Growth-induced Residual Stresses Research Articles

A relaxed growth modeling framework for controlling growth-induced residual stresses

BackgroundConstitutive models of the mechanical response of soft tissues have been established and are widely accepted, but models of soft tissues remodeling are more controversial. Specifically for growth, one important question arises pertaining to residual stresses: existing growth models inevitably introduce residual stresses, but it is not entirely clear if this is physiological or merely an artifact of the modeling framework. As a consequence, in simulating growth, some authors have chosen to keep growth-induced residual stresses, and others have chosen to remove them. MethodsIn this paper, we introduce a novel “relaxed growth” framework allowing for a fine control of the amount of residual stresses generated during tissue growth. It is a direct extension of the classical framework of the multiplicative decomposition of the transformation gradient, to which an additional sub-transformation is introduced in order to let the original unloaded configuration evolve, hence relieving some residual stresses. We provide multiple illustrations of the framework mechanical response, on time-driven constrained growth as well as the strain-driven growth problem of the artery under internal pressure, including the opening angle experiment. FindingsThe novel relaxed growth modeling framework introduced in this paper allows for a better control of growth-induced residual stresses compared to standard growth models based on the multiplicative decomposition of the transformation gradient. InterpretationGrowth-induced residual stresses should be better handled in soft tissues biomechanical models, especially in patient-specific models of diseased organs that are aimed at augmented diagnosis and treatment optimization.

Influence of Initial Residual Stress on Growth and Pattern Creation for a Layered Aorta

Residual stress is ubiquitous and indispensable in most biological and artificial materials, where it sustains and optimizes many biological and functional mechanisms. The theory of volume growth, starting from a stress-free initial state, is widely used to explain the creation and evolution of growth-induced residual stress and the resulting changes in shape, and to model how growing bio-tissues such as arteries and solid tumors develop a strategy of pattern creation according to geometrical and material parameters. This modelling provides promising avenues for designing and directing some appropriate morphology of a given tissue or organ and achieve some targeted biomedical function. In this paper, we rely on a modified, augmented theory to reveal how we can obtain growth-induced residual stress and pattern evolution of a layered artery by starting from an existing, non-zero initial residual stress state. We use experimentally determined residual stress distributions of aged bi-layered human aortas and quantify their influence by a magnitude factor. Our results show that initial residual stress has a more significant impact on residual stress accumulation and the subsequent evolution of patterns than geometry and material parameters. Additionally, we provide an essential explanation for growth-induced patterns driven by differential growth coupled to an initial residual stress. Finally, we show that initial residual stress is a readily available way to control growth-induced pattern creation for tissues and thus may provide a promising inspiration for biomedical engineering.

MODELING OF ANISOTROPIC GROWTH AND RESIDUAL STRESSES IN ARTERIAL WALLS

Based on the multiplicative decomposition of the deformation gradient, a local formulation for anisotropic growth in soft biological tissues is formulated by connecting the growth tensor to the main anisotropy directions. In combination with an anisotropic driving force, the model enables an effective stress reduction due to growth-induced residual stresses. A method for the imitation of opening angle experiments in numerically simulated arterial segments, visualizing the deformations related to residual stresses, is presented and illustrated in a numerical example.

Growing skin: A computational model for skin expansion in reconstructive surgery

The goal of this manuscript is to establish a novel computational model for stretch-induced skin growth during tissue expansion. Tissue expansion is a common surgical procedure to grow extra skin for reconstructing birth defects, burn injuries, or cancerous breasts. To model skin growth within the framework of nonlinear continuum mechanics, we adopt the multiplicative decomposition of the deformation gradient into an elastic and a growth part. Within this concept, we characterize growth as an irreversible, stretch-driven, transversely isotropic process parameterized in terms of a single scalar-valued growth multiplier, the in-plane area growth. To discretize its evolution in time, we apply an unconditionally stable, implicit Euler backward scheme. To discretize it in space, we utilize the finite element method. For maximum algorithmic efficiency and optimal convergence, we suggest an inner Newton iteration to locally update the growth multiplier at each integration point. This iteration is embedded within an outer Newton iteration to globally update the deformation at each finite element node. To demonstrate the characteristic features of skin growth, we simulate the process of gradual tissue expander inflation. To visualize growth-induced residual stresses, we simulate a subsequent tissue expander deflation. In particular, we compare the spatio-temporal evolution of area growth, elastic strains, and residual stresses for four commonly available tissue expander geometries. We believe that predictive computational modeling can open new avenues in reconstructive surgery to rationalize and standardize clinical process parameters such as expander geometry, expander size, expander placement, and inflation timing.

On the theory of reactive mixtures for modeling biological growth

Mixture theory, which can combine continuum theories for the motion and deformation of solids and fluids with general principles of chemistry, is well suited for modeling the complex responses of biological tissues, including tissue growth and remodeling, tissue engineering, mechanobiology of cells and a variety of other active processes. A comprehensive presentation of the equations of reactive mixtures of charged solid and fluid constituents is lacking in the biomechanics literature. This study provides the conservation laws and entropy inequality, as well as interface jump conditions, for reactive mixtures consisting of a constrained solid mixture and multiple fluid constituents. The constituents are intrinsically incompressible and may carry an electrical charge. The interface jump condition on the mass flux of individual constituents is shown to define a surface growth equation, which predicts deposition or removal of material points from the solid matrix, complementing the description of volume growth described by the conservation of mass. A formulation is proposed for the reference configuration of a body whose material point set varies with time. State variables are defined which can account for solid matrix volume growth and remodeling. Constitutive constraints are provided on the stresses and momentum supplies of the various constituents, as well as the interface jump conditions for the electrochemical potential of the fluids. Simplifications appropriate for biological tissues are also proposed, which help reduce the governing equations into a more practical format. It is shown that explicit mechanisms of growth-induced residual stresses can be predicted in this framework.

Effect of sputtering target power density on topography and residual stress during growth of nanocomposite nc-TiN/a-SiN x thin films

Nanocrystalline TiN (nc-TiN) has been imbedded in amorphous silicon nitride (a-SiN x ) matrix to form a nanocomposite thin film (nc-TiN/a-SiN x ) via magnetron sputtering. Adjusting Ti and Si 3N 4 target power ratio alters film composition, morphology, microstructure and residual stresses. The effects of silicon nitride target power density on surface morphology and residual stress have been studied. Atomic force microscopy (AFM) shows that the film surface becomes increasingly smoother and the roughness ( R a) drops from about 6 nm down to less than 1 nm as the silicon nitride target power density increases from 1.1 to 5.5 W cm −2. The surface topography is quantitatively described by calculation of two parameters: the interface width ( w) and the lateral correlation length ( ξ). The interface width ( w) describes the vertical growth, and the lateral correlation length ( ξ) characterizes the lateral growth of the surface. The growth-induced residual stress during the sputtering deposition of nc-TiN/a-SiN x nanocomposite thin film increases with sputtering power density, but only reaches about −1 GPa even at the highest experimented Si 3N 4 target power density of 5.5 W cm −2. The growth-induced stress is opposite in sign and roughly equal in quantity to the thermal stress induced by the difference in the coefficient of thermal expansion (CTE) between the films and the silicon wafer substrate. As a result, the magnetron sputtered nc-TiN/a-SiN x nanocomposite films become almost stress-free (as low as 0 to −0.15 GPa residual stress).

Stress-modulated growth, residual stress, and vascular heterogeneity.

A simple phenomenological model is used to study interrelations between material properties, growth-induced residual stresses, and opening angles in arteries. The artery is assumed to be a thick-walled tube composed of an orthotropic pseudoelastic material. In addition, the normal mature vessel is assumed to have uniform circumferential wall stress, which is achieved here via a mechanical growth law. Residual stresses are computed for three configurations: the unloaded intact artery, the artery after a single transmural cut, and the inner and outer rings of the artery created by combined radial and circumferential cuts. The results show that the magnitudes of the opening angles depend strongly on the heterogeneity of the material properties of the vessel wall and that multiple radial and circumferential cuts may be needed to relieve all residual stress. In addition, comparing computed opening angles with published experimental data for the bovine carotid artery suggests that the material properties change continuously across the vessel wall and that stress, not strain, correlates well with growth in arteries.

Quantitative evaluation of growth-induced residual stress in InP epitaxial micromechanical structures

We have investigated the residual stress distribution and its relation to the presence of impurity atoms in InP layers grown by metalorganic vapor phase epitaxy (MOVPE) on top of lattice-matched GaInAs layers. The homogeneous and gradient components of the stress have been directly measured using highly sensitive micromechanical test structures. Secondary ion mass spectrometry depth profiling of the composition evidenced that residual As and Ga atoms are present throughout the relatively thick (>0.5 μm) InP layers. Our investigations show that the distribution of these impurities accounts for the measured residual stress in the layers and, at the same time, it is specific to the MOVPE growth technique.