Surface growth in deformable solids using an Eulerian formulation

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.

Standard measures of local deformation such as deformation gradient, strain, elastic deformation, and plastic deformation are dimensionless. However, many macroscopic and submacroscopic geometrical changes observed in continuous bodies result in the formation of zones across whose boundaries significant changes in geometry can occur. In order to predict the sizes of such zones and their influence on material response, theories of elasticity and plasticity have been employed in which second gradients of deformation, gradients of strain, as well as gradients of elastic or of plastic deformation are taken into account. The theory of structured deformations provides additive decompositions of first deformation gradient and of second deformation gradient, valid for large deformations of any material, in which each term has a multiscale geometrical interpretation corresponding to the presence or absence of submacroscopic disarrangements (non-smooth geometrical changes such as slips and void formation). This article provides a field theory that broadens the earlier field theory, elasticity with disarrangements, by including energetic contributions from submacroscopic “gradient-disarrangements” (limits of averages of jumps in gradients of approximating deformations) and by treating particular kinematical conditions as internal constraints. An explicit formula is obtained showing the manner in which submacroscopic gradient-disarrangements determine a defectiveness density analogous to the dislocation density in theories of plasticity. A version of the new field theory incorporates this defectiveness density to obtain a counterpart of strain-gradient plasticity, while another instance of elasticity with gradient-disarrangements recovers an instance of strain-gradient elasticity with symmetric Cauchy stress. All versions of the new theory included here are compatible with the Second Law of Thermodynamics.

<p>We present a versatile open-source FEM-based multi-physics numerical code GALES for volcanic and general-purpose problems. The code is developed/applied to a suite of problems in magma and volcano dynamics. The software is written in modern C++ and is parallelized using OpenMPI and  Trilinos libraries. GALES comprises several advanced solvers for 2D and 3D problems dealing with heat transfer, compressible to incompressible mono and multi-fluid flows in Eulerian and Arbitrary Lagrangian-Eulerian (ALE) formulations, Elastic (static and dynamic) deformation of solids and fluid-solid interaction. Fluid solvers account for both Newtonian and non-Newtonian rheologies. Solvers account for transient as well as steady problems. Non-linear problems are linearized using Newton's method. All solvers have been thoroughly verified and validated on standard benchmarks. The software is regularly used for high-performance computing (HPC) on our local cluster machines at INGV, Pisa, Italy. Recently, we analyzed the performance of the code by a series of strong-scaling tests conducted on the Marenostrum supercomputer at the Barcelona Supercomputing Centre (BSC) up to 12288 cores. The results revealed a computational speedup close to ideal and above satisfactory levels as long as the element/core ratio is sufficiently large, making GALES an excellent choice for utilizing HPC resources efficiently for complex magma flow and rock dynamics problems.</p>

It is now a rather common approach to perform patient-specific stress analyses of arterial walls using finite-element models reconstructed from gated medical images. However, this requires to compute for every Gauss point the deformation gradient between the current configuration and a stress-free reference configuration. It is technically difficult to define such a reference configuration, and there is actually no guarantee that a stress-free configuration is physically attainable due to the presence of internal stresses in unloaded soft tissues. An alternative framework was proposed by Bellini et al. (Ann Biomed Eng 42(3):488-502, 2014). It consists of computing the deformation gradients between the current configuration and a prestressed reference configuration. We present here the first finite-element results based on this concept using the Abaqus software. The reference configuration is set arbitrarily to the in vivo average geometry of the artery, which is obtained from gated medical images and is assumed to be mechanobiologically homeostatic. For every Gauss point, the stress is split additively into the contributions of each individual load-bearing constituent of the tissue, namely elastin, collagen, smooth muscle cells. Each constituent is assigned an independent prestretch in the reference configuration, named the deposition stretch. The outstanding advantage of the present approach is that it simultaneously computes the in situ stresses existing in the reference configuration and predicts the residual stresses that occur after removing the different loadings applied onto the artery (pressure and axial load). As a proof of concept, we applied it on an ideal thick-wall cylinder and showed that the obtained results were consistent with corresponding experimental and analytical results of the well-known literature. In addition, we developed a patient-specific model of a human ascending thoracic aneurysmal aorta and demonstrated the utility in predicting the wall stress distribution in vivo under the effects of physiological pressure. Finally, we simulated the whole process preceding traditional in vitro uniaxial tensile testing of arteries, including excision from the body, radial cutting, flattening and subsequent tensile loading, showing how this process may impact the final mechanical properties derived from these in vitro tests.

A soft solid is said to be initially stressed if it is subjected to a state of internal stress in its unloaded reference configuration. In physical terms, its stored elastic energy may not vanish in the absence of an elastic deformation, being also dependent on the spatial distribution of the underlying material inhomogeneities. Developing a sound mathematical framework to model initially stressed solids in nonlinear elasticity is key for many applications in engineering and biology. This work investigates the links between the existence of elastic minimizers and the constitutive restrictions for initially stressed materials subjected to finite deformations. In particular, we consider a subclass of constitutive responses in which the strain energy density is taken as a scalar-valued function of both the deformation gradient and the initial stress tensor. The main advantage of this approach is that the initial stress tensor belongs to the group of divergence-free symmetric tensors satisfying the boundary conditions in any given reference configuration. However, it is still unclear which physical restrictions must be imposed for the well-posedness of this elastic problem. Assuming that the constitutive response depends on the choice of the reference configuration only through the initial stress tensor, under given conditions we prove the local existence of a relaxed state given by an implicit tensor function of the initial stress distribution. This tensor function is generally not unique, and can be transformed according to the symmetry group of the material at fixed initial stresses. These results allow one to extend Ball's existence theorem of elastic minimizers for the proposed constitutive choice of initially stressed materials. This article is part of the theme issue 'Rivlin's legacy in continuum mechanics and applied mathematics'.

In this study, we address the computational modeling of biological soft tissue growth, focusing on the development of epithelial tissues. The formulation of the corresponding constitutive growth law using non-linear continuum mechanics and its implementation within a total-Lagrangian-type finite element method are described. In describing the growth law, we use multiplicative decomposition of the deformation gradient into a growth part and an elastic part. We propose two surface growth deformation gradients; isotropic surface growth and anisotropic surface growth with relative shrinkage in the principal direction of maximum stress at the initial state. We first apply our laws to a hollow thick-walled hemiellipsoid that idealizes a structure generally observed in the early development of epithelial tissues. Our simulation shows the following: (i) under isotropic surface growth, the hemiellipsoid becomes sphere-like, i.e., the ratio between the longest and shortest axial length tends to 1; (ii) in contrast, under anisotropic surface growth, the tissue elongates and flattens, i.e., the ratio between the longest and shortest axial length is enhanced. These results motivate us to apply the latter growth law to the realistic example of a vertebrate limb bud, which shows similar elongation and flattening during development. As expected, our numerical simulation succeeded in reproducing the essential aspects of morphological change in the limb bud, providing a new hypothesis for the vertebrate limb development.

The changing mass of biomaterials can either be modelled at the constitutive level or at the kinematic level. This contribution attends on the description of growth at the kinematic level. The deformation gradient will be multiplicatively split into a growth part and an elastic part. Hence, in addition to the material and the spatial configuration, we consider an intermediate configuration or grown configuration without any elastic deformations. With such an ansatz at hand, contrary to the modelling of mass changes at the constitutive level, both a change in density and a change in volume can be modelled. The algorithmic realisation of this framework within a finite element setting constitutes the main contribution of this paper. To this end the key kinematic variable, i.e. the isotropic stretch ratio, is introduced as internal variable at the integration point level. The consistent linearisation of the stress update based on an implicit time integration scheme is developed. Basic features of the model are illustrated by means of representative numerical examples.

A new mathematical framework for modelling the biomechanics of growing trees with rod theory

Modeling anisotropic inelastic effects in sheet metal forming using microstructural vectors—Part I: Theory

A continuum theory for the mechanical response of solid bodies subjected to potentially finite deformation is further developed and applied to solve several new problems in the context of the micromechanics of crystalline solids. The theory invokes concepts from Finsler differential geometry, and it provides a diffuse interface description of fracture surfaces. The director or internal state vector is associated with an order parameter describing degradation of the solid. Here, the deformation gradient between pseudo-Finsler reference and spatial configuration spaces is decomposed into a product of two terms, neither necessarily integrable to a vector field. The first is the recoverable elastic deformation, the second is the residual deformation attributed to changes in free volume in failure zones. The latter is restricted to spherical or isotropic symmetry; resulting Euler–Lagrange equations for mechanical and state variable equilibrium are derived. Metric tensors and volume elements depend on the internal state via a conformal transformation, i.e., Weyl scaling. This version of the theory is first applied to tensile fracture of magnesium. Analytical solutions demonstrate the model's capability to predict ductile versus brittle fracture depending on incorporation of Weyl scaling, with results aligned with molecular dynamics (MD) simulations. The second application is shear fracture in boron carbide: solutions depict weakening and tensile pressure in conjunction with structural collapse in shear transformation zones, as suggested by experiments, quantum mechanics, and/or MD simulations.

A constitutive material spin tensor in the case of purely elastic finite-strain deformation is introduced for a two-dimensional orthotropic media using the minimizing principle applied to obtain the reloaded configuration of the material volume. This material spin explains the rotation of the orthonormal vector frame which coincides with the material symmetry axes in the initial configuration of the material volume and uniquely corresponds to a set of these axes in the current configuration although it does not coincide with the latter. The given definition is followed by the exact expression which includes the deformation gradient tensor, unit vectors of the initial material anisotropy axes and their axial parameters. This definition allows obtaining a new variant of decomposing any elastic finite-strain motion onto rigid and deformational parts and introducing the material corotational rate. The latter is used for the formulation of the anisotropic rate-type elastic law in the current configuration based on the strain measure which does not belong to the Seth-Hill family. For isotropic as well as for tetragonal media, the introduced material rotation tensor coincides with the rotation tensor from the polar decomposition of a deformation gradient.

Growing avascular tumours as elasto-plastic bodies by the theory of evolving natural configurations

We describe a method for computing time-dependent solutions to the compressible Navier-Stokes equations coupled to a hyperelastic Neo-Hookean membrane model. The deforming domain is handled by introducing a continuous mapping between a fixed reference configuration and the time varying domain, and rewriting the Navier-Stokes equations as a conservation law for the independent variables in the reference configuration. The spatial discretization is carried out using the Discontinuous Galerkin method on unstructured meshes of triangles, and the membrane model is discretized with regular finite elements. The method uses an explicit smooth mapping for the entire domain which is dierentiated to obtain accurate grid velocities and deformation gradients. This mapping is constructed by a combination of spline interpolation and high order blending functions. Various examples are shown to illustrate the methods, and elastic and rigid membranes are compared.

Experiments on the kinematics of large plastic strain in ordered solids

Mapping the nonlinear crack tip deformation field in soft elastomer with a particle tracking method