A dynamic flow rule for viscoplasticity in polycrystalline solids under high strain rates

International Journal of Non-Linear Mechanics 95 (2017) 10–18

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A dynamic flow rule for viscoplasticity in polycrystalline solids under high strain rates Md. Masiur Rahaman a , Bensingh Dhas a , D. Roy a , J.N. Reddy b, * a b

Computational Mechanics Lab, Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India Advanced Computational Mechanics Lab, Department of Mechanical Engineering,Texas A&M University, College Station, TX 77843-3123, United States

a r t i c l e

i n f o

Keywords: Micro-inertia High strain rate Visco-plasticity Hyperbolic flow rule Micro-inertial length scale

a b s t r a c t For visco-plasticity in polycrystalline solids under high strain rates, we introduce a dynamic flow rule (also called the micro-force balance) that has a second order time derivative term in the form of micro-inertia. It is revealed that this term, whose physical origin is traced to dynamically evolving dislocations, has a profound effect on the macro-continuum plastic response. Based on energy equivalence between the micro-part of the kinetic energy and that associated with the fictive dislocation mass in the continuous dislocation distribution (CDD) theory, an explicit expression for the micro-inertial length scale is derived. The micro-force balance together with the classical momentum balance equations thus describes the viscoplastic response of the isotropic polycrystalline material. Using rational thermodynamics, we arrive at constitutive equations relating the thermodynamic forces (stresses) and fluxes. A consistent derivation of temperature evolution is also provided, thus replacing the empirical route. The micro-force balance, supplemented with the constitutive relations for the stresses, yields a locally hyperbolic flow rule owing to the micro-inertia term. The implication of micro-inertia on the continuum response is explicitly demonstrated by reproducing experimentally observed stress–strain responses under high strain-rate loadings and varying temperatures. An interesting finding is the identification of micro-inertia as the source of oscillations in the stress–strain response under high strain rates. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Viscoplasticity is a very intricate manifestation of highly nonlinear dynamical processes associated with microscopic defects (e.g. dislocations, micro-voids etc.). Typically, the underlying microscopic defects, like dislocations for metals, tend to self organize in the form of patterns and render the microscale deformation field heterogeneous, even as the macroscopic field of interest in continuum viscoplasticity may remain homogeneous [1]. Despite the demonstrable success of classical viscoplasticity models for a large class of problems (especially with length scale above several microns), their inadequacy in modelling problems that involve mesoscopic length scales (typically in the range of a tenth of micron to a few tens of micron) or lower is well documented. Such failure has been put down to the absence of microstructure-related length scale information in the model to capture the size dependency of the meso scale viscoplastic response. As a remedy, several length scale dependent plasticity models [2–9] have also been developed in the last few decades. Unfortunately, a rationally grounded continuum model with the capability of reproducing viscoplastic response in accord

with experimental observations under high strain-rates and varying temperatures still remains elusive and attempts in this direction are often mired in controversies. To quote an instance, most existing continuum viscoplastic models are unable to capture the experimentally observed oscillations in the stress–strain response of metals (e.g. molybdenum, tantalum etc.) under high strain rates and such phenomena are sometimes dismissed as mere experimental artefacts [10]. Indeed, the fact that certain transient features of viscoplasticity under high strain rates may not be revealed without experiments aided by an adequately time-resolved sensing facility also underlines the importance of rational underpinnings of a predictive model. In introducing the desired capability within continuum viscoplasticity, one must start by grasping the limitations of the existing approaches in incorporating changes in the mechanical behaviour of materials (e.g., yield strength, ductility, etc.) under different strain-rates and temperatures. There are both phenomenological and physically motivated models to predict material behaviour under high strain rates and elevated temperatures. Initial attempts were in developing

* Corresponding author.

E-mail addresses: [email protected] (Md.M. Rahaman), [email protected] (B. Dhas), [email protected] (D. Roy), [email protected] (J.N. Reddy). http://dx.doi.org/10.1016/j.ijnonlinmec.2017.05.010 Received 18 April 2017; Received in revised form 21 May 2017; Accepted 21 May 2017 0020-7462/© 2017 Elsevier Ltd. All rights reserved.

Md.M. Rahaman et al.

International Journal of Non-Linear Mechanics 95 (2017) 10–18

in the equation of motion for dislocation, especially at high strain rates, may significantly alter the predicted micro-structural evolution and the macroscopic response compared to the oft-used over-damped equation of motion. This motivates us to trace the dislocation inertia to its physical roots and build up a rational set-up to investigate its effect on the continuum response. Specifically, by extending the micro force balance of Gurtin [23], we propose a dynamic flow rule that contains a micro inertia term to account for dislocation inertial effects on the viscoplastic response of polycrystalline materials under high strain rates. A thermodynamically consistent evolution equation for the temperature associated with the plastic flow is also derived. A probable expression for the micro-inertial length scale is arrived at based on energy equivalence between the micro-kinetic energy and the kinetic energy pertaining to the dislocation mass in the CDD framework [21]. We demonstrate that the inclusion of the micro-inertia term enables capturing the experimentally observed oscillations in the stress–strain responses of molybdenum and highstrength low-alloy steel (HSLA-65) at high strain rates. We also briefly comment on how the micro-inertia term could be seamlessly incorporated in several other existing models.

empirical models using simple uniaxial stress–strain data [11] and one dimensional stress wave propagation approaches taking into account the effect of temperature, strain and strain rate on the flow stress. For the von Mises type of materials, these one dimensional models could be used within the general three dimensional case by replacing the one dimensional quantities by equivalent invariant measures of stress, strain and strain rate tensors. Johnson and Cook [12] proposed a phenomenological model to predict the material flow stress subjected to high strain, strain rate and temperature. Although, the Johnson–Cook (JC) model is very popular thanks to its simplicity and known parameter values for various materials of interest, the assumed uncoupling of strain, strain rate and temperature effects on the flow stress is rarely applicable to certain materials (e.g. molybdenum) where rate sensitivity changes with varying temperature. Moreover, the JC model does not account for thermal or strain history effects, accounting of which underlines the relative benefits of an internal-variable-based theory [13]. In order to consider the coupling among strain, strain rate and temperature, Zerilli and Armstrong [14] have proposed a dislocation mechanics based constitutive model, which incorporates a thermal activation analysis for overcoming obstacles to dislocation motion. The main idea in the Zerilli–Armstrong (ZA) model is that the constitutive behaviour could be different depending on the rate-controlling mechanism specific to the material structure type [body-centred cubic (bcc), face-centred cubic (fcc) and so on]. A very strong dependence of the yield stress on strain rate and temperature is observed in bcc metals, even though the yield stress of fcc metals is mainly affected by strain hardening. The rationale behind this disparity is given based on the dislocation characteristics. Specifically, while the cutting of dislocation forests is the main mechanism in fcc metals, it is the overcoming of Peierls–Nabarro barriers in bcc metals. Although, the ZA model provides a very good correlation of the yield stress of metals with experiments, prediction of work-hardening behaviour for bcc metals is not very accurate. This drawback arises from the inaccurate assumption that work-hardening behaviour in bcc metals does not depend on the strain rate and temperature [15]. Moreover, material parameters in the ZA model lose their physical meaning at high strain rates and temperatures because of the approximation used in the derivation of the constitutive relations (ln(1 + 𝑥) ≈ 𝑥, 𝑥 a function of strain rate and temperature). Voyiadjis and Abed [16] have proposed a constitutive relation for metals based on a modified ZA model, wherein the exact value of the function ln(1 + 𝑥) replaces its truncated expansion. The Voyiadjis–Abed (VA) model reportedly predicts the flow stress better at relatively higher strain rates. In addition to thermally activated dislocation interactions, the effect of dislocation drag at very high strain-rates of order 107 has been considered important, resulting in the Preston–Tonks–Wallace (PTW) model. However, even after incorporating additional physics associated with dislocation motion, these models do not always show a very good agreement with the experiment, viz. for molybdenum over a wide range of strain, strain rate and temperature [17]. Many other such attempts at continuum modelling of response of materials across different strain rate and temperature regimes have also been reported [18– 20]. Even so, to our knowledge, few are able to offer a rationally grounded route to capturing certain finer features in the experimentally observed stress–strain plots, e.g. oscillations in the stress–strain response of polycrystalline materials (e.g. molybdenum, high-strength low-alloy steel etc.) under high strain rates. Thus the question arises as to whether these models have consistently overlooked any important mechanism related to dislocation motion which could be very important at high strain-rate loading and possibly responsible for oscillations in the stress– strain response. In search for an answer to this question, we recognize an important aspect of dislocation motion, viz. dislocation inertia, which is not incorporated in existing viscoplasticity models. Kosevich [21] has identified that the dislocation inertia plays a pivotal role in the evolution of micro structure when the dislocation acceleration is large. More recently, Wang et al. have shown that [22] the inclusion of dislocation inertia

2. Micro-inertial length scale We begin by describing a continuum representation of the dislocation inertia to be included in our thermoviscoplasticity formulation. In order to incorporate dislocation acceleration effects in the continuum response, an additional form of kinetic energy, called the micro-kinetic energy K𝑚𝑖𝑐𝑟𝑜 , is introduced. It is given by 1 2 2 𝜌 𝑙 𝛾̇ (1) 2 0𝑚 𝑝 where 𝛾𝑝 is the equivalent plastic strain introduced as an internal variable, 𝛾̇ 𝑝 is the rate of the equivalent plastic strain, 𝑙𝑚 is the micro-inertial length scale and 𝜌0 is the reference material density. In the present formulation, 𝑙𝑚 𝛾̇ 𝑝 is a continuum scalar representation of dislocation velocity. To characterize 𝑙𝑚 , we take recourse to the CDD theory wherein dislocations are represented using line like objects. There, stress and strain fields generated by dislocations are typically described using linear elasticity [21,24]. The force on a dislocation f is thus given by the Peach–Koehler equation.

K𝑚𝑖𝑐𝑟𝑜 =

f = −l × 𝝈b.

(2)

Here 𝝈 is the stress field, b the Burgers vector and l the line direction associated with the dislocation. As the CDD theory is adopted within a linear set up, it allows an additive decomposition of the stress into an external component 𝝈 𝑒 and a self-stress 𝝈 𝑠 . The external stress 𝝈 𝑒 is caused by the applied boundary conditions and body forces, whereas the self-stress is due to the presence of dislocations and their motion. The self-stress is again additively decomposed into a static part 𝝈 0 and a dynamic part 𝝈 𝑑 . The force on the dislocation loop due to the dynamic part of the self-stress is given by, f𝑑 =

∮

𝝁(l, l′ )v̇ 𝑑 (l′ )𝑑l.

(3)

In the above expression, 𝝁 is the mass per unit length of the dislocation and v𝑑 its velocity. l and l′ are the local tangents to the dislocation line respectively at points 𝑠 and 𝑠′ . The expression for mass per unit length of the dislocation line is given by (see [21]), ( ) ( )4 𝑐𝑡 1 𝝁 = 𝜌0 ‖b‖2 [I − (l ⊗ l′ )] 1 + sin2 𝜃 ∫𝛤 ∫𝛤 8𝜋 𝑐𝑙 ) ( 𝜔(𝜷)𝜔(𝜷 ′ ) × 𝑑𝑎𝑑𝑎′ 𝑑𝑙𝑑𝑙′ (4) ∫𝐴 ∫𝐴 ‖𝜷 − 𝜷 ′ ‖ 𝜔 is an averaging kernel with the property ∫𝐴 𝜔𝑑𝐴 = 1, 𝐴 is the dislocation core area often assumed to√be circular with diameter √ 2𝑟0 and 𝛤 is the dislocation line. 𝑐𝑙 = (𝜆 + 2𝐺)∕𝜌0 and 𝑐𝑡 = 𝐺∕𝜌0 11

Md.M. Rahaman et al.

International Journal of Non-Linear Mechanics 95 (2017) 10–18

are the longitudinal and transverse wave velocities in the media. 𝜷 and 𝜷 ′ are dislocation core cross section variables respectively at 𝑠 and 𝑠′ . 𝜃 = b.(x(𝑠) − x(𝑠′ ))∕‖(x(𝑠) − x(𝑠′ )‖ ‖b‖), x(𝑠) and x(𝑠′ ) are the position vector of the dislocation line 𝑠 and 𝑠′ respectively. I is the second order identity tensor. The averaging kernel is introduced to avoid the singularity of the elastic energy associated with the zero thickness of dislocation line by smearing the Burgers vector over an area 𝐴. The non-local nature of the integral in Eq. (4) poses difficulties in finding a closed form expression for the dislocation mass. To make the integral tractable, the following assumptions are made. (a) The dislocation line is assumed to be straight, i.e. l = l′ . (b) The Burgers vector associated with the dislocation is uniformly smeared over an area of radius 𝑟0 . In [25], a cubic kernel is used to smear the Burgers vector over the dislocation core area so as to avoid the unboundedness of the elastic energy. Such kernels would require numerical methods for calculating the non-local dislocation mass. With the uniform smearing adopted in this work, the mass of dislocation per unit length for a straight dislocation is found to be, ( ) ( )4 𝑐𝑡 𝜌 ‖b‖2 (I − l ⊗ l) 1 + sin2 𝜃 log(1∕𝑟0 ). (5) 𝝁𝑠𝑡 = 0 4𝜋 𝑐𝑙

Comparison of Eq. (12) with the proposed micro-kinetic energy given by Eq. (1) yields the expression for the micro inertial length scale as, ) ( ( )4 𝑐𝑡 1 2 2 sin 𝜃 log(1∕𝑟0 ). (13) 1+ 𝑙𝑚 = 4𝜋 𝑐𝑙

Having discussed the notion of mass associated with dislocation, we now proceed to calculate the kinetic energy associated with its motion. The expression for the kinetic energy per unit length of the dislocation line is given by,

3.1. Kinematics

K𝐶𝐷𝐷 =

1 𝑑 v .𝝁𝑠𝑡 v𝑑 . 2

It is important to note that this expression for the micro-inertial length scale is limited to straight dislocations. This assumption can be relaxed by adopting a geometric theory for curved dislocations presented in [26,27]. Moreover the calculation of dislocation mass is based on the infinitesimal strain assumption; such assumptions are not required if one adopts a geometric setting for describing dislocations [28]. As our current interest is mainly to show the effect of micro inertia on plastic deformation, derivation of a micro-inertial length scale using a geometric theory of curved dislocations is not attempted here. 3. Thermoviscoplasticity model We now describe a thermoviscoplasticity model with the aid of a micro inertia driven dynamic flow rule. The theoretical background needed for this development could be found in the work of Gurtin [23], though the latter is essentially limited to the isothermal quasi-static case.

Consider a body , the reference configuration of which could be identified with an open set 𝑈 ⊂ R3 . The deformation of the body at time 𝑡 ∈ R+ is defined by 𝝋𝑡 ∶ 𝑈 → 𝑉 ⊂ R3 where the deformed configuration of the body is identified with the open set 𝑉 . The spatial derivative of the deformation 𝝋𝑡 is called the deformation gradient, which is given by F = 𝜕𝝋∕𝜕X in a Cartesian co-ordinate system with X denoting the co-ordinates of the reference configuration. The deformation gradient F should satisfy the constraint 𝑑𝑒𝑡(F) > 0 to preserve the orientation and local invertibility of the deformation. In order to characterize the plastic deformation, Kröner–Lee decomposition of the deformation gradient into an elastic distortion F𝑒 and a plastic distortion F𝑝 is adopted here.

(6)

The present continuum formulation is related to the CDD through the following relation, Sym(J) = D𝑝

(7)

J is the dislocation flux density tensor and Sym(.) denotes the symmetric part of a tensor. In the CDD theory, J is expressed in terms of the dislocation velocity, line direction and the Burgers vector b as,

F = F𝑒 F𝑝 . J = (l × v𝑑 ) ⊗ b

(8)

(14)

D𝑝 , the plastic strain rate tensor (see Section 3 for details), is the rate of plastic slip occurring in a slip plane. The slip plane and the direction of slip are given by m and s respectively. The magnitude of the plastic slip rate is given by the equivalent plastic strain rate 𝛾̇ 𝑝 , so that we have

It should be noted that even though the deformation gradient is integrable, the elastic and plastic parts of it are in general non-integrable; that is one cannot find deformations whose derivatives will be F𝑒 and F𝑝 . For incompressible plastic deformation, F𝑝 needs to satisfy the constraint 𝑑𝑒𝑡(F𝑝 ) = 1 which implies that 𝑑𝑒𝑡(F) = 𝑑𝑒𝑡(F𝑒 ). The rate of deformation is the velocity field v, gradient of which is defined as,

D𝑝 = 𝛾̇ 𝑝 Sym(s ⊗ m).

̇ −1 . L = FF

(9)

In the equation above, it is assumed that plastic slip occurs only in one slip plane at a given instant of time. However, the slip plane and the slip direction could be time dependent which is accounted for in the proposed formulation (see Section 3 for details). Expressing the dislocation velocity in the orthonormal basis {l, m, s}, we may write, v 𝑑 = 𝑣𝑙 l + 𝑣𝑚 m + 𝑣𝑠 s

Using Kröner–Lee decomposition in Eq. (15) leads to the following relation, L = L𝑒 + F𝑒 L𝑝 F−1 𝑒 .

𝑣𝑠 = 𝛾̇ 𝑝 ∕‖b‖.

(16)

For brevity, we have introduced the following notations

(10)

L𝑒 ∶= Ḟ 𝑒 F−1 𝑒 ;

𝑣𝑙 , 𝑣𝑚 and 𝑣𝑠 are the velocity components of dislocation in the l, m and s directions respectively. The slip direction is typically taken the same as Burgers vector direction. Employing Eqs. (9) and (10) in Eq. (7), we arrive at the following equation, v𝑑 = 𝑣𝑠 s;

(15)

L𝑝 ∶= Ḟ 𝑝 F−1 𝑝 .

(17)

The elastic strain rate and elastic spin tensors are defined to be the symmetric and antisymmetric parts of L𝑒 . 1 1 (L + L𝑇𝑒 ); W𝑒 = (L𝑒 − L𝑇𝑒 ). (18) 2 𝑒 2 The symmetric and anti-symmetric parts of L𝑝 are respectively called the plastic strain rate tensor and the plastic spin tensor. D𝑒 =

(11)

From these expressions, it may be seen that 𝛾̇ 𝑝 is related to the dislocation velocity in the slip direction s. Using Eq. (11), we may express K𝐶𝐷𝐷 in terms of the internal variable 𝛾̇ 𝑝 . ( ) ( )4 𝑐𝑡 1 2 K𝐶𝐷𝐷 = 𝜌 1+ sin 𝜃 log(1∕𝑟0 )𝛾̇ 𝑝2 . (12) 8𝜋 0 𝑐𝑙

1 1 (L + L𝑇𝑝 ); W𝑝 = (L𝑝 − L𝑇𝑝 ). (19) 2 𝑝 2 In the present formulation, plastic flow is assumed to be only due to the gliding motion of dislocation. Thus the plastic spin tensor W𝑝 is set to be zero, which implies that L𝑝 = D𝑝 . Now, from the second part of

D𝑝 =

12

Md.M. Rahaman et al.

International Journal of Non-Linear Mechanics 95 (2017) 10–18

Eq. (17), one can arrive at the evolution rule for F𝑝 given by Ḟ 𝑝 = D𝑝 F𝑝 , the solution of which is the matrix exponential function given below. F𝑝 (𝑡) = exp((𝑡 − 𝑡0 )D𝑝 )F0𝑝 ;

F𝑝 (𝑡0 ) = F0𝑝 .

plastic deformation. Associated with the micro-traction 𝝌(n), a vectorvalued stress called the micro-stress 𝝃 is introduced. Now, using the second transport theorem and using 𝝌(n) = 𝝃 ⋅ n in Eq. (25), one may arrive at,

(20)

Since D𝑝 is symmetric, exp((𝑡 − 𝑡0 )D𝑝 ) is symmetric and positive definite. We accordingly conclude that the plastic distortion tensor remains symmetric and positive definite for all time if F0𝑝 = I. Thus, the assumption of irrotational plastic flow i.e. W𝑝 = 0 renders the plastic distortion tensor a pure stretch and allows us to characterize the plastic deformation through the plastic strain rate tensor D𝑝 , which is written as a product of its magnitude 𝛾̇ 𝑝 and a unit tensor N𝑝 . D𝑝 = 𝛾̇ 𝑝 N𝑝 ;

𝛾̇ 𝑝 = ‖D𝑝 ‖; N𝑝 ∶=

D𝑝 ‖D𝑝 ‖

∫

3.3. Thermodynamics first law and internal energy evolution In the framework of rational thermodynamics, existence of an internal energy density is assumed and the evolution of it is derived based on the conservation of energy. The first law of thermodynamics, a version of the law of energy conservation, could be stated as follows (see Reddy [29]):

The internal variable 𝛾𝑝 , which is positive and non-decreasing by definition, is taken as an additional field variable in this theory. In order to quantify the elastic part of total deformation, a strain tensor E𝑒 , which is a measure of the change in length of an infinitesimal material line element between the intermediate and deformed configurations, is introduced on the intermediate configuration. Recall that the intermediate configuration is the collection of tangent spaces pulled back using F−1 𝑒 . The expression for E𝑒 in terms of the Green Lagrangian strain E = 12 (F𝑇 F − I) and the plastic distortion F𝑝 is given as,

𝑑𝐸 = 𝛿𝑊 + 𝛿𝑄;

( ) 1 −𝑇 −1 F F −I . (23) 2 𝑝 𝑝 It is interesting to observe that the assumption of incompressible plastic deformation i.e. 𝑑𝑒𝑡(F𝑝 ) = 1 implies that Tr(D𝑝 )(𝑡) = 0 which can be seen from the following arguments. From Eq. (20), it is easy to find that 𝑑𝑒𝑡(F𝑝 ) = 1 implies det(exp(((𝑡 − 𝑡0 )D𝑝 ))) = 1. Also, we know that exp(Tr((𝑡 − 𝑡0 )D𝑝 )) = det(exp(((𝑡 − 𝑡0 )D𝑝 ))) is the solution to the scalar differential equation 𝑦̇ = Tr(D𝑝 )𝑦, with initial condition 𝑦0 = 1. Now, consider the following cases. If Tr(D𝑝 ) > 0, the solution 𝑦(𝑡) → ∞ as 𝑡 → ∞. On the other hand, if Tr(D𝑝 ) < 0, 𝑦(𝑡) → 0 as 𝑡 → ∞. However, for the case of Tr(D𝑝 ) = 0, every point 𝑦 in the phase space is a neutral fixed point of the differential equation and the solution 𝑦(𝑡) = 𝑦0 , ∀𝑡. Thus, the condition det F𝑝 (𝑡) = 1 is satisfied by imposing Tr(D𝑝 )(𝑡) = 0 on D𝑝 for all time 𝑡.

𝑈=

∫

(29)

𝜌0 𝑒𝑑𝑉 .

𝑑𝑈 = 𝜌 𝑒𝑑𝑉 ̇ . ∫ 0 𝑑𝑡

(30)

Similarly the rate of the kinetic energy could be written, considering contributions from its conventional macroscopic and the presently introduced microscopic constituents. First, we write 𝐾 in terms of its macro and micro components: ( ) 1 1 2 2 𝐾= 𝜌 v.v + 𝜌0 𝑙𝑚 𝛾̇ 𝑝 𝑑𝑉 . (31) ∫ 2 0 2

With the kinematics available entirely from geometrical considerations without a reference to the external forcing, (macro) momentum balance laws enable a description of the continuum dynamics under the action of the external forces. Local forms of linear and angular momentum balance, which hold for each material point in the reference configuration, are as follows (see Reddy [29]):

One thus obtains the rate of the kinetic energy as, ( ) 1 1 2 𝑑𝐾 ̇ + 𝜌0 𝑙 𝑚 = 𝜌 v.v 𝛾̈ 𝑝 𝛾̇ 𝑝 𝑑𝑉 . ∫ 2 0 𝑑𝑡 2

(32)

Now an expression for the external power 𝑊 ◦ may be written for an arbitrary sub-domain  ⊂  with boundary 𝜕, considering the macroscopic part corresponding to macro-traction Pn and body force b0 , and the microscopic part corresponding to the micro-traction 𝝌(n).

(24)

P is the first Piola stress tensor, b0 the body force and ∇X the gradient operator in the reference configuration. A micro force balance is proposed based on the premise that the ‘‘rate of change of momentum associated with the rate of plastic deformation equals all forces arising from (and maintaining) the effect of plastic deformation’’ [30]. For any arbitrary sub-domain  ⊂  with boundary 𝜕, we may write, 𝑑 𝝌(n)𝑑𝐴 + 𝜋𝑑𝑉 ̄ 𝜌 𝑙2 𝛾̇ 𝑑𝑉 = ∫𝜕 ∫ 𝑑𝑡 ∫ 0 𝑚 𝑝

(28)

The rate of the internal energy is given by

3.2. Balance laws

FP𝑇 = PF𝑇

𝑑𝐸 = 𝑊 ◦ + 𝑄◦ 𝑑𝑡

where 𝑊 is the work done by the external forces on the system and 𝑄 the heat supply to the system. Note that 𝑊 and 𝑄 are path dependent and hence not state functions; (.)◦ denotes the rate of an inexact differential, signifying path dependence of 𝑊 and 𝑄. The total energy 𝐸 equals the sum of the internal energy 𝑈 and the kinetic energy 𝐾, that is, 𝐸 = 𝐾+𝑈 . For an arbitrary subdomain  ⊂ , 𝑈 may be written in terms of the specific internal energy 𝑒 as,

−1 E𝑒 = F−𝑇 𝑝 EF𝑝 +

̇ ∇X .P + b0 = 𝜌0 v;

(27)

The local micro-force balance given by Eq. (27) acts as a dynamic flow rule and provides for the evolution of the equivalent plastic strain in a yield-free set up.

(21)

(22)

𝛾̇ 𝑝 (𝑠)𝑑𝑠.

(26)

(∇X .𝝃 + 𝜋)𝑑𝑉 ̄ .

2 ∇X .𝝃 + 𝜋̄ = 𝜌0 𝑙𝑚 𝛾̈ 𝑝 .

𝑡

∫−∞

∫

Applying the localization theorem on Eq. (26) leads to the following local form for the micro-force balance.

𝛾̇ 𝑝 is treated as an internal variable. It may be observed that 𝛾̇ 𝑝 ≥ 0 by its definition itself and N𝑝 determines the direction of the plastic flow. The equivalent plastic strain is defined as, 𝛾𝑝 =

2 𝜌0 𝑙𝑚 𝛾̈ 𝑝 𝑑𝑉 =

𝑊◦ =

∫𝜕

Pn.v𝑑𝐴 +

∫

b0 .v𝑑𝑉 +

∫𝜕

𝝌(n).𝛾̇ 𝑝 𝑑𝐴.

(33)

Employing the relation 𝝌(n) = 𝝃.n and invoking Stokes’ theorem for the boundary terms in Eq. (33), the following is obtained. 𝑊◦ =

∫

( ) (∇X .P + b0 ).v + P ∶ Ḟ + ∇X .𝝃 𝛾̇ 𝑝 + 𝝃.∇X 𝛾̇ 𝑝 𝑑𝑉 .

(34)

Considering the heat source ℎ and the heat flux vector q, the thermal power 𝑄◦ may be expressed as follows.

(25)

where n is the unit normal to the boundary 𝜕, 𝝌(n) is the micro-traction and 𝜋̄ is the net micro body force which acts as the driving force for the

𝑄◦ = 13

∫𝜕

( ) 𝜌0 ℎ − ∇X .q 𝑑𝑉 .

(35)

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International Journal of Non-Linear Mechanics 95 (2017) 10–18

Substituting Eqs. (32) and (34)–(36) in Eq. (28) and using macro and micro force balances given by Eqs. (24) and (27), we obtain the following equation. ∫

𝜌0 𝑒𝑑𝑉 ̇ =

∫

( ) P ∶ Ḟ − 𝜋̄ 𝛾̇ 𝑝 + 𝝃.∇X 𝛾̇ 𝑝 + 𝜌0 ℎ − ∇X .q 𝑑𝑉 .

In terms of the Helmholtz free energy 𝜓 ∶= 𝑒 − 𝜃𝑠, Eq. (44) is rewritten as, ) ( ̇ − q .∇x 𝜃 ≥ 0. − 𝜌0 𝜓̇ + P ∶ Ḟ − 𝜋̄ 𝛾̇ 𝑝 + 𝝃.∇x 𝛾̇ 𝑝 − 𝜃𝑠 𝜃

(36)

Employing Eqs. (16), (17) and (43) into Eq. (45) and using the relation 𝑇 Ė 𝑒 = (1∕2)(Ḟ 𝑒 F𝑒 + F𝑇𝑒 Ḟ 𝑒 ) we arrive at, ( ) −𝜌0 F𝑒 𝜕E𝑒 𝜓 + PF𝑇𝑝 ∶ Ḟ 𝑒 + (𝝃 − 𝜌0 𝜕∇𝛾𝑝 .𝜓).∇X 𝛾̇𝑝 ( ) ( ) q (46) − 𝜌0 𝜕𝜃 𝜓 + 𝜌0 𝑠 𝜃̇ + F𝑇𝑒 PF𝑇𝑝 ∶ N𝑝 − 𝜋̄ 𝛾̇ 𝑝 − .∇X 𝜃 ≥ 0. 𝜃

An evolution equation for the internal energy density results upon localization of Eq. (36) which can be written as 𝜌0 𝑒̇ = P ∶ Ḟ − 𝜋̄ 𝛾̇ 𝑝 + 𝝃.∇X 𝛾̇ 𝑝 + 𝜌0 ℎ − ∇X .q.

(37)

Applying the Coleman–Noll procedure [31] to the equation above, we obtain the following constitutive relations.

3.4. Material frame indifference and constitutive restrictions Material frame indifference is an important principle in continuum mechanics to impose restrictions on constitutive functions [29]. A change of reference frame is given by an element from the Euclidean group, which is characterized by a rotation Q ∈ SO(3), a translation in space c ∈ R3 and a translation in time 𝑎 ∈ R. In general, Q and c are functions of time. The change of reference frame is given by, x∗ = Qx + c;

𝑡∗ = 𝑡 + 𝑎

(38)

P = 𝜌0 F𝑒 𝜕E𝑒 𝜓F−𝑇 𝑝

(47)

𝝃 = 𝜌0 𝜕∇𝛾𝑝 .𝜓

(48)

𝑠 = −𝜕𝜃 𝜓.

(49)

Then the inequality given by Eq. (46) boils down to requiring that the last two terms are greater than or equal to zero, i.e. ( ) q (50) F𝑇𝑒 PF𝑇𝑝 ∶ N𝑝 − 𝜋̄ 𝛾̇ 𝑝 − .∇X 𝜃 ≥ 0. 𝜃

x∗ and 𝑡∗ are the co-ordinate and time measured from an arbitrary moving frame. A variable with the superscript ∗ denotes a quantity from the arbitrary moving frame. The transformation rule for the deformation gradient under a change of reference frame is given by,

This constraint can be satisfied by choosing

F∗ = QF.

(39) F𝑇𝑒 PF𝑇𝑝 ∶ N𝑝 − 𝜋̄ = g1 𝛾̇ 𝑝 ;

As F𝑝 is defined on the reference configuration, F∗𝑃 = F𝑝 . From Eq. (39) it follows that F∗𝑒 = QF𝑒 . The velocity vector in the arbitrary rotating frame is given by, ẋ ∗ − Qẋ = ċ + W(x∗ − c);

̇ 𝑇. W = QQ

q = −g2 ∇X 𝜃;

L∗𝑝 = L𝑝 ;

L∗𝑒 = W + QL𝑒 Q𝑇 .

(41)

Since the Cauchy stress T is a frame indifferent quantity, its transformation under a change of reference frame is given by T∗ = QTQ𝑇 . Using this, the transformation rule for the first Piola stress tensor may be deduced as P∗ = QP. With the vector micro stress 𝝃 defined on the reference configuration, frame indifference demands 𝝃 ∗ = 𝝃. Suppose ̃ 𝑒 , 𝜶) is a constitutive function that depends on the elastic that 𝚵(F distortion F𝑒 and an array of internal variables 𝜶. Then the requirement of the constitutive function being frame indifferent is reduced to the ̂ 𝑒 , 𝜶). existence of the function 𝚵(E

N𝑝 =

We now present the constitutive equations obtained upon imposing the second law of thermodynamics. The Helmholtz free energy 𝜓 is assumed to depend constitutively on the elastic strain E𝑒 , the gradient of the equivalent plastic strain ∇X 𝛾𝑝 and temperature 𝜃. With this, the free energy is expressed as,

(52)

Tdev 𝑝 ‖Tdev 𝑝 ‖

;

𝜏 ∶= F𝑇𝑒 PF𝑇𝑝 ∶ N𝑝 = ‖Tdev 𝑝 ‖.

(54)

3.6. Temperature evolution The evolving visco-plastic response typically entails significant changes in the resulting temperature and a visco-plasticity model is ideally required to reproduce these changes faithfully. We accordingly aim at deriving a coupled, thermodynamically consistent temperature evolution equation. We start by considering the free energy as the sum of the elastic energy 𝜌0 𝜓 el , a defect energy 𝜌0 𝜓 de and the free energy due to temperature 𝜓 𝜃 . The decomposition, so accomplished, may be thought of as a generalization over Gurtin’s [23] isothermal model.

(42)

Using chain rule, the rate of Helmholtz free energy is written as, ̇ 𝜓̇ = 𝜕E𝑒 𝜓 ∶ Ė 𝑒 + 𝜕∇𝛾𝑝 𝜓.∇𝛾̇𝑝 + 𝜕𝜃 𝜓 𝜃.

g2 ≥ 0

(51)

Assuming co-directionality between the plastic flow and Tdev = T𝑝 − 𝑝 (1∕3)Tr(T𝑝 ), we arrive at the following conclusion.

3.5. Second law of thermodynamics and constitutive modelling

𝜓 = 𝜓(E ̂ 𝑒 , ∇X 𝛾𝑝 , 𝜃).

g1 ≥ 0

where g1 and g2 are constitutive functions. Since N𝑝 is symmetric, only the symmetric part of F𝑇𝑒 PF𝑇𝑝 does work on D𝑝 . Moreover the pressure part of F𝑇𝑒 PF𝑇𝑝 does no work on D𝑝 ; this is because D𝑝 is deviatoric. The validity of the last statement may be gauged from the fact that the inner product between a deviatoric tensor and an arbitrary tensor equals the inner product between the deviatoric tensor and the deviatoric part of the arbitrary tensor. For clarity of notation, we introduce the following stress, ( ) T𝑝 ∶= Sym F𝑇𝑒 PF𝑇𝑝 . (53)

(40)

The rate quantities transform according to the following rules. L∗ = W + QLQ𝑇 ;

(45)

(43)

In the equation above, 𝜕(.) with a suffix represents the derivative of a function with respect to one of its arguments keeping the others fixed. The second law of thermodynamics requires that the rate of entropy production be greater than or equal to zero, which is enforced using the local form of Clausius–Duhem inequality, (q) 𝜌 ℎ 𝜌0 𝑠̇ + ∇X . − 0 ≥ 0. (44) 𝜃 𝜃

𝜌0 𝜓 = 𝜌0 𝜓 el + 𝜌0 𝜓 de + 𝜓 𝜃 .

(55)

Using the Legendre transform 𝜓 ∶= 𝑒 − 𝜃𝑠 along with Eqs. (49) and (55), the internal energy can be written as, ) ( 𝑑𝜓 𝜃 𝜌0 𝑒 = 𝜌0 𝜓 el + 𝜌0 𝜓 de + 𝜓 𝜃 − 𝜃 . (56) 𝑑𝜃 14

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For zero mechanical strains i.e. F𝑒 = I and F𝑝 = I, considerations of the first law of thermodynamics as in Eqs. (28) and (56) give the following:

stress–strain response, so that any reasonable choice of the dynamic yield function should suffice. As we know that such a function should contain terms reflecting strain hardening, strain rate hardening and temperature softening, we choose the functional form of g1 as, ( ) ( 𝛾̇ 𝑝 )𝑚1 −1 ( ) 1 ̄ 𝑚2 g1 = 𝑆0 + 𝐻0 𝛾𝑝𝑛 𝜃 (64) 𝑑0 𝑑0 ( ) 𝜃−𝜃ref where 𝐻0 is the hardening modulus, 𝜃̄ = 𝜃 −𝜃 the homologous melt ref temperature, 𝑛 the hardening parameter, 𝑑0 the reference strain rate, 𝑚1 the strain rate hardening parameter and 𝑚2 the temperature softening parameter which depends on the initial temperature 𝜃0 . In this study 𝑚2 is chosen as a quadratic function of 𝜃0 , i.e. 𝑚2 = 𝑎𝜃02 + 𝑏𝜃0 + 𝑐. Use of Eqs. (54) and (64) in Eq. (52) provides the expression of 𝜋. ̄ ) ( 𝛾̇ 𝑝 )𝑚1 ( 𝑛 𝜋̄ = ‖Tdev 𝜃̄ 𝑚2 . (65) 𝑝 ‖ − 𝑆0 + 𝐻0 𝛾𝑝 𝑑0

𝑑2𝜓 𝜃 𝑑𝜃 = 𝛿𝑄 ∶= 𝜌0 𝐶𝑉 𝑑𝜃 (57) 𝑑𝜃 2 where 𝐶𝑉 is the specific heat. Consideration of Eq. (57) in the differential form of Eq. (56) leads to the following:

𝜌0 𝑑𝑒 = −𝜃

̇ 𝜌0 𝑒̇ = 𝜌0 𝜓̇ el + 𝜌0 𝜓̇ de + 𝜌0 𝐶𝑉 𝜃.

(58)

Employing Eqs. (37), (47), (48), (60) and (62) in Eq. (58), the temperature evolution may be obtained as, 𝜃̇ =

] 1 [ (𝜏 − 𝜋) ̄ 𝛾̇ 𝑝 + 𝜌0 ℎ − ∇X .q . 𝜌0 𝐶 𝑉

(59)

3.7. Specialization of constitutive functions Our model so far is quite general. However, a quantification of the thermodynamic forces requires a specific choice for the constitutive functions, viz. 𝜓 el , 𝜓 de , g1 and g2 . From the application perspective, quadratic forms for 𝜓 el and 𝜓 de could be used to quantify the first Piola stress and the vector micro-stress respectively. Computation of the driving force 𝜋̄ is accomplished through a proper choice of g1 based on the physical understanding of the governing factors (e.g. applied strain rate, temperature etc.) which affect the plastic flow. The constitutive function g2 is taken as the thermal conductivity 𝜅 (a material constant) so Eq. (52) is identifiable as the Fourier law of heat conduction.

3.8. Dynamic flow rule Augmentation of the constitutive equations for the micro-forces given by Eqs. (63) and (65) with the micro-force balance as in Eq. (27) leads to the following second order partial differential equation. ( ) ( 𝛾̇ 𝑝 )𝑚1 2 𝑛 𝜌0 𝑙 𝑚 𝛾̈ 𝑝 = 𝑆0 𝑙12 ∇2X 𝛾𝑝 + ‖Tdev 𝜃̄ 𝑚2 . ‖ − 𝑆 + 𝐻 𝛾 (66) 0 0 𝑝 𝑝 𝑑0 Eq. (66) acts as a flow rule which incorporates the dislocation acceleration effect through the term involving second order time derivative of the equivalent plastic strain. Although, for low strain rate loading, the left hand side of Eq. (66) may be negligible, it assumes a much more important role under high strain rate loading as the dislocation acceleration affects the plastic flow. For an inhomogeneous plastic flow, Eq. (66) can be seen as a generalization of the micro-force balance by Gurtin [23] for high strain rates. In general, solution to Eq. (66) requires specified initial conditions on 𝛾𝑝 and 𝛾̇ 𝑝 and boundary conditions. However, for homogeneous plastic deformation, the first term on the right hand side of Eq. (66) vanishes and hence Eq. (66) reduces to a nonlinear ordinary differential equation for 𝛾𝑝 . In such a case, Eq. (66) may be thought as a generalization of the flow rule of the consistency visco-plasticity theory [32].

3.7.1. Elastic free energy and constitutive equation for macro stress Presently, we limit our attention to isotropic materials and consider the following quadratic form of the elastic free energy. 1 𝜆Tr(E𝑒 )2 (60) 2 where 𝐺 and 𝜆 are the Lamé parameters. Now, utilizing Eqs. (47) and (55), we obtain the expression for the first Piola stress tensor P in terms of the kinematic variables as, 𝜌0 𝜓 el = 𝐺(E𝑒 ∶ E𝑒 ) +

( ) P = F𝑒 2𝐺E𝑒 + 𝜆Tr(E𝑒 )I F−𝑇 𝑝 .

(61)

Substituting Eq. (61) in Eq. (53), we can estimate T𝑝 , the deviatoric part of which determines the plastic flow direction (see Eq. (54)).

4. Numerical results and discussions

3.7.2. Defect free energy and constitutive equation for vector micro-stress Free energy associated with the defect is assumed to be of the form,

We now undertake an assessment of the predictive capability of our model through numerical simulations. Specifically, we validate the simulated results against high strain rate experimental data for molybdenum and high-strength low-alloy steel (HSLA-65) [33,34]. Our focus will be on how the proposed model captures the experimentally observed oscillations in the stress–strain response. The relevant macroscopic experiments correspond to a uniform state of stress and homogeneous plastic deformation. Uniform state of stress implies that the macroscopic force balance equation (24) is trivially satisfied in the absence of body force. Homogeneous plastic deformation also implies that the first term of the right hand side of Eq. (66) is zero. Moreover, experiments under high strain rate loading typically correspond to the adiabatic condition; so the divergence of the heat flux term appearing in Eq. (59) vanishes. Thus, in practice, we solve a set of non-linear ordinary differential equations in order to compute the uni-axial strain and stress respectively using the kinematic as well as constitutive relations. Computation of stress–strain response is done by choosing the following deformation gradient which corresponds to a homogeneous deformation.

1 𝑆 𝑙2 ‖∇ X 𝛾𝑝 ‖2 (62) 2 01 where 𝑆0 is the initial yield strength and 𝑙1 is an energetic length scale. Employing Eqs. (48) and (55), the vector micro-stress 𝝃 may be obtained as, 𝜌0 𝜓 de =

𝝃 = 𝑆0 𝑙12 ∇ X 𝛾𝑝 .

(63)

This vector micro-stress may be identified as the back stress that enables the model to capture the Bauschinger effect. 3.7.3. Constitutive equation for 𝜋̄ Quantification of the driving force 𝜋̄ for plastic deformation needs a specific choice of the constitutive function g1 . It is greatly interesting and useful to observe that g1 may be identified with the so called dynamic yield stress. This identification makes possible to interface the most suitable dynamic yield function with the proposed model for a given strain rate and temperature. Thus, with the best available evidence and/or physical understanding of the dynamic yield, g1 may be judiciously chosen from amongst the existing flow-stress models (e.g. JC, ZA, VA etc.). This, by itself, constitutes an interesting study that we leave out of the scope of this work. Our present interest is only in assessing the extent to which micro-inertia influences the continuum

F = (1 − 𝛼)(e1 ⊗ e1 ) + (e2 ⊗ e2 )

(67)

𝛼 is a time dependent parameter, e1 and e2 are the standard basis vectors. Since we are interested in constant strain rate loading, 𝛼 is assumed to be a linearly varying function of time. The generated 15

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(a) Molybdenum.

(b) HSLA-65.

Fig. 1. Stress–strain response for different strain rate loading 𝜖.̇ The solid line represents the stress–strain response for non-zero 𝑙𝑚 and the line with circular marker for 𝑙𝑚 = 0. Table 1 Material parameters for Molybdenum and HSLA-65.a

Mo HSLA-65

𝐶𝑣 (J/K)

𝐻0 (MPa)

𝑆0 (MPa)

𝑛

𝑚1

𝑎

𝑏

𝑐

𝑑0 (1∕s)

𝜃ref (K)

𝑙𝑚 (m)

277 460

300 500

448 790

0.45 0.3

0.1 0.1

−2.56𝑒−7 1.47𝑒−7

4.24𝑒−4 −6.71𝑒−5

0.11 0.18

100 100

398 77

1.0𝑒−3 3.0𝑒−3

a The parameters mentioned in the table cannot be obtained by fitting the flow curve with the constitutive function for 𝜋. ̄ Since 𝜋̄ depends on the rate of equivalent plastic strain which is only obtainable through solution of dynamic flow. Hence, an inverse problem is needed to determine the material parameters.

rates. We now aim at assessing how the computed response through our model corresponds with available experimental evidence, specifically for molybdenum and HSLA-65 at high strain rates with varying temperature. These are shown in Figs. 2 and 3. Fig. 2(b), whilst providing adequate evidence of a good correspondence of the oscillation patterns in the predicted with 𝑙𝑚 ≠ 0 and experimental response of molybdenum, is perhaps also indicative of a slightly degraded performance of our model for lower temperature. As anticipated and reported in Fig. 2(a), the predicted oscillations disappear for 𝑙𝑚 = 0. Indeed, for HSLA-65, the prediction 𝑙𝑚 ≠ 0 versus experiment match-up appears to be still better; see Fig. 3(b) and (a), the latter showing the predicted solutions with 𝑙𝑚 = 0. Prior to concluding this section, a word about the discrepancies in the prediction–experiment match-up especially at low temperatures. In such cases, neglect of the heat flux consequent upon the assumed adiabatic condition may no longer hold. This is also clearly suggestive of a future possibility to improve upon the current model.

stress associated with the deformation is computed based on the elastic part of the deformation gradient. The elastic and plastic parts of the deformation gradient are determined based on the flow rule Eq. (66), which is a nonlinear ordinary differential equation for 𝛾𝑝 . A second order implicit integration scheme is used to integrate the flow rule from which the stress–strain response for molybdenum and HSLA-65 is recovered. In high temperature applications, a frequently used metal is molybdenum (𝜌0 = 10 200 kg∕m3 ) thanks to its ability to maintain reasonably high strength even at high temperatures. Melting temperature of molybdenum is as high as 2885 K which makes it useful even in nonoxidizing conditions (above 1273 K). HSLA-65 (𝜌0 = 7850 kg∕m3 ) is extensively used in naval surface vessels and submarines owing to its higher strength, better impact toughness and easier weldability. Material parameters for molybdenum and HSLA-65 used for our simulations are given in Table 1. Nemat-Nasser et al. [33,34] have carried out uniaxial compression tests on molybdenum and HSLA-65 at high strain rates and temperatures and these data are used here for assessing the predictive capability of the proposed model. We investigate the source of oscillations in the continuum stress– strain response at high strain rates. For a given micro-inertial length scale 𝑙𝑚 , stress–strain response is computed for different strain rate loading. Fig. 1 shows the stress–strain response for HSLA-65 and molybdenum for different strain rates. The graphs are plotted with 𝑙𝑚 = 0 as well as with non-zero 𝑙𝑚 . As shown in Fig. 1, while non-zero 𝑙𝑚 does not make much of a difference in the response for low strain rates, it does bring in the additional feature of oscillations in the stress–strain response at high strain rates. This is proof positive of the significance the micro-inertia term has as a continuum representation of dislocation inertia. This is also in line with Kosevich’s [21] observation that, though the material response is mainly influenced by the dissipative forces for small dislocation acceleration, dislocation inertia starts playing an important role as the dislocation acceleration is larger. It is thus clear that, for small strain rate loading, our model reproduces the same result as given by a flow rule without the microinertia, the effect micro-inertia showing up only for higher applied strain

5. Conclusions In an effort to incorporate the effect of dislocations acceleration in the continuum response of isotropic polycrystalline solids under high strain rate loading, we have presented a micro-inertia driven dynamic flow rule in the framework of rational thermodynamics. The dynamic flow rule is in the form of a second order partial differential equation, solution to which gives the evolution of the equivalent plastic strain. Towards this, the equivalent plastic strain rate is expressed through the average dislocation velocity in the slip direction. This enables writing the kinetic energy due to dislocation motion in terms of the equivalent plastic strain rate. We are then led to an expression for the micro-inertial length scale by an energy equivalence argument involving our proposed micro-kinetic energy and the kinetic energy due to dislocation motion, as in the CDD theory. Through numerical simulations and a comparative assessment of predicted solutions with the experimental evidence for molybdenum and HSLA-65, we have explicated the significant role 16

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Fig. 2. Stress–strain response of molybdenum at strain rate 8000/s. Solid lines indicate the prediction from the proposed model; the circular marker denotes the experimental data reproduced from [33]. (a) Comparison of stress–strain response computed using flow rule with 𝑙𝑚 = 0 m against experimental data. (b) Comparison of stress–strain response computed using flow rule with 𝑙𝑚 = 1 × 10−3 m against experimental data.

Fig. 3. Stress–strain response of HSLA-65 at strain rate 3000/s. Solid lines indicate prediction from the proposed model, circular markers represent the experimental data reproduced from [34]. (a) Comparison of stress–strain response computed with flow rule 𝑙𝑚 = 0 m against experimental data. (b) Comparison of stress–strain response computed with flow rule 𝑙𝑚 = 3 × 10−3 m against experimental data.

of micro-inertia in reproducing the oscillatory stress–strain responses under high strain rates. However, the significance of the micro-inertia term recedes as the applied strain rate tends lower. Thus, augmentation with a micro-inertia term enables the flow rule to be applicable at higher strain rates and hence should qualify as a non-trivial addition that has been overlooked so far. Although our micro-inertia based flow rule is in the set-up of a micro-force balance in line with the work of Gurtin and his colleagues [7,23,35], our identification of the constitutive function g1 as the so called dynamic yield stress affords an accommodation of the proposed micro-inertia term within several other flow models, viz. ZA, VA, mechanical threshold stress (MTS) or Preston–Tonks–Wallace (PTW) models. Specifically, as our numerical work indicates, the microinertia does a good job of replicating the oscillatory pattern in the experimental stress–strain response if the flow model reproduces well the overall flow curve minus the oscillations. A good agreement with the overall flow curve is in turn possible by choosing g1 appropriately, which is determined by the flow model itself (such as ZA, VA, PTW, MTS etc.). Indeed, given the material, applied strain rate and temperature,

choosing the right flow model prior to the application of micro-inertia should in itself be a useful study. Acknowledgements This work is funded by the Defence Research and Development Organization, Government of India, through Grant No. DRDO/0642. The last author gratefully acknowledges the support by the Oscar S Wyatt Endowed Chair in Mechanical Engineering at Texas A&M University, College Station. The authors express their sincere thanks to Dr. Kari Santaoja of Aalto University, Finland, for his constructive comments on the contents of the paper. References [1] H.M. Zbib, T.D. de la Rubia, A multiscale model of plasticity, Int. J. Plast. 18 (9) (2002) 1133–1163. [2] O. Dillon, J. Kratochvil, A strain gradient theory of plasticity, Int. J. Solids Struct. 6 (12) (1970) 1513–1533.

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International Journal of Non-Linear Mechanics 95 (2017) 10–18 [19] S. Nemat-Nasser, T. Okinaka, L. Ni, A physically-based constitutive model for bcc crystals with application to polycrystalline tantalum, J. Mech. Phys. Solids 46 (6) (1998) 1009–1038. [20] C. Gao, L. Zhang, Constitutive modelling of plasticity of fcc metals under extremely high strain rates, Int. J. Plast. 32 (2012) 121–133. [21] A. Kosevich, Dynamical theory of dislocations, Phys.-Usp. 7 (6) (1965) 837–854. [22] Z. Wang, I. Beyerlein, R. LeSar, Dislocation motion in high strain-rate deformation, Phil. Mag. 87 (16) (2007) 2263–2279. [23] M.E. Gurtin, On the plasticity of single crystals: free energy, microforces, plasticstrain gradients, J. Mech. Phys. Solids 48 (5) (2000) 989–1036. [24] T. Mura, Micromechanics of Defects in Solids, Springer Science & Business Media, 2013. [25] W. Cai, A. Arsenlis, C.R. Weinberger, V.V. Bulatov, A non-singular continuum theory of dislocations, J. Mech. Phys. Solids 54 (3) (2006) 561–587. [26] T. Hochrainer, M. Zaiser, P. Gumbsch, A three-dimensional continuum theory of dislocation systems: kinematics and mean-field formulation, Phil. Mag. 87 (8–9) (2007) 1261–1282. [27] T. Hochrainer, Thermodynamically consistent continuum dislocation dynamics, J. Mech. Phys. Solids 88 (2016) 12–22. [28] A. Yavari, A. Goriely, Riemann–Cartan geometry of nonlinear dislocation mechanics, Arch. Ration. Mech. Anal. 205 (1) (2012) 59–118. [29] J.N. Reddy, An Introduction to Continuum Mechanics with Applications, second ed., Cambridge University Press, 2013. [30] P. Naghdi, A. Srinivasa, A dynamical theory of structured solids. I basic developments, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 345 (1677) (1993) 425–458. [31] B.D. Coleman, W. Noll, The thermodynamics of elastic materials with heat conduction and viscosity, Arch. Ration. Mech. Anal. 13 (1) (1963) 167–178. [32] W. Wang, L. Sluys, R. De Borst, Viscoplasticity for instabilities due to strain softening and strain-rate softening, Internat. J. Numer. Methods Engrg. 40 (20) (1997) 3839– 3864. [33] S. Nemat-Nasser, W. Guo, M. Liu, Experimentally-based micromechanical modeling of dynamic response of molybdenum, Scr. Mater. 40 (7) (1999) 859–872. [34] S. Nemat-Nasser, W.-G. Guo, Thermomechanical response of HSLA-65 steel plates: experiments and modeling, Mech. Mater. 37 (2) (2005) 379–405. [35] M.E. Gurtin, On a framework for small-deformation viscoplasticity: free energy, microforces, strain gradients, Int. J. Plast. 19 (1) (2003) 47–90.

[3] N. Fleck, J. Hutchinson, A phenomenological theory for strain gradient effects in plasticity, J. Mech. Phys. Solids 41 (12) (1993) 1825–1857. [4] N. Fleck, J. Hutchinson, A reformulation of strain gradient plasticity, J. Mech. Phys. Solids 49 (10) (2001) 2245–2271. [5] H. Gao, Y. Huang, W. Nix, J. Hutchinson, Mechanism-based strain gradient plasticityi. Theory, J. Mech. Phys. Solids 47 (6) (1999) 1239–1263. [6] H. Gao, Y. Huang, Taylor-based nonlocal theory of plasticity, Int. J. Solids Struct. 38 (15) (2001) 2615–2637. [7] M.E. Gurtin, A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations, J. Mech. Phys. Solids 50 (1) (2002) 5–32. [8] Y. Huang, S. Qu, K. Hwang, M. Li, H. Gao, A conventional theory of mechanism-based strain gradient plasticity, Int. J. Plast. 20 (4) (2004) 753–782. [9] G.Z. Voyiadjis, D. Faghihi, Thermo-mechanical strain gradient plasticity with energetic and dissipative length scales, Int. J. Plast. 30 (2012) 218–247. [10] D. Jia, K. Ramesh, E. Ma, L. Lu, K. Lu, Compressive behavior of an electrodeposited nanostructured copper at quasistatic and high strain rates, Scr. Mater. 45 (5) (2001) 613–620. [11] J. Campbell, A. Eleiche, M. Tsao, Strength of metals and alloys at high strains and strain rates, in: Fundamental Aspects of Structural Alloy Design, Springer, 1977, pp. 545–563. [12] G.R. Johnson, W.H. Cook, A constitutive model and data for metals subjected to large strains, high strain rates and high temperatures, in: Proceedings of the 7th International Symposium on Ballistics, Vol. 21, 1983, The Hague, The Netherlands, 1983, pp. 541–547. [13] D.J. Bammann, An internal variable model of viscoplasticity, Internat. J. Engrg. Sci. 22 (8–10) (1984) 1041–1053. [14] F.J. Zerilli, R.W. Armstrong, Dislocation-mechanics-based constitutive relations for material dynamics calculations, J. Appl. Phys. 61 (5) (1987) 1816–1825. [15] R. Liang, A.S. Khan, A critical review of experimental results and constitutive models for BCC and FCC metals over a wide range of strain rates and temperatures, Int. J. Plast. 15 (9) (1999) 963–980. [16] G.Z. Voyiadjis, F.H. Abed, Microstructural based models for bcc and fcc metals with temperature and strain rate dependency, Mech. Mater. 37 (2) (2005) 355–378. [17] J.-B. Kim, H. Shin, Comparison of plasticity models for tantalum and a modification of the PTW model for wide ranges of strain, strain rate, and temperature, Int. J. Impact Eng. 36 (5) (2009) 746–753. [18] P. Follansbee, U. Kocks, A constitutive description of the deformation of copper based on the use of the mechanical threshold stress as an internal state variable, Acta Metall. 36 (1) (1988) 81–93.

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