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A Geometric Theory of Growth Mechanics

Arash Yavari

arXiv:0911.4671v1math.DGmath-phmath.MP

TL;DR

Growing solids can change mass and shape under mechanical loading, which motivates a theory beyond the usual mass-conserving continuum framework. This paper models growth with an evolving Riemannian material metric and derives related governing equations and stress-free growth conditions. It reports that shape changes can still produce residual stress and identifies stress-free distributions in nonlinear and linearized settings.

  • Problem

    Classical continuum mechanics usually models mass-conserving bodies, while growth can add or remove mass during deformation and is coupled to mechanical loads in biological systems.

    Source: §1 · p. 2

  • Method

    The paper represents bulk growth by a Riemannian material manifold with an evolving metric and derives governing equations through covariance of energy balance.

    Source: §1 · p. 3, §5 · p. 33

  • Results

    The authors find that growth involving only shape change may produce residual stresses and identify stress-free isotropic growth distributions in nonlinear and linearized settings.

    Source: §5 · pp. 33–34

  • Takeaways & Limitations

    Material curvature provides a way to study stress-free isotropic growth distributions in two and three dimensions.

    Source: §5 · p. 34

    A time-dependent Riemannian metric with its Levi-Civita connection is not guaranteed to model all growth; torsion and non-metricity are left for future work.

    Source: §1 · p. 3

Abstract

from arXiv · show

In this paper we formulate a geometric theory of the mechanics of growing solids. Bulk growth is modeled by a material manifold with an evolving metric. Time dependence of metric represents the evolution of the stress-free (natural) configuration of the body in response to changes in mass density and "shape". We show that time dependency of material metric will affect the energy balance and the entropy production inequality; both the energy balance and the entropy production inequality have to be modified. We then obtain the governing equations covariantly by postulating invariance of energy balance under time-dependent spatial diffeomorphisms. We use the principle of maximum entropy production in deriving an evolution equation for the material metric. In the case of isotropic growth, we find those growth distributions that do not result in residual stresses. We then look at Lagrangian field theory of growing elastic solids. We will use the Lagrange-d'Alembert's principle with Rayleigh's dissipation functions to derive all the governing equations. We make an explicit connection between our geometric theory and the conventional multiplicative decomposition of deformation gradient $\mathbf{F}=\mathbf{F}_e\mathbf{F}_g$ into growth and elastic parts. We linearize the nonlinear theory and derive a linearized theory of growth mechanics. Finally, we obtain the stress-free growth distributions in the linearized theory.

1 Introduction

Because growth couples changing mass and shape to mechanical loading, the paper models growing solids through an evolving Riemannian material metric rather than treating the natural configuration as fixed. It positions this geometric framework alongside multiplicative deformation-gradient models while restricting its scope to fixed material points and torsion-free Riemannian growth.

1 Introduction

  • Growth and remodeling can add or remove mass during deformation, and biological growth is coupled to mechanical loads, motivating theories beyond mass-conserving continuum mechanics.Large deformations in soft tissues also make linear elasticity inadequate.Source: §1 · p. 2
  • The model keeps the number of material points fixed and represents bulk growth through evolving geometry, while restricting the theory to monophase continua and torsion-free Levi-Civita connections.The authors say a time-dependent Riemannian metric and its Levi-Civita connection are not guaranteed to model every kind of growth; torsion and non-metricity are left for future work.Source: §1 · pp. 2–3
  • Conventional growth models use F = FeFg, while the geometric approach treats the material manifold as non-Euclidean to represent its geometry directly.The paper does not question the usefulness of the multiplicative decomposition, but argues that existing models based on it lack a rigorous mathematical foundation.Source: §1 · pp. 2–3
  • The paper models bulk growth with a Riemannian material manifold whose metric evolves, making the changing natural configuration explicit and connecting the approach to thermoelasticity.Unlike the temperature-dependent metric in the cited thermoelasticity theory, biological growth has no simple prescribed dependence for the material metric.Source: §1 · pp. 2–3
  • The paper develops modified energy balance and entropy-production conditions, derives governing equations, studies stress-free growth, connects to multiplicative decomposition, and linearizes the theory.It also applies the Lagrange-d’Alembert principle with Rayleigh dissipation functions in a Lagrangian field theory.Source: §1 · p. 3

2 Evolving Material Metrics and Bulk Growth

Bulk growth is represented by a fixed material manifold whose metric evolves to model changes in the stress-free configuration. The resulting mass balance and metric evolution modify energy accounting, while covariance of the energy balance yields governing balance laws.

2 Evolving Material Metrics and Bulk Growth

  • The material manifold stays fixed while its evolving metric represents local changes in the relaxed body's volume and shape.Figure 2.1 depicts deformation as a map from a material manifold with a dynamic metric to an ambient space with a fixed background metric.Source: §2 · p. 5, Figure 2.1 · p. 5

2.1 Energy Balance.

  • With a time-dependent material metric, the material mass form can change even when material mass density is fixed, so growth requires a mass balance.The metric's rate of change is also a kinematical variable that contributes to power in the modified energy balance.Source: §2.1 · p. 6

2.2 Covariance of Energy Balance.

  • Even when mass is conserved at a point, a change in shape can contribute to energy balance in this formulation.The text notes this contribution in comparison with formulations discussed by Epstein and Maugin [2000] and Lubarda and Hoger [2002].Source: §2.2 · p. 6
  • Energy-balance covariance under time-dependent spatial diffeomorphisms gives the balance laws, with mass balance replacing mass conservation.The material metric also enters divergence and acceleration, affecting the governing equations.Source: §2.2 · pp. 6–7

2.3 Local Form of Energy Balance.

The section localizes the energy balance and treats material-metric evolution as a kinetic variable. It also modifies the entropy production inequality for growth and derives constitutive restrictions using thermodynamic arguments and covariance.

2.3 Local Form of Energy Balance.

  • The local form of energy balance is obtained using the integral balance together with the balances of linear and angular momentum.The text also gives an equivalent expression in terms of the first Piola-Kirchhoff stress.Source: §2.3 · p. 8
  • Material-metric evolution is assumed to follow a kinetic equation, though its specific form is not stated here.The passage frames metric evolution as a modeling assumption.Source: §2.3 · p. 8

2.4 The Second Law of Thermodynamics and Restrictions on Constitutive Equations

  • With a time-dependent material metric, the Clausius-Duhem inequality is modified using the balance of mass.The unmodified material-coordinate inequality is stated for the absence of growth.Source: §2.4 · p. 8
  • Ricci flow is given as an example of metric evolution that smooths a distorted 3-sphere.The cited description identifies R as the Ricci curvature of the metric G.Source: §2.4 · p. 8
  • Applying the Coleman-Noll and Marsden-Hughes arguments yields restrictions on constitutive equations.The supplied passage does not specify the resulting restrictions.Source: §2.4 · p. 9
  • Covariance of energy balance is presented as an alternative route to the constitutive restrictions, without reference to the entropy production inequality.The discussion relates this approach to energy-balance and invariance arguments in classical elasticity.Source: §2.4 · p. 9

2.5 Restrictions on Constitutive Equations Using a Thermomechanical Covariance of Energy Balance

The authors use invariance of the integral energy balance under spatial diffeomorphisms and temperature rescalings to obtain balance laws and constitutive restrictions. Their proposition states that these covariance requirements recover both sets of relations.

2.5 Restrictions on Constitutive Equations Using a Thermomechanical Covariance of Energy Balance

  • Invariance of energy balance under spatial diffeomorphisms and monotonically increasing temperature rescalings yields all balance laws and the constitutive restrictions imposed by the Clausius-Duhem inequality.The approach starts from the integral form of energy balance.Source: §2.5 · pp. 9, 11

2.6 Covariance of the Entropy Production Inequality

The entropy production inequality is considered under simultaneous spatial diffeomorphisms and temperature rescalings. Under the stated transformations, the inequality is trivially covariant and identical to its earlier form.

2.6 Covariance of the Entropy Production Inequality

  • Under the stated spatial diffeomorphisms and temperature rescalings, the entropy production inequality is trivially covariant and identical to its earlier form.The section also identifies the relation ∂g◦ϕ = τ as the Doyle-Ericksen formula.Source: §2.6 · p. 11

2.7 Principle of Maximum Entropy Production

The section applies the maximum entropy production principle to derive a kinetic equation for the evolving material metric. It maximizes entropy production with respect to the metric rate subject to a constraint.

2.7 Principle of Maximum Entropy Production

  • The maximum entropy production principle is used to obtain a kinetic equation for the material metric’s evolution.The principle states that a constrained non-equilibrium system evolves to maximize its entropy production.Source: §2.7 · p. 12
  • The entropy production rate is defined for a subbody of a growing body and maximized with respect to the material metric rate under a constraint.The derivation introduces a Lagrange multiplier and uses the energy balance and constraint in the maximization.Source: §2.7 · p. 12
  • The derivation gives the Lagrange multiplier λ = −2.Source: §2.7 · p. 12

2.8 Isotropic Growth

Isotropic growth is represented by a time-dependent conformal family of material metrics. In this case, growth changes volume without changing shape, and the deformation-gradient decomposition uses a scalar field and the identity map.

2.8 Isotropic Growth

  • For isotropic growth, the evolving material metric is a family of conformal metrics.Source: §2.8 · p. 13
  • The section relates the changing material metric to mass balance and the evolution of volume elements over time.Source: §2.8 · p. 13
  • Isotropic growth entails no shape change from mass addition or removal, and its deformation-gradient decomposition uses the identity map and a scalar field.Source: §2.8 · p. 13

2.9 Examples of Bulk Growth

The section develops analytical examples of bulk growth and residual stresses, using annuli and a hollow sphere. It also examines when rotationally symmetric grown metrics can be represented as surfaces of revolution.

2.9 Examples of Bulk Growth

  • The examples provide analytical solutions for residual stresses under both isotropic and non-isotropic bulk growth.Source: §2.9 · p. 13
  • An incompressible neo-Hookean annulus is analyzed under isotropic growth, with traction boundary conditions used to determine the unknown radial mapping.The stress components are then obtained from the solution.Source: §2.9 · pp. 13–14
  • A growth that changes shape without changing volume can still produce residual stresses in an anisotropically growing annulus.Source: §2.9 · p. 16
  • A third example treats radial growth of an incompressible neo-Hookean hollow sphere and derives its equilibrium equation under spherical symmetry.Source: §2.9 · pp. 16–17
  • A rotationally symmetric metric may lack a surface-of-revolution embedding when M^2 < dotN^2; the isotropic profiles Ω(R) = −R and Ω(R) = −R^2 are illustrated on restricted radial ranges.Source: §2.9 · p. 18, Figure 2.2 · p. 19
  • The anisotropic annulus examples visualize material manifolds for the profiles Ω(R) = cos^2 R, Π(R) = 0 and Ω(R) = 0, Π(R) = −ln R^2.Source: §2.9 · p. 18, Figure 2.3 · p. 19

2.11 Stress-Free Isotropic Growth

Stress-free isotropic growth is characterized through flatness of the grown material metric. In two dimensions, harmonic growth fields meet the local flatness condition, while global topology can still restrict whether a whole body can lie stress-free in a plane.

2.11 Stress-Free Isotropic Growth

  • For an initially flat two-dimensional body, stress-free growth requires the grown material metric to be Euclidean, and the local condition is that Ω be harmonic.The section treats time as a parameter because growth is slow compared with elastic deformation.Source: §2.11 · pp. 20–21
  • Local flatness does not guarantee a global stress-free planar configuration: a full cylinder cannot be laid flat without tearing, folding, or stretching, although finite pieces can.Source: §2.11 · p. 21
  • For a radial growth profile on an annulus, an intrinsically flat grown metric can still cause stress in the full annulus, whereas a simply-connected piece can expand stress-free.The annulus grows to a cone, while the simply-connected piece can lie between the rigid planes without stress.Source: §2.11 · p. 21, Figure 2.4 · p. 21
  • Every smooth curved two-dimensional stress-free solid admits a unique growth distribution that makes its grown configuration flat and stress-free.Equivalently, a flat stress-free sheet can be grown into any smooth curved shape without residual stress.Source: §2.11 · p. 21
  • In three dimensions, local flatness requires the Ricci tensor to vanish; the section derives a nonlinear PDE system for the isotropic growth field.Source: §2.11 · p. 22
  • Some three-dimensional stress-free growth solutions make the material metric Euclidean in a transformed radial coordinate, which can invert radii; other cases recover uniform growth or a metric conformal to the Poincaré half space.The inversion may not be possible for a solid ball without tearing, but is possible for a piece of one.Source: §2.11 · p. 22

2.12 Lagrangian Field Theory of Growing Bodies

The Lagrangian formulation treats the material metric as a dynamical variable alongside the deformation. Because growth is dissipative and mass density varies with the metric, the authors use a Lagrange-d’Alembert principle with a Rayleigh dissipation function and a mass-balance constraint.

2.12 Lagrangian Field Theory of Growing Bodies

  • The material metric is a dynamical variable in the Lagrangian, together with the deformation, its velocity, the deformation gradient, and the spatial metric.The action is formed from this Lagrangian over the material volume.Source: §2.12 · p. 23
  • Growth is dissipative, so the conservative Hamilton principle is insufficient; the formulation instead uses a Rayleigh dissipation function in the Lagrange-d’Alembert principle.The dissipation function depends on the material metric and its rate.Source: §2.12 · p. 23
  • Separate variations of deformation and metric yield their corresponding Euler-Lagrange equations under the Lagrange-d’Alembert principle.The metric evolution equation is noted to be similar to the one obtained using maximum entropy production.Source: §2.12 · pp. 23–24
  • Variations of density and material metric are coupled by the nonholonomic mass-balance constraint, unlike in classical nonlinear elasticity.The density variation is therefore not independent of the metric variation.Source: §2.12 · p. 24
  • With a quadratic Rayleigh function, the metric dissipation potential is specified in terms of the metric rate, yielding a particular form of its evolution equation.The example takes R(G, Ġ) = ω tr Ġ².Source: §2.12 · p. 25

3 Connection between F = FeFg and the Geometric Theory

The geometric theory encodes growth through a material metric, while the conventional multiplicative decomposition encodes the same metric through a growth deformation gradient. In an orthonormal material basis, the elastic part corresponds to the deformation gradient, and the two approaches admit equivalent constitutive descriptions.

3 Connection between F = FeFg and the Geometric Theory

  • A Riemannian material manifold with a metric related to nonuniform growth represents the relaxed state against which strains are measured.Constitutive relations depend on the material metric, spatial metric, and deformation gradient.Source: §3 · pp. 25, 29
  • The local-relaxation construction allows F_e and F_g to be incompatible while still prescribing them from deformation and growth to calculate stresses.In one material dimension, the decomposition of the deformation mapping is always possible.Source: §3 · p. 26
  • The deformation gradient expressed in an orthonormal material basis has components equal to those of F_e in the F = F_e F_g decomposition.The growth part supplies the orthonormal frame in the material manifold.Source: §3 · p. 27
  • The same material metric can arise from different growth deformation gradients related by an orthogonal transformation, giving an SO(3) ambiguity in choosing F_g.The metric is unique even though the local stress-free configurations are not.Source: §3 · pp. 25, 27
  • A constitutive relation in the geometric approach can be converted to the local-relaxation approach by changing to an orthonormal basis and omitting constant metric components.The reverse conversion, from the multiplicative-decomposition free energy to a geometric free energy, is also possible.Source: §3 · p. 27
  • The approaches differ geometrically: the geometric theory uses a Riemannian material manifold, whereas F = F_e F_g implicitly uses the same metric in an absolutely parallelizable, non-Riemannian manifold.The geometric approach avoids introducing torsion, though its material manifold generally has curvature.Source: §3 · p. 29

4 Linearized Theory of Growth Mechanics

The linearized theory studies changes in equilibrium and stress caused by a small change in the material metric. It relates the linearized strain to geometric variation and identifies stress-free growth distributions under specified assumptions.

4 Linearized Theory of Growth Mechanics

  • A small change in the material metric generally moves the body away from equilibrium, and the linearized equations determine the resulting change in stress or configuration.The setup uses a one-parameter family of metrics, equilibrium configurations, and stresses around a reference motion.Source: §4 · p. 29
  • The linearization treats growth as slow relative to elastic deformation, so inertial effects are ignored and time is treated as a parameter.The reference material metric and ambient space are assumed Euclidean for the stated stress-free classification.Source: §4 · pp. 29, 32
  • The linearized strain is the push-forward of the linearized Lagrangian strain, and it vanishes when the variation field is a Killing field of the spatial metric.For Euclidean ambient space in Cartesian coordinates, covariant derivatives reduce to partial derivatives.Source: §4 · p. 30
  • In three dimensions, isotropic growth from an initially Euclidean material manifold is stress-free in the linearized theory if and only if β is linear in the initial Cartesian coordinates.The curvature conditions reduce to vanishing second derivatives of β.Source: §4 · pp. 32–33

5 Concluding Remarks

The paper presents a geometric theory of elastic solids with bulk growth, using an evolving material metric to represent changes in density and shape. It derives balance laws and thermodynamically consistent metric evolution, connects the theory to multiplicative decomposition, and identifies stress-free growth distributions in the linearized theory.

5 Concluding Remarks

  • An evolving material metric models bulk growth while material points remain fixed, and its time dependence requires modifying energy balance and entropy production.Covariance of energy balance yields balance laws, while the Clausius-Duhem inequality imposes constitutive restrictions.Source: §5 · p. 33
  • The principle of maximum entropy production provides thermodynamically consistent evolution equations for the material metric.The material metric is the variable whose evolution represents growth.Source: §5 · p. 33
  • The theory obtains analytical residual-stress solutions for three radially symmetric growing-body examples, and mass conservation does not preclude residual stress when growth changes shape.The paper also studies isotropic stress-free distributions using material curvature in two and three dimensions.Source: §5 · pp. 33–34
  • The geometric theory connects to F = F_e F_g, and its linearization yields eigenstrains similar to classical linear thermoelasticity and stress-free growth distributions in two and three dimensions.The stated linearization assumes Euclidean ambient space and an initially Euclidean material manifold.Source: §5 · p. 34
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