## Abstract

We present a constitutive model for particle-binder composites that accounts for finite-deformation kinematics, nonlinear elastoplasticity without apparent yield, cyclic hysteresis, and progressive stress-softening before the attainment of stable cyclic response. The model is based on deformation mechanisms experimentally observed during quasi-static monotonic and cyclic compression of mock plastic-bonded explosives (PBX) at large strain. An additive decomposition of strain energy into elastic and inelastic parts is assumed, where the elastic response is modeled using Ogden hyperelasticity while the inelastic response is described using yield-surface-free endochronic plasticity based on the concepts of internal variables and of evolution or rate equations. Stress-softening is modeled using two approaches; a discontinuous isotropic damage model to appropriately describe the softening in the overall loading–unloading response, and a material scale function to describe the progressive cyclic softening until cyclic stabilization. A nonlinear multivariate optimization procedure is developed to estimate the elastoplastic model parameters from nominal stress–strain experimental compression data. Finally, a correlation between model parameters and the unique stress–strain response of mock PBX specimens with differing concentrations of aluminum is identified, thus establishing a relationship between model parameters and material composition.

## 1 Introduction

Particle-binder composite materials consist of a large concentration of hard particles, called fillers, randomly dispersed in the matrix of a soft material. Generally, fillers are used to enhance the mechanical properties of the soft material. For example, filled elastomers such as carbon-black and silica filled rubbers [1,2] have been shown to possess superior stiffness, strength, and damping properties over natural rubber, making them suitable for application in automotive parts such as tires and bearing seals and in structures providing vibration and shock isolation to mechanical systems.

Another class of particle-binder composites, called energetic composite materials or plastic-bonded explosives (PBX), consists of explosive crystals and, in some formulations, metal fuel powder, embedded in a binder composed mainly of a soft polymer and a plasticizer. Since their initial development at Los Alamos National Laboratory in 1947, PBXs have been commonly used as ammunition in defense applications. Examples of explosives used include cyclotrimethylenetrinitramine (RDX) and cyclotetra-minetetranitramine (HMX), while the binder composition includes polymers like hydroxyl-terminated polybutadiene (HTPB) and Estane, and plasticizers like dioctyl adipate (DOA) and isodecyl pelargonate (IDP), along with smaller concentrations of various additives [3]. In a PBX binder, the polymer, apart from providing structural integrity to the explosive, reduces their impact and vibrational sensitivity [46], while the plasticizer improves their mechanical properties and processability [7]. The metal fuel, usually aluminum, is used to enhance the blast effects. Therefore, aluminized PBXs are typically used in naval weapons and missile warheads [8].

Energetic composites are commonly used in applications requiring high mechanical confinement or compression [9,10]. Additionally, these materials may be subjected to diverse loading conditions, ranging from low to high strain-rate cyclic, vibrational and impact loading during transport, storage and handling over their operational life. Since PBXs are designed to detonate under specific external energy stimulus, such loading conditions may alter their mechanical response, rendering them unpredictable and unsafe [11]. Therefore, understanding and predicting the mechanical response of PBX under different loading conditions is of particular interest to the defense and propulsion communities.

Earlier experimental studies on the mechanical behavior of PBX under uniaxial load have shown a dependence on strain rate and temperature [1214]. Several constitutive models in the context of small and large deformation mechanics have been developed to model their rate and temperature-dependent behavior. For instance, Bardenhagen et al. [15] proposed a large deformation viscoelastic model for PBX binder materials using a Blatz-Ko hyperelasticity formulation and a Maxwell element. Clements and Mas [16] proposed a viscoelastic constitutive theory for filled polymers using the Boltzmann superposition principle, where the filler particles were modeled as randomly positioned elastic ellipsoidal particles. Composite stress relaxation functions were developed with their prony series coefficients dependent on filler concentration, ellipsoidal aspect ratio, and polymer moduli. The viscoelastic cracking constitutive model developed by Bennet et al. [17] combined a linear viscoelastic Maxwell’s model with the isotropic damage model by Addessio and Johnson [18]. It was based on the mechanism of microcracking and derived from the statistical crack mechanics approach proposed by Dienes [19] for brittle materials. The model was developed to predict the nonlinear stressstrain response of the material, as well as the strain-softening and nonlinearity observed due to extensive cracking at larger deformations. The model has since been extensively implemented in finite element analyses and used to predict the thermo-mechanical behavior of PBX and study hotspot generation [6,20]. In the context of PBX undergoing low-frequency vibrational loading, nonlinear viscoelastic models for a mass-material system undergoing base excitation were proposed by Paripovic and Davies [21,22].

While the majority of constitutive formulations proposed for the PBX model the material as viscoelastic due to the rubbery nature of the polymer binder, it has been shown that these materials may exhibit considerable plastic deformation with increasing confinement. Uniaxial confined compression tests carried out on inert mock sugar-based specimens for PBX 9501 [9,10] under different confining pressures revealed increasing plastic deformation and strain hardening with increasing pressure. Recently, Agarwal and Gonzalez [23] conducted unconfined compression tests at room temperature and low strain rate (10−3/s) on cylindrical specimens of three mock explosive formulations (Fig. 1) based on PBXN-109 [24], with sugar as a substitute for the explosive HMX crystals. The three formulations contained 85% w/w of solids (sugar and aluminum) but differed from each other by the amount of aluminum content in the solids composition, namely, 85–00 (0% Al in solids, 0% in total), 85–15 (15% Al in solids, 12.75% in total) and 85–30 (30% additive in solids, 25.5% in total). The monotonic nominal stressstrain response of the 85–00 specimen (Fig. 2(a)) exhibited a quasi-brittle behavior [25], with an initial nonlinear increase in stress until a peak stress level at around 10–11% strain, followed by strain-softening due to the evolution of microcracks into larger transgranular fracture leading to a loss of strength. Such behavior has been recorded and studied extensively for many non-aluminized PBX formulations [6,12,26]. However, the aluminized specimens, i.e., 85–15 (Fig. 2(b)) and 85–30 (Fig. 2(c)) exhibited a more ductile plastic flow behavior with strain hardening, indicating that the presence of aluminum could cause the material to deform plastically even in the absence of confinement.

Additionally, plastic deformation and damage in PBX have been shown to be associated with hotspot mechanisms [27,28], thus directly impacting their ignition characteristics. Therefore, the development of a constitutive model capable of characterizing the plastic behavior of explosives is extremely important for proper modeling and prediction of their mechanical behavior. Constitutive models for PBX incorporating plasticity have been proposed for cyclic deformation [29] and mild impact [30]. However, the models have been developed using small deformation theory and, therefore, are inapplicable to the large deformation mechanical behavior of explosives.

It is apparent that the observed mechanical response of mock PBX specimens exhibits similar characteristics to filled elastomers at large deformations. Constitutive models of filled elastomers have been proposed by adopting both phenomenological [3237] and micromechanical [3840] frameworks. For instance, Ayoub et al. [33] proposed a Zener-type visco-hyperelastic constitutive model of rubber-like materials that accounts for the Mullins effect [41,42], continuous stress-softening, and permanent residual strains by utilizing the network alteration theory [43,44]. Raghunath et al. [39] extended the micromechanical dynamic flocculation model developed by Klüppel [45] to include time-dependent effects typical of filled elastomers. It is worth mentioning that the majority of these models attribute the observed hysteresis and other inelastic phenomena to time-dependent viscous overstress in the rubber matrix. In contrast, only a limited number of models consider these phenomena as time-independent plastic deformation mechanisms. A case in point is the phenomenological elastoplastic constitutive model proposed by Kaliske and Rothert [46] that captures the rate-independent process of internal material friction due to irreversible polymer slippage on the filler surface and plastic deformation of the filler particles. This multi-yield-surface rheological model is comprised of a finite number of elastoplastic Prandtl elements arranged in parallel with an elastic member. More recently, yield-surface-free endochronic plasticity [47] was utilized by Netzker et al. [48] to model the smooth and hysteretic cyclic stress–strain response of carbon-black filled rubbers, with higher computational efficiency and fewer parameters.

In this paper, we present an elastoplastic constitutive formulation with isotropic damage capable of modeling the cyclic response of mock energetic composite materials. Specifically, a Lagrangian finite-deformation formulation based on the additive decomposition of strain energy [4953] into elastic and plastic parts is adopted. The formulation uses Ogden’s hyperelastic model [54,55] to predict the rubber-like nonlinear elastic response of the polymeric binder and a hereditary (path-dependent) yield-surface-free endochronic plasticity theory [47], based on the concept of internal state variables [56], to account for irreversible deformations. A discontinuous isotropic damage model [57,58] is utilized to model the stress-softening that occurs during unloading, and an endochronic material scale function is utilized to model progressive cyclic stress-softening and attainment of stable cyclic response. It is worth mentioning that while the material scale function has been utilized in previous works to describe the cyclic hardening or softening behavior of metals and alloys [47,5961], its application to filled polymeric materials with complex microstructure like particle-binder composites has not been demonstrated in the open literature, and thus, it is one of the primary contributions of this paper. The number of model parameters is a function of the number of active Ogden terms and endochronic branches; therefore, a significant number of parameters may need to be identified. Accordingly, we develop a parameter identification method based on a nonlinear multivariate least-squares problem, which is expected to be non-convex and affected by multiple local optima. The range of behavior predicted by the proposed model is demonstrated by calibrating the model parameters for the three mock PBX material compositions [23] under both monotonic and cyclic compressive loading. Finally, the versatility and the capability of the model to represent the material, specifically, the material composition, is demonstrated by identifying a correlation between the calibrated model parameters and the unique stress–strain response of each mock PBX specimen.

The paper is organized as follows. The constitutive model is presented in Sec. 2, followed by Sec. 3 which provides a discrete numerical procedure to solve for stresses along a loading path. Section 4 presents the model parameter identification procedure and the calibrated material properties of the three mock PBX formulations. Section 5 shows a detailed comparative analysis of the three specimen formulations and presents the correlation identified between the estimated mechanical properties and the mechanical behavior, as well as the composition of the specimens. Finally, a summary and concluding remarks are presented in Sec. 6.

## 2 Constitutive Model

### 2.1 General Framework.

The finite-deformation constitutive law is based on an additive decomposition of the Helmholtz free energy density function [4953] into elastic and plastic parts, and it employs a local multiplicative decomposition of the deformation gradient into isochoric and volumetric contributions. This approach is in contrast to the constitutive formulations based on the multiplicative elastic–plastic decomposition of the deformation gradient [6265] or additive decomposition of the rate of deformation tensor [6668]. For an isothermal elastoplastic deformation process, the free energy density function Ψ is given by
$Ψ(C,ζ1,…,ζN)=Ψvole(J)+Ψisoe(C¯)+∑j=1NΓji(C,ζj)$
(1)
where $Ψvole(J)$ and $Ψisoe(C¯)$ are the volumetric and isochoric elastic strain energy density functions, dependent respectively, on the Jacobian of the deformation $J=det(F)$ and on the isochoric right Cauchy-Green deformation tensor $C¯=F¯TF¯$ (where $F¯=J−1/3F$). The inelastic strain energy $∑j=1NΓji(C,ζj)$ is a set of configurational free energy functions $Γji(j=1,…,N)$ that characterizes the inelastic deformation behavior, namely, irreversible slip at particle–particle and particle–binder contacts, plastic deformation, and fracture of particles, among other dissipative mechanisms typical of particle-binder composites. This path-dependent dissipative potential depends on the right Cauchy-Green deformation tensor $C$ and a set of inelastic strain-like second-order tensorial internal variables $ζj(j=1,…,N)$ that represent the inelastic deformation history of the material.

Discontinuous damage models, specifically, the pseudoelastic material models [75,77,83], are based on the application of suitable damage or softening function φ to the strain energy in a hypothetical, undamaged material. The evolution of this damage function is discontinuous and depends on the deformation history of the material. For a given deformation measure ϕ and its maximum value $ϕmax=maxτ∈[−∞,t]ϕ(τ)$ throughout the material’s deformation history until the current time t, the damage function φ is given by
$φ{=1,ϕ=ϕmax∈(0,1),ϕ<ϕmax$
(2)

Therefore, during the initial loading of the virgin material when the current value of the deformation remains equal to its maximum value throughout the material’s history, the damage function does not evolve, and the material’s strain energy remains unchanged. However, during unloading–reloading from a previous maximum deformation, the current deformation measure is lower than the maximum value, and the damage function evolves monotonically between 0 and 1. This in turn reduces the effective strain energy in the material and produces the softening effect.

With the application of the damage function, the effective strain energy density function $Ψ__$ is given by
$Ψ__(C,ζ1,…,ζN,φ)=φ(ϕ,ϕmax)Ψ(C,ζ1,…,ζN)$
(3)
To explore the thermodynamic consistency of the constitutive model by means of fulfillment of the second law of thermodynamics, we utilize the Clausius–Duhem inequality for an isothermal process, given by
$D:=12S:C˙−Ψ__˙≥0$
(4)
where $D$ is the dissipation and $S$ is the second Piola-Kirchhoff stress tensor. By applying the definition of effective strain energy density function $Ψ__$ according to Eqs. (3) and (1), the inequality becomes
$[S−2φ∂Ψ∂C]:C˙2−Ψφ˙−∑j=1N∂Γji∂ζj:ζ˙j≥0$
(5)
From standard arguments [84,85], the first term yields the definition of the second Piola-Kirchhoff stress, that is
$S=2φ∂Ψ(C,ζ1,…,ζN)∂C=φ[Svole+Sisoe+∑j=1NSji]$
(6)
while the second and third terms yield the dissipation
$D=−Ψφ˙−∑j=1N∂Γji∂ζj:ζ˙j≥0$
(7)
where the volumetric elastic stress $Svole$, the isochoric elastic stress $Sisoe$, the inelastic stress $∑j=1NSji$, the thermodynamic force q (conjugate to damage function φ), and the stress-like internal variables $S~ji$ (thermodynamically conjugate to $ζj$) are given by
$Svole=JdΨvole(J)dJC−1$
(8)
$Sisoe=J−2/3DEV{2∂Ψisoe(C¯)∂C¯}$
(9)
$∑j=1NSji=2∑j=1N∂Γji(C,ζj)∂C$
(10)
$q=∂Ψ__∂φ=Ψ$
(11)
$S~ji=−∂Ψ__∂ζj=−∂Γji(C,ζj)∂ζj(j=1,…,N)$
(12)
In Eq. (9), $DEV{⋅}={⋅}−(1/3)[{⋅}:C]C−1$ provides the deviator of a tensor in the reference configuration. It is worth noting that the second term of the dissipation $D$ from Eq. (7) (now given by $∑j=1NS~ji:ζ˙j$ using Eq. (12)) becomes positive by defining a flow rule of the following form
$ζ˙j=BBj(C,ζj):S~ji$
(13)
where $BBj$ is a positive definite fourth-order tensor [52].

### 2.2 Elastic Strain Energy.

The nonlinear elastic constitutive behavior of rubber-like materials, such as the binder in the application studied here, is commonly described through hyperelastic material models (see, e.g., the finite-deformation formulations presented in Refs. [54,55,8688]). Ogden’s hyperelastic model [54,55] is one of the most extensively used models for describing complex nonlinear responses, mainly due to its flexible series representation with the capability to introduce several model parameters. Specifically, the isochoric elastic strain energy density $Ψisoe$ is given by
$Ψisoe(C¯)=∑k=1Mμkαk(λ¯1αk+λ¯2αk+λ¯3αk−3)$
(14)
where the isochoric principal stretches are given by $λ¯a=J−1/3λa(a=1,2,3)$ for principal stretches λa (a = 1, 2, 3). Constants μk and αk (k = 1, …, M) are material parameters that satisfy the following inequality to enforce stability in the material response
$μkαk>0∀k∈[1,M]$
(15)
From consistency conditions with respect to linear elasticity at small strain [89], the reference (ground-state) shear modulus of the material is given by
$μ=∑k=1Mμkαk2$
(16)

### 2.3 Discontinuous Isotropic Damage.

As seen in Sec. 2.1, a discontinuous isotropic damage model is utilized to model the stress-softening that occurs during unloading, i.e., the Mullins effect [41,42] for polymeric materials, with damage function φ(ϕ, ϕmax). In the spirit of the damage function proposed by Zúñiga and Beatty [77] and pseudoelastic damage functions proposed by Ogden and Roxburgh [75] and Dorfmann and Ogden [76], we specifically define the scalar damage variable as follows:
$φ(ϕ,ϕmax)=tanhp(ϕmax−ϕm)$
(17)
where the deformation measure ϕ is taken as the one proposed by Beatty and coworkers [77,90], given by
$ϕ=|C|=C:C$
(18)
and where ϕmax is the maximum deformation level given by
$ϕmax(t)=maxτ∈[−∞,t]ϕ(|C(τ)|)$
(19)

The damage function is then a monotonic function of the deformation measure in the interval $ϕ∈[3,ϕmax]$, with positive damage parameters m and p.

### 2.4 Yield-Surface-Free Endochronic Plasticity and Evolution Equation.

As explained previously, particle-binder composites, specifically plastic-bonded explosives (PBX), exhibit a nonlinear elastoplastic response without apparent yield. Classical plasticity theories require a yield surface, and thus, their applicability is limited for modeling these composites. In this study, we follow the work of Holzapfel and Simo [51] and assume a quadratic relationship between free energy functions $Γji(C,ζj)$ and the internal variables $ζj$, i.e.,
$∂Γji∂ζjζk=2μjδjkII(jnotsummed)$
(20)
where μj are the reference (ground-state) shear moduli related to j inelastic processes, δmn is the Kronecker delta and $II$ is the fourth-order identity tensor. Therefore, by integrating Eq. (13) twice, the following inelastic strain energy functions are obtained
$Γji(C,ζj)=μj|ζj|2+Sj*(C):ζj+Ψj*(C)(j=1,…,N)$
(21)
where $S*(C)$ and $Ψ*(C)$ are unknown tensor and scalar-valued functions, respectively. Similarly, an evolution or rate equation for the stress-like internal variables is obtained from Eqs. (12), (13), and (21), and it is given by
$S~˙ji+2μjBBj(C,ζj):S~ji=−S˙*(C)(j=1,…,N)$
(22)
The inelastic strain energy without apparent yield is realized by adopting the endochronic plasticity theory developed by Valanis [47]. This theory is strain path-dependent in nature, and it does not require the definition of a yield surface. It has been applied successfully to develop both infinitesimal-strain and finite-deformation plastic formulations to describe many metals and alloys [59,91,92], concrete [93], powder compaction [9496] and, more recently, filled elastomers [37,48]. In accordance with this theory, we specifically assume the fourth-order tensor $BBj$ to be given by
$BBj(C,ζj)=z˙(|C˙|)2μjγjII(j=1,…,N)$
(23)
where the intrinsic time scale z is a monotonically increasing measure of the deformation history, and the memory kernel is given by a set of positive dimensionless material parameters γj (j = 1, …, N) [95,97] that defines the path-dependent behavior of the formulation. It is worth noting that μjγj ≥ 0 for $BBj$ to be a positive definite tensor. In addition, we assume
$Sj*(C)=−Siso,je(C)=−J−2/3DEV{2∂Ψiso,je(C¯)∂C¯}(j=1,…,N)$
(24)
where $Ψiso,je$ are a set of isochoric elastic strain energy functions defined according to Ogden’s model as
$Ψiso,je=∑i=1Pμijαij(λ¯1αij+λ¯2αij+λ¯3αij−3)(j=1,…,N)$
(25)
with constants μij and αij such that μijαij > 0 (Eq. (15)). It is worth noting that after substituting $BBj$ and $Sj*$ according to Eqs. (23) and (24), respectively, the evolution or rate Eq. (22) simplifies to
$S~˙ji+z˙(|C¯˙|)γjS~ji=S˙iso,je(j=1,…,N)$
(26)
which can be regarded as the nonlinear extension of the one-dimensional isothermal evolution equation for a standard linear solid proposed by Valanis [98] (see Sec. 6 of the reference). Furthermore, by eliminating time dependence on both sides of the differential Eq. (26) and integrating with respect to the intrinsic time scale z, the classical hereditary or convolution form is recovered, that is
$S~ji=S~j,0i+∫0zS˙iso,jee−z−z′γjdz′(j=1,…,N)$
(27)
with $S~ji(z=0)=S~j,0i$. Finally, for completeness, we set $Ψj*(C)=Ψiso,je(C¯)$ and, therefore, the reference shear moduli μj in Eq. (21) are given by
$μj=∑i=1Pμijαij2>0(j=1,…,N)$
(28)

### 2.5 Intrinsic Time Scale and Material Scale Function.

The intrinsic time scale z, used in the evolution or rate Eq. (26) for internal variables $S~ji$, is defined as
$z˙(|C˙|)=|C˙|f(z)$
(29)
where f(z) is a material scale function that represents the transient cyclic behavior of the material until attainment of a stable response. In its simplest form, f(z) is a monotonically increasing function for cyclic hardening materials, and a monotonically decreasing function for cyclic softening materials, that is asymptotic to a saturation value at steady-state. A simple scale function proposed by Wu and Yip [59] is given by
$f(z)=c−(c−1)e−βcz$
(30)
where βc controls the rate of evolution, and the value at steady-state is c > 1, for hardening materials, and c < 1, for softening materials. Yeh [60] and Lin et al. [61] proposed the following improvements to the scale function
$f(z)=cs−(s−1)e−βszref−(cs−(s−1)e−βszref−1)e−βc(z−zref)$
(31)

In the above improved scale function, the additional parameter zref, defined as the value of z at which the last reversal of the strain path occurred, adds the capability to model “fading memory” effects exhibited by materials with memory [99]. In turn, the saturation value at steady-state depends on the reference time scale zref. It is equal to c for zref = 0, i.e., before the first reversal, and approaches c/s for zref → ∞, i.e., at cyclic stabilization. It is worth noting that for c > s > 1, the initial saturation value c is greater than the saturation value c/s at cyclic stabilization, i.e., the scale function describes the progressive cyclic softening typically observed in particle-binder composites. Figure 4 shows a schematic representation of the scale function proposed by Lin et al. [61]. It is evident from the figure that the rate of evolution of the material scale function is controlled by βc, whereas the rate of evolution of the saturation value is controlled by βs.

### 2.6 Inelastic Stress.

The inelastic stress contributions $Sji(j=1,…,N)$ are determined by using Eq. (10) with the inelastic strain energy function defined by Eq. (21) and by expressing the internal variables $ζj$ in terms of their conjugates $S~ji$, i.e., using Eq. (12). The result of this mathematical manipulation is given by
$Sji=Siso,je−1μj∂Siso,je∂C:(Siso,je−S~ji)$
(32)
where the internal variables $S~ji$ are determined using the evolution or rate Eq. (26).

## 3 Incremental Solution Procedure

The constitutive model presented in the previous section is integrated by using an incremental solution procedure with time or loading intervals [n, n + 1]. We assume that the state of the material, $Cn$, $S~j,ni$, zn, zref,n, $ϕnmax$, is known at loading step n and that $Cn+1$ at loading step n + 1 is given. The problem is then to determine $S~j,n+1i$, zn+1, zref,n+1, and $ϕn+1max$ at loading step n + 1, and the value of the second Piola-Kirchhoff stress $Sn+1$, which is given by
$Sn+1=φn+1[Svol,n+1e+Siso,n+1e+∑j=1NSiso,j,n+1e−∑j=1N12μjC¯j,n+1:(Siso,j,n+1e−S~j,n+1i)]$
(33)
where
$C¯j,n+1=2[∂Siso,je∂C]n+1$
(34)
and
$S~j,n+1i=(1−Δz/2γj)S~j,ni+(Siso,j,n+1e−Siso,j,ne)(1+Δz/2γj)$
(35)
In the previous equations, all quantities other than φn+1 and Δz are either known or depend on the right Cauchy-Green deformation tensor through Eqs. (8), (9), and (24). The scalar damage function φn+1 is computed after updating $ϕn+1max$ using Eq. (19). The stress-like internal variables $S~j,n+1i$ defined by Eq. (35) above are obtained by using a midpoint rule to discretize the evolution or rate Eq. (26), i.e., from
$S~j,n+1i−S~j,ni+(Δzγj)(S~j,ni+S~j,n+1i−S~j,ni2)=Siso,j,n+1e−Siso,j,ne$
(36)

The value of intrinsic time scale increment, Δz, is obtained by using a midpoint rule to discretize Eq. (29) together with Eq. (31), i.e., from

$Δz[cs−(s−1)e−βszref,n+1−(cs−(s−1)e−βszref,n+1−1)e−βc(zn+Δz2−zref,n+1)]−|C¯n+1−C¯n|=0$
(37)
with zn+1 = zn + Δz, and
$zref,n+1={znif(ϕn+1−ϕn)⋅(ϕn−ϕn−1)<0zref,notherwise$
(38)

The previous nonlinear equation is solved numerically using a Newton-Raphson method.

## 4 Model Parameter Identification of Mock Plastic-Bonded Explosives

The range of behavior predicted by the proposed constitutive model is demonstrated by calibrating the model parameters for the three mock PBX formulations (85–00, 85–15, and 85–30) under monotonic and cyclic compression [23]. Therefore, Sec. 4.1 presents the incremental procedure for uniaxial loading under unconfined lateral conditions and incompressible material assumptions. Section 4.2 presents a parameter identification procedure based on a nonlinear multivariate minimization problem and the least-squares principle. Finally, Sec. 4.3 shows the results of the parameter identification.

The material is assumed to be incompressible, i.e., Jn = 1, and thus $C¯n=Cn$, due to the rubber-like behavior of the particle-binder composite. The cylindrical mock PBX specimen is loaded along its axial direction which is coincidental with the $e3$ of a Cartesian coordinate system $eI$, with I = 1, 2, 3. Therefore, the right Cauchy-Green deformation tensor has principal stretches λI are related by
$λ1,n=λ2,n=1λ3,n$
(39)
and principal directions aligned with the Cartesian axes, i.e., $Cn=∑I=13λI,n2eI⊗eI$. The specimen is laterally unconfined and the platens that apply the load are frictionless. Therefore, the only component of the second Piola-Kirchhoff stress that is different from zero is S33,n.
The elastic strain energy functions involved in the formulation reduce to
$Ψe(λ1,λ2,λ3)=∑k=1Mμkαk(λ1αk+λ2αk+λ3αk−3)Ψje(λ1,λ2,λ3)=∑i=1Pμijαij(λ1αij+λ2αij+λ3αij−3)(j=1,…,N)Ψvole(J)=−po(J−1)$
(40)
where the last term is used to augment the energy density and enforce the material internal or kinematic constraint imposed by incompressibility, i.e., J = 1, with po being the hydrostatic pressure.

Following the incremental procedure presented in Sec. 3, we assume that the state of the material, $Cn$, $S~j,ni$, zn, zref,n, and $ϕnmax$, is known at loading step n, and that $Cn+1$ at loading step n + 1 is given. The maximum deformation level $ϕn+1max$ is determined from Eqs. (18) and (19) using

$ϕn+1=λ3,n+14+2λ3,n+12$
(41)
The intrinsic time scale increment Δz is obtained using Eq. (37) with
$|C¯n+1−C¯n|=2(λ3,n+1−1−λ3,n−1)2+(λ3,n+12−λ3,n2)2$
(42)
and zref,n+1 is updated using Eq. (38) and zn+1 = zn + Δz. The stress components in Eq. (33) simplify to
$S11,iso,n+1e=S22,iso,n+1e=∑k=1Mμkλ3,n+11−αk/2S11,vol,n+1e=S22,vol,n+1e¯=−po,n+1λ3,n+1S11,iso,j,n+1e=S22,iso,j,n+1e=∑i=1Pμijλ3,n+11−αij/2C¯1111,j,n+1=C¯2222,j,n+1=∑i=1Pμij(αij−2)λ3,n+12−αij/2S33,iso,n+1e=∑k=1Mμkλ3,n+1αk−2S33,vol,n+1e=−po,n+1λ3,n+1−2S33,iso,j,n+1e=∑i=1Pμijλ3,n+1αij−2C¯3333,j,n+1=∑i=1Pμij(αij−2)λ3,n+1αij−4$
(43)
where the hydrostatic pressure po,n+1 is obtained from the traction free boundary condition, i.e., from S11,n+1 = S22,n+1 = 0, and it is given by
$po,n+1=1λ3,n+1[S11,iso,n+1e+∑j=1NS11,iso,j,n+1e−∑j=1N12μjC¯1111,j,n+1:(S11,iso,j,n+1e−S~11,j,n+1i)]$
(44)
and the stress-like internal variables $S~j,n+1i$ are readily computed using Eq. (35).

### 4.2 Parameter Identification Method.

We have developed a parameter identification method based on a nonlinear multivariate minimization problem. The proposed constitutive model has 2M + 2P × N + N + 6 material parameters, namely

1. 2M parameters in the elastic branch: μk and αk (k = 1, …, M).

2. 2P × N elastic parameters in the yield-surface-free endochronic branches: μij and αij (i = 1, …, P; j = 1, …, N).

3. N kernel parameters in the yield-surface-free endochronic branches: γj(j = 1, …, N).

4. Four material scale function parameters: c, s, βc, and βs.

5. Two isotropic damage parameters: m and p.

The method is based on the least-squares principle, for experimental monotonic and cyclic nominal stress values $Tkexp(k=1,…,K)$ and $Tlexp(l=1,…,L)$, respectively, and corresponding model first Piola-Kirchhoff stress values λ3,kS33,k and λ3,lS33,l. It is given by
$minv∈R2M+2P×N+N+6,εo1K[wmono∑k=1K(Tkexp−λ3,kS33,k)2]+1L[wcyclic∑l=1L(Tlexp−λ3,lS33,l)2]subjectto−μkαk≤0fork=1,…,M−μijαij≤0fori=1,…,P;j=1,…,Ns−c≤0v∈[vmin,vmax]$
(45)
where wmono and wcyclic are appropriate weights applied to the monotonic and cyclic least-squares functions, respectively, to balance the accuracy of model predictions for these tests. The inequality constraints come from Ogden’s stability conditions (e.g., from Eq. (19) for the elastic branch) and from the requirement of progressive cyclic softening. Since the nonlinear optimization problem is expected to be non-convex and affected by multiple local optima, appropriate bounds are applied to the set of material parameters $v$. Lastly, the experimental stress–strain response of the mock PBX specimens suggests that the initial loading response is affected by machine/specimen mismatch (Figs. 2 and 3). Therefore, the initial part of the response is omitted from the parameter estimation by applying a strain offset ɛo to the experimental data and setting stresses for the initial mismatch to be zero.

An iterative procedure is followed to determine the number of terms M and P in the Ogden strain energy series functions for the elastic branch and the endochronic branches, respectively, as well as to determine the number of endochronic branches N. The procedure starts from an initial value of M = P = N = 1 and calculates the least-squares error given by Eq. (45). During each iteration, three error values are computed by independently incrementing the values of M, P, and N by 1. The increment corresponding to the minimum of the three errors is adopted, only if the minimum error is reduced by a specified percentage tolerance (taken as 5% in this study) as compared to the minimum error computed in the previous iteration. Otherwise, the values of M, P, and N obtained during the previous iteration are adopted and the procedure is concluded. This procedure aims to improve the estimation of the model without overly increasing its complexity.

### 4.3 Parameter Identification of Mock Plastic-Bonded Explosives.

The constitutive model and parameter identification method presented earlier were used to calibrate the experimental monotonic (Fig. 2) and cyclic (Fig. 3) compressive response of 85–00, 85–15, and 85–30 mock PBX cylindrical specimens. The nonlinear multivariate minimization problem was solved in matlab® [100] using the constrained optimization function fmincon with the default interior-point algorithm. The model-calibrated nominal stress–strain curves are presented in Figs. 5 (monotonic) and 6 (cyclic), and the estimated material parameters for each specimen formulation are listed in Table 1. Comparison of experimental and model curves suggests an excellent agreement between the two for all three formulations. The strain offset was calculated as 2.25% for 85–00, 1.03% for 85–15, and 2.51% for 85–30, and it is denoted in the figures by a dashed line.

For detailed comparative analysis, experimental and model-estimated values of the cumulative energy dissipation (Fig. 7), peak stress (Fig. 8), and apparent modulus (Fig. 9) are compared for cyclic compression of the specimens. The apparent modulus [23,37] is calculated as the slope of the line connecting peak and valley stresses in each cycle, while the cyclic energy dissipation [23] is calculated as the difference between the energy supplied to the material (i.e., the area under the loading path) and the energy recovered after a cycle (i.e., the area under the unloading path). Again, an excellent agreement between the experimental and model values is obtained for the three specimens, demonstrating the capability of the proposed model to describe the strength, stiffness, and softening characteristics of particle-binder composites.

## 5 Comparative Analysis of Mock Plastic-Bonded Explosives Formulations and the Relationship Between Model Parameters and Material Composition

From the observation of calibrated properties of mock PBX specimens in Table 1, it is possible to identify the correlation between the property values and the unique mechanical behavior of each specimen and, therefore, establish the relationship between material properties and material composition. For instance, the 85–00 specimen is accurately modeled by two endochronic branches, while 85–15 and 85–30 specimens require three endochronic branches to fully represent their mechanical response, which signifies the higher material nonlinearity and more extensive inelastic behavior of the aluminized specimens. Specific trends in individual property values are also observed, which are discussed in detail below.

### 5.1 Elastic and Endochronic Branches.

In the context of representing the deformation mechanics of particle-binder composites by a linear rheological model (Fig. 10) with respect to the constitutive formulation presented in Sec. 2, the elastic branch can be seen as a spring of stiffness 2μe acting in parallel with N endochronic branches, each of which consists of a spring of stiffness $2μji$ and a friction element with threshold strain γj, where j = 1, …, N. The total stiffness μ of the material is represented by a sum of the elastic and endochronic moduli, i.e., $μ=μe+∑j=1Nμji$, while the threshold strains represent the macroscopic strains at the activation of inelastic processes, such as irreversible slipping of binder molecular chains on the filler (solid) surface and plastic deformation of the solid particles [46,48].

From the calibrated parameter values listed in Table 1, the total stiffness of 85–00, 85–15, and 85–30 specimens is found to be 5.1941 MPa, 12.3132 MPa, and 8.2868 MPa, respectively. This suggests that the addition of aluminum in the PBX composition increases material stiffness; however, the stiffness starts to reduce with an increase in the amount of aluminum in the composition. This postulation is also supported by the evolution of the apparent modulus during the cyclic loading of the three compositions (Fig. 9). The figures show that the apparent modulus of the cyclic loops ranges between 5 MPa and 7 MPa for 85–00, 6 MPa and 11 MPa for 85–15, and 4 MPa and 11 MPa for 85–30. Therefore, the experimentally determined average modulus follows the same order with respect to the material composition as the model-estimated modulus. Another interesting observation lies in the contribution to the total modulus by the endochronic branches, which from Table 1 is 0.0013 MPa (0.025%) for 85–00, 8.2364 MPa (66.89%) for 85–15 and 7.5831 MPa (91.51%) for 85–30. Therefore, it is evident that with the activation of inelastic deformation mechanisms, the elastic material stiffness decreases with increasing aluminum content, which is also consistent with the observation of predominantly nonlinear mechanical response and ductile plastic flow to larger strain levels during monotonic compression in 85–15 and 85–30 as compared to 85–00 (Fig. 2).

With regard to the threshold strains γj, the value of strain γ1 in the first endochronic branch is very small and comparable in magnitude for all PBX compositions (0.0017 for 85–00, 0.001 for 85–15, and 0.0018 for 85–30). This suggests that mock PBX material starts exhibiting inelastic behavior from almost the beginning of the loading process regardless of aluminum content, which rationalizes the selection of a yield surface-free plasticity theory to model these materials. However, the value of threshold strain in the second endochronic branch (γ2) is much larger for 85–00 (0.1723) as compared to those of 85–15 (0.0358) and 85–30 (0.0622), indicating that more extensive inelastic processes start occurring at smaller macroscopic strains in the aluminized specimens. When comparing 85–15 and 85–30 specimens, we observe that although the second threshold strain is smaller for 85–15, the threshold strain in the third branch (γ3) is smaller for 85–30 (0.2388) as compared to 85–15 (0.3413). Therefore, it can be concluded that higher aluminum content leads to the material exhibiting predominant inelastic deformations at a smaller strain. This conclusion is also supported by observations of higher residual strain in 85–30 (∼2%) as compared to 85–15 (∼1.57%) after the recovery period following the virgin cyclic loading tests as observed by Agarwal and Gonzalez [23].

### 5.2 Material Scale Function.

As explained in Sec. 2.5, the material scale function f(z), which is a function of the intrinsic time scale z, is used in the constitutive model to represent the progressive cyclic stress-softening behavior observed during cyclic compression of particle-binder composites. The function is given by Eq. (31) and consists of four material parameters, namely, c, s, βc, and βs. The parameter c is the saturation value of the function during initial loading from zero deformation when the reference intrinsic time scale, zref, is equal to zero. If the loading remains monotonic, the simpler scale function proposed by Wu and Yip [59] and given by Eq. (30) is recovered. According to this simpler function, the value of f(z) evolves from 1 as z increases and approaches c at z → ∞, with the rate of evolution governed by βc. If c > 1, then f(z) ≥ 1. Therefore, during a loading step, the increment in the intrinsic time scale z is greater than the increment in deformation measure $|C|$ according to Eq. (29), which produces a strain-hardening behavior. Conversely, if c < 1, a strain-softening response is obtained. This is also true during cyclic loading–unloading; however, the saturation value changes as zref increases, and approaches c/s at zref → ∞ which signifies cyclic stabilization. If c/s < c, then a progressive stress-softening response is obtained, which, as already stated, is one of the primary deformation mechanisms observed in mock PBX.

The value of parameter c for the studied mock PBX specimens is seen to increase with increasing aluminum content in the composition. Specifically, it is much higher for 85–30 as compared to 85–00 and 85–15 specimens. The reason for this difference lies in the monotonic compression response of the specimens. Figure 11 shows the influence of variation in c from its estimated value on the monotonic compression response of 85–00 specimen. It is evident that c controls the occurrence of maximum or ultimate stress in the response, as well as the magnitude of the ultimate stress, which stems from its representative role as the magnitude of strain hardening in the material. The ultimate stress is observed in the monotonic response of 85–00 and 85–15 specimens, with the value of the stress and the strain at which it occurs is higher for 85–15 as compared to that for 85–00. Therefore, the estimated value of parameter c for 85–15 is comparatively higher than for 85–00. However, the 85–30 specimen does not show an ultimate stress and continues to harden until the maximum applied deformation of $30%$; hence, in accordance with Fig. 11, the estimated value of c for 85–30 is much higher than both 85–00 and 85–15. With regard to parameter βc, a correlation between the estimated values and PBX aluminum content is not observed; rather, the values are closely related to the evolution of the scale function during monotonic loading and the estimated values of c. Figure 12 shows the influence of variation in βc on the monotonic compression response as well as the evolution of scale function f(z) with intrinsic time z for the three specimens. It is evident that for 85–00 and 85–15 specimens, the scale function f(z) reaches its saturation value c during the monotonic loading process, while the same does not occur for the 85–30 specimen as the material continues to harden beyond the maximum applied strain. For 85–15 (Figs. 12(c) and 12(d)), the saturation value is reached much before the attainment of ultimate stress, and therefore, any variation in βc has only a slight influence on the loading response at small strains (∼1%–5%). For 85–00 (Figs. 12(a) and 12(b)), the saturation value is reached near the end of the loading process, and therefore, the influence of variation in βc on the stress–strain response is comparatively larger than 85–15. For 85–30 (Figs. 12(e) and 12(f)), however, any variation in βc has a significant impact on the stress–strain response. It is specifically observed that βc controls only the evolution of stress, with the stress at any strain increasing with increasing βc and vice versa. With regard to the magnitude of βc, a higher value of βc increases the rate of evolution of f(z) and therefore the rate of attainment of the saturation value. Hence, 85–15 has the highest value of βc (944.2924), followed by 85–00 (125.3399), while 85–30 assumes a much lower value (2.0906).

As discussed before, the role of parameters s and βs in the context of PBX material is to reduce the saturation value of the scale function f(z) from c to c/s during the cyclic loading process in order to achieve progressive stress-softening until cyclic stabilization. While s primarily controls the magnitude of the softening, βs controls the softening rate or the rate of attainment of cyclic stabilization. From the estimated values of s for the three specimens, the value of c/s is obtained as 11.9027, 14.4811, and 99.0748 for 85–00, 85–15, and 85–30 respectively, resulting in a reduction of 20.58%, 50.94%, and 83.24% in their respective saturation values at cyclic stabilization as compared to c. This implies that the magnitude of softening is the highest for 85–30, followed by 85–15, and the least for 85–00. Concurrently, the estimated values of βs for the three specimens indicate the highest rate of stabilization for 85–00 (212.6798), followed by 85–15 (16.7109), and the lowest for 85–30 (5.7283). These findings with reference to the model parameters are in agreement with the experimental data, as shown in Fig. 13, where the ratio of cyclic peak stresses with respect to the first peak stress (Fig. 13(a)) drops the highest for 85–30, followed by 85–15 and then 85–00, while the change in peak stress with each cycle (Fig. 13(b)) approaches zero (i.e., cyclic stabilization) fastest for 85–00, followed by 85–15 and then 85–30.

## 6 Summary and Discussion

We have presented a constitutive model for particle-binder composites that accounts for finite-deformation kinematics, nonlinear elastoplasticity without apparent yield, cyclic hysteresis, and progressive stress-softening before the attainment of stable cyclic response. The model is based on an additive decomposition of strain energy into elastic and inelastic parts, where the elastic response is modeled using Ogden hyperelasticity while the inelastic response is described using yield-surface-free endochronic plasticity based on the concepts of internal variables and of evolution or rate equations. Stress-softening is modeled using two approaches; a discontinuous isotropic damage model to appropriately describe the overall softening between loading and unloading responses, and a material scale function to describe the progressive cyclic softening until the attainment of stable response. The constitutive model is then based on the deformation mechanisms experimentally observed during cyclic compression of PBX at large strain.

Furthermore, we have proposed a discrete numerical procedure to solve for stresses along a loading path, and we have developed a parameter identification method, based on a nonlinear multivariate minimization problem, to determine material properties from experimental nominal stress–strain data. Specifically, we have used monotonic and cyclic compression data for three mock PBX formulations based on PBXN-109 with varying aluminum content [23] to demonstrate the range of behavior predicted by the proposed constitutive model and the effectiveness of the parameter identification method. This is evident from the remarkable model calibration results that yielded an excellent agreement between experimental data and model response. Finally, we have identified the correlation between the mechanical properties and the unique stress–strain response of each specimen, which contributes toward establishing and validating specific trends observed in the property values with respect to the specimen composition, specifically, the amount of aluminum in the mock PBX. We, therefore, conclude that each of the estimated material parameters is more than a mere number and has a significant contribution toward the establishment of the highly complex and nonlinear mechanical behavior of particle-binder composites under finite strain. Based on these conclusions, the constitutive model presented in this paper serves as a reliable tool to characterize different compositions of particle-binder composites, and it especially serves the defense and propulsion communities in their efforts to characterize the mechanical properties of both existing and new energetic material formulations.

We close by pointing out some limitations of our approach and possible directions for the future extension of our analysis. First, we have restricted our attention to quasi-static, i.e., low strain-rate behavior of particle-binder composites, and thereby neglected any strain-rate-dependent effects in the constitutive model. However, it is well known that viscous effects become dominant in these materials at moderate to high strain rates. Second, the proposed constitutive model carries an assumption of elastoplastic incompressibility, which limits the analysis to particle-binder composites with moderate levels of compressibility. Therefore, the extension of the model to rate-dependent viscous effects and compressible behavior is a worthwhile direction for future research.

## Acknowledgment

The authors wish to acknowledge Professor Patricia Davies, Professor Jeff Rhoads, Professor Steve Son, Professor Stuart Bolton, Jelena Paripovic, Allison Range, Nick Cummock, and Tim Manship for their feedback and interesting discussions.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The data sets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request. Data provided by a third party are listed in Acknowledgment.

This research is supported by the Air Force Research Laboratory through Grant No. FA8651-16-0287 entitled “Near-Resonant Thermomechanics of Energetic and Mock Energetic Composite Materials,” Grant No. FA8651-16-D-0287 entitled “Near-Resonant Thermomechanics of Energetic and Mock Energetic Composite Materials, Part II,” and Grant No. FA8651-17-S-0003 entitled “Exploring the Thermomechanics of Energetic and Mock Energetic Composite Materials Under Quasi-Static and Near-Resonant Excitations.”

## Distribution Statement

Approved for public release. Distribution unlimited (AFRL-2021-2732).

## References

1.
Rattanasom
,
N.
,
Saowapark
,
T.
, and
Deeprasertkul
,
C.
,
2007
, “
Reinforcement of Natural Rubber With Silica/Carbon Black Hybrid Filler
,”
Polym. Test.
,
26
(
3
), pp.
369
377
.
2.
Omnès
,
B.
,
Thuillier
,
S.
,
Pilvin
,
P.
,
Grohens
,
Y.
, and
Gillet
,
S.
,
2008
, “
Effective Properties of Carbon Black Filled Natural Rubber: Experiments and Modeling
,”
Compos. Pt. A-Appl. Sci. Manuf.
,
39
(
7
), pp.
1141
1149
.
3.
Akhavan
,
J.
,
2011
,
Chemistry of Explosives
,
Royal Society of Chemistry
,
Cambridge, UK
.
4.
Palmer
,
S. J. P.
,
Field
,
J. E.
, and
Huntley
,
J. M.
,
1993
, “
Deformation, Strengths and Strains to Failure of Polymer Bonded Explosives
,”
Proc. R. Soc. London, A
,
440
(
1909
), pp.
399
419
.
5.
Rae
,
P. J.
,
Goldrein
,
H. T.
,
Palmer
,
S. J. P.
,
Field
,
J. E.
, and
Lewis
,
A. L.
,
2002
,
Quasi-static Studies of the Deformation and Failure of β-HMX Based Polymer Bonded Explosives
,
Proc. Math. Phys. Eng. Sci.
,
458
(
2019
), pp.
743
762
.
6.
Rangaswamy
,
P.
,
Thompson
,
D. G.
,
Liu
,
C.
, and
Lewis
,
M. W.
,
2010
, “
Modeling the Mechanical Response of PBX 9501
,”
Proceedings of the 14th International Detonation Symposium
,
Coeur D’Alene, ID
,
Apr. 11–16
, pp.
174
183
.
7.
Yılmaz
,
G. A.
,
Şen
,
D.
,
Kaya
,
Z. T.
, and
Tinçer
,
T.
,
2014
, “
Effect of Inert Plasticizers on Mechanical, Thermal, and Sensitivity Properties of Polyurethane-Based Plastic Bonded Explosives
,”
J. Appl. Polym. Sci.
,
131
(
20
), p.
40907
.
8.
Kumar
,
A.
,
Rao
,
V.
,
Sinha
,
R.
, and
Rao
,
A.
,
2010
, “
Evaluation of Plastic Bonded Explosive (PBX) Formulations Based on RDX, Aluminum, and HTPB for Underwater Applications
,”
Propellants, Explos., Pyrotech.
,
35
(
4
), pp.
359
364
.
9.
Wiegand
,
D. A.
,
2000
, “
The Influence of Confinement on the Mechanical Properties of Energetic Materials
,”
AIP Conf. Proc.
,
505
(
1
), pp.
675
678
.
10.
Wiegand
,
D. A.
, and
Reddingius
,
B.
,
2005
, “
Mechanical Properties of Confined Explosives
,”
J. Energ. Mater.
,
23
(
2
), pp.
75
98
.
11.
Drodge
,
D. R.
, and
Williamson
,
D. M.
,
2016
, “
Understanding Damage in Polymer-Bonded Explosive Composites
,”
J. Mater. Sci.
,
51
(
2
), pp.
668
679
.
12.
Idar
,
D. J.
,
Thompson
,
D. G.
,
Gray
,
G. T.
,
Blumenthal
,
W. R.
,
,
C. M.
,
Peterson
,
P. D.
,
Roemer
,
E. L.
,
Wright
,
W. J.
, and
Jacquez
,
B. J.
,
2002
, “
Influence of Polymer Molecular Weight, Temperature, and Strain Rate on the Mechanical Properties of PBX 9501
,”
AIP Conf. Proc.
,
620
(
1
), pp.
821
824
.
13.
Thompson
,
D. G.
,
Idar
,
D. J.
,
Gray
,
G.
,
Blumenthal
,
W. R.
,
,
C. M.
,
Roemer
,
E. L.
,
Wright
,
W. J.
, and
Peterson
,
P. D.
,
2002
, “
Quasi-static and Dynamic Mechanical Properties of New and Virtually-Aged PBX 9501 Composites as a Function of Temperature and Strain Rate
,”
Proceedings of the 12th International Detonation Symposium
,
San Diego, CA
,
Aug. 11–16
, pp.
363
368
.
14.
Williamson
,
D. M.
,
Siviour
,
C. R.
,
Proud
,
W. G.
,
Palmer
,
S. J. P.
,
Govier
,
R.
,
Ellis
,
K.
,
Blackwell
,
P.
, and
Leppard
,
C.
,
2008
, “
Temperature-Time Response of a Polymer Bonded Explosive in Compression (EDC37)
,”
J. Phys. D: Appl. Phys.
,
41
(
8
), p.
085404
.
15.
Bardenhagen
,
S.
,
,
E.
,
Maudlin
,
P.
,
Gray
,
G.
, and
Foster Jr
,
J.
,
1998
, “
Viscoelastic Models for Explosive Binder Materials
,”
AIP Conf. Proc.
,
429
(
1
), pp.
281
284
.
16.
Clements
,
B. E.
, and
Mas
,
E. M.
,
2001
, “
Dynamic Mechanical Behavior of Filled Polymers. I. Theoretical Developments
,”
J. Appl. Phys.
,
90
(
11
), pp.
5522
5534
.
17.
Bennett
,
J. G.
,
Haberman
,
K. S.
,
Johnson
,
J. N.
, and
Asay
,
B. W.
,
1998
, “
A Constitutive Model for the Non-shock Ignition and Mechanical Response of High Explosives
,”
J. Mech. Phys. Solids
,
46
(
12
), pp.
2303
2322
.
18.
,
F. L.
, and
Johnson
,
J. N.
,
1990
, “
A Constitutive Model for the Dynamic Response of Brittle Materials
,”
J. Appl. Phys.
,
67
(
7
), pp.
3275
3286
.
19.
Dienes
,
J. K.
,
1996
, “A Unified Theory of Flow, Hot Spots, and Fragmentation With an Application to Explosive Sensitivity,”
High-Pressure Shock Compression of Solids II: Dynamic Fracture and Fragmentation
,
L.
Davison
,
D. E.
, and
M
.
Shahinpoor
, eds.,
Springer
,
New York
, pp.
366
398
.
20.
Buechler
,
M. A.
, and
Luscher
,
D. J.
,
2014
, “
A Semi-implicit Integration Scheme for a Combined Viscoelastic-Damage Model of Plastic Bonded Explosives
,”
Int. J. Numer Methods Eng.
,
99
(
1
), pp.
54
78
.
21.
Paripovic
,
J.
, and
Davies
,
P.
,
2013
, “
Identification of the Dynamic Behavior of Surrogate Explosive Materials
,”
International Design Engineering Technical Conferences and Computers and Information in Engineering Conference; 22nd Reliability, Stress Analysis, and Failure Prevention Conference; 25th Conference on Mechanical Vibration and Noise
,
Portland, OR
,
Aug. 4–7
, ASME, p. V008T13A066.
22.
Paripovic
,
J.
, and
Davies
,
P.
,
2016
, “
A Model Identification Technique to Characterize the Low Frequency Behaviour of Surrogate Explosive Materials
,”
J. Phys. Conf. Ser.
,
744
, p.
012124
.
23.
Agarwal
,
A.
, and
Gonzalez
,
M.
,
2020
, “
Effects of Cyclic Loading and Time-Recovery on Microstructure and Mechanical Properties of Particle-Binder Composites
,”
ASME J. Appl. Mech.
,
87
(
10
), p.
101008
.
24.
Lochert
,
I. J.
,
Dexter
,
R. M.
, and
Hamshere
,
B. L.
,
2002
,
Evaluation of Australian RDX in PBXN-109, DSTO-TN-0440., Defence Science
,
Technology Organisation, Weapons Systems Division
,
Edinburgh (SA), Australia
.
25.
Pijaudier-Cabot
,
G.
,
Bittnar
,
Z.
, and
Gérard
,
B.
,
1999
,
Mechanics of Quasi-brittle Materials and Structures
,
Hermes Science Publications
,
Paris, France
.
26.
Idar
,
D.
,
Peterson
,
P.
,
Scott
,
P.
, and
Funk
,
D.
,
1998
, “
Low Strain Rate Compression Measurements of PBXN-9, PBX 9501, and Mock 9501
,”
AIP Conf. Proc.
,
429
(
1
), pp.
587
590
.
27.
Trumel
,
H.
,
Lambert
,
P.
, and
Biessy
,
M.
,
2012
, “
Mechanical and Microstructural Characterization of a HMX-Based Pressed Explosive: Effects of Combined High Pressure and Strain Rate
,”
EPJ Web Conf.
,
26
, p.
02005
.
28.
Keyhani
,
A.
,
Kim
,
S.
,
Horie
,
Y.
, and
Zhou
,
M.
,
2019
, “
Energy Dissipation in Polymer-Bonded Explosives With Various Levels of Constituent Plasticity and Internal Friction
,”
Comput. Mater. Sci.
,
159
, pp.
136
149
.
29.
Le
,
V. D.
,
Gratton
,
M.
,
Caliez
,
M.
,
Frachon
,
A.
, and
Picart
,
D.
,
2010
, “
Experimental Mechanical Characterization of Plastic-Bonded Explosives
,”
J. Mater. Sci.
,
45
(
21
), pp.
5802
5813
.
30.
Yang
,
K.
,
Wu
,
Y.
, and
Huang
,
F.
,
2018
, “
Numerical Simulations of Microcrack-Related Damage and Ignition Behavior of Mild-Impacted Polymer Bonded Explosives
,”
J. Hazard. Mater.
,
356
, pp.
34
52
.
31.
Tang
,
M.
,
Pang
,
H.
,
Lan
,
L.
,
Wen
,
M.
, and
Ming
,
L.
,
2016
, “
,”
Chin. J. Energ. Mater.
,
24
(
9
), pp.
832
837
.
32.
Laiarinandrasana
,
L.
,
Piques
,
R.
, and
Robisson
,
A.
,
2003
, “
Visco-hyperelastic Model With Internal State Variable Coupled With Discontinuous Damage Concept Under Total Lagrangian Formulation
,”
Int. J. Plast.
,
19
(
7
), pp.
977
1000
.
33.
Ayoub
,
G.
,
Zaïri
,
F.
,
Naït-Abdelaziz
,
M.
,
Gloaguen
,
J. M.
, and
Kridli
,
G.
,
2014
, “
A Visco-hyperelastic Damage Model for Cyclic Stress-Softening, Hysteresis and Permanent Set in Rubber Using the Network Alteration Theory
,”
Int. J. Plast.
,
54
, pp.
19
33
.
34.
Österlöf
,
R.
,
Wentzel
,
H.
, and
Kari
,
L.
,
2016
, “
A Finite Strain Viscoplastic Constitutive Model for Rubber With Reinforcing Fillers
,”
Int. J. Plast.
,
87
, pp.
1
14
.
35.
Guo
,
Q.
,
Zaïri
,
F.
, and
Guo
,
X.
,
2018
, “
A Thermo-viscoelastic-Damage Constitutive Model for Cyclically Loaded Rubbers. Part I: Model Formulation and Numerical Examples
,”
Int. J. Plast.
,
101
, pp.
106
124
.
36.
Guo
,
Q.
,
Zaïri
,
F.
, and
Guo
,
X.
,
2018
, “
A Thermo-Viscoelastic-Damage Constitutive Model for Cyclically Loaded Rubbers. Part II: Experimental Studies and Parameter Identification
,”
Int. J. Plast.
,
101
, pp.
58
73
.
37.
Chen
,
Y.
,
Kang
,
G.
,
Yuan
,
J.
, and
Yu
,
C.
,
2018
, “
Uniaxial Ratchetting of Filled Rubber: Experiments and Damage-Coupled Hyper-viscoelastic-Plastic Constitutive Model
,”
ASME J. Appl. Mech.
,
85
(
6
), p.
061013
.
38.
Dargazany
,
R.
,
Khiêm
,
V. N.
, and
Itskov
,
M.
,
2014
, “
A Generalized Network Decomposition Model for the Quasi-static Inelastic Behavior of Filled Elastomers
,”
Int. J. Plast.
,
63
, pp.
94
109
.
39.
Raghunath
,
R.
,
Juhre
,
D.
, and
Klüppel
,
M.
,
2016
, “
A Physically Motivated Model for Filled Elastomers Including Strain Rate and Amplitude Dependency in Finite Viscoelasticity
,”
Int. J. Plast.
,
78
, pp.
223
241
.
40.
Plagge
,
J.
, and
Klüppel
,
M.
,
2017
, “
A Physically Based Model of Stress Softening and Hysteresis of Filled Rubber Including Rate- and Temperature Dependency
,”
Int. J. Plast.
,
89
, pp.
173
196
.
41.
Mullins
,
L.
,
1948
, “
Effect of Stretching on the Properties of Rubber
,”
Rubber Chem. Technol.
,
21
(
2
), pp.
281
300
.
42.
Mullins
,
L.
,
1969
, “
Softening of Rubber by Deformation
,”
Rubber Chem. Technol.
,
42
(
1
), pp.
339
362
.
43.
Marckmann
,
G.
,
Verron
,
E.
,
Gornet
,
L.
,
Chagnon
,
G.
,
Charrier
,
P.
, and
Fort
,
P.
,
2002
, “
A Theory of Network Alteration for the Mullins Effect
,”
J. Mech. Phys. Solids
,
50
(
9
), pp.
2011
2028
.
44.
Chagnon
,
G.
,
Verron
,
E.
,
Marckmann
,
G.
, and
Gornet
,
L.
,
2006
, “
Development of New Constitutive Equations for the Mullins Effect in Rubber Using the Network Alteration Theory
,”
Int. J. Solids Struct.
,
43
(
22
), pp.
6817
6831
.
45.
Klüppel
,
M.
,
2003
, “The Role of Disorder in Filler Reinforcement of Elastomers on Various Length Scales,”
Filler-Reinforced Elastomers Scanning Force Microscopy
, 1, Vol.
164
,
B.
Capella
,
M.
Geuss
,
M.
Klüppel
,
M.
Munz
,
E.
Schulz
, and
H.
Sturm
, eds.,
Springer
,
Berlin/Heidelberg
, pp.
1
86
.
46.
Kaliske
,
M.
, and
Rothert
,
H.
,
1998
, “
Constitutive Approach to Rate-Independent Properties of Filled Elastomers
,”
Int. J. Solids Struct.
,
35
(
17
), pp.
2057
2071
.
47.
Valanis
,
K. C.
,
1970
,
A Theory of Viscoplasticity Without a Yield Surface Part 1—General Theory
,
Air Force Office of Scientific Research, Office of Aerospace Research, US Air Force
,
Arlington, VA
.
48.
Netzker
,
C.
,
Dal
,
H.
, and
Kaliske
,
M.
,
2010
, “
An Endochronic Plasticity Formulation for Filled Rubber
,”
Int. J. Solids Struct.
,
47
(
18–19
), pp.
2371
2379
.
49.
Simo
,
J. C.
,
1987
, “
On a Fully Three-Dimensional Finite-Strain Viscoelastic Damage Model: Formulation and Computational Aspects
,”
Comput. Methods Appl. Mech. Eng.
,
60
(
2
), pp.
153
173
.
50.
Govindjee
,
S.
, and
Simo
,
J.
,
1991
, “
A Micro-mechanically Based Continuum Damage Model for Carbon Black-Filled Rubbers Incorporating Mullins’ Effect
,”
J. Mech. Phys. Solids
,
39
(
1
), pp.
87
112
.
51.
Holzapfel
,
G. A.
, and
Simo
,
J. C.
,
1996
, “
A New Viscoelastic Constitutive Model for Continuous Media at Finite Thermomechanical Changes
,”
Int. J. Solids Struct.
,
33
(
20
), pp.
3019
3034
.
52.
Holzapfel
,
G. A.
,
1996
, “
On Large Strain Viscoelasticity: Continuum Formulation and Finite Element Applications to Elastomeric Structures
,”
Int. J. Numer Methods Eng.
,
39
(
22
), pp.
3903
3926
.
53.
Holzapfel
,
G. A.
,
2002
, “
Nonlinear Solid Mechanics: A Continuum Approach for Engineering Science
,”
Meccanica
,
37
(
4
), pp.
489
490
.
54.
Ogden
,
R. W.
,
1972
, “
Large Deformation Isotropic Elasticity on the Correlation of Theory and Experiment for Incompressible Rubberlike Solids
,”
Proc. R. Soc. London, Ser. A
,
326
(
1567
), pp.
565
584
.
55.
Ogden
,
R. W.
,
1972
, “
Large Deformation Isotropic Elasticity: On the Correlation of Theory and Experiment for Compressible Rubberlike Solids
,”
Proc. R. Soc. London, Ser. A
,
328
(
1575
), pp.
567
583
.
56.
Horstemeyer
,
M. F.
, and
Bammann
,
D. J.
,
2010
, “
Historical Review of Internal State Variable Theory for Inelasticity
,”
Int. J. Plast.
,
26
(
9
), pp.
1310
1334
.
57.
Kachanov
,
L. M.
,
1986
,
Introduction to Continuum Damage Mechanics
,
Springer
,
Netherlands
.
58.
Lemaitre
,
J.
,
Chaboche
,
J.-L.
, and
Germain
,
P.
,
1985
,
Mécanique Des Matériaux Solides
,
Dunod
,
Paris, France
.
59.
Wu
,
H.
, and
Yip
,
M.
,
1981
, “
Endochronic Description of Cyclic Hardening Behavior for Metallic Materials
,”
ASME J. Eng. Mater. Technol.
,
103
(
3
), pp.
212
217
.
60.
Yeh
,
W.-C.
,
1995
, “
Verification of the Endochronic Theory of Plasticity Under Biaxial Load
,”
J. Chin. Inst. Eng.
,
18
(
1
), pp.
25
34
.
61.
Lin
,
H.-Y.
,
Yeh
,
W.-C.
, and
Lee
,
W.-J.
,
2007
, “
A Material Function of Endochronic Theory and Its Application to Test Under Axisymmetrically Cyclic Loading Conditions
,”
J. Mech.
,
23
(
2
), pp.
135
148
.
62.
Kröner
,
E.
,
1959
, “
Allgemeine Kontinuumstheorie Der Versetzungen Und Eigenspannungen
,”
Arch. Ration. Mech. Anal.
,
4
(
1
), pp.
273
334
.
63.
Lee
,
E.
, and
Liu
,
D.
,
1967
, “
Finite-Strain Elastic-Plastic Theory With Application to Plane-Wave Analysis
,”
J. Appl. Phys.
,
38
(
1
), pp.
19
27
.
64.
Lee
,
E. H.
,
1969
, “
Elastic-Plastic Deformation at Finite Strains
,”
ASME J. Appl. Mech.
,
36
(
1
), pp.
1
6
.
65.
Gurtin
,
M. E.
, and
Anand
,
L.
,
2005
, “
The Decomposition F = FeFp, Material Symmetry, and Plastic Irrotationality for Solids That Are Isotropic-Viscoplastic or Amorphous
,”
Int. J. Plast.
,
21
(
9
), pp.
1686
1719
.
66.
Hill
,
R.
,
1950
,
The Mathematical Theory of Plasticity
,
Clarendon Press
,
Oxford, UK
.
67.
Hill
,
R.
,
1958
, “
A General Theory of Uniqueness and Stability in Elastic-Plastic Solids
,”
J. Mech. Phys. Solids
,
6
(
3
), pp.
236
249
.
68.
Prager
,
W.
,
1961
, “
An Elementary Discussion of Definitions of Stress Rate
,”
Q. Appl. Math.
,
18
(
4
), pp.
403
407
.
69.
Blanchard
,
A. F.
, and
Parkinson
,
D.
,
1952
, “
Breakage of Carbon-Rubber Networks by Applied Stress
,”
Ind. Eng. Chem.
,
44
(
4
), pp.
799
812
.
70.
Bueche
,
F.
,
1960
, “
Molecular Basis for the Mullins Effect
,”
J. Appl. Polym. Sci.
,
4
(
10
), pp.
107
114
.
71.
Houwink
,
R.
,
1956
, “
Slipping of Molecules During the Deformation of Reinforced Rubber
,”
Rubber Chem. Technol.
,
29
(
3
), pp.
888
893
.
72.
Clément
,
F.
,
Bokobza
,
L.
, and
Monnerie
,
L.
,
2001
, “
On the Mullins Effect in Silica-Filled Polydimethylsiloxane Networks
,”
Rubber Chem. Technol.
,
74
(
5
), pp.
847
870
.
73.
Kraus
,
G.
,
Childers
,
C. W.
, and
Rollmann
,
K. W.
,
1966
, “
Stress Softening in Carbon Black-Reinforced Vulcanizates. Strain Rate and Temperature Effects
,”
J. Appl. Polym. Sci.
,
10
(
2
), pp.
229
244
.
74.
Klüppel
,
M.
, and
Schramm
,
J.
,
2000
, “
A Generalized Tube Model of Rubber Elasticity and Stress Softening of Filler Reinforced Elastomer Systems
,”
Macromol. Theory Simul.
,
9
(
9
), pp.
742
754
.
75.
Ogden
,
R. W.
, and
Roxburgh
,
D. G.
,
1999
, “
A Pseudoelastic Model for the Mullins Effect in Filled Rubber
,”
Proc. R. Soc. London, Ser. A
,
455
(
1988
), pp.
2861
2877
.
76.
Dorfmann
,
A.
, and
Ogden
,
R. W.
,
2003
, “
,”
Int. J. Solids Struct.
,
40
(
11
), pp.
2699
2714
.
77.
Zúñiga
,
A. E.
, and
Beatty
,
M. F.
,
2002
, “
A New Phenomenological Model for Stress-Softening in Elastomers
,”
J. Appl. Math. Phy.
,
53
(
5
), pp.
794
814
.
78.
Miehe
,
C.
,
1995
, “
Discontinuous and Continuous Damage Evolution in Ogden-Type Large-Strain Elastic Materials
,”
Eur. J. Mech. A Solids
,
14
(
5
), pp.
697
720
.
79.
Kaliske
,
M.
,
Nasdala
,
L.
, and
Rothert
,
H.
,
2001
, “
On Damage Modelling for Elastic and Viscoelastic Materials at Large Strain
,”
Comput. Struct.
,
79
(
22–25
), pp.
2133
2141
.
80.
Lin
,
R. C.
, and
Schomburg
,
U.
,
2003
, “
A Finite Elastic–Viscoelastic–Elastoplastic Material Law With Damage: Theoretical and Numerical Aspects
,”
Comput. Methods Appl. Mech. Eng.
,
192
(
13–14
), pp.
1591
1627
.
81.
Rickaby
,
S. R.
, and
Scott
,
N. H.
,
2013
, “
Cyclic Stress-Softening Model for the Mullins Effect in Compression
,”
Int. J. Non Linear Mech.
,
49
, pp.
152
158
.
82.
Diani
,
J.
,
Fayolle
,
B.
, and
Gilormini
,
P.
,
2009
, “
A Review on the Mullins Effect
,”
Eur. Polym. J.
,
45
(
3
), pp.
601
612
.
83.
Naumann
,
C.
, and
Ihlemann
,
J.
,
2015
, “
On the Thermodynamics of Pseudo-elastic Material Models Which Reproduce the Mullins Effect
,”
Int. J. Solids Struct.
,
69–70
, pp.
360
369
.
84.
Coleman
,
B. D.
, and
Noll
,
W.
,
1963
, “
The Thermodynamics of Elastic Materials With Heat Conduction and Viscosity
,”
Arch. Ration. Mech. Anal.
,
13
(
1
), pp.
167
178
.
85.
Coleman
,
B. D.
, and
Gurtin
,
M. E.
,
1967
, “
Thermodynamics With Internal State Variables
,”
J. Chem. Phys.
,
47
(
2
), pp.
597
613
.
86.
Mooney
,
M.
,
1940
, “
A Theory of Large Elastic Deformation
,”
J. Appl. Phys.
,
11
(
9
), pp.
582
592
.
87.
Rivlin
,
R.
,
1948
, “
Large Elastic Deformations of Isotropic Materials IV. Further Developments of the General Theory
,”
Philos. Trans. Royal Soc.
,
241
(
835
), pp.
379
397
.
88.
Boyce
,
M. C.
, and
Arruda
,
E. M.
,
2000
, “
Constitutive Models of Rubber Elasticity: A Review
,”
Rubber Chem. Technol.
,
73
(
3
), pp.
504
523
.
89.
Ogden
,
R. W.
,
1984
,
Non-linear Elastic Deformations
,
Dover Publications
,
Mineola, NY.
90.
Beatty
,
M. F.
, and
Krishnaswamy
,
S.
,
2000
, “
A Theory of Stress-Softening in Incompressible Isotropic Materials
,”
J. Mech. Phys. Solids
,
48
(
9
), pp.
1931
1965
.
91.
Valanis
,
K. C.
, and
Lee
,
C.
,
1984
, “
Endochronic Theory of Cyclic Plasticity With Applications
,”
ASME J. Appl. Mech.
,
51
(
2
), pp.
367
374
.
92.
Guo
,
J.
,
Wu
,
L.
,
Yang
,
X.
, and
Zhong
,
S.
,
2016
, “
Elastic-Plastic Endochronic Constitutive Model of 0Crl7Ni4Cu4Nb Stainless Steels
,”
Math. Probl. Eng.
,
2016
, pp.
1
16
.
93.
Valanis
,
K.
, and
,
H.
,
1986
, “
An Endochronic Plasticity Theory for Concrete
,”
Mech. Mater.
,
5
(
3
), pp.
277
295
.
94.
Bakhshiani
,
A.
,
Khoei
,
A.
, and
Mofid
,
M.
,
2002
, “
An Endochronic Plasticity Model for Powder Compaction Processes
,”
J. Mater. Process. Technol.
,
125–126
, pp.
138
143
.
95.
Khoei
,
A.
,
Bakhshiani
,
A.
, and
Mofid
,
M.
,
2003
, “
An Endochronic Plasticity Model for Finite Strain Deformation of Powder Forming Processes
,”
Finite Elem. Anal. Des.
,
40
(
2
), pp.
187
211
.
96.
Khoei
,
A.
, and
Bakhshiani
,
A.
,
2004
, “
A Hypoelasto-Plastic Finite Strain Simulation of Powder Compaction Processes With Density-Dependent Endochronic Model
,”
Int. J. Solids Struct.
,
41
(
22–23
), pp.
6081
6110
.
97.
Hsu
,
S. Y.
,
Jain
,
S. K.
, and
Griffin
,
O. H.
,
1991
, “
,”
J. Eng. Mech.
,
117
(
1
), pp.
110
131
.
98.
Valanis
,
K. C.
,
1972
,
Irreversible Thermodynamics of Continuous Media: Internal Variable Theory
, Vol.
1
,
Springer
,
Vienna
.
99.
Coleman
,
B. D.
,
1964
, “
Thermodynamics of Materials With Memory
,”
Arch. Ration. Mech. Anal.
,
17
(
1
), pp.
1
46
.
100.
MATLAB®, Version 9.4.0
,
2018
,
Natick, Massachusetts: The Mathworks Inc., 1984–2018
.