Thermodynamic framework for damage-healing-plasticity of granular materials and net damage variable

Abstract

Based on the meso-structured Voronoi cell model for discrete particle assembly and the derived meso-mechanically informed constitutive relations of anisotropic Cosserat continuum, thermodynamic framework of isothermal meso-mechanically informed damage-healing and plastic process for granular materials is presented. The accumulated net (effective) damage factor tensor combining both material damage and healing effects is defined in terms of the initial (undamaged) and current (damaged) elastic moduli tensors of the meso-structured Voronoi cell attributed to the material point. According to the non-negativity of thermodynamic energy dissipations, the net damage variable is separated into the two component internal state variables; i.e. the damage and healing variables, which are accumulated in terms of incremental damage and healing variables, respectively. The meso-mechanically informed macroscopic damage-healing and plastic characterization are achieved without the need to specify macroscopic phenomenological damage, healing and plastic criteria, and their evolution laws. The merit of the proposed tensorial net damage and healing variables in modeling healing effects on initial weakened elastic stiffness (i.e. initial material defects) is demonstrated in terms of their isotropic scalar forms and integrated into the continuum damage-healing mechanics. The numerical results conceptually illustrate the performance of the proposed definitions of meso-mechanically informed net damage, damage, and healing variables. The coupled damage-healing and plastic process in anisotropic Cosserat continuum for granular materials is characterized in terms of densities of thermodynamic dissipations that make effects of the damage-healing and the plastic component processes on the material failure quantitatively comparable.

Keywords

Granular materials Cosserat continuum discrete particle assembly meso-mechanically informed damage-healing and plasticity thermodynamic framework anisotropic net damage and damage-healing variables

Introduction

Granular material is highly heterogeneous and discontinuous medium at the grain scale. Geological matters, soils and clays in nature, geo-structure, concrete, etc. are its practical examples. The discrete nature of granular material makes its behavior highly nonlinear, dissipative, and intimately related to its meso-structure. To study the complex mechanical behaviors of granular materials, multi-scale methods have been proposed to bridge their two distinct scales ranging from the particle scale to the continuum scale.

In the frame of the meso–macro homogenization, granular material is homogenized with the Cosserat continuum rather than the Cauchy continuum due to the nature of the granular medium as a discrete particle assembly at the meso-scale (cf. Alonso-Marroquin, 2011; Chang and Kuhn, 2005; d'Addetta et al., 2004; Ehlers et al., 2003; Kruyt, 2003; Li et al., 2010, 2013; Pasternak and Muhlhaus, 2005). The micro-rotations defined as independent degrees of freedom (DOF) at each mathematical point in Cosserat continuum play an important role in properly transiting the effect of rotations of particles within the representative meso-structure via its boundary to the macroscopic continuum and vice versa. The discrete particle assembly - Cosserat continuum modeling of granular material in the two distinct scales leads to the construction of meso-mechanically informed macroscopic constitutive relations.

As failure phenomena of granular material are concerned, it is further required to quantitatively describe material degradations of macroscopic effective Cosserat continuum such as stiffness and strength reductions by means of mesoscopic dissipative mechanisms. On the other hand, when significant meso-structural evolution in the granular material develops, severe dissipative sliding and rolling frictions and loss of contacts among immediate neighboring particles occur, and mesoscopic irreversible energy dissipations should be characterized in the effective continuum at the macro-scale.

The continuum theory in the thermodynamic framework describing macroscopic material damage and irreversible deformation phenomena has been studied and reported (Chaboche, 1988; Chow and Wang, 1987; Ju, 1989; Ladeveze and LeDantec, 1992; Simo and Ju, 1987; Voyiadjis and Deliktas, 2000). It is also realized for instance for the composites that macroscopic plasticity resulting in reduction of the material strength describes the microscopic dissipative slips of material whereas the damage resulting in reduction of the material stiffness defined in continuum damage mechanics (Kachanov, 1958) provides a macroscopic representation of the microcrack and void distribution (Barbero et al., 2005).

The material damage defined in classical theory of continuum damage mechanics describes an irreversible and dissipative process (Kachanov, 1958; Lemaitre and Chaboche, 1990; Voyiadjis and Kattan, 1999). However, it has been observed that many classes of engineering materials, such as biomaterials, composites, polymers, glass materials, and granular materials particularly related to the present work, possess the potential to heal and retrieve part of their stiffness under certain conditions (cf. Brown et al., 2005; Darabi et al., 2012; Herbst and Luding, 2008; Ju et al., 2012, Ju and Yuan, 2012; Plaisted and Nemat-Nasser, 2007; White et al., 2001; Xu et al., 2014). Hence it is required to generalize classical theory of continuum damage mechanics to take into account the healing process with non-dissipative nature (Barbero et al., 2005). Most constitutive models for analyzing healing process followed a phenomenological approach (Voyiadjis et al., 2012). Ju et al. firstly proposed a novel phenomenological strain based coupled elastoplastic damage and healing model for geomaterials (Ju et al., 2012; Ju and Yuan, 2012).

Healing effects may be caused by different phenomena in different materials, leading to a reduction of internal material defects, and consequently rehabilitation of the structure. It was reported that composites can be healed due to chemical, physical or biological phenomena (Barbero et al., 2005). In this paper, it is assumed that the damage-healing process is only caused by mechanical phenomena, ignoring chemical, thermal and wetting effects, etc. in the dry densely packed particle assembly of granular materials.

In the frame of macroscopic phenomenological theories for continuum damage-healing mechanics and elastoplasticity, the phenomenological criteria to govern thresholds of both damage-healing and plastic processes and the evolution laws to govern their developments are usually required, and assumed to determine the internal state variables such as the damage-healing variables and the effective plastic strain, defined at a material point (Barbero et al., 2005; Herbst and Luding, 2008). Voyiadjis et al. (2011) proposed a computational method based on a thermodynamic consistent damage and healing model for self-healing materials. While in the frame of meso-mechanically based macroscopic damage-healing and plasticity, the internal state variables characterizing macroscopic material degradation are to be defined and evaluated according to the current meso-structure and mesoscopic energy dissipation attributed to the material point. Few attempts have been made in the literature for meso-mechanically informed constitutive modeling of the damage-healing and plasticity in materials without need to specify macroscopic phenomenological constitutive relations, damage-healing, and plastic models and their evolution laws. Dartois et al. (2012) developed a multi-scale model to investigate interfacial damage and subsequent effects in highly filled particulate composites such as solid propellants. Three criteria formulated at the microscale exploit the explicit microstructure representation and the knowledge of the local displacement field as a function of local morphology around the interfaces. The damage-healing process considered in this work is attributed to describe the mesoscopic dissipative deformations; i.e. relative dissipative sliding and rolling movements between every two particles in contact followed by loss of contacts, resulting in the reduction of macroscopic elastic stiffness (damage), and contrarily the mesoscopic non-dissipative deformations, resulting in the enhancement of macroscopic elastic stiffness (healing). The mesoscopic mechanisms of healing described in this paper for granular materials are different from those in the literature, such as by Barbero et al. (2005) for a self-healing fiber-reinforced lamina and Dartois et al. (2012) for highly filled particulate composites.

To achieve meso-mechanically informed constitutive modeling of the coupled damage-healing and plastic process for granular materials, it is required to link meso-structural mechanisms, such as the meso-slip between every two particles in contact, the loss and the generation of contacts, re-orientation of contacts, etc. detected in discrete particle assembly as the deformation proceeds, to the macroscopic failure phenomena of granular materials. In this regard, one should perceive local character of those models to be developed due to the heterogeneity and anisotropy of meso-structures of granular materials in order to accurately capture the weakened location, where the real failure process is triggered, and the weakened orientation, along which the failure process is developed.

Indeed, bearing in mind the local character of typical meso-structure to be selected from discrete particle assembly for granular materials, the Voronoi cell model (Li et al., 2013) has been proposed to establish such meso-mechanically informed macroscopic constitutive relation of anisotropic Cosserat continuum. The concept of Voronoi cell (Oda and Iwashita, 1999; Walsh et al., 2007) is introduced for each reference particle and its associated void space. The proposed Voronoi cell model involves not only the reference particle laid inside the Voronoi cell but also the immediate neighboring particles around the reference particle. The Voronoi cell represents an effective Cosserat continuum element with the volume assigned to the reference particle and possessing a meso-structure composed of a small group of particles; i.e. a reference particle and its immediate neighboring particles, in which voids and discontinuities exist. With the meso-structure described by the Voronoi cell model the void ratio around the reference particle and the contact topology of the reference particle with its immediate neighboring particles can be identified. The deformation of the reference Voronoi cell modeled as a continuum element is linked to translational movements and rotations of not only the reference particle but also its immediate neighboring particles defined by the Voronoi cell model.

In “Meso-mechanically informed constitutive relation and net damage tensor of anisotropic Cosserat continuum for granular materials” section, based on the Voronoi cell model characterizing fundamental meso-structures of granular materials, the meso-mechanically informed constitutive relation of anisotropic Cosserat continuum for granular materials is briefly introduced. Subsequently, the net (effective) elastic damage factor tensor representing the effect of both damage and healing for anisotropic Cosserat continuum is defined in terms of changes in meso-structural parameters of granular materials.

In “Thermodynamic framework of damage-healing and plasticity of Cosserat continuum for granular materials” section, a thermodynamic framework for the constitutive modeling of meso-mechanically informed macroscopic damage-healing and plastic process in anisotropic Cosserat continuum for granular materials is presented. The tensorial net damage, damage, and healing variables termed the net damage tensor, the damage tensor, and the healing tensor, respectively, hereafter in this paper, are defined. The densities of incremental and accumulated thermodynamic dissipations are taken as the internal state variables to quantitatively characterize the coupled anisotropic damage-healing and plastic process.

The isotropic scalar forms of the proposed net damage, damage, and healing variables, termed the net damage factor, the damage factor, and the healing factor, respectively, hereafter in this paper, which distinguish from existing definitions of those variables in the literature, are discussed in “Net (effective) damage variables, damage, and healing variables” section. The merit of the proposed net damage variable and healing variable in modeling the healing effects on initially weakened elastic stiffness, which reflects initial material defects, is also demonstrated in “Net (effective) damage variables, damage, and healing variables” section and further conceptually illustrated by numerical examples shown in “Numerical results” section. The meso-mechanically informed coupled anisotropic damage-healing and plastic process in a boundary value problem (BVP) for granular materials are characterized with densities of thermodynamic energies as scalar internal state variables and their evolutions with respect to time in “Numerical results” section. “Discussions and concluding remarks” section summarizes the characteristics and advantages of the proposed meso-mechanically informed approach in defining the net damage variable and the healing variable, and in characterizing coupled net elastic damage and plasticity for granular materials in the frame of thermodynamics of anisotropic Cosserat continuum.

Meso-mechanically informed constitutive relation and net damage tensor of anisotropic Cosserat continuum for granular materials

Based on the Voronoi cell model (Li et al., 2013), the meso-mechanically informed macroscopic constitutive relation for the Cauchy stress ${\bar{σ}}_{ji}^{A}$ and the couple stress ${\bar{μ}}_{j 3}^{A}$ exerted on a reference Voronoi cell A as a two dimensional effective Cosserat continuum element can be derived by means of the homogenization procedure and written in the form

{\bar{σ}}_{ji}^{A} = {\bar{σ}}_{ji}^{Ae} - {\bar{σ}}_{ji}^{Ap}

(1)

{\bar{μ}}_{j 3}^{A} = {\bar{μ}}_{j 3}^{Ae} - {\bar{μ}}_{j 3}^{Ap}

(2)

in which, with the strain

{\bar{ɛ}}_{lk}^{A}

and the curvature

{\bar{κ}}_{l}^{A}

defined for the Voronoi cell A,

{\bar{σ}}_{ji}^{Ae}

{\bar{μ}}_{j 3}^{Ae}

{\bar{σ}}_{ji}^{Ap}

, and

{\bar{μ}}_{j 3}^{Ap}

are expressed as

{\bar{σ}}_{ji}^{Ae} = D_{jilk}^{e σ ɛ} {\bar{ɛ}}_{lk}^{A} + D_{jil}^{e σ κ} {\bar{κ}}_{l}^{A}

(3)

{\bar{μ}}_{j 3}^{Ae} = D_{j 3 mi}^{e μ ɛ} {\bar{ɛ}}_{mi}^{A} + D_{j 3 i}^{e μ κ} {\bar{κ}}_{i}^{A}

(4)

{\bar{σ}}_{ji}^{Ap} = \frac{r A}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) f_{t}^{cp} n_{j}^{c} t_{i}^{c}

(5)

{\bar{μ}}_{j 3}^{Ap} = \frac{r A}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) m_{3}^{cp} n_{j}^{c} + \frac{r_{A}^{2}}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) f_{t}^{cp} n_{j}^{c}

(6)

The elastic moduli tensors in equations(3) and (4) are given by

D_{jilk}^{e σ ɛ} = \frac{r A}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) ((k_{s}^{c} (r A + r_{B}^{c}) + k_{r}^{c} (r A - r_{B}^{c})) t_{i}^{c} n_{j}^{c} t_{k}^{c} n_{l}^{c} + k_{n}^{c} (r A + r_{B}^{c}) n_{i}^{c} n_{j}^{c} n_{k}^{c} n_{l}^{c})

(7)

D_{jil}^{e σ κ} = \frac{r A}{V A} [\sum_{c = 1}^{m} h (u_{n}^{cAt}) (k_{s}^{c} - k_{r}^{c}) r_{B}^{c} (r A + r_{B}^{c}) t_{i}^{c} n_{j}^{c} n_{l}^{c}]

(8)

D_{j 3 mi}^{e μ ɛ} = \frac{r_{A}^{2}}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) (k_{s}^{c} (r A + r_{B}^{c}) + k_{r}^{c} (r A - r_{B}^{c})) t_{i}^{c} n_{m}^{c} n_{j}^{c}

(9)

D_{j 3 i}^{e μ κ} = \frac{r A}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) [r A r_{B}^{c} (r A + r_{B}^{c}) (k_{s}^{c} - k_{r}^{c}) + k_{θ}^{c} (r A + r_{B}^{c})] n_{j}^{c} n_{i}^{c}

(10)

where

j, i = 1, 2

. r_A and

r_{B}^{c}

(

c = 1 ~ m

) are the radius of the reference particle A and the radii of its immediate neighboring particles, respectively.

u_{n}^{cAt}

is the overlap of the reference particle A with one of its immediate neighboring particles at the current time instant,

h (u_{n}^{cAt})

is the Heaviside unit function depending on the value of

u_{n}^{cAt}

and embodies loss and generation of particle contacts and their evolution with respect to time.

k s, k r, k n, k θ

are stiffness coefficients between two particles in contact corresponding to the sliding, rolling tangential friction forces, the normal contact force, and the rolling friction moment, respectively.

The “plastic” tangential force $f_{t}^{cp}$ and rolling moment $m_{3}^{cp}$ between two particles in contact, shown in equations (5) and (6) are accumulated with their increments and given by

(f_{t}^{cp}) t n + 1 = (f_{t}^{cp}) t n + Δ f_{t}^{cp} with Δ f_{t}^{cp} = - k_{s}^{c} Δ u_{s}^{cp} - k_{r}^{c} Δ u_{r}^{cp}

(11)

(m_{3}^{cp}) t n + 1 = (m_{3}^{cp}) t n + Δ m_{3}^{cp} with Δ m_{3}^{cp} = - k_{θ}^{c} Δ θ_{r}^{cp}

(12)

It is noted that $h (u_{n}^{cAt})$ also implicitly controls the resetting of $f_{t}^{cp}$ and $m_{3}^{cp}$ , which are accumulated with their increments due to “plastic” parts of relative tangential sliding and rolling translational displacement increments and relative rolling angular displacement increment; i.e. $Δ u_{s}^{cp}$ , $Δ u_{r}^{cp}$ , $Δ θ_{r}^{cp}$ , as exhibited by equations (11) and (12).

Equations (1) to (4) can be re-written in a matrix-vector form rendered by

Σ = D t \cdot E e = D t \cdot (E - E p) = D t \cdot E - Σ p

(13)

with

Σ = {{\bar{σ}}_{(ji)}^{A} {\bar{μ}}_{(j 3)}^{A}}, E = {{\bar{ɛ}}_{(ji)}^{A} {\bar{κ}}_{(j 3)}^{A}} = E e + E p, E p = D_{t}^{- 1} Σ p, Σ p = {{\bar{σ}}_{(ji)}^{Ap} {\bar{μ}}_{(j 3)}^{Ap}}

(14)

D t = D e (t) = [D_{t}^{σ ɛ} D_{t}^{σ κ} D_{t}^{μ ɛ} D_{t}^{μ κ}] = [D_{(ji) (lk)}^{e σ ɛ} D_{(ji) (l)}^{e σ κ} D_{(j 3) (mi)}^{e μ ɛ} D_{(j 3) (i)}^{e μ κ}]

(15)

where

E e, E p

are the elastic and plastic portions of additive decomposition of the strain vector E, respectively,

Σ

is the stress vector, and

D t

is the elastic moduli matrix for the Voronoi cell modeled as a Cosserat continuum element at the current time instant t.

Let $D t$ and $D 0$ denote current (damaged) and initial (undamaged) elastic moduli tensors, respectively; i.e. $D 0 = D t (t = 0, d = 0)$ , where d denotes the damage tensor of the second order reflecting the effect of anisotropic material damage for the anisotropic Cosserat continuum. According to the concept of continuum damage mechanics the material damage is defined as a reduction in the elastic stiffness of material identified with the change in the elastic moduli tensor; i.e. $d = d (D t, D 0)$ .

Let $\tilde{Σ}$ denote the effective stress of the stress vector $Σ$ . The stress–strain relation for undamaged material characterized by $D 0 = D t (t = 0, d = 0)$ can be expressed as

\tilde{Σ} = D 0 \cdot E e

(16)

while the stress–strain relation for damaged material characterized by

D t

has been written by equation (13). Based on the hypothesis of strain equivalence (Lemaitre, 1985) to measure the damage in materials, substitution of equation (13) into equation (16) gives

\tilde{Σ} = (D 0 \cdot D_{t}^{- 1}) \cdot Σ

(17)

For the anisotropic damage, equation (17) may be expressed in a generalized form given by (cf. Chaboche, 1982; Chow and Wang, 1987)

\tilde{Σ} = M \cdot Σ = (I - d) - 1 \cdot Σ

(18)

where M is known as damage effect tensor represented by a second order tensor for the anisotropic Cosserat continuum. The second equality sign in equation (18) setting

M = (I - d) - 1

is based on the generalization of the original theories by Kachanov (1958) and Rabotnov (1969) to accommodate anisotropic damage. Further,

M = (I - d) - 1

was also employed by Chaboche (1987) and Jarić et al. (2013). Substitution of equation (18) into equation (17) gives

d = d (D t, D 0) = I - D t \cdot (D 0) - 1

(19)

As a typical time interval $Δ t = t i - t i - 1$ in the simulation is concerned, the incremental internal state variable $δ d$ can be evaluated by

δ d = - δ D t \cdot (D 0) - 1 = - [D i - D i - 1] \cdot (D 0) - 1

(20)

where

D i

and

D i - 1

denote the elastic moduli tensors at time instants t_i and

t i - 1

, respectively.

It is remarked that in the frame of meso-mechanically based macroscopic damage mechanics, the damaged elastic moduli tensors $D t$ and $D i, D i - 1$ are directly determined with the homogenization and evaluated by using equations (15) and (7) to (10) according to varying local meso-structures at times $t, t i, t i - 1$ , respectively. The damage tensor $d = d (D t, D 0)$ and its increment $δ d$ within the time interval $Δ t$ will be determined by using equations (19) and (20) with given initial (undamaged) and current (damaged) elastic moduli tensors, or the two successive elastic moduli tensors $D i$ and $D i - 1$ , for the reference Voronoi cell assigned at a material point.

It is observed from equations (15), (7) to (10) that the reduction in the elastic stiffness, i.e. the material damage may be caused by the three microscopic mechanisms characterizing the evolution of the meso-structure of the Voronoi cell; i.e. (1) loss of contacts of the reference particle with its immediate neighboring particles, (2) change in contact orientations towards weakening the elastic stiffness of the Voronoi cell, and (3) the concomitant volumetric dilatation of $V A$ . The meso-mechanically informed macroscopic damage tensor d at the current configuration of the Voronoi cell is measured from its initial meso-structure characterized by $D 0$ . However, $D 0$ is not certainly most stiff among all of possible particle collocation patterns describing the meso-structure attributed to the reference particle. Namely, the evolution of the meso-structure of the Voronoi cell may also be characterized by (1) generation, instead of loss, of contacts of the reference particle with its immediate neighboring particles; (2) change in contact orientations towards strengthening, instead of weakening, the elastic stiffness of the Voronoi cell; (3) the volumetric shrinkage, instead of volumetric dilatation, of $V A$ . Therefore, under the above mesoscopic deformations it is possible to result in a negative definite damage tensor d ( $δ d$ ) for a deformed Voronoi cell as it is subjected to certain external conditions (mainly compression). In summary, the resulting meso-mechanically informed damage tensors d and $δ d$ may be positive or negative definite, characterizing the reduction or recovery of elastic stiffness, i.e. material damage or healing, respectively, at the meso-structured material point. In addition, it is noted that the gross amount of the elastic stiffness, which may be healed (enhanced) in comparison with $D 0$ during the loading history, is indeterminate for granular materials.

When healing is included, the elastic damage tensors d and $δ d$ defined in the classical theory of continuum damage mechanics and discussed above will be interpreted as a generalized internal state variable representing the net effects of both accumulated and incremental damage (i.e. the “positive definite damage”) and healing. d and $δ d$ expressed by equations (19) and (20) will be termed as the accumulated and incremental net (effective) elastic damage tensors in continuum damage-healing mechanics.

The details of incremental and accumulated net damage, damage, and healing tensors, along with their isotropic scalar forms will be discussed in “Thermodynamic framework of damage-healing and plasticity of Cosserat continuum for granular materials” and “Net (effective) damage variables, damage, and healing variables” sections in the frame of thermodynamics of meso-mechanically informed Cosserat continuum for granular materials.

Thermodynamic framework of damage-healing and plasticity of Cosserat continuum for granular materials

Combining the first and the second laws of thermodynamics in the isothermal condition, incremental form of energy conservation of the Voronoi cell modeled as a Cosserat continuum element equivalent to a small group of compacted particles in the homogenization sense can be written as (Walsh and Tordesillas, 2004)

\tilde{W} = δ ϕ + \tilde{Φ}

(21)

with the second law of thermodynamics in the form of the Clausius-Duhem inequality

\tilde{Φ} = - δ ϕ + \tilde{W} \geq 0

(22)

where

\tilde{W}

is the incremental mechanical work done on the Voronoi cell and can be expressed by

\tilde{W} = Σ \cdot δ E

(23)

The elastic damage-healing Helmholz free energy density function ϕ is postulated to take the form

ϕ = ϕ (E e, d) = \frac{1}{2} E_{e}^{T} \cdot D t (D 0, d) \cdot E e

(24)

where d is the net (effective) damage tensor characterizing the net effect combining anisotropic damage and healing of Cosserat continuum at time t. With the use of equation (19), the current elastic moduli tensor

D t

can be expressed in terms of

D 0

and the net damage tensor d; i.e.

D t = (I - d) \cdot D 0

(25)

According to equation (24), the increment of the elastic damage-healing Helmholz free energy is then expressed by

δ ϕ = \frac{\partial ϕ}{\partial E e} \cdot δ E e + \frac{\partial ϕ}{\partial d} : δ d

(26)

The incremental non-negative energy dissipation $\tilde{Φ}$ is assumed to be contributed from two decoupled sources: the “plastic” and the “net elastic damage” that may be understood from the meso-mechanically informed constitutive equations (13) to (15) of Cosserat continuum derived by the homogenization procedure for granular materials.

Under the hypothesis of decoupling between plastic and net damage processes and according to the Clausius-Duhem inequality, the density of incremental thermodynamic dissipation $\tilde{Φ} (E p, δ E p, d, δ d)$ can be additively decomposed into the plastic and the net elastic damage components $\tilde{Φ} p (E p, δ E p)$ and $\tilde{Φ} d (d, δ d)$ given by

\tilde{Φ} = \tilde{Φ} p + \tilde{Φ} d = \frac{\partial \tilde{Φ} p}{\partial δ E p} \cdot δ E p + \frac{\partial \tilde{Φ} d}{\partial δ d} : δ d \geq 0

(27)

\tilde{Φ} p = \frac{\partial \tilde{Φ} p}{\partial δ E p} \cdot δ E p \geq 0, \tilde{Φ} d = \frac{\partial \tilde{Φ} d}{\partial δ d} : δ d

(28)

Substitution of equations (23), (26), (27) into equation (21) renders

(\bar{Σ} - \frac{\partial ϕ}{\partial E e}) \cdot δ E e + (\bar{Σ} - \frac{\partial \tilde{Φ} p}{\partial δ E p}) \cdot δ E p + (- \frac{\partial ϕ}{\partial d} - \frac{\partial \tilde{Φ} d}{\partial δ d}) : δ d = 0

(29)

From equation (29) and equation (24) it is observed that

\bar{Σ} = \frac{\partial ϕ}{\partial E e} = D t \cdot E e

(30)

\bar{Σ} = \frac{\partial \tilde{Φ} p}{\partial δ \bar{E} p}

(31)

One may define the thermodynamic generalized force $Ξ d$

Ξ d = - \frac{\partial ϕ}{\partial d}

(32)

conjugated with the incremental net damage tensor

δ d

. From equation (29) it is also observed that

Ξ d = \frac{\partial \tilde{Φ} d}{\partial δ d}

(33)

With the use of equations (31) and (33), the densities of incremental thermodynamic dissipations $\tilde{Φ}$ , $\tilde{Φ} p$ , and the density of incremental net damage energy $\tilde{Φ} d$ exhibited in equations (27) and (28) can be expressed in the forms

\tilde{Φ} = \tilde{Φ} p + \tilde{Φ} d \geq 0, \tilde{Φ} p = Σ \cdot δ E p \geq 0, \tilde{Φ} d = Ξ d : δ d

(34)

in which the incremental plastic internal state variable

δ \bar{E} p

can be evaluated with the use of equations (13), (5), (6) and expressed in terms of mesoscopic quantities as

δ E p = D_{t}^{- 1} δ Σ p = D_{t}^{- 1} {\frac{r A}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) δ f_{t}^{cp} n_{j}^{c} t_{i}^{c} \frac{r A}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) δ m_{3}^{cp} n_{j}^{c} + \frac{r_{A}^{2}}{V A} \sum_{c = 1}^{m} h (u_{n}^{cAt}) δ f_{t}^{cp} n_{j}^{c}}

(35)

Using equation (25) and noticing the following expression

\frac{\partial (I - d)}{\partial d} = - (I - d) \cdot \frac{\partial (I - d) - 1}{\partial d} \cdot (I - d)

(36)

one may substitute equation (24) into equation (32) and obtain the following

Ξ d = - \frac{\partial ϕ}{\partial d} = - \frac{1}{2} E_{e}^{T} \cdot \frac{\partial D t}{\partial d} \cdot E e = \frac{1}{2} E_{e}^{T} \cdot (I - d) \cdot \frac{\partial (I - d) - 1}{\partial d} \cdot Σ

(37)

To derive the expression of $Ξ d$ in terms of $D t$ and $D 0$ , we apply equation (25) and the expressions below

\frac{\partial (I - d) - 1}{\partial d} = - (I - d) - 2 \cdot \frac{\partial (I - d)}{\partial d}

(38)

\frac{\partial (I - d)}{\partial d} = \frac{\partial (- d)}{\partial d} = - \frac{\partial d ab}{\partial d cd} e a \otimes e b \otimes e c \otimes e d = - δ ac δ bd e a \otimes e b \otimes e c \otimes e d

(39)

and notice the asymmetry of the elastic moduli tensor

D t

. Then further deduction of equation (37) results in

Ξ d = - \frac{1}{2} Σ T \cdot (D_{t}^{- T}) \cdot (I - d) - 1 \cdot \frac{\partial (I - d)}{\partial d} \cdot Σ = - \frac{1}{2} Σ T \cdot (D_{t}^{- T} \cdot D 0 \cdot D_{t}^{- 1}) \cdot \frac{\partial (I - d)}{\partial d} \cdot Σ = - \frac{1}{2} Σ i (D_{t}^{- T} \cdot D 0 \cdot D_{t}^{- 1}) ij e j \cdot [- δ ac δ bd e a \otimes e b \otimes e c \otimes e d] \cdot Σ k e k = \frac{1}{2} Σ i (D_{t}^{- T} \cdot D 0 \cdot D_{t}^{- 1}) ij δ ja [δ ac δ bd e b \otimes e c] δ dk Σ k = \frac{1}{2} Σ i (D_{t}^{- T} \cdot D 0 \cdot D_{t}^{- 1}) ij δ jb δ ak Σ k e a \otimes e b

(40)

With the derived expression for thermodynamic generalized force $Ξ d$ given by equation (40), the negative definiteness of $δ d$ is defined by

\tilde{Φ} d = Ξ d : δ d = \frac{\partial \tilde{Φ} d}{\partial δ d} : δ d = - \frac{\partial ϕ}{\partial d} : δ d < 0

(41)

which characterizes the healing evolution at a local meso-structured material point and describes non-dissipative nature of the healing process.

For a particular time interval a local meso-structured material point will undergo the material damage if

\tilde{Φ} d = Ξ d : δ d > 0

(42)

or the material healing if

\tilde{Φ} d = Ξ d : δ d < 0

(43)

The condition distinguishing between the material damage and the material healing can be written by

\tilde{Φ} d = Ξ d : δ d = 0

(44)

It implies that neither material damage nor material healing occurs in view of no change in the density of thermodynamic damage energy within the time interval.

Physically, a negative definite net damage tensor d (or $δ d$ ) implies a re-collocation of the particles for the Voronoi cell, which results in a reinforced current meso-structure measured with $D t$ when compared with its initial (or previous) meso-structure measured with $D 0$ (or $D t - Δ t$ ), and leads to an increase in the elastic stiffness of material. Then, it characterizes the healing evolution in granular materials, opposite to damage in a damage-healing process. Both healing and damage provide a macroscopic representation of recovery (or enhancement) and reduction of elastic stiffness, respectively. For an isotropic damage-healing process, the negative definite net damage tensor d ( $δ d$ ) will degrade to a negative scalar net damage factor d ( $δ d$ ).

It should be pointed out that even though $\tilde{Φ} d$ may be positive or negative, characterizing the damage and healing, respectively, $\tilde{Φ} = \tilde{Φ} p + \tilde{Φ} d$ due to the coupled damage-healing and plastic process is always positive. The healing process in the meso-structure referred to a single Voronoi cell of granular material is activated by dissipative relative sliding and rolling movements between the reference particle with its immediate neighboring particles in contact. The thermodynamic dissipations of the damage-healing and plastic process occurring in granular materials will also be illustrated by numerical results in “Numerical results” section.

It is understood from equations (42) to (44) that only one phenomenon, either material damage or material healing, may occur at one time instant for a meso-structured material point in the macroscopic continuum, though the material damage and healing evolution may alternately occur for the material point from one instant to its successive instant. Nevertheless, to distinguish incremental damage and healing evolutions, the incremental healing and damage tensors denoted by $δ h i$ and $δ d_{i}^{+}$ , respectively, for a typical time interval $Δ t = t i - t i - 1$ are defined by

δ h i = H (- Ξ_{d}^{i} : δ d i) (- δ d i) = H (- Ξ_{d}^{i} : δ d i) (D i - D i - 1) \cdot (D 0) - 1

(45)

δ d_{i}^{+} = H (Ξ_{d}^{i} : δ d i) (δ d i) = H (Ξ_{d}^{i} : δ d i) (D i - 1 - D i) \cdot (D 0) - 1

(46)

where

H (*)

is the Heaviside unit function, depending on the value of the argument *.

The incremental net damage tensor rendered by equation (20) representing the net effect of both damage and healing can be rephrased in term of $δ d +$ and $δ h$ by

δ d = δ d + - δ h

(47)

equation (34) − 1 for

\tilde{Φ}

is then replaced by

\tilde{Φ} = \tilde{Φ} p + \tilde{Φ} d = \tilde{Φ} p + \tilde{Φ} d + - \tilde{Φ} h \geq 0

(48)

in which the density of incremental dissipation

\tilde{Φ} d + (d, δ d)

due to the elastic damage and the density of incremental healing energy

\tilde{Φ} h (h, δ h)

are defined as

\tilde{Φ} d + = H (Ξ d : δ d) Ξ d : δ d + \geq 0

(49)

\tilde{Φ} h = H (Ξ d : δ d) Ξ d : δ h \geq 0

(50)

As observed by equation (48), $\tilde{Φ}$ is additively decomposed into the three components: the plastic dissipation $\tilde{Φ} p (E p, δ E p)$ , the elastic damage dissipation $\tilde{Φ} d + (d, δ d +)$ and the healing energy density $\tilde{Φ} h (d, δ h)$ .

The densities of accumulated thermodynamic dissipations and the density of accumulated healing energy at time t_n for a meso-structured material point can be expressed as

Φ = Φ p + Φ d = Φ p + Φ d + - Φ h = Φ pd + - Φ h \geq 0

(51)

Φ p = \sum_{i = 1}^{n} {\tilde{Φ}}_{p}^{i} \geq 0

(52)

Φ d + = \sum_{i = 1}^{n} {\tilde{Φ}}_{d +}^{i} \geq 0

(53)

Φ h = \sum_{i = 1}^{n} {\tilde{Φ}}_{h}^{i} \geq 0

(54)

Φ d = Φ d + - Φ h = \sum_{i = 1}^{n} {\tilde{Φ}}_{d +}^{i} - \sum_{i = 1}^{n} {\tilde{Φ}}_{h}^{i}

(55)

Φ pd + = Φ p + Φ d + \geq 0

(56)

Net (effective) damage variables, damage, and healing variables

The anisotropic nature of irregular meso-structure of granular materials leads to a tensorial description of the damage and healing variables. The accumulated damage and healing tensors $d_{n}^{+}, h n$ at time t_n for a material point can be defined, with their increments given by equations (45) and (46), as

d_{n}^{+} = \sum_{i = 1}^{n} H ({\tilde{Φ}}_{d +}^{i}) δ d_{i}^{+} = \sum_{i = 1}^{n} H ({\tilde{Φ}}_{d +}^{i}) (- δ D i \cdot D_{0}^{- 1}) = \sum_{i = 1}^{n} H ({\tilde{Φ}}_{d}^{i}) (D i - 1 - D i) \cdot D_{0}^{- 1}

(57)

h n = \sum_{i = 1}^{n} H ({\tilde{Φ}}_{h}^{i}) δ h i = \sum_{i = 1}^{n} H (- {\tilde{Φ}}_{d}^{i}) (- δ d i) = \sum_{i = 1}^{n} H (- {\tilde{Φ}}_{d}^{i}) (D i - D i - 1) \cdot D_{0}^{- 1}

(58)

It is noted that $d_{n}^{+}, h n$ defined by equations (57) and (58) refer to the initial (undamaged) elastic moduli tensor $D 0$ . Then the anisotropic net damage tensor d to include the net effect of both damage and healing at time t_n for a material point can be defined by

d = d + - h = \sum_{i = 1}^{n} H ({\tilde{Φ}}_{d}^{i}) (D i - 1 - D i) \cdot D_{0}^{- 1} - \sum_{i = 1}^{n} H (- {\tilde{Φ}}_{d}^{i}) (D i - D i - 1) \cdot D_{0}^{- 1}

(59)

where for clarity in expression, subscripts n to express the damage and healing tensors in equations (57) and (58) at time t_n are omitted. The physical significance of the net damage tensor d proposed in equation (59) will be demonstrated later with its scalar form.

To interpret the differences of the proposed definitions of the net damage tensor, the damage and the healing tensors given by equations (57) to (59) with existing definitions of those variables in the literature, the following discussions will be focused on the isotropic scalar counterparts of the proposed definitions.

For isotropic damage-healing evolution, equations (42) to (44) to distinguish material damage and healing for a time interval $t \in [t i - Δ t, t i]$ can be simplified to the forms below:

δ d i > 0 damage, δ d i < 0 healing, δ d i = 0 neitherdamagenorhealing

(60)

The incremental damage and healing factors $δ d_{i}^{+}$ and $δ h i$ for the time interval $Δ t = t i - t i - 1$ are defined by

δ d_{i}^{+} = H (δ d i) (δ d i) = H (D i - 1 - D i) (D i - 1 - D i) / D 0

(61)

δ h i = H (- δ d i) (- δ d i) = H (D i - D i - 1) (D i - D i - 1) / D 0

(62)

i.e.

δ d_{i}^{+} > 0

, if

D i - 1 > D i

; otherwise,

δ d_{i}^{+} = 0

; and

δ h i > 0

, if

D i > D i - 1

; otherwise,

δ h i = 0

The accumulated damage and healing factors $d_{n}^{+}, h n$ at time t_n for a meso-structured material point can be calculated by

d_{n}^{+} = \sum_{i = 1}^{n} δ d_{i}^{+} = \sum_{i = 1}^{n} H (D i - 1 - D i) (D i - 1 - D i) / D 0

(63)

h n = \sum_{i = 1}^{n} δ h i = \sum_{i = 1}^{n} H (D i - D i - 1) (D i - D i - 1) / D 0

(64)

Using equations (63) and (64) the scalar form of equation (59) to describe the net effect of isotropic damage and healing can be expressed by

d = d + - h = \sum_{i = 1}^{n} δ d_{i}^{+} - \sum_{i = 1}^{n} δ h i

(65)

where for clarity in expression, subscripts n in equations (63) and (64) is omitted.

Based on the definitions of the net damage tensor d and its increment $δ d$ given by equations (19) and (20), the net damage factor d calculated by using equations (63) to (65) at the time instant t_n can be also expressed by

d = d + - h = \frac{D 0 - D 1}{D 0} + \dots \dots + \frac{D i - 1 - D i}{D 0} + \frac{D i - D i + 1}{D 0} + \dots . . \frac{D n - 1 - D n}{D 0} = \frac{D 0 - D n}{D 0}

(66)

where

D 0, D 1 \dots D i - 1, D i, D i + 1, \dots D n - 1, D n

are the scalar elastic moduli at a series of successive time instants in the simulation. It is ensured that the net damage factor given by equation (65) is bounded by a maximum value of one since the inequality

D 0 - D n \leq D 0

always holds.

However, the resulting net damage factor d shown in equation (66) may be negative; i.e. it is not bounded by a minimum value of zero. The negative value of d implies that the current elastic modulus D_n is greater than the initial elastic modulus D₀ at a local material point. This situation may arise and has to be taken into account in the development of continuum damage-healing mechanics, particularly for intrinsically or deformation-induced heterogeneous and discontinuous granular materials such as geomaterials in the nature. For those materials initial defects at local material points may generally exist and healing may tend to occur in the sense of enhancing initial elastic stiffness under certain conditions such as compression (compaction) even no material damage occurs in the simulation of the damage-healing evolution.

The definition of the net damage variable proposed in this paper is based on the fundamental concept of continuum damage mechanics, which is mathematically expressed by equation (19) or equation (59) in the tensorial form for anisotropic continuum and equation (66) in the scalar form for isotropic continuum. Physically the net damage variable describes a reduction or an increase in elastic stiffness of material in comparison with initial elastic stiffness of material at a meso-structured material point. The physical significance of the net damage tensor d proposed in this paper is clearly demonstrated by its scalar form equation (66). It reflects the integrated effect of material damage and healing, particularly the reduction (if $D 0 > D n$ , $0 < d \leq 1$ ) or the strengthening (if $D 0 < D n$ , $d < 0$ ) of current elastic moduli $D n$ relative to the initial elastic moduli $D 0$ .

With the concept of reduced area, healing or damage will cause an increase of the healed area or the damaged area of cross-section of material, respectively, the definition of the net damage factor expressed by equation (66) can be rephrased as

d = \frac{A 0 - A n}{A 0} = \frac{A d - A h}{A 0} = d + - h

(67)

d + = \frac{A d}{A 0}, A d = \sum_{i = 1}^{n} H (A i - 1 - A i) (A i - 1 - A i)

(68)

h = \frac{A h}{A 0}, A h = \sum_{i = 1}^{n} H (A i - A i - 1) (A i - A i - 1)

(69)

where A₀ is the cross-sectional area in the nominal (initial) configuration, A_d is the accumulated damaged area, A_h is the accumulated healed area, and

A i - 1

and A_i are cross-sectional areas of configurations at the two successive time instants

t i - 1

and t_i, respectively.

It is rendered by equations (66) and (67) that though the net damage factor d combining both damage and healing effects is bounded by a maximum value of one, the accumulated damage factor $d +$ and healing factor h proposed in this work are not bounded by maximum value of one.

Namely, one has the bounds $0 \leq d +$ , $0 \leq h$ , and $d \leq 1$ . Though d has the upper bound of one (unity), there are no determinate upper bounds for h and d⁺. On the other hand, there is no determinate lower bound for d (d may be negative). The negative value of the proposed net damage factor is meaningful, and implies the enhancement of initial elastic stiffness. Particularly in a cyclic damage-healing evolution, a damaged area may be healed and then damaged again and this repeated damage-healing evolution at a local material point make the physical area damaged and healed for a number of times. This will also be numerically demonstrated in the next section of this paper.

The definition of the scalar net damage variable d introduced in the existing literature of continuum damage-healing mechanics was given by (Darabi et al., 2012; Ju et al., 2012; Ju and Yuan, 2012; Voyiadjis and Kattan, 2014)

\tilde{d} = \tilde{d} + (1 - R)

(70)

where

\tilde{d} + = \frac{A d}{A 0}, R = \frac{A h}{A d}

(71)

The healing factor R defined in the literature is referred to the total (accumulated) damaged area A_d, instead of the cross-sectional area A₀ in the initial (nominal) configuration. In comparison of equation (69) − 1 with equation (71) − 2, it is realized that the healing factor h defined in this paper represents the healed fraction referred to the total initial area, while the healing factor R defined in the existing literature represents the healed fraction referred to the accumulated damaged area A_d. The limitation of the definition of the net damage variable in the literature (as given by equation (70) in the present paper) in its applications to the damage-healing process is that $A d \neq 0$ (i.e. $\tilde{d} + > 0$ ) is required; i.e. the healing can only be applied to heal the damaged area, and is incapable of healing initially weakened area. It is also exhibited by equations (70) and (71) that if there is no material damage (i.e. $\tilde{d} + = 0$ or $A d = 0$ ), no healing effect on strengthening the weakened initial area can be represented.

One may substitute equation (71) into equation (70) and re-express equation (70) as below

\tilde{d} = \tilde{d} + (1 - R) = \tilde{d} + - \tilde{d} + R = \frac{A d}{A 0} - \frac{A d}{A 0} \cdot \frac{A h}{A d} = \frac{A d}{A 0} - \frac{A h}{A 0}

(72)

in which the two terms after the last equal mark are as same as those defined in equation (67). However, the precondition of existence of the last equal mark in equation (72) is that

A d \neq 0

To make the healing factor defined in continuum damage-healing mechanics applicable not only to the damaged area but also to the initially weakened area, the definition of the healing factor h proposed in this paper is referred to the initial (nominal) cross-sectional area A₀, instead of the accumulative damaged area A_d. Actually the net damage factor defined by equation (65) can be re-written as

d = d + (1 - h) - h (1 - d +)

(73)

in which the first term at the right hand side represents the healing effect on the reduction of the damaged area and the second term at the right hand side represents the healing effect on the effective (residual) cross-sectional area

\bar{A}

(

\bar{A} = A 0 - A d

As compared with other net damage variable available in the literature, the distinct difference of the proposed net damage variable lies in the definition of the associated healing variable defined in this paper. The healing variable defined in this paper refers to the initial (undamaged) area and represents the healed fraction relative to the total initial area A₀, while the healing variables defined in the literature referred to the total damaged area and represent the healed fraction referred to the accumulated damaged area A_d. Even if no damage occurs (i.e. $d + = 0$ or $A d = 0$ ; $\bar{A} = A 0$ ), a negative value of the net damage variable $d = - h$ defined in this paper is meaningful and implies to enhance the initial elastic stiffness at a local material point.

Then the net damage variable proposed in the present paper may describe not only the healing of the damaged area (when $d + > h$ , $D 0 > D n$ ), but also the strengthening of the initial defect of material (when $d + < h$ , $D 0 < D n$ ).

It is noted that damage and healing evolutions may occur simultaneously within one time step of the simulation in different principal directions of anisotropic damage tensor for one material point of anisotropic continuum. As the two dimensional Cosserat continuum is concerned, the six principal net damage values of the anisotropic $6 \times 6$ incremental net damage tensor $δ d$ and the principal values of the conjugated thermodynamic generalized force tensor $Ξ d$ may be denoted with $δ \bar{d} i (i = 1 ~ 6)$ and $\bar{Ξ} i (i = 1 ~ 6)$ , respectively.

The material damage and healing evolutions are defined with the principal values $δ \bar{d} i$ of incremental net damage factor tensor and associated $\bar{Ξ} i$ as below:

material damage in the $ith$ principal direction:

\bar{Ξ} i = - \frac{\partial ϕ}{\partial \bar{d} i} \geq 0, δ \bar{d} i \geq 0, \tilde{Φ} di = \bar{Ξ} i δ \bar{d} i \geq 0 (nosummationofindex i)

(74)

material healing in the

ith

principal direction:

\bar{Ξ} i = - \frac{\partial ϕ}{\partial \bar{d} i} \geq 0, δ \bar{d} i \leq 0, \tilde{Φ} di = \bar{Ξ} i δ \bar{d} i \leq 0 (nosummationofindex i)

(75)

Therefore, the incremental thermodynamic energy density related to damage-healing evolution can be evaluated by

\tilde{Φ} d = \sum_{i = 1}^{6} \tilde{Φ} di

(76)

Nevertheless, principal directions of the anisotropic net damage tensor usually vary at a meso-structured material point of granular materials from one to its successive time instant. The study of the characterization for this complex meso-mechanically based damage-healing process in the anisotropic Cosserat continuum is not within the scope of the present paper and will be carried out in our forthcoming work.

Numerical results

The definitions of net damage, damage, and healing variables proposed in the present work are conceptually reviewed and validated in terms of their scalar forms by using a numerical example problem. The example considers a hypothetical system with a number of parallel springs as rendered in Figure 1. Firstly, the system with four parallel springs as shown in Figure 1(a) presented by Ju et al. (2012) and Ju and Yuan (2012) is used to test the definitions and to compare the results obtained by the proposed definitions with the definitions presented by Ju et al. (2012). The test is carried out with a series of conceptual cyclic tension-compression loadings on the system. The four successive loading steps termed as T1–C1–T2–C2, in which the capital letters T and C denote tension and compression, respectively, are rendered in Figure 2. It illustrates the damage-healing evolutions in the system of the four springs, in which a spring with the cross mark stands for a broken spring while a spring with the circle mark stands for a healed broken spring. Table 1 lists the results of incremental and accumulated damage and healing factors and the net damage factor at the end of each loading step obtained by the proposed definitions and those presented by Ju et al. (2012). It is observed that both of them give the same results for both (incremental and accumulated) the net damage factor and the damage factor, but different results for the (incremental and accumulated) healing factor. This is because the healing factor defined in Ju et al. (2012) represents the healed fraction relative to the accumulative damaged area, whereas the healing factor defined in the present work represents the healed fraction relative to the total initial area.

Table 1.

Comparisons of results of damage, healing, and net damage variables at the end of each loading step for a hypothetical conceptual system with four parallel springs, obtained by Li’s present work and Ju et al. (2012).

Loading order No.	Loading	$Δ d +$ ( $Δ d$ )		$d +$ (d)		$Δ h$ ( $Δ R$ )		h(R)		$d (d net)$
Loading order No.	Loading	Ju	Li	Ju	Li	Ju	Li	Ju	Li	Ju	Li
1	T1	0.5	0.5	0.5	0.5	0.0	0.0	0.0	0.0	0.5	0.5
2	C1	0.0	0.0	0.5	0.5	0.5	0.25	0.5	0.25	0.25	0.25
3	T2	0.5	0.5	1.0	1.0	0.0	0.0	0.5	0.25	0.75	0.75
4	C2	0.0	0.0	1.0	1.0	1.0	0.75	1.5	1.0	0.0	0.0

Subsequently, the example problem of the hypothetical system with a number of parallel springs presented by Ju et al. (2012) is extended to include an additional initially broken spring as shown in Figure 1(b) to conceptually take into account the initial defect of material at a local material point. The example is performed for five successive cyclic compression-tension loading steps termed as C0–T1–C1–T2–C2 on the spring system. Figure 3 illustrates the damage-healing process in the system of the five springs subjected to the C0–T1–C1–T2–C2 loadings. Table 2 lists the results of incremental and accumulated damage and healing factors and the net damage factor at the end of each step of the damage-healing evolutions shown by Figure 3, obtained by the proposed definitions.

Table 2.

The results of damage, healing, and net damage variables at the end of each loading step for a hypothetical conceptual system with five parallel springs including an initially broken spring, obtained by the present work.

Loading order No.	Loading	$Δ d +$	$d +$	$Δ h$	h	d
1	C0	0.0	0.0	0.25	0.25	−0.25
2	T1	0.75	0.75	0.0	0.25	0.5
3	C1	0.0	0.75	0.25	0.5	0.25
4	T2	0.5	1.25	0.0	0.5	0.75
5	C2	0.0	1.25	0.75	1.25	0.0

The results show that the proposed definitions of net damage, damage, and healing variables are capable of taking into account the healing effects not only on the damaged part of elastic stiffness but also on initially weakened elastic stiffness (initial material defects).

The second example concerns a rectangular panel with $25 \times 44.3 cm 2$ . The purpose of the example is to demonstrate the meso-mechanically informed damage-healing and plastic evolutions in a BVP of geo-structures. The panel is subjected to a uniaxial compression between two rigid plates applied by a vertical displacement control, specified by the vertical velocity $61.8 cm / s$ and the time step size $Δ t = 1.618 \times 10 - 6 s$ , as shown in Figure 4(a). The granular assembly is generated by 1249 homogeneous particles with radius of 0.5 cm collocated in a regular manner. The gravitational forces are neglected. The particles on the left and the right boundaries are free in the vertical and horizontal directions. The contact between the particles on the top and the bottom boundaries with the plates is modeled by a vertical sticking. The vertical displacements of the particles on the top and the bottom boundaries are specified as uniformly prescribed values under the displacement control. The horizontal movements of the particles on the top and the bottom boundaries relating to the plates are allowed with the sliding coefficient 0.5 being used. The material parameters of the granular assembly used in the example are listed in Table 3. The numerical example is performed by using the discrete element method (DEM) solver (Li et al., 2005) developed based on the pioneering work of Cundall and Strack (1979) for the DEM. The meso-mechanical behavior is identified with the DEM solutions for the BVP of the panel, in which the discontinuity (loss and re-generation of contacts) and dissipative inter-particle movement are involved, and their evolutions with respect to the loading history are taken into account.

Table 3.

The material parameters of the granular assembly used in the rectangular panel example.

Parameter	Selected value
Particle density (ρ)	2000 (kg/m³)
Stiffness coefficient of normal force (k_n)	$6.0 \times 108 (N / m)$
Stiffness coefficient of sliding force (k_s)	$4.0 \times 108 (N / m)$
Stiffness coefficient of rolling force $(k r)$	$2.0 \times 103 (N / m)$
Stiffness coefficient of rolling moment ( $k θ$ )	$7.0 \times 102 (N / m / rad)$
Sliding friction force coefficient ( $μ s$ )	0.5
Rolling friction force coefficient ( $μ r$ )	0.05
Rolling friction moment coefficient ( $μ θ$ )	0.0001

The load–displacement curve to show the load history applied on the top of the granular assembly with increasing displacements of the top surface of the panel is shown in Figure 4(b). The curve illustrates the evolution of the load-carrying capability of the panel, particularly, the two softening stages of the panel; i.e. the main softening stage bounded from the point A to the point B on the curve and the secondary softening stage bounded from the point C to the point E on the curve.

Figures 5 –7 illustrate distributions of the densities of accumulated thermodynamic energies at the four successive states of the panel marked with the four points A, B, C, and D, respectively, on the load–displacement curve exhibited in Figure 4(b). They are: (1) distributions of the density of accumulated plastic dissipation for Figure 5; (2) distributions of the density of accumulated damage-healing energy for Figure 6; (3) distributions of the density of accumulated damage-healing and plastic dissipation for Figure 7. It is observed from Figures 5 –7 that the great portions of thermodynamic dissipations in the panel occur at the main softening stage first and then at the secondary softening stage. In addition, the dissipations occurring at the main softening stage concentrate to a great extent in the central region of the panel. After the main softening stage marked with the point B on the curve in Figure 4(b), the incremental thermodynamic dissipations will concentrate in the narrow strain localization bands as shown in Figure 8.

The curves in Figure 9 illustrate developments of densities of accumulated damage and plastic dissipations and the density of accumulated non-dissipative healing energy with respect to time for the three local meso-structured material points marked with the red, the blue and the green colors, respectively, shown in Figure 4(a). The six curves shown in Figure 9(a), (b), (c) represent evolutions of $Φ pd +, Φ, Φ p, Φ d +, Φ d, Φ h$ calculated according to equations (51) to (56). It is also illustrated by Figure 9 that the thermodynamic dissipations due to coupled damage-healing and plastic evolutions mostly occur in the main softening stage of the panel bounded by the two points A and B on the curve shown in Figure 4(b), and then more energy dissipations occur around the secondary softening stage of the panel bounded from the point C to the point E on the curve in Figure 4(b). It is observed that the plastic dissipation first develop, that is obviously observed for the meso-structured material point marked with the red color in the panel, followed by the developments of the damage dissipation. The observed phenomenon indicates that as the irreversible material failure at local meso-structured material points is concerned mesoscopic damage evolution characterizing loss of contacts, volumetric dilatation in meso-scale is often triggered after mesoscopic plastic evolution characterizing the dissipative relative sliding and rolling movement of the reference particle with each of its immediate neighbors. The effects of the healing variable on healing the mesoscopic damage and alleviating the thermodynamic dissipations are also illustrated in Figure 9.

Discussions and concluding remarks

Along the meso-mechanically informed approach with the use of the constitutive relation derived from the proposed meso-structured Voronoi cell model, macroscopic internal state variables and thermodynamic dissipations to characterize the coupled damage-healing and plastic process in anisotropic Cosserat continuum are defined in the frame of thermodynamics of granular materials.

The characteristics of the proposed meso-mechanically informed approach for the characterization distinguishing from most of existing approaches, particularly the phenomenologically based approaches in the literature lie in the following aspects:

There is no need to specify macroscopic phenomenological constitutive models. Neither damage-healing and plastic criteria nor their evolution laws are required.

The net damage variable combining effects of both material damage and healing is defined, prior to the definitions of damage and healing variables, according to the fundamental concept of continuum damage-healing mechanics, instead of with an intuitive manner.

The proposed healing variable is defined to be capable of healing both the damaged area and initial defects of material, the latter of which is often indeterminate and cannot be provided in advance in the simulation. While existing healing variables proposed in the literature were defined to only heal the damaged area, i.e. if there is no material damage ( $\tilde{d} + = 0$ ), no healing effect on strengthening the weakened initial area can be represented.

The proposed net damage factor combining the effects of both damage and healing is bounded by a maximum value of one, but no longer bounded by a minimum value of zero. The negative value of the proposed net damage factor is meaningful and implies the enhancement of initial elastic stiffness. The absolute value of the negative net damage factor represents the fraction area of healed initial defect referred to the total initial area.

The meso-mechanically informed coupled damage-healing and plasticity in macroscopic anisotropic Cosserat continuum for granular materials is characterized in terms of the thermodynamic dissipations in view of anisotropy of damage-healing variables of granular materials. In addition, the characterization using the thermodynamic dissipations makes the effects of meso-mechanically informed damage-healing and plastic component processes on the material failure quantitatively comparable.

Footnotes

Acknowledgments

The authors are pleased to acknowledge the support of this work by the National Natural Science Foundation of China through contract/grant number 11372066, 11072046 and the National Key Basic Research and Development Program (973 Program) through contract number 2010CB731502.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China through contract/grant number 11372066, 11072046 and the National Key Basic Research and Development Program (973 Program) through contract number 2010CB731502.

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