Surface growth in deformable solids using an Eulerian formulation

Naghibzadeh, S. Kiana; Walkington, Noel J.; Dayal, Kaushik

doi:10.1016/j.jmps.2021.104499

Cited by 11 publications

(6 citation statements)

References 44 publications

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“…An important challenge in the Eulerian formulation is to know the location of the boundaries to be able to apply the boundary conditions. We refer to the brief discussion in Section 3.3.1 of [24], as well as [29,30], and, noticing that the velocity field can be used to locate the boundary of the body for the parts of the boundary without growth, we follow the characteristic lines of particles located on the boundary; for the parts with growth, we use equation ( 3) together with the path lines to determine the location of the growing boundaries.…”

Section: Discussionmentioning

confidence: 99%

“…Following [24], we model surface growth through the introduction of sources that are localized on the growing surface. The source terms can be understood as a coarse-grained approach to treat the complex process of growth without considering the fine details of the processes in the ambient environment outside the growing body; the growth is defined only in terms of net mass and linear momentum transfer.…”

Section: Physical Model and Governing Equationsmentioning

confidence: 99%

“…To avoid the difficulties of dealing with an evolving reference configuration, our previous work [24] developed an Eulerian description of surface growth. This has the advantage that we do not need to calculate the reference configuration explicitly, e.g.…”

Section: Introductionmentioning

confidence: 99%

“…Section 2 summarizes the approach proposed in [24]. We start with a discussion on balance laws and jump or boundary conditions, using the view of surface growth as a localized source on the growing surface.…”

Section: Introductionmentioning

confidence: 99%

“…e rr F e rθ F e rφ F e θr F e θθ F e θφ F e φr F e φθ F e φφ  (23)F e (r, t = 0) given, F e (r = r 0 , t) = I(24) σ (F e ) = −pI + GF e F T e , G is the shear modulus (25). …”

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Accretion and ablation in deformable solids with an Eulerian description: examples using the method of characteristics

Naghibzadeh

Walkington

Dayal

2021

Mathematics and Mechanics of Solids

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Accretion and ablation, i.e., the addition and removal of mass at the surface, are important in a wide range of physical processes, including solidification, growth of biological tissues, environmental processes, and additive manufacturing. The description of accretion requires the addition of new continuum particles to the body, and is therefore challenging for standard continuum formulations for solids that require a reference configuration. Recent work has proposed an Eulerian approach to this problem, enabling side-stepping of the issue of constructing the reference configuration. However, this raises the complementary challenge of determining the stress response of the solid, which typically requires the deformation gradient that is not immediately available in the Eulerian formulation. To resolve this, the approach introduced the elastic deformation as an additional kinematic descriptor of the added material, and its evolution has been shown to be governed by a transport equation. In this work, the method of characteristics is applied to solve concrete simplified problems motivated by biomechanics and manufacturing. Specifically, (1) for a problem with both ablation and accretion in a fixed domain and (2) for a problem with a time-varying domain, the closed-form solution is obtained in the Eulerian framework using the method of characteristics without explicit construction of the reference configuration.

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Section: Discussionmentioning

confidence: 99%

Section: Physical Model and Governing Equationsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Accretion and ablation in deformable solids with an Eulerian description: examples using the method of characteristics

Naghibzadeh

Walkington

Dayal

2021

Mathematics and Mechanics of Solids

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Phase‐Field Modeling of Fracture Under Compression and Confinement in Anisotropic Geomaterials

Hakimzadeh,

Mora‐Corral,

Walkington

et al. 2024

Num Anal Meth Geomechanics

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Strongly anisotropic geomaterials, such as layered shales, have been observed to undergo fracture under compressive loading. This paper applies a phase‐field fracture model to study this fracture process. While phase‐field fracture models have several advantages—primarily that the fracture path is not predetermined but arises naturally from the evolution of a smooth non‐singular damage field—they provide unphysical predictions when the stress state is complex and includes compression that can cause crack faces to contact.Building on a recently developed phase‐field model that accounts for compressive traction across the crack face, this paper extends the model to the setting of anisotropic fracture. The key features of the model include the following: (1) a homogenized anisotropic elastic response and strongly anisotropic model for the work to fracture; (2) an effective damage response that accounts consistently for compressive traction across the crack face, that is derived from the anisotropic elastic response; (3) a regularized crack normal field that overcomes the shortcomings of the isotropic setting, and enables the correct crack response, both across and transverse to the crack face.To test the model, we first compare the predictions to phase‐field fracture evolution calculations in a fully resolved layered specimen with spatial inhomogeneity, and show that it captures the overall patterns of crack growth. We then apply the model to previously reported experimental observations of fracture evolution in laboratory specimens of shales under compression with confinement, and find that it predicts well the observed crack patterns in a broad range of loading conditions. We further apply the model to predict the growth of wing cracks under compression and confinement. Prior approaches to simulate wing cracks have treated the initial cracks as an external boundary, which makes them difficult to apply to general settings. Here, the effective crack response model enables us to treat the initial crack simply as a nonsingular damaged zone within the computational domain, thereby allowing for easy and general computations.

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On the Stability of Surface Growth: The Effect of a Compliant Surrounding Medium

Abeyaratne

Puntel

Tomassetti

2022

J Elast

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In a recent paper Abeyaratne et al. (J. Mech. Phys. Solids 167:104958, 2022) concerning the stability of surface growth of a pre-stressed elastic half-space with surface tension, it was shown that steady growth is never stable, at least not for all wave numbers of the perturbations, when the growing surface is traction-free. On the other hand, steady growth was found to be always stable when growth occurred on a flat frictionless rigid support and the stretch parallel to the growing surface was compressive. The present study is motivated by these somewhat unexpected and contrasting results.In this paper the stability of a pre-compressed neo-Hookean elastic half-space undergoing surface growth under plane strain conditions is studied. The medium outside the growing body resists growth by applying a pressure on the growing surface. At each increment of growth, the incremental change in pressure is assumed to be proportional to the incremental change in normal displacement of the growing surface. It is shown that surface tension stabilizes a homogeneous growth process against small wavelength perturbations while the compliance of the surrounding medium stabilizes it against large wavelength perturbations. Specifically, there is a critical value of stretch, $\lambda _{\mathrm{cr}} \in (0,1)$ λ cr ∈ ( 0 , 1 ) , such that growth is linearly stable against infinitesimal perturbations of arbitrary wavelength provided the stretch parallel to the growing surface exceeds $\lambda _{\mathrm{cr}}$ λ cr . This stability threshold, $\lambda _{\mathrm{cr}}$ λ cr , is a function of the non-dimensional parameter $\sigma \kappa /G^{2}$ σ κ / G 2 , which is the ratio between two length-scales $\sigma /G$ σ / G and $G/\kappa $ G / κ , where $G$ G is the shear modulus of the elastic body, $\sigma $ σ is the surface tension, and $\kappa $ κ is the stiffness of the surrounding compliant medium.It is shown that $(a)$ ( a ) $\lambda _{\mathrm{cr}} \to 1$ λ cr → 1 as $\kappa \to 0$ κ → 0 and $(b)$ ( b ) $\lambda _{\mathrm{cr}} \to 0^{+}$ λ cr → 0 + as $\kappa \to \infty $ κ → ∞ , thus recovering the results in Abeyaratne et al. (J. Mech. Phys. Solids 167:104958, 2022) pertaining to the respective limiting cases where growth occurs $(a)$ ( a ) on a traction-free surface and $(b)$ ( b ) on a frictionless rigid support. The results are also generalized to include extensional stretches.

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Surface growth in deformable solids using an Eulerian formulation

Cited by 11 publications

References 44 publications

Accretion and ablation in deformable solids with an Eulerian description: examples using the method of characteristics

Accretion and ablation in deformable solids with an Eulerian description: examples using the method of characteristics

Phase‐Field Modeling of Fracture Under Compression and Confinement in Anisotropic Geomaterials

On the Stability of Surface Growth: The Effect of a Compliant Surrounding Medium

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