scholarly journals Homogenized Balance Equations for Nonlinear Poroelastic Composites

2021 ◽  
Vol 11 (14) ◽  
pp. 6611
Author(s):  
Laura Miller ◽  
Raimondo Penta
Keyword(s):  
Deformation Gradient ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Type Model ◽  
Nonlinear Behaviour ◽  
Pore Scale ◽  
Balance Equations ◽  
The Matrix ◽  
Elastic Composites ◽  
Porous Composites

Within this work, we upscale the equations that describe the pore-scale behaviour of nonlinear porous elastic composites, using the asymptotic homogenization technique in order to derive the macroscale effective governing equations. A porous hyperelastic composite can be thought of as being comprised of a matrix interacting with a number of subphases and percolated by a fluid flowing in the pores (which is chosen to be Newtonian and incompressible here). A general nonlinear macroscale model is derived and is then specified for a particular choice of strain energy function, namely the de Saint-Venant function. This leads to a macroscale system of PDEs, which is of poroelastic type with additional terms and transformations to account for the nonlinear behaviour of the material. Our new porohyperelastic-type model describes the effective behaviour of nonlinear porous composites by prescribing the stress balance equations, the conservation of mass and Darcy’s law. The coefficients of these macroscale equations encode the detailed microstructure of the material and are to be found by solving pore-scale differential problems. The model reduces to the following limit cases of (a) linear poroelastic composites when the deformation gradient approaches the identity, (b) nonlinear composites when there are no pores and (c) nonlinear poroelasticity when only the matrix–fluid interaction is considered. This model is applicable when the interactions between various hyperelastic solid phases occur at the pore-scale, as in biological tissues such as artery walls, the myocardium, lungs and liver.

A STUDY ON THE EFFECT OF COLLAGEN FIBER ORIENTATION ON MECHANICAL RESPONSE OF SOFT BIOLOGICAL TISSUES

2015 ◽  
Vol 76 (7) ◽  
Author(s):  
Farshid Fathi ◽  
Shahrokh Shahi ◽  
Soheil Mohammadi
Keyword(s):  
Finite Element ◽  
Soft Tissues ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Soft Materials ◽  
Biological Tissues ◽  
Soft Biological Tissues ◽  
Proposed Model ◽  
The Matrix ◽  
Element Approach

Extensive research has been performed in the past decades to study the behavior of soft biological tissues in order to reduce the need for practical experiments. The applicability of these researches, particularly for skin, ligament, muscles and the heart, brings up its importance in various biological science and technology disciplines such as surgery and medicine. Softness and large deformation govern the behavior of soft materials and prohibit the use of small strains solutions in finite element method.In this work, the focus is set on a strain energy function which has the advantage of accurately representing the behavior of a variety of soft tissues with only a few parameters in a finite element approach. The numerical results are verified with a set of tensile experiments to demonstrate the performance of the proposed model. The parameters include the matrix and collagen bundles and their orientation. Different cases are analyzed and discussed for better prediction of different soft tissue responses.  

A Structurally Motivated Constitutive Model for Bat Wing Skin

2014 ◽  
Author(s):  
Alyssa J. Skulborstad ◽  
N. C. Goulbourne
Keyword(s):  
Constitutive Model ◽  
Deformation Gradient ◽  
Compressive Loading ◽  
Strain Energy Function ◽  
Tissue Response ◽  
Material Behavior ◽  
Elastin Fiber ◽  
The Matrix ◽  
Wing Skin ◽  
Bat Wing

Unique among animal flyers, bats have highly flexible and stretchable thin wing membranes. The connection between the structural constituents of bat wing skin, its material behavior, and flight abilities is not yet known. In this work we propose a structurally motivated constitutive model for the wing skin. Within a continuum mechanics framework, the proposed strain energy function for the wing skin is the sum of contributions due to the matrix and two mesoscopic fiber families, one oriented primarily spanwise consisting of elastin fiber bundles and the other family oriented chordwise consisting of muscle fibers. While the fibers are flat and straight when the wing is somewhat open, the matrix exhibits corrugations due to compressive loading from the pre-stretched spanwise fibers. This mismatch in the natural configurations of components is accounted for in the model by a decomposition of the deformation gradient of the spanwise fibers. The material parameters are fit with a procedure motivated by the underlying deformation mechanisms of the tissue corresponding to the regions of the j-shaped constitutive curves. The proposed model is fit to the first set of biaxial experimental stress-strain data for bat wing skin and captures the general features of the tissue response well.

Effect of membrane bending stiffness on the deformation of capsules in simple shear flow

2001 ◽  
Vol 440 ◽  
pp. 269-291 ◽  
Author(s):  
C. POZRIKIDIS
Keyword(s):  
Curvature Tensor ◽  
Deformation Gradient ◽  
Bending Stiffness ◽  
Boundary Element Methods ◽  
Simple Shear Flow ◽  
Normal Vector ◽  
Membrane Thickness ◽  
Bending Moments ◽  
Balance Equations ◽  
Capsule Deformation

The effect of interfacial bending stiffness on the deformation of liquid capsules enclosed by elastic membranes is discussed and investigated by numerical simulation. Flow-induced deformation causes the development of in-plane elastic tensions and bending moments accompanied by transverse shear tensions due to the non-infinitesimal membrane thickness or to a preferred configuration of an interfacial molecular network. To facilitate the implementation of the interfacial force and torque balance equations involving the hydrodynamic traction exerted on either side of the interface and the interfacial tensions and bending moments developing in the plane of the interface, a formulation in global Cartesian coordinates is developed. The balance equations involve the Cartesian curvature tensor defined in terms of the gradient of the normal vector extended off the plane of the interface in an appropriate fashion. The elastic tensions are related to the surface deformation gradient by constitutive equations derived by previous authors, and the bending moments for membranes whose unstressed shape has uniform curvature, including the sphere and a planar sheet, arise from a constitutive equation that involves the instantaneous Cartesian curvature tensor and the curvature of the resting configuration. A numerical procedure is developed for computing the capsule deformation in Stokes flow based on standard boundary-element methods. Results for spherical and biconcave resting shapes resembling red blood cells illustrate the effect of the bending modulus on the transient and asymptotic capsule deformation and on the membrane tank-treading motion.

scholarly journals Spatial-Frequency Azimuthally Stable Cartography of Biological Polycrystalline Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
V. A. Ushenko ◽  
N. D. Pavlyukovich ◽  
L. Trifonyuk
Keyword(s):  
Spatial Frequency ◽  
Optical Activity ◽  
Polarization Plane ◽  
Biological Tissues ◽  
Uterine Wall ◽  
Beam Polarization ◽  
Linear Birefringence ◽  
Histological Sections ◽  
The Matrix ◽  
Anisotropic Structures

A new azimuthally stable polarimetric technique processing microscopic images of optically anisotropic structures of biological tissues histological sections is proposed. It has been used as a generalized model of phase anisotropy definition of biological tissues by using superposition of Mueller matrices of linear birefringence and optical activity. The matrix elementM44has been chosen as the main information parameter, whose value is independent of the rotation angle of both sample and probing beam polarization plane. For the first time, the technique of concerted spatial-frequency filtration has been used in order to separate the manifestation of linear birefringence and optical activity. Thereupon, the method of azimuthally stable spatial-frequency cartography of biological tissues histological sections has been elaborated. As the analyzing tool, complex statistic, correlation, and fractal analysis of coordinate distributions ofM44element has been performed. The possibility of using the biopsy of the uterine wall tissue in order to differentiate benign (fibromyoma) and malignant (adenocarcinoma) conditions has been estimated.

Micromechanical Analysis of Nonlinear Response of Unidirectional Composites: A Fundamental Approach

2001 ◽  
Author(s):  
Virendra R. Jadhav ◽  
Srinivasan Sridharan
Keyword(s):  
Mechanical Model ◽  
Nonlinear Response ◽  
Experimental Studies ◽  
Matrix Cracking ◽  
Type Model ◽  
Unidirectional Composites ◽  
The Matrix ◽  
Experimental Response ◽  
Square Cells ◽  
Experimental Comparisons

Abstract Micromechanical models with different representative volume elements have been developed to study their ability to predict nonlinear response of unidirectional composites. A simple, square cells type micro-mechanical model similar to those widely used by other researchers is compared with a more advanced 3-phase finite element based micro-mechanical model. The models utilize the “bulk” properties of the matrix without attempting to “tune” the model to fit with experimental response of laminae. This is a more fundamental approach and constitutes a departure from current practice. The models account for shear softening, matrix cracking and the presence of residual stresses. A smeared cracking approach was used to characterize the micro-cracking in matrix. Experimental studies were performed on laminae, laminates and cylinders made from carbon epoxy composites. Experimental comparisons show that the more accurate micro-mechanical model with proper partial cracking options provides good bounds on experimental response with consistent accuracy. A square cells type model however is not consistent in its predictions, thus raising questions about its applicability in any general micro-mechanics based analysis.

scholarly journals A Computational Model of Viscoelastic Composite Materials for Ligament or Tendon Prostheses

2000 ◽  
Vol 9 (3) ◽  
pp. 096369350000900
Author(s):  
P. Vena
Keyword(s):  
Composite Materials ◽  
Constitutive Relations ◽  
Strain Energy Function ◽  
Time Dependent ◽  
Element Formulation ◽  
Viscoelastic Composite ◽  
Uniaxial Test ◽  
The Matrix ◽  
Tissue Reconstruction ◽  
Constitutive Formulation

A constitutive model and a finite element formulation for viscoelastic anisotropic materials subject to finite strains is expounded in this paper. The composite material is conceived as a matrix reinforced with stiff fibres. The constitutive relations are obtained by defining a strain energy function and a relaxation function for each constituent. By means of this approach, the viscoelastic properties of the material constituents can be taken into account and therefore different time dependent behaviour can be assigned to the matrix and to the reinforcing fibres. The response provided by this kind of constitutive formulation allows for the description of mechanical behaviour for either natural anisotropic tissues (such as tendons and ligaments) and for the composite materials which are currently adopted for tissue reconstruction. The main features of those mechanical properties observed in an ideal uniaxial test are: a non linear stress-strain response and a time dependent response which is observed in relaxation of stresses for a prescribed constant stretch and in a moderate strain rate dependence of the measured response.

scholarly journals Pore-scale investigation of selective plugging mechanism in immiscible two-phase flow using phase-field method

2019 ◽  
Vol 74 ◽  
pp. 78 ◽  
Author(s):  
Ehsan Sabooniha ◽  
Mohammad-Reza Rokhforouz ◽  
Shahab Ayatollahi
Keyword(s):  
Oil Recovery ◽  
Phase Field ◽  
Water Injection ◽  
Flow Patterns ◽  
Field Method ◽  
Phase Field Method ◽  
Phase Flow ◽  
Computational Domain ◽  
Pore Scale ◽  
The Matrix

Biotechnology has had a major effect on improving crude oil displacement to increase petroleum production. The role of biopolymers and bio cells for selective plugging of production zones through biofilm formation has been defined. The ability of microorganisms to improve the volumetric sweep efficiency and increase oil recovery by plugging off high-permeability layers and diverting injection fluid to lower-permeability was studied through experimental tests followed by multiple simulations. The main goal of this research was to examine the selective plugging effect of hydrophobic bacteria cell on secondary oil recovery performance. In the experimental section, water and aqua solution of purified Acinetobacter strain RAG-1 were injected into an oil-saturated heterogeneous micromodel porous media. Pure water injection could expel oil by 41%, while bacterial solution injection resulted in higher oil recovery efficiency; i.e., 59%. In the simulation section, a smaller part of the heterogeneous geometry was employed as a computational domain. A numerical model was developed using coupled Cahn–Hilliard phase-field method and Navier–Stokes equations, solved by a finite element solver. In the non-plugging model, approximately 50% of the matrix oil is recovered through water injection. Seven different models, which have different plugging distributions, were constructed to evaluate the influences of selective plugging mechanism on the flow patterns. Each plugging module represents a physical phenomenon which can resist the displacing phase flow in pores, throats, and walls during Microbial-Enhanced Oil Recovery (MEOR). After plugging of the main diameter route, displacing phase inevitably exit from sidelong routes located on the top and bottom of the matrix. Our results indicate that the number of plugs occurring in the medium could significantly affect the breakthrough time. It was also observed that increasing the number of plugging modules may not necessarily lead to higher ultimate oil recovery. Furthermore, it was shown that adjacent plugs to the inlet caused flow patterns similar to the non-plugging model, and higher oil recovery factor than the models with farther plugs from the inlet. The obtained results illustrated that the fluids distribution at the pore-scale and the ultimate oil recovery are strongly dependent on the plugging distribution.

scholarly journals A double-layer Boussinesq-type model for highly nonlinear and dispersive waves

2009 ◽  
Vol 465 (2108) ◽  
pp. 2319-2346 ◽  
Author(s):  
F. Chazel ◽  
M. Benoit ◽  
A. Ern ◽  
S. Piperno
Keyword(s):  
Double Layer ◽  
Deep Water ◽  
Type Model ◽  
Nonlinear Behaviour ◽  
Approximate Inverse ◽  
Dispersive Waves ◽  
Final Model ◽  
Water Value ◽  
Highly Nonlinear ◽  
Model Derivation

We derive and analyse, in the framework of the mild-slope approximation, a new double-layer Boussinesq-type model that is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential, thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. ( Agnon et al. 1999 J. Fluid Mech. 399 , 319–333) and enhanced by Madsen et al. ( Madsen et al. 2003 Proc. R. Soc. Lond. A 459 , 1075–1104), which consists of constructing infinite-series Taylor solutions to the Laplace equation, to truncate them at a finite order and to use Padé approximants, and the double-layer approach of Lynett & Liu ( Lynett & Liu 2004 a Proc. R. Soc. Lond. A 460 , 2637–2669), which allows lowering the order of derivatives. We formulate the model in terms of a static Dirichlet–Neumann operator translated from the free surface to the still-water level, and we derive an approximate inverse of this operator that can be built once and for all. The final model consists of only four equations both in one and two horizontal dimensions, and includes only second-order derivatives, which is a major improvement in comparison with so-called high-order Boussinesq models. A linear analysis of the model is performed, and its properties are optimized using a free parameter determining the position of the interface between the two layers. Excellent dispersion and shoaling properties are obtained, allowing the model to be applied up to the deep-water value k h =10. Finally, numerical simulations are performed to quantify the nonlinear behaviour of the model, and the results exhibit a nonlinear range of validity reaching at least k h =3π.

Finite Elastic Deformation for a Class of Soft Biological Tissues

1977 ◽  
Vol 99 (2) ◽  
pp. 98-103
Author(s):  
Han-Chin Wu ◽  
R. Reiss
Keyword(s):  
Energy Function ◽  
Annulus Fibrosus ◽  
Strain Energy ◽  
Soft Tissues ◽  
Broad Class ◽  
Strain Energy Function ◽  
Biological Tissues ◽  
Tension Test ◽  
Soft Biological Tissues ◽  
Precise Form

The stress response of soft biological tissues is investigated theoretically. The treatment follows the approach of Wu and Yao [1] and is now extended for a broad class of soft tissues. The theory accounts for the anisotropy due to the presence of fibers and also allows for the stretching of fibers under load. As an application of the theory, a precise form for the strain energy function is proposed. This form is then shown to describe the mechanical behavior of annulus fibrosus satisfactorily. The constants in the strain energy function have also been approximately determined from only a uniaxial tension test.

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