Mass transport in morphogenetic processes: A second gradient theory for volumetric growth and material remodeling

In this work, we derive a novel thermo-mechanical theory for growth and remodeling of biological materials in morphogenetic processes. This second gradient hyperelastic theory is the first attempt to describe both volumetric growth and mass transport phenomena in a single-phase continuum model, where both stress- and shape-dependent growth regulations can be investigated. The diffusion of biochemical species (e.g. morphogens, growth factors, migration signals) inside the material is driven by configurational forces, enforced in the balance equations and in the set of constitutive relations. Mass transport is found to depend both on first- and on second-order material connections, possibly withstanding a chemotactic behavior with respect to diffusing molecules. We find that the driving forces of mass diffusion can be written in terms of covariant material derivatives reflecting, in a purely geometrical manner, the presence of a (first-order) torsion and a (second-order) curvature. Thermodynamical arguments show that the Eshelby stress and hyperstress tensors drive the rearrangement of the first- and second-order material inhomogeneities, respectively. In particular, an evolution law is proposed for the first-order transplant, extending a well-known result for inelastic materials. Moreover, we define the first stress-driven evolution law of the second-order transplant in function of the completely material Eshelby hyperstress.The theory is applied to two biomechanical examples, showing how an Eshelbian coupling can coordinate volumetric growth, mass transport and internal stress state, both in physiological and pathological conditions. Finally, possible applications of the proposed model are discussed for studying the unknown regulation mechanisms in morphogenetic processes, as well as for optimizing scaffold architecture in regenerative medicine and tissue engineering.     

Micromechanical analysis of volumetric growth in the context of open systems thermodynamics and configurational mechanics. Application to tumor growth

We adopt in this paper the physically and micromechanically motivated point of view that growth (resp. resorption) occurs as the expansion (resp. contraction) of initially small tissue elements distributed within a host surrounding matrix, due to the interfacial motion of their boundary. The interface motion is controlled by the availability of nutrients and mechanical driving forces resulting from the internal stresses that built in during the growth. A general extremum principle of the zero potential for open systems witnessing a change of their mass due to the diffusion of nutrients is constructed, considering the framework of open systems thermodynamics. We postulate that the shape of the tissue element evolves in such a way as to minimize the zero potential among all possible admissible shapes of the growing tissue elements. The resulting driving force for the motion of the interface sets a surface growth models at the scale of the growing tissue elements, and is conjugated to a driving force identified as the interfacial jump of the normal component of an energy momentum tensor, in line with Hadamard’s structure theorem. The balance laws associated with volumetric growth at the mesoscopic level result as the averaging of surface growth mechanisms occurring at the microscopic scale of the growing tissue elements. The average kinematics has been formulated in terms of the effective growth velocity gradient and elastic rate of deformation tensor, both functions of time. This formalism is exemplified by the simulation of the avascular growth of multicell spheroids in the presence of diffusion of nutrients, showing the respective influence of mechanical and chemical driving forces in relation to generation of internal stresses.     

A kinematically and thermodynamically consistent volumetric growth model based on the stress-free configuration

A continuum volumetric growth theory is presently elaborated based on the stress-free configuration. The total deformation gradient is the composition of a growth mapping followed by an elastic mapping ensuring the compatibility of the body. A measure of the growth rate fully lying in the stress-free configuration is elaborated, and the balance of momentum accounting for mass changes due to growth is written in the same configuration, highlighting measures of kinematic incompatibilities. An Eshelby stress is identified as the driving force for growth, in the sense that it appears as the true dissipative thermodynamic force conjugated to the growth rate. Examples of incompatible cylindrical growth with residual stress generation and tumor growth with mechanics coupled to diffusion of nutrients illustrate the proposed theory.     

The Eshelby stress tensor,angular momentum tensor and dilatation flux in gradient elasticity

The Eshelby (static energy momentum) stress tensor, the angular momentum tensor and the dilatation flux are derived for anisotropic linear gradient elasticity in non-homogeneous materials. The divergence of these tensors gives the configurational forces, moments and work terms in gradient elasticity. There are several types of configurational forces, acting on the dislocation density and its gradient, on the inhomogeneities, proportional to the distortion, and linear and quadratic in the distortion gradient, and on the body force.     

A continuum treatment of growth in biological tissue: the coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a fluid phase, and (ii) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.     

Thermomechanics of volumetric growth in uniform bodies

A theory of material growth (mass creation and resorption) is presented in which growth is viewed as a local rearrangement of material inhomogeneities described by means of first- and second-order uniformity “transplants”. An essential role is played by the balance of canonical (material) momentum where the flux is none other than the so-called Eshelby material stress tensor. The corresponding irreversible thermodynamics is expanded. If the constitutive theory of basically elastic materials is only first-order in gradients, diffusion of mass growth cannot be accommodated, and volumetric growth then is essentially governed by the inhomogeneity velocity “gradient” (first-order transplant theory) while the driving force of irreversible growth is the Eshelby stress or, more precisely, the “Mandel” stress, although the possible influence of “elastic” strain and its time rate is not ruled out. The application of various invariance requirements leads to a rather simple and reasonable evolution law for the transplant. In the second-order theory which allows for growth diffusion, a second-order inhomogeneity tensor needs to be introduced but a special theory can be devised where the time evolution of the second-order transplant can be entirely dictated by that of the first-order one, thus avoiding insuperable complications. Differential geometric aspects are developed where needed.     

The energy of a growing elastic surface

The potential energy of the elastic surface of an elastic body which is growing by the coherent addition of material is derived. Several equivalent expressions are presented for the energy required to add a single atom, also known as the chemical potential. The simplest involves the Eshelby stress tensors for the bulk medium and for the surface. Dual Lagrangian/Eulerian expressions are obtained which are formally similar to each other. The analysis employs two distinct types of variations to derive the governing bulk and surface equations for an accreting elastic solid. The total energy of the system is assumed to comprise bulk and surface energies, while the presence of an external medium can be taken into account through an applied surface forcing. A detailed account is given of the various formulations possible in material and current coordinates, using four types of bulk and surface stresses: the Piola-Kirchhoff stress, the Cauchy stress, the Eshelby stress and a fourth, called the nominal energy-momentum stress. It is shown that inhomogeneity surface forces arise naturally if the surface energy density is allowed to be position dependent.     

Phase equilibrium in non-fluids and non-fluid mixtures

A criterion of phase equilibrium for mixtures of materials with arbitrary symmetry (e.g. between solid and fluid or two solid mixture phases) is deduced using a rational thermodynamics approach. This criterion, known also as Maxwell relation, is expressed via the difference of chemical potential tensors (Eshelby tensors) on the singular surface dividing the bulk phases.The thermomechanical balance equations, the entropy inequality and the Maxwell relation for phase equilibrium are given first for the case of pure (one-constituent) materials of arbitrary symmetry and then for the case of mixtures (including chemically reacting ones) of arbitrary symmetry.In the special case of fluids it is shown that the chemical potential tensors reduce to the classical scalar chemical potentials and the Maxwell relations to the classical thermodynamic criterion for the phase equilibrium.     

Configurational forces and couples in fracture mechanics accounting for microstructures and dissipation

Configurational forces and couples acting on a dynamically evolving fracture process region as well as their balance are studied with special emphasis to microstructure and dissipation. To be able to investigate fracture process regions preceding cracks of mode I, II and III we choose as underlying continuum model the polar and micropolar, respectively, continuum with dislocation motion on the microlevel. As point of departure balance of macroforces, balance of couples and balance of microforces acting on dislocations are postulated. Taking into account results of the second law of thermodynamics the stress power principle for dissipative processes is derived.Applying this principle to a fracture process region evolving dynamically in the reference configuration with variable rotational and crystallographic structure, the configurational forces and couples are derived generalizing the well-known Eshelby tensor. It is shown that the balance law of configurational forces and couples reflects the structure of the postulated balance laws on macro- and microlevel: the balance law of configurational forces and configurational couples are coupled by field variable, while the balance laws of configurational macro- and microforces are coupled only by the form of the free energy. They can be decoupled by corresponding constitutive assumption.Finally, it is shown that the second law of thermodynamics leads to the result that the generalized Eshelby tensor for micropolar continua with dislocation motion consists of a non-dissipative part, derivable from free and kinetic energy, and a dissipative part, derivable from a dissipation pseudo-potential.     

An illustration of the equivalence of the loss of ellipticity conditions in spatial and material settings of hyperelasticity

The loss of ellipticity indicated through the rank-one-convexity condition is elaborated for the spatial and material motion problem of continuum mechanics. While the spatial motion problem is characterized through the classical equilibrium equations parametrised in terms of the deformation gradient, the material motion problem is driven by the inverse deformation gradient. As such, it deals with material forces of configurational mechanics that are energetically conjugated to variations of material placements at fixed spatial points. The duality between the two problems is highlighted in terms of balance laws, linearizations including the consistent tangent operators, and the acoustic tensors. Issues of rank-one-convexity are discussed in both settings. In particular, it is demonstrated that if the rank-one-convexity condition is violated, the loss of well-posedness of the governing equations occurs simultaneously in the spatial and in the material motion context. Thus, the material motion problem, i.e. the configurational force balance, does not lead to additional requirements to ensure ellipticity. This duality of the spatial and the material motion approach is illustrated for the hyperelastic case in general and exemplified analytically and numerically for a hyperelastic material of Neo-Hookean type. Special emphasis is dedicated to the geometrical representation of the ellipticity condition in both settings.     

Electroelastic behavior modeling of piezoelectric composite materials containing spatially oriented reinforcements

In this work, a modeling of electroelastic composite materials is proposed. The extension of the heterogeneous inclusion problem of Eshelby for elastic to electroelastic behavior is formulated in terms of four interaction tensors related to Eshelby’s electroelastic tensors. Analytical formulations of interaction tensors are presented for ellipsoidal inclusions. These tensors are basically used to derive the self-consistent model, Mori–Tanaka and dilute approaches. Numerical solutions are based on numerical computations of these tensors for various types of inclusions. Using the obtained results, effective electroelastic moduli of piezoelectric multiphase composites are investigated by an iterative procedure in the context of self-consistent scheme. Generalised Mori–Tanaka’s model and dilute approach are re-formulated and the three models are deeply analysed. Concentration tensors corresponding to each model are presented and relationships of effective coefficients are given. Numerical results of effective electroelastic moduli are presented for various types of piezoelectric inclusions and for various orientations and compared to existing experimental and theoretical ones.     

Solution of Eshelby's inclusion problem with a bounded domain and Eshelby's tensor for a spherical inclusion in a finite spherical matrix based on a simplified strain gradient elasticity theory

A solution for Eshelby's inclusion problem of a finite homogeneous isotropic elastic body containing an inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). An extended Betti's reciprocal theorem and an extended Somigliana's identity based on the SSGET are proposed and utilized to solve the finite-domain inclusion problem. The solution for the disturbed displacement field is expressed in terms of the Green's function for an infinite three-dimensional elastic body in the SSGET. It contains a volume integral term and a surface integral term. The former is the same as that for the infinite-domain inclusion problem based on the SSGET, while the latter represents the boundary effect. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is not considered. The problem of a spherical inclusion embedded concentrically in a finite spherical elastic body is analytically solved by applying the general solution, with the Eshelby tensor and its volume average obtained in closed forms. This Eshelby tensor depends on the position, inclusion size, matrix size, and material length scale parameter, and, as a result, can capture the inclusion size and boundary effects, unlike existing Eshelby tensors. It reduces to the classical Eshelby tensor for the spherical inclusion in an infinite matrix if both the strain gradient and boundary effects are suppressed. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing as the inclusion becomes large enough, and the boundary effect is vanishing as the inclusion volume fraction gets sufficiently low.     

A theoretical model for tissue growth in confined geometries

It is known that cells proliferate and produce extracellular matrix in response to biochemical and mechanical stimuli. Constitutive models considering these phenomena are needed to quantitatively describe the process of tissue growth in the context of tissue engineering and regenerative medicine. In this paper we re-examine the theoretical framework provided by Ambrosi and Guana (2007) and Ambrosi and Guillou (2007). We show how a volumetric growth rate term can be obtained (both in a large and small strain setting), which is consistent with the laws of thermodynamics and then apply the model to a simple geometry of tissue growth within a circular pore. The model, despite its simplicity, is comparable with experimental measurements of tissue growth and highlights the contribution of the mechanical stresses produced during tissue growth on the growth rate itself.     

Eshelby tensors for an ellipsoidal inclusion in a microstretch material

Eshelby tensors for an ellipsoidal inclusion in a microstretch material are derived in analytical form, involving only one-dimensional integral. As micropolar Eshelby tensor, the microstretch Eshelby tensors are not uniform inside of the ellipsoidal inclusion. However, different from micropolar Eshelby tensor, it is found that when the size of inclusion is large compared to the characteristic length of microstretch material, the microstretch Eshelby tensor cannot be reduced to the corresponding classical one. The reason for this is analyzed in details. It is found that under a pure hydrostatic loading, the bulk modulus of a microstretch material is not the same as the one in the corresponding classical material. A modified bulk modulus for the microstretch material is proposed, the microstretch Eshelby tensor is shown to be reduced to the modified classical Eshelby tensor at large size limit of inclusion. The fully analytical expressions of microstretch Eshelby tensors for a cylindrical inclusion are also derived.     

Covariant formulation of the tensor algebra of non-linear elasticity

This work aims at obtaining a covariant representation of the elasticity tensor of a hyperelastic material when the elastic strain energy potential is written employing the volumetric–distortional decomposition of the deformation. This requires the careful definition of some fundamental fourth-order tensors: the identity, the spherical operator, and the deviatoric operator, which appear in the material and spatial expressions of the elasticity tensor. These operators can be defined in the spatial or the material setting and admit pulled-back and pushed-forward forms, respectively. These forms are intimately related to the pulled-back and pushed-forward metric tensors, and are somewhat awkward to derive in Cartesian coordinates, because of the loss of the distinction between a vector space and its dual, and therefore between objects having contravariant and covariant components, which obey to different transformation laws. The relationship between the deformation and the various forms of the identity, spherical, and deviatoric operators can be entirely clarified only within a covariant theory, where the central role played by the spatial and material metric tensors, and their pulled-back and pushed-forward counterparts, which are deformation tensors, can be emphasised.     

The fundamental role of nonlocal and local balance laws of material forces in finite elastoplasticity and damage mechanics

In this paper the fundamental role of independent balance laws of material forces acting on dislocations and microdefects is shown. They enable a thermodynamically consistent formulation of dissipative deformation processes of continua with dislocation motion and defect evolution in the material space on meso- and microlevel.The balance laws of material forces together with the classical balance laws of physical forces and couples, first and second laws of thermodynamics for physical and material space and general constitutive equations are the basis to develop a thermodynamically consistent framework of nonlocal finite elastoplasticity and brittle and ductile damage.It is shown that a weakly-nonlocal formulation of the balance laws of material forces leads to gradient theories, where local theories are obtained, if all gradient contributions are assumed to be small. In this case the local balance laws of material forces together with the constitutive equations represent evolution laws of the material forces. In the classical approach of internal variables they are assumed from the outset with the result that there is a large number of different propositions in the literature.The well-known splitting test of a circular cylinder of concrete is simulated numerically, where the process of deformation in the physical space and defect and plastic evolution in the material space is represented.     

Interface balance laws,phase growth and nucleation conditions for multiphase solids with inhomogeneous surface stress

In this paper, the thermodynamic configurational force associated with a moving interface is used to derive the conditions for phase growth and nucleation in bodies with multiple diffusing species and arbitrary surface stress at the phase interface. First, the mass, momentum and energy balances are derived on the evolving phase interface. The thermodynamic conditions that result from free energy inequality at the interface are derived leading to the analytical form of the configurational force for bodies subject to mechanical loads, heat and multiple diffusing species. The derived second law condition naturally extends the Eshelby energy–momentum tensor to include species diffusion terms. The above second law restriction is then used to derive the condition for the growth of new phases in a body undergoing finite deformation subject to inhomogeneous as well as anisotropic interface stress, and multiple diffusing species. The growth conditions are derived in both current and reference configurations. The statistical temperature-dependent growth velocity is next derived using the Boltzmann distribution. The derived finite deformation form of growth requirement is simplified to obtain the small deformation diffusive void growth condition. Next, a general, finite deformation, arbitrary surface stress form of phase nucleation condition is derived by considering uncertainty in growth of a small nucleus. The probability of nucleation is shown to naturally depend on a theoretical estimate of critical volumetric energy density, which is directly related to the surface stress. The classical nucleation theory is shown to result from a simplified special case of the general criterion. As an application of the developed theory, the classical Blech electromigration experiment is simulated to estimate the critical energy density corresponding to the onset of electromigration voids at Al–TiN interface.     

Strain gradient solution for the Eshelby-type polyhedral inclusion problem

The Eshelby-type problem of an arbitrary-shape polyhedral inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains a material length scale parameter. The Eshelby tensor for a polyhedral inclusion of arbitrary shape is obtained in a general analytical form in terms of three potential functions, two of which are the same as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as volume integrals over the polyhedral inclusion, are evaluated by dividing the polyhedral inclusion domain into tetrahedral duplexes, with each duplex and the associated local coordinate system constructed using a procedure similar to that employed by Rodin (1996. J. Mech. Phys. Solids 44, 1977–1995). Each of the three volume integrals is first transformed to a surface integral by applying the divergence theorem, which is then transformed to a contour (line) integral based on Stokes' theorem and using an inverse approach different from those adopted in the existing studies based on classical elasticity. The newly derived SSGET-based Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson's ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. This SSGET-based Eshelby tensor reduces to that based on classical elasticity when the strain gradient effect is not considered. For homogenization applications, the volume average of the new Eshelby tensor over the polyhedral inclusion is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its volume average, three types of polyhedral inclusions – cubic, octahedral and tetrakaidecahedral – are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for each of the three inclusion shapes vary with both the position and the inclusion size, while their counterparts based on classical elasticity only change with the position. It is found that when the inclusion is small, the contribution of the gradient part is significantly large and should not be neglected. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size: the smaller the inclusion, the smaller the components. When the inclusion size becomes sufficiently large, these components are seen to approach (from below) the values of their classical elasticity-based counterparts, which are constants independent of the inclusion size.     

Laws of flow with a limiting gradient in anisotropic porous media

The equations of viscoplastic fluid flow through a porous medium are written for all types of anisotropy. It is shown that in anisotropic media the flows with a limiting gradient are characterized by two material tensors: the tensor of permeability (flow resistance) coefficients and the tensor of limiting gradients. A complex of laboratory measurements for determining the tensors of permeability coefficients and limiting gradients is considered for all types of anisotropic media. It is shown that the tensors of permeability coefficients and limiting gradients are coaxial. Conditions of flow onset and fluid flow laws are formulated for media with monoclinic and triclinic symmetries of flow characteristics.