]. With a standard generalized model, the complementary equations yield:

$$ \epsilon^p = \dot{\lambda} \frac{\partial f}{\partial s} = \dot{\lambda} \frac{\partial \bar{\sigma}}{\partial s} \quad \text{and} \quad \dot{\epsilon} = -\dot{\lambda} \frac{\partial f}{\partial r} = \dot{\lambda}(1 - \alpha h). \qquad (38) $$

Using the same arguments as for the previous example, the dissipation is recast as

$$ \mathcal{D} = \dot{\lambda}(\bar{\sigma}(s) - (1 - \alpha h)r) \qquad (39) $$

and the Clausius-Duhem inequality satisfied if, and only if:

$$ \alpha h \le 1. \qquad (40) $$

Owing to the chosen parametrization, the condition is identical to that of the Drucker-Prager model. A comparison of the two formulations is proposed in figure 3, where the Spitzig-Richmond model hardening appears as a widening of the initial cone with an unchanged vertex position. The model is associated regardless of the pressure sensitivity.

# 5. Generalizations

The theory was unfolded in the practical case of rate independent isochoric plasticity that covers a wide range of application. The development may nonetheless be propagated towards more general plasticity theories in order to analyse the limits of the assumptions.
## 5.1. Rate dependence: visco-plasticity

Isochoric plasticity with rate dependence is classically standard generalized, described using a dissipation pseudo-potential formulated based on the yield function: $\Omega(f)$. The complementary equations are then given by (14) and the dissipation follows immediately by substitution in (12).

$$ \mathcal{D} = \frac{\partial \Omega}{\partial f} \left( s : \frac{\partial f}{\partial s} + A_q \frac{\partial f}{\partial A_q} \right). \qquad (41) $$

This expression is identical to (27), except $\dot{\lambda}$ is replaced by $\frac{\partial \Omega}{\partial f}$. There is no more restrictions than in the classical theories: $\Omega$ must be positive, convex and nil at the origin. The conclusions regarding hydrostatic stress are therefore identical to those of section 3.

## 5.2. Non-isochoric plasticity

In order to elaborate on materials not limited to deform in isochoric plasticity the theory should be expanded to plasticity volume changes. The developments of section 3 can be undertaken without enforcing traceless plastic strains. Let $\varepsilon^p$ denote the plastic strain, not restricted to any tensorial subspace. The specific free energy takes the form

$$ \psi(e - \operatorname{dev} \varepsilon^p, v - \operatorname{tr} \varepsilon^p, V_q) \Rightarrow \begin{bmatrix} -A_1 = s = \rho \frac{\partial \psi}{\partial \operatorname{dev} \varepsilon^p} \\ -A_2 = h = \rho \frac{\partial \psi}{\partial \operatorname{tr} \varepsilon^p} \end{bmatrix} \qquad (42) $$

This indicates that the plastic strain necessarily splits into independent contributions. In other words, the independence of hydrostatic–deviatoric contributions to the free energy implies the independence of the hydrostatic–deviatoric plastic strains. The two internal variables are then written:

$$ V_1 = \operatorname{dev} \varepsilon^p = \varepsilon^p, \quad V_2 = \operatorname{tr} \varepsilon^p = v^p. \qquad (43) $$

In the general case, the complementary equations will provide two independent models for the volumic and deviatoric contributions. But a simply associated material verifies, by definition,

$$ \dot{e}^p = \dot{\lambda} \frac{\partial f}{\partial s} \quad \text{and} \quad \dot{v}^p = \dot{\lambda} \frac{\partial f}{\partial h}. \qquad (44) $$

It follows that the plastic strain rate reduces to:

$$ \varepsilon^p = \dot{\lambda} \frac{\partial f}{\partial \sigma}; \qquad (45) $$
since

$$ \dot{\epsilon}^p = \dot{e}^p + (\partial^p/3)\mathbf{I} = \lambda \frac{\partial f}{\partial \sigma} : \left( \frac{\partial \sigma}{\partial \mathbf{s}} + \frac{1}{3} \frac{\partial \sigma}{\partial \mathbf{h}} \mathbf{I} \right) = \lambda \frac{\partial f}{\partial \sigma} : (\mathbf{P}^D + \mathbf{P}^H) = \lambda \frac{\partial f}{\partial \sigma} : \mathbf{I}_4 \quad (46) $$

where $\mathbf{I}_4$ is the fourth order symmetric identity, $\mathbf{P}^H = \mathbf{I} \otimes \mathbf{I}/3$ the hydrostatic projector and $\mathbf{P}^D = \mathbf{I}_4 - \mathbf{P}^H$ is the deviatoric projector.

Therefore in the case of non-isochoric plasticity, the present assumption (44) falls back to the classical thermodynamic description (45). The proposed definition of associativity remains consistent in this case. Indeed, a strict normality in the dissipative stress space (now including the hydrostatic direction) is here needed for the model to be associated.

## 6. Discussion

In this section, the paper analysis is put in perspective with the evolution of the theory of plasticity. Chronology is an important aspect: the potential theory of plastic flow was developed mainly on metals, earlier than the formalization of modern rational thermodynamics, which happened before the pressure sensitivity of isochoric plastic materials was put in evidence in steels. It is for these reasons that the topics addressed in this paper have not, as far as the authors research reached, been considered together in the literature.

### 6.1. On the principle of maximum plastic work and Drucker-stability

The principle of maximum plastic work (PMW) is formalized by Hill [21] for plastic solids, after independent contributions of von Mises and Taylor [41, 38]. It is later generalized by Mandel [31] to elasto-plastic solids. It expresses as follows: given an actual state characterized by $\{\sigma, \dot{\epsilon}^p, A_q\}$, and considering any other stress tensor $\sigma^*$ such that $f(\sigma^*, A_q) \le 0$ (plastically admissible), then the material must verify

$$ (\sigma - \sigma^*) : \dot{\epsilon}^p \ge 0. \quad (47) $$

This means that the actual stress tensor is the one that maximizes the plastic work within all plastically admissible stresses. Without pressure sensitivity, it is also synonym to simple associativity.

Then from a different perspective, Drucker [14] examines the consequences of this inequation only supposing the existence of a yield surface. He concludes on the necessity of (i) a convex yield
surface (in the six-dimensional stress space), (ii) a strain rate orthogonal to the yield surface (in the six-dimensional stress space), and (iii) a positive second order incremental plastic work ($\dot{\sigma} : \dot{\varepsilon}^p \ge 0$). This last item defines material stability in the sense of Drucker [15].

But in general, isochoric plastic materials with pressure sensitivity are Drucker-unstable and violate the principle of maximum plastic work as stated above. Let us show that this is expectable from the essence of the PMW and does not lead to any conceptual contradiction.

Why is it expectable? The PMW was initially thought of for polycrystals assuming the sole effect of shear stresses on their deformation. The assumption is that the gliding resistance along a slip plane is insensitive to the normal stress applied onto it [7]. It is then natural that pressure sensitive materials violate the principle since they drop the above simplifying assumption. The PMW is inherently built outside of their scope of application. Why does it not lead to any contradiction? The PMW concerns a specific class of materials, yet is neither sufficient nor necessary for the Clausius-Duhem inequality to be verified. This statement is illustrated by the two examples treated in section 4 which entirely comply with the SPT. This question was also addressed by Green and Naghdi [19] who analyzed another class of materials (with possible volumetric plastic strains and pressure sensitivity) and also conclude that “Drucker postulate is not essential for a consistent theory of an elastic/plastic continuum”. In short, the PMW as well as Drucker-stability or normality in the six-dimensional stress space, are all faces of a same coin designed for a restricted class of materials; posing them as a requirement would be irrelevant for the models presented in this paper.

Instead (as has already been done [12]) one must restate a maximum work principle relevant to pressure sensitive materials by simply splitting it into a hydrostatic and a deviatoric part—after hypothesis 1. It expresses as follows: given an actual state characterized by {$s, h, \dot{\varepsilon}^p, A_q$}, and a yield function $f(s, h, A_q)$, then the material must verify

$$ (s - s^*) : \dot{\varepsilon}^p \ge 0, \quad \forall s^*, f(s^*, h, A_q) \le 0, \qquad (48) $$

$$ (h - h^*)I : \dot{\varepsilon}^p \ge 0, \quad \forall h^*, f(s, h^*, A_q) \le 0. \qquad (49) $$

This statement simply consists in uncoupling the hydrostatic and deviatoric contributions, then maximizing one while regarding the other like any other variable $A_q$ that we fix to apply the PMW. In classical isochoric plasticity, it becomes usable with and without pressure sensitivity and returns an undetermined value for $h$ since $I : \dot{\varepsilon}^p = 0$. Without the isochoric assumption, it reduces to the classical principle of equation (47). While this modification of the PMW is given as an illustration
<figure><img src="image_5.png" /><figcaption>Figure 4: Associativity is to be observed in the space of thermodynamic forces that conjugate with dissipative variables. In the considered cases, the hydrostatic axis is to be disregarded.</figcaption></figure>

more than a novel tool, it also shows how associativity always leads to a power maximization.

## 6.2. On the concept of associativity

It must be noted that the notion of associativity was introduced in the early sixties [23], while its connection with work-conjugacy in rational thermodynamics was clarified years later [29, 17]. Its definition has certainly not always been identical for all authors, but it has converged throughout the years. Today, the consistent understanding of the thermodynamics of plasticity leads to an unequivocal meaning.

The developments of section 3 lead to a necessary discrimination of the notions of associativity and normality: the associativity of a model refers to its normality in the space of all thermodynamic
forces *that conjugate with dissipative variables* (figure 4). This is needed since the yield function
depends upon all thermodynamic forces; and in isochoric plasticity:

(i) the hydrostatic stress is a thermodynamic force (derives from the state potential),

(ii) but it does not conjugate with any dissipative variable (does not appear in Clausius-Duhem inequality);

(iii) hence it does not intervene in the associative character of a model.

This can be counter-intuitive because thermodynamic forces ordinarily all conjugate with dissipative
variables, which is why dependencies are ordinarily obvious enough not to be stated. Normality is
a property that only refers to geometrical properties with no direct physical implications. On the
contrary, associativity is a property that helps verifying the SPT by exploiting characteristics of
the yield function—see under (27)—instead of fully expanding the Clausius-Duhem inequality. It
carries the physical meaning that an internal variable and its conjugate maximize their contribution
to dissipation (as seen in section 6.1).

**7. Conclusion**

In this work, the thermodynamics of continuum media are retraced from the Clausius–Duhem inequality to the complementary equations of rate-(in)dependent plasticity. A single shift is introduced when the stress dependencies (hydrostatic and deviatoric) are chosen to be accounted for independently: hypothesis 1. Every step of the development of rational thermodynamics is retraced in order to ensure the consistency of the analysis. It is logically deduced that

* the isochoric plastic strains do not conjugate with the whole stress tensor, but only with its deviatoric part as long as the isochoric condition is imposed.

* Any isochoric plastic model with pressure dependence is associated *in the classical sense* if the flow rule is normal *in the deviatoric plane only*.

The latter refers to the notion of deviatoric associativity [3], that is here demonstrated to be
equivalent to simple associativity, in the classical sense (remarks 4 and 5). Consequences on
associativity, maximum plastic work or material stability are clarified. An entire class of pressure-
dependent materials is reconciled with these fundamental concepts in plasticity, which strengthens
their physical relevance.
<figure><img src="image_6.png" /><figcaption>Figure A.5: Relative frequency of occurrence of the words "associative" and "associated" when prepending the words "flow" or "plasticity", in Google Ngram's English corpus from 1950 to 2019.</figcaption></figure>

## Acknowledgements

The authors are grateful to POSCO for their generous support. The data for figure 1 was kindly provided by Shin-Yeong Lee (orcid: 0000-0001-5719-756X). The first author thanks Vasudevan Kamasamudram (orcid: 0000-0001-6967-5987) for his helpful comments.

## Appendix A. Notations

Every notation used throughout the paper is detailed in table A.1. The third column is the tensorial order which distinguishes scalars (0) from vectors (1) from second or fourth order tensors (2, 4). The fourth column is the unit, where x designates a variable unit. Note that Pa and J/m³ are equivalent, and that the unit of the yield function is only imposed when the model has at least one associated variable. Otherwise it does not appear in the dissipation and any transformation can be used. For example: substituting *f* for *f*' = 2*f*³ is still admissible in the Kuhn–Tucker conditions.

Finally, it is noted that the words "associative" and "associated" are both used with the same definition in the scientific literature. Looking for the most popular, figure A.5 illustrates Google's ngram data for the frequency of occurrence of each of them in the context of the mechanics of plasticity. To do so, their occurrences are here accounted for only when they are followed by the word "flow" or the word "plasticity". The word "associated" seems to be more commonly used.
<table><caption>Table A.1: Notations. The order refers to the tensorial one.</caption><thead><tr><th>Symbol</th><th>Name</th><th>Order</th><th>Unit</th></tr></thead><tbody><tr><td>D<sub>tot</sub></td><td>volumic total dissipation</td><td>0</td><td>J/m<sup>3</sup>/s</td></tr><tr><td>D</td><td>volumic mechanical dissipation</td><td>0</td><td>J/m<sup>3</sup>/s</td></tr><tr><td>D<sub>th</sub></td><td>volumic thermal dissipation</td><td>0</td><td>J/m<sup>3</sup>/s</td></tr><tr><td>σ</td><td>Cauchy stress</td><td>2</td><td>Pa</td></tr><tr><td>ε</td><td>Linearized Green-Lagrange strain</td><td>2</td><td>1</td></tr><tr><td>ρ</td><td>mass density</td><td>0</td><td>kg/m<sup>3</sup></td></tr><tr><td>ψ</td><td>specific free energy</td><td>0</td><td>J/kg</td></tr><tr><td>η</td><td>specific entropy</td><td>0</td><td>J/kg/K</td></tr><tr><td>θ</td><td>temperature</td><td>0</td><td>K</td></tr><tr><td>q</td><td>heat flux</td><td>1</td><td>J/m<sup>2</sup>/s</td></tr><tr><td>s</td><td>stress deviator</td><td>2</td><td>Pa</td></tr><tr><td>h</td><td>hydrostatic stress</td><td>0</td><td>Pa</td></tr><tr><td>e</td><td>strain deviator</td><td>2</td><td>1</td></tr><tr><td>v</td><td>volumetric ratio</td><td>0</td><td>1</td></tr><tr><td>V<sub>k</sub></td><td>dissipative internal variables</td><td></td><td>x</td></tr><tr><td>A<sub>k</sub></td><td>dissipative forces</td><td></td><td>J/m<sup>3</sup>/x</td></tr><tr><td>M<sub>k</sub></td><td>constitutive model</td><td></td><td>J/m<sup>3</sup>/x</td></tr><tr><td>Ω</td><td>dissipation pseudo-potential</td><td>0</td><td>J/m<sup>3</sup>/s</td></tr><tr><td>ε<sup>p</sup></td><td>plastic strain</td><td>2</td><td>1</td></tr><tr><td>f</td><td>yield function</td><td>0</td><td>J/m<sup>3</sup></td></tr><tr><td>λ̇</td><td>plastic multiplier</td><td>0</td><td>1/s</td></tr><tr><td>F</td><td>plastic potential</td><td>0</td><td>J</td></tr><tr><td>σ̄</td><td>homogeneous equivalent stress</td><td>0</td><td>Pa</td></tr><tr><td>σ<sub>y</sub></td><td>initial yield stress</td><td>0</td><td>Pa</td></tr><tr><td>α</td><td>pressure coefficient</td><td>0</td><td>Pa<sup>-1</sup></td></tr></tbody></table>
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