This article proposes a continuum thermomechanical homogenization method inspired by the Irving–Kirkwood procedure relating the atomistic equations of motion to the balance laws of continuum mechanics. This method yields expressions for the macroscopic stress and heat flux in terms of microscopic kinematic and kinetic quantities. The resulting equation for macroscopic stress affords a rational comparison with the widely used Hill–Mandel stress-deformation condition, while the one for heat flux reduces, under certain assumptions, to a Hill–Mandel-like condition involving heat flux and the gradient of temperature.
The macroscopic thermomechanical behaviour of heterogeneous media may depend strongly on their microstructure. In such cases, the constitutive behaviour of the microstructure needs to be considered when modelling the bulk material response. When there exists sufficient length- and time-scale separation between the macroscopic body and its microstructural constituents (e.g. in a polycrystal or a collection of granules embedded in a matrix), it is possible to employ multi-scale methods that yield macroscopic variables, such as stress and heat flux, by homogenization of related microscopic variables (Hill 1972; Kouznetsova et al. 2001; Miehe et al. 2002; Özdemir et al. 2008; Ladeveze et al. 2010). The values of the latter are determined from initial/boundary-value problems formulated over a space- and time-domain commensurate with the characteristic dimensions of the microscale, and with boundary/initial conditions consistent with the local macroscale deformation and temperature fields.
In this study, a novel homogenization method is proposed under the fundamental assumption that continuum mechanical modelling is permissible at both scales. The method extends to the continuum-on-continuum setting the celebrated approach of Irving & Kirkwood (1950), which forms the basis for upscaling atomistic variables of classical statistical mechanics, such as position, momentum and interatomic forces to continuum variables, such as stress and heat flux. A crucial property of the Irving–Kirkwood procedure, which is inherited in the proposed method, is that the connection between the two scales is effected through the homogenization of extensive quantities, namely mass, linear momentum and total energy. Extensive quantities are those that depend directly on the amount of material in the body. These are a natural choice for homogenization, because their densities in the macroscale are volumetric averages of the corresponding densities in the microscale. Moreover, mass, linear momentum and internal energy are the three physical quantities whose rates of change are paramount in formulating the three balance laws of continuum thermomechanics. By extension of the Irving–Kirkwood procedure, the macroscopic mass, linear momentum and energy densities are determined as volume averages of their microscopic counterparts. These are subsequently employed in conjunction with the continuum balance laws to derive expressions for the macroscopic stress and heat flux, thereby forming the constitutive laws in the macroscale. The proposed method is general in that it does not depend on any simplifying assumptions, such as mechanical or thermal equilibrium in the microscale, zero body force or heat supply, or particular choices of boundary conditions for the microscale problem. In addition, the method is applicable to both solids and fluids regardless of the constitutive law posited in the microscale. There is a relation between the proposed method and large eddy simulation (LES) methods in turbulence modelling (Lesieur & Métais 1996), where spatial averaging of the velocity is employed. However, unlike LES methods, the deviations of the velocity from the average are not modelled (which is necessary in turbulence for closure), but rather exactly calculated from the microscale problem.
The proposed homogenization method is contrasted to the widely employed method of Hill (1972), which, under certain restrictions, implies that the average of the product of stress and deformation gradient equals the product of the average first Piola–Kirchhoff stress times the average deformation gradient. This so-called Hill–Mandel condition, in turn, ensures that the average stress (equated, by assumption, to the macroscopic stress) is determined purely by surface kinematic and kinetic data. The preceding property, as remarked by Hill (1972, section 3), constitutes a practical starting point for homogenization owing to the analytical simplicity that it affords. It is shown here that the proposed method may be sufficiently specialized to yield the results of Hill's method for the stress power. In fact, it is argued that neglecting the deviation of the microscopic velocities from their macroscopic counterpart is a necessary condition for this reduction. Upon admitting certain constitutive simplifications, a condition is also derived that relates the microscale to the macroscale heat fluxes, and which is shown to follow the structure of Hill's stress-power condition, namely relating the average product of the flux and temperature gradient to their product of the averages, as assumed by Ostoja-Starzewski (2002) and Özdemir et al. (2008).
The organization of the paper is as follows: In §2, global and local forms of the balance laws of mass, linear momentum and energy are written at both scales with respect to the spatial configuration and relations are postulated between macroscopic variables and microscopic extensive quantities. These are subsequently exploited within the Irving–Kirkwood procedure to deduce expressions for the macroscopic stress and heat flux in the spatial configuration. The referential formulation of this method is next discussed in §3. This is followed in §4 by a derivation, as a special case, of the Hill–Mandel condition and a related condition involving heat flux and temperature gradient.
2. Homogenization in spatial form
In this section, balance laws are written in spatial form for both scales, and a homogenization procedure is proposed that yields expressions for macroscopic stress and heat flux as a function of microscopic quantities.
(a) Microscopic and macroscopic balance laws
Consider a body which occupies the region in the spatial configuration, and let each point y in be associated in a microscopic scale with a region whose typical point is denoted by x (figure 1).
The macroscopic balances of mass, linear momentum and energy may be written, respectively, for a region with smooth-orientable boundary as 2.1 2.2 and 2.3Here, ρM(y,t) is the mass density, is the velocity, tM(y,t;nM) is the traction on with nM being the outward unit normal to , bM(y,t) is the body force per unit mass, eM(y,t) is the total energy (including kinetic energy) per unit mass, hM(y,t;nM) is the heat flux per unit area of and rM(y,t) is the heat supply per unit mass. Note that herein the superscript ‘M’ signifies quantities in the macroscopic scale. Neglecting, in the interest of brevity, the explicit statement of functional dependencies, corresponding local forms of the preceding balance laws may be deduced using standard arguments of continuum mechanics as 2.4 2.5 and 2.6where TM and qM are the macroscopic Cauchy stress tensor and heat-flux vector, defined, respectively, from tM(y,t;nM)=TM(y,t)nM and hM(y,t;nM)=qM(y,t)⋅nM. Here, ‘∂/∂y ⋅’ denotes the divergence operator relative to y, whereas denotes the material time derivative of (•).
Likewise, spatial balances of mass, linear momentum and energy can be stated in the microscopic scale as 2.7 2.8 and 2.9Here, is any arbitrary microscopic subset of containing x, and all physical quantities are defined in complete analogy to their macroscopic counterparts with the superscript ‘m’ used to emphasize their microscopic character. Ignoring, again, the explicit statement of the functional dependence of all physical quantities, the local forms of these balance laws can be deduced as 2.10 2.11 and 2.12where ‘∂/∂x ⋅’ now denotes the divergence operator relative to x.
(b) Relations between microscopic and macroscopic quantities
In this section, explicit relations are established between physical quantities in the two scales using formulae motivated from particle-to-continuum homogenization methods originally proposed by Irving & Kirkwood (1950) and Hardy (1982). Specifically, recalling that mass, linear momentum and energy are extensive quantities, the pointwise macroscopic mass density, linear momentum and energy at a macroscopic point y and time t are defined according to 2.13 2.14 and 2.15Here, g(y,x) is a weighting function which quantifies the relative contribution of the microscopic point x in the definition of the pointwise quantities at y. The underlying assumption in equations (2.13)–(2.15) is that the pointwise macroscopic mass density, linear momentum and energy are weighted volumetric averages of mass, linear momentum and energy over the microscopic points around point y, respectively. The relations (2.13)–(2.15) ensure that the macroscopic quantities at y are obtained by averaging their microscopic counterparts over a neighbourhood of the point. While this averaging is applicable to the description of both solids and fluids, it is particularly relevant to most fluids, where averaging relative to some reference state is not practical. A referential version of (2.13)–(2.15) geared specifically to solids is introduced in §3.
The weighting function g(y,x) in (2.13)–(2.15) is assumed to be continuous and to possess the following additional properties: (i) , (ii) , where ‘supp’ denotes the support of a function and ℓm is the characteristic microstructural length, (iii) g(y,x)=0 on when and (iv) g(y,x) attains a maximum when x=y. Assumption (i) is a standard normalization condition that enforces consistency of the relations (2.13)–(2.15). In addition, assumptions (ii) and (iv) signify the local nature of the macroscopic quantities, whereas assumption (iii) implies that g(y,x) affects only interior averaging.
An additional assumption made here is that the function g(y,x) is invariant under superposed rigid motions. This implies that 2.16where the superscript ‘+’ denotes a quantity defined in a configuration which differs from the current configuration by an arbitrary rigid-body displacement. Writing y+=Qy+c and x+=Qx+c, where Q is a time-dependent rotation tensor and c is a time-dependent translation vector, it follows from (2.16) that 2.17Choosing now Q=i and c=−x, where i is the spatial identity tensor, it follows from (2.17) that g is exclusively a function of y−x. Furthermore, because g is a scalar function of a vector argument, it can depend only on the magnitude |y−x| (because the direction would change arbitrarily by the choice of the rotation Q), which implies that 2.18For the sake of notational simplicity, in the remainder of this article, the weighting function is written as g(y,x) with the understanding that it is a function of |y−x|.
In the following sections, the relations (2.13)–(2.15) are used along with the balance laws to derive expressions for the macroscopic stress and heat flux in terms of the corresponding microscopic quantities.
(c) Homogenization: balance of mass
In this section, it is shown that balance of mass in the macroscopic scale is satisfied exactly using balance of mass in the microscopic scale and the properties of the weighting function g(y,x). Indeed, taking the time derivative of (2.13) and applying the Reynolds transport theorem, it follows that 2.19Invoking mass balance in the microscale given by equation (2.10) and the relation (2.18), equation (2.19) leads to 2.20Taking into account the definitions (2.13) and (2.14), equation (2.20) assumes the form 2.21which is the conventional statement of mass balance in the macroscale. Hence, the definition for macroscopic mass density given by (2.13) in conjunction with the property (2.18) of the weighting function is consistent with balances of mass at both scales.
(d) Homogenization: balance of linear momentum
The procedure used in §2c for mass balance is now repeated for linear momentum. In particular, taking the time derivative of (2.14) yields 2.22Using the microscopic balances of mass and linear momentum given by (2.10) and (2.11), respectively, as well as the identity (2.18), equation (2.22) is reduced to 2.23The divergence part in the first term on the right-hand side of (2.23) may be rewritten as 2.24where use is made of the divergence theorem, the identity (2.18) and property (iii) of the weighting function. Using (2.24), equation (2.23) becomes 2.25Next, the third term on the right-hand side of (2.25) is written as 2.26where the last two terms of (2.26)2 cancel each other by virtue of the definitions in (2.13) and (2.14), respectively. Hence, equation (2.25) may be rewritten as 2.27With the aid of (2.14), it can be easily seen that the last two terms on the right-hand side of (2.27) can be reduced to −ρMvM(∂/∂y)⋅vM. In addition, the left-hand side of (2.27) can be expanded by means of (2.4) as . Therefore, equation (2.27) can be finally reduced to 2.28which is analogous in form to the balance of macroscale linear momentum in (2.5). Comparing the two terms with the right-hand side of (2.28) and (2.5), it follows that body force at the macroscale is given by 2.29whereas the stress tensor is defined as 2.30to within a divergence-free tensor function of (y,t). At any given point y and time t, this function may be set to zero, provided that the balance laws and assorted boundary conditions are satisfied simultaneously in both scales.
Equation (2.30) shows that the macroscopic stress comprises two terms: the first is equal to the volumetric average of the microscopic stress, whereas the second is due to the momentum induced by the velocity of the microscale relative to the macroscale. Equation (2.30) may be viewed as a continuum extension to the one derived by Irving & Kirkwood (1950) for the case of a microscale consisting of interacting particles.
(e) Homogenization: balance of energy
Following the derivation in §2d, the time derivative of the relation between macroscopic and microscopic total energies (2.15) yields 2.31Using equation (2.12), the balance of energy at the microscale (2.13), and the identity (2.18), equation (2.31) can be reduced to 2.32Again, using the divergence theorem, property (ii) for the weighting function g, and condition (2.18), the third and fourth terms on the right-hand side of (2.32) can be rewritten as 2.33 and 2.34respectively. In addition, the fifth term on the right-hand side of (2.32) can be rewritten as 2.35where the second term in (2.35)2 is obtained by appealing to the relation between macroscopic and microscopic energies in (2.15). The previous relation may be also invoked to rewrite the last term on the right-hand side of (2.32) as 2.36Substituting (2.33)–(2.36) into (2.32) leads to 2.37Defining the total energy per unit mass in the microscale as 2.38where ϵm is the internal energy per unit mass, the last integral term on the right-hand side of (2.37) can be rewritten with the aid of (2.14) as 2.39Here, is given by 2.40and represents the total energy per unit mass in the microscale, where the kinetic energy is measured relative to a coordinate frame that is convected by the macroscopic velocity. Upon substituting (2.39)3 into the right-hand side of (2.37), recalling (2.4), and rearranging terms, it follows that 2.41Taking into account the definition of the macroscopic stress in (2.30), and then rewriting the first integral term on the right-hand side so as to yield the power of the macroscopic body force defined in (2.29), equation (2.41) becomes 2.42Comparing equations (2.42) and (2.6), it may be concluded that the rate of internal heating and the heat flux at the macroscale are given by 2.43and, to within a divergence-free term, 2.44respectively. From a strictly mathematical viewpoint, the preceding definitions assign to the macroscopic rate of internal heating and to the macroscopic heat flux all the outstanding terms in (2.42) that assume non-divergence and divergence forms, respectively. However, a physical argument may also be made in support of these definitions. Specifically, in addition to the weighted volumetric average of the microscopic rate of internal heating, the macroscopic rate of internal heating in (2.43) should include the rate of work performed by the body forces owing to fluctuations in microscopic velocity relative to its macroscopic counterpart. Likewise, the macroscopic heat flux in (2.44) consists of its microscopic counterpart, as well as of contributions owing to: (i) the convection of total energy owing to fluctuations of the microscale velocity around the macroscopic value and (ii) the rate of work performed by the internal forces when going over fluctuations of the microscopic velocity around the macroscopic value. The negative sign of the latter contribution is due to the fact that the work performed by the internal forces tends to increase the energy, thereby reducing the flux of heat leaving the region around the macroscopic point y.
3. Homogenization theory in referential form
For multi-scale problems in solids (which typically involve polycrystals or composites with granules embedded in a matrix), it is often advantageous to write the balance laws in referential form. For this reason, expressions are derived in this section for referential measures of macroscopic stress and heat flux using balance laws and averaging relations formulated in the reference configuration. Further, it is shown that the macroscopic first Piola–Kirchhoff stress tensor and the referential heat flux vector derived using the referential configurations are consistent with the macroscopic Cauchy stress tensor and the spatial heat flux vector derived in §2.
(a) Macroscopic and microscopic balance laws
In this section, the balance laws for both the macroscopic and microscopic scales are written with respect to the referential configuration. The integral forms of these laws for the macroscopic scale are 3.1 3.2 and 3.3where is any arbitrary subset of the region occupied by the body in its reference configuration, is the referential density, is the velocity, is the Piola traction with N being the outward unit normal to , is the body force per unit mass, is the total energy per unit mass, is the heat flux per unit referential area and is the internal heating per unit mass. A change of variables relates the following quantities: , , and . Also, if FM(Y,t) is the deformation gradient in the macroscopic scale, then dvM=JM(Y,t)dVM, where and . Omitting, again, the explicit mention of the dependencies of the physical quantities, the local forms of (3.1)–(3.3) are given by 3.4 3.5 and 3.6where PM is the first Piola–Kirchhoff tensor and is the referential heat-flux vector defined from the standard relations pM(Y,t;N)=PM(Y,t)N and , respectively. These quantities are related to their spatial counterparts by PM=JMTM(FM)−T and , respectively.
Finally, the local forms of the balance laws in the microscale are simply recorded here as 3.7 3.8 and 3.9where all the quantities in (3.7)–(3.9) are functions of the microscopic referential position X and time. The integral forms of the microscopic referential balance laws are omitted in the interest of brevity.
(b) Homogenization of the balance laws
In this section, macroscopic quantities such as density, linear momentum and internal energy are obtained by homogenization with respect to a region occupied by the body in the reference configuration. These quantities are now defined at a referential macroscopic point Y as 3.10 3.11 and 3.12where g0 is the referential weighting function. The latter may be obtained from the spatial weighting function g(y,x) by relating any one of the three spatial definitions (2.13)–(2.15) to the corresponding referential definitions (3.10)–(3.12). For instance, mapping the definition (2.13) of mass density to the reference configuration leads to 3.13Comparing (3.13) with (3.10), it follows that 3.14Equation (3.14) implies that the referential weighting function g0 depends explicitly on X, Y and time t. Moreover, unlike its spatial counterpart, the dependence of g0 on X, Y cannot be reduced to a single dependence on Y−X by appealing to invariance under superposed rigid motions.
Starting with the relation (3.10), and taking the time derivative of both sides, it follows that 3.15where use is made of (3.14) and (2.18), the identity and the Piola identity (∂/∂Y)⋅(JMFM−T)=0. Hence, the referential statement of mass balance (3.4) in the macroscale is recovered from its microscale counterpart (3.7) and the relation (3.10) between the macroscopic and microscopic referential densities.
In an entirely analogous manner, one may use the relations (3.11) and (3.12) in connection with the macroscopic and microscopic balances of linear momentum and energy (3.5), (3.8), (3.6) and (3.9), respectively, to derive expressions for the macroscopic first Piola–Kirchhoff stress and the referential heat flux. In the interest of brevity, these derivations are omitted, and the final results are merely recorded as 3.16 and 3.17Equations (3.16) and (3.17) reveal that the first Piola–Kirchhoff stress tensor and the referential heat flux in the macroscale are consistent with their usual definitions when determined by referential homogenization.
4. A reflection on the Hill–Mandel condition
The purpose of this section is to compare the assumptions made in the derivation of the widely used Hill–Mandel condition (Hill 1972) with those of the proposed homogenization method, and to show that the former may be obtained from the latter by introducing a set of additional assumptions.
By way of background, recall that the Hill–Mandel condition may be derived by starting with the equilibrium equation (∂/∂X)⋅Pm=0 in the microscale in the absence of body forces, and employing the divergence theorem to first conclude that 4.1where and are averages over a microscale region with boundary ∂ω0 and volume V , namely 4.2In the practice of homogenization, the region ω0 is typically referred to as a representative volume element. Subsequently, appropriate boundary conditions (e.g. of Dirichlet type: , or Neumann type: ) are chosen on ∂ω0 to eliminate the boundary integral term on the right-hand side of (4.1), thus leading to the classical Hill–Mandel condition 4.3
The proposed homogenization method may be specialized to deduce a condition similar to (4.3). Indeed, starting from (3.16)1, recast here as 4.4let the fluctuation of the microscopic velocity vm relative to the macroscopic velocity vM be small enough for the kinetic part of the stress in (4.4) to be negligible compared with the contribution of the microscopic stress. This is a reasonable assumption for solids and reduces (4.4) to 4.5Next, let the weighting function g(y,x) for a point y∈ω be specified as 4.6where v is the volume of the image of a referential region under the microscale motion. This choice of the weighting function is obviously consistent with the general properties (i), (iii) and (iv) stipulated in §2b, as well as the invariance requirement in (2.16). Substituting (4.6) in (4.5) yields 4.7Lastly, let property (ii) of §2b be invoked, which requires , thereby leading to . Hence, equation (4.7) becomes 4.8which, upon contracting the tensor terms on each side, results in 4.9Equation (4.9) is analogous to the Hill–Mandel condition (4.3), with the volume averages of stress and deformation gradient replaced by the corresponding macroscopic quantities.
In comparing the derivations of (4.3) and (4.9), it is noted that the former is based on the a priori volumetric averaging of stress and deformation, as well as on the assumption of equilibrium in the microscale. In addition, it relies on the application of certain microscale boundary conditions that eliminate the right-hand side of equation (4.1). In contrast, more general homogenization conditions (3.16) and (3.17) are derived in the proposed method without imposing any microscale boundary conditions, applying averaging on intensive quantities or relying on microscale equilibrium, but rather by assuming the homogenization of extensive quantities. Of course, upon placing appropriate restrictions that limit sufficiently the scope of the present method, an analogue of the Hill–Mandel condition is readily recovered in the form of equation (4.9). However, this recovered condition, in contrast to the original Hill–Mandel condition, is not predicated upon microscopic equilibrium or vanishing of the microscopic body forces.
Note that the definitions (4.2), which are paramount in the derivation of the Hill–Mandel condition may be recovered from the proposed method under further specialization. To this end, starting from (4.8) and using the divergence theorem, it can be easily shown that 4.10Now, if microscale Dirichlet boundary conditions are applied in the form x=FMX on ∂ω0, and microscale equilibrium is assumed to hold in the absence of body forces, the first and second integral terms on the right-hand side of (4.10) vanish identically. In addition, upon taking the tensor product of the Dirichlet boundary condition with the normal Nm, integrating by parts, and using the divergence theorem, it follows that . Therefore, equation (4.10) is reduced to 4.11hence, . Therefore, under the preceding assumptions of microscale equilibrium in the absence of body forces, the Dirichlet boundary conditions lead to the assumptions (4.2) used by Hill (1972) to obtain the Hill–Mandel condition (4.3). An analogous procedure may be followed to conclude that the relations (4.2) are recovered by applying the Neumann boundary conditions pm=PMNm on ∂ω0 in the microscale, along with microscale equilibrium and absence of body forces.
Turning to the homogenization of the heat flux, it is noted that equation (3.17) may be again simplified by neglecting the two kinetic terms and admitting the special form of the weighting function in (4.6). In this case, it is seen from (3.17) that 4.12Unlike the stress power, a Hill–Mandel-like condition relating the average of the product between the heat flux and the temperature gradient at the microscale to the product of the heat flux and temperature gradient averages may be established only upon introducing further restrictions. Preliminary to the discussion of such assumptions, one may appeal once more to the divergence theorem to deduce that 4.13where θm and θM are the temperatures, and ∇M=∂/∂Y, ∇m=∂/∂X are the referential gradient operators in the microscale and macroscale, respectively. Upon applying the thermal Dirichlet boundary conditions θm=θM+∇MθM⋅(X−Y) on ∂ω0, it follows that , hence equation (4.13) reduces to 4.14At this stage, imposing the Dirichlet boundary conditions x=FMX on ∂ω0 in the microscale and invoking the divergence theorem, equation (4.12) can be rewritten as 4.15Furthermore, assuming a steady state in the microscale with zero internal heating and stress power, that is, , equation (4.15) reduces to 4.16The steady-state assumption in conjunction with (4.16) simplifies equation (4.14) to 4.17which is a Hill–Mandel-like condition for heat flux. Note that, as already argued, the derivation of this condition is predicated upon neglecting the internal heating and stress power in the microscale. This is a strong constitutive assumption that may be mollified by scale-separation arguments, which result in flux-divergence being the dominant term in the microscale energy equation (Bensoussan et al. 1978; Sanchez-Palencia 1980).
A continuum-on-continuum homogenization method has been presented that relies on a minimal set of assumptions relating the microscopic and macroscopic counterparts of the three fundamental extensive quantities of continuum mechanics (mass, linear momentum and total internal energy). Upon employing the Irving–Kirkwood procedure, these relations yield general expressions for macroscopic stress and heat flux in terms of microscopic variables. These expressions do not rely on any assumptions regarding the nature of microscopic body force or heat supply, nor do they require the imposition of special boundary conditions on the microscopic thermomechanical problem.
The proposed method may be readily specialized to solid materials by neglecting the fluctuations of the microscopic velocities relative to the macroscopic one. A further assumption on the nature of the weighting function yields a stress-deformation Hill–Mandel-like condition without relying on the identification of the macroscopic stress or deformation gradient as volumetric averages of their microscopic counterparts. Such identifications are, in fact, recovered by further assuming equilibrium under no body force and imposing Dirichlet or Neumann boundary conditions in the microscale. A similar set of conditions is shown to apply to the product of heat flux and temperature gradient, although a Hill–Mandel analogue is valid only under fairly restrictive constitutive assumptions in the microscale. In addition to their theoretical importance, these results have direct implications in computational homogenization, where the extraction of consistent macroscopic quantities from microscopic ones is of essence when designing multi-scale algorithms for modelling the thermomechanical behaviour of systems made of heterogeneous materials.
- Received September 21, 2011.
- Accepted January 24, 2012.
- This journal is © 2012 The Royal Society