Growth, mass transfer, and remodeling in fiber-reinforced, multi-constituent materials☆
Highlights
► Multi-constituent, fiber-reinforced materials as representation of biological tissues. ► Growth, mass transfer and remodeling. ► Anealastic deformations and fiber reorientation in statistical composites. ► Standard and non-standard balance laws. ► Study of dissipation.
Introduction
Growth and remodeling are aspects of the evolution of a body: growth is the either accretive or resorptive variation of the body mass, and remodeling is the change of the body structure. In the context of Biomechanics, these concepts apply to biological tissues [34], [73], e.g., bone, articular cartilage, blood vessels and tumors. In their theory of adaptive elasticity, Cowin and Hegedus [19] speak of “remodeling” as a collective term capturing growth, reinforcement and resorption of living bone.
Both growth and remodeling are responses of the body to a variety of processes that involve several scales, or levels, of observation. If we select a fine, an intermediate, and a coarse scale, we may propose the following description: the fine scale frames molecular processes; the intermediate scale characterizes the internal structure of the body; and the coarse scale is the one with respect to which the properties of the homogenized body are defined. Continuum Mechanics usually focuses on aspects of growth and remodeling captured by the intermediate and coarse scales. The motivation for pursuing investigations of these processes from the perspective of mechanics dates back to Wolff's studies [76], where mechanical stress plays a crucial role for describing the adaptation of bone. Later, Skalak [71] proposed the idea that growth is accompanied by incompatible deformations and residual stresses. In the case of growth in arteries, residual stresses were studied by Guillou and Ogden [44]. Rodriguez et al. [68] suggested to decompose deformation into an elastic (accommodating) and a growth (anelastic) part. The resulting decomposition offers a formal analogy with the method introduced in studies of continuous distributions of dislocations [14], [55], [56], [58].
Several authors adopted the multiplicative decomposition of deformation in proposing theories of growth and remodeling in one-component continua (see, for example, [25], [23], [1], [2], [39], [62]). These theories have two important features: (i) the part of deformation related to growth and remodeling, here denoted by and referred to as “anelastic deformation” or “distortion”, individuates an evolving relaxed configuration of body elements (this configuration is also said to be “intermediate”), and represents an unknown of the theory; and (ii) the rate of anelastic deformation, , can be related to the rate of production, or depletion, of mass in such a way that the mass density of the body is constant when measured with respect to the intermediate configuration. Accordingly, the description of the evolution of the body necessitates the determination of the overall deformation, , and its anelastic part, . The former characterizes the change of shape of the body, and the latter supplies information about the structural changes and variation of mass of the body. In other words, tensor keeps track of processes which cover the intermediate and the coarse scale of the body. Moreover, since by virtue of the feature (ii), the trace of is constrained to equal the rate at which mass is produced, or depleted, in the body [25], [60], we might conclude that the spherical part of defines growth, while the non-spherical part accounts for remodeling. Lastly, we remind that is, in general, an incompatible deformation.
While the determination of relies on the standard laws of Continuum Mechanics, suitably modified in order to account for the presence of , the determination of has become the main issue of several investigations dealing with growth and remodeling. To the best of our knowledge and understanding, two major criteria can be mentioned. The first one, based on the Tensor Representation Theorem [72], relates with some measure of stress (for example, Eshelby or Mandel stress) [25], [52]. If a further analogy with the Theory of Plasticity is allowed, the resulting expression might be thought of as originating from some “flow rule”. Alternatively, one may assign an evolution law to and then find extrema of the anelastic working [61], which would lead to generally non-associative “flow rules”. The second criterion attempts to determine directly from the dissipation inequality [35], [3], [4], [5], [59], [1], [7], [40], [41], [42]. Most of the cited papers employ the idea put forward by DiCarlo and Quiligotti in [23], where the evolution law for bulk growth is obtained as a constitutively augmented balance, named “accretive force balance”, which is independent of the standard force balance. DiCarlo and Quiligotti [23] construct their theory on a “two-layer” kinematics, described by standard velocity and the rate (with the formalism of this work), and a “two-layer” dynamics, whose “starring characters” are a standard force (named brute), dual to , and a non-standard remodeling force, dual to . “Force” is understood as a continuous, linear, real-valued functional on the space of test velocities. The remodeling force, a tensor field of configurational nature [45], couples with Eshelby tensor and allows for the definition of an extra-stress, which is constitutively related with .
Some chemical aspects of growth have been mathematically modeled by viewing biological tissues as multiphasic media whose phases interact through chemo-mechanical processes [6], [11], [16], [17], [21], [26]. In all these papers, a biological tissue is regarded as a fluid-solid mixture, and growth is modeled as the mass uptake of the solid phase at the expenses of the fluid phase. The growth rate is governed by the concentration of chemical agents, which is the solution of a system of advection-diffusion-reaction equations coupled with the balance laws of the fluid and solid phases. Publications that proposed unified models for describing deformation, growth, and transport processes by means of the Theory of Mixtures are, for example, [37], [53], [54], [35], [59], [39], [40], [41], [42], [7]. Moreover, on the basis of a pore-scale analysis of the processes occurring in a multi-constituent, biphasic, growing medium [48], [49], [12], a conceptual distinction between growth, that occurs in some phases, and interphase mass transfer, that involves all phases, was proposed in [40]. The same distinction was presented in [59] without having recourse to pore-scale analysis.
Our contribution aims to summarize and improve some of the results presented in [40], [41], [42]. With respect to these papers, the novelties are the following:
- (a)
The balance laws characterizing the non-standard dynamics of DiCarlo and Quiligotti [23] are retrieved by means of the Extended Hamilton's Principle, which we present here in the framework of the Theory of Mixtures (the tools for doing that are taken from [70]) in order to study a mixture comprising a fiber-reinforced solid phase. These laws determine the balance of external and internal non-standard forces (cf. (4.22)), which produce work to change the configuration of the solid phase of the mixture in response to mass transfer, growth, remodeling, and reorientation of the fibers.
- (b)
The rates of anelastic deformation associated with mass transfer and growth are written in a form that takes into account how these processes are coupled with mechanical stress as well as with the orientation of the fibers in the solid phase of the mixture. This is done in three steps. Firstly, the anelastic deformation is written as the product of two tensors, i.e. , where and represent the anelastic contributions of growth and mass transfer, respectively. Secondly, the power done by the internal non-standard forces (which are power-conjugate to the time derivatives of and ) is presented in the dissipation inequality. Then, two auxiliary stress-like variables (denoted by and in (5.16), (5.15)) are introduced to determine the dissipative parts of the internal non-standard forces. These dissipative forces are assumed to be constitutively determinable through invertible functions of the rates of anelastic deformation associated with and . Thirdly, the constitutive expressions of the dissipative forces are inserted in the non-standard balance laws, and the rates of anelastic deformation are determined by inverting the constitutive relations. Thus, the rates of anelastic deformation are expressed as functions of and , which, in turn, are re-written in terms of the external non-standard forces.
- (c)
We adapt the model of fiber reorientation put forward by Olsson and Klarbring [62] to the case of a multi-constituent solid with statistical distribution of fibers. The theory presented in [62] is thus used as a point of departure for our treatment. We conceive the solid phase of the mixture as a composite material whose fibers are distributed according to some probability density distribution. The procedure for determining the evolution laws of the variables describing the reorientation of fibers is similar to that used for mass transfer and growth. The novelty, however, is conceptual since we view these variables either as the parameters specifying the probability density distribution or as the probability density distribution itself.
The paper is organized as follows: in Section 2 we introduce the concepts of Mixture Theory necessary for our purposes; in Section 3 we discuss mass balances and kinematics of anelastic processes; in Section 4 we determine the standard and non-standard force balances; in Section 5 we show the procedure followed to obtain the expression of residual dissipation; in Section 6 we study the residual dissipation inequality; in Section 7, we summarize the results presented in our manuscript; in Section 8 we discuss our results and propose an outlook for possible further research.
Section snippets
General description of the mixture
We consider a mixture consisting of a fluid phase and a solid phase, which comprises the matrix and the fibers of a porous, fiber-reinforced material. Each phase of the mixture is assumed to comprise several constituents, which can be reciprocally exchanged between the phases.
For brevity, we use the conventions , , and throughout the manuscript, unless otherwise specified. The formalism adopted in this section follows [48], [49], [12].
Balance of mass and anelastic deformations
In this section, we present the balances of mass of all constituents and phases of the mixture and their link with anelastic deformations. In the following, the mass balances are presented in local form. A procedure for obtaining these laws from an integral formulation is given, for example, in [66].
Extended Hamilton's principle
In order to determine the equations of motion of our mixture, we employ the Extended Hamilton's Principle [9], [57]. In the following, we modify the variational procedure shown in [70]. In order to account for growth, mass transfer and reorientation of fibers, we bring together and slightly generalize the ideas put forward in [23], [62].
The variational approach requires to specify the Lagrangian parameters with respect to which the variation is performed. We require that there exist two sets of
Dissipation
We require that every constituent of the mixture is characterized by its Helmholtz free energy density . Adding these quantities over all constituents of the kth phase yields , while the addition over all phases leads to the Helmholtz free energy density of the mixture, i.e. .
In the case of constant and uniform temperature, the energy imbalance is postulated, for any part of , in the formwhere the time
Study of the residual dissipation inequality
Eq. (5.31) collects all dissipative processes taken into account in the formulated problem. The first two terms on the right-hand side represent the dissipation due to fluid filtration through the porous medium and diffusion of species, respectively. Analogously to and , the fields , and are generalized forces conjugate to the generalized velocities , and , which describe growth, mass transfer and reorientation of fibers. The last two terms on the right-hand side of
Summary
The general equations presented in the manuscript are rewritten according to the modeling assumptions introduced so far, and gathered together in order to give clarity to the closed mathematical problem they form. Before doing that, we consider the identitywhich allows for rewriting the balance of momentum (4.15) asFrom here on we neglect the inertial forces of the fluid and solid phase, and approximate the stress
Conclusions and outlook
We proposed a theoretical study of growth, mass transfer, and remodeling in multi-constituent, fiber-reinforced materials. As representative of this class of materials, we took a system comprising a fluid and a fiber-reinforced solid. We referred to these as to phases, and allowed them to exchange mass reciprocally through the transfer of their constituents. We viewed growth and remodeling as processes pertaining the solid phase only.
The study of the above delineated system was based on the
Acknowledgments
The authors gratefully acknowledge the support of the Goethe-Universität Frankfurt (Frankfurt am Main, Germany), German Ministry for Economy and Technology (BMWi), contract 02E10326 (A. Grillo and G .Wittum), and the AIF New Faculty Programme (Alberta Innovates—Technology Futures, formerly Alberta Ingenuity Fund, Canada) as well as the NSERC Discovery Programme (Natural Science and Engineering Research Council of Canada) (S. Federico).
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- ☆
In honor of Prof. Ray W. Ogden, recipient of the 2010 William Prager Medal of the Society of Engineering Science.