Poroelastic Composites Research Articles

Homogenised governing equations for pre-stressed poroelastic composites

AbstractWe propose the governing equations for a pre-stressed poroelastic composite material. The structure that we investigate possesses a porous elastic matrix with embedded elastic subphases with an incompressible Newtonian fluid flowing in the pores. Both the matrix and individual subphases are assumed to be linear elastic and pre-stressed. We are able to apply the asymptotic homogenisation technique by exploiting the length-scale separation that exists between the porescale and the overall size of the material (the macroscale). We derive the novel macroscale model which describes a poroelastic composite material where the elastic phases possess a pre-stress. We extend the current literature for poroelastic composites by addressing the role of the pre-stresses in the functional form of the new system of derived partial differential equations and its coefficients. The latter are computed by solving appropriate periodic cell differential problems which encode the specific contribution related to the pre-stresses. The model in the first instance is derived in the most general scenario and then specified for a variety of particular cases which are associated with different macroscale behaviour of materials.

Investigating the effects of microstructural changes induced by myocardial infarction on the elastic parameters of the heart

Within this work, we investigate how physiologically observed microstructural changes induced by myocardial infarction impact the elastic parameters of the heart. We use the LMRP model for poroelastic composites (Miller and Penta in Contin Mech Thermodyn 32:1533–1557, 2020) to describe the microstructure of the myocardium and investigate microstructural changes such as loss of myocyte volume and increased matrix fibrosis as well as increased myocyte volume fraction in the areas surrounding the infarct. We also consider a 3D framework to model the myocardium microstructure with the addition of the intercalated disks, which provide the connections between adjacent myocytes. The results of our simulations agree with the physiological observations that can be made post-infarction. That is, the infarcted heart is much stiffer than the healthy heart but with reperfusion of the tissue it begins to soften. We also observe that with the increase in myocyte volume of the non-damaged myocytes the myocardium also begins to soften. With a measurable stiffness parameter the results of our model simulations could predict the range of porosity (reperfusion) that could help return the heart to the healthy stiffness. It would also be possible to predict the volume of the myocytes in the area surrounding the infarct from the overall stiffness measurements.

Micromechanical analysis of the effective stiffness of poroelastic composites

Within this work we investigate the role that the microstructure of a poroelastic material has on the resulting elastic parameters. We are considering the effect that multiple elastic and fluid phases at the same scale (LMRP model (L. Miller and R. Penta, 2020)) have on the estimation of the materials elastic parameters when compared with a standard poroelastic approach. We present a summary of both the LMRP model and the comparable standard poroelastic approach both derived via the asymptotic homogenization approach. We provide the 3D periodic cell problems with associated boundary loads that are required to be solved to obtain the effective elasticity tensor for both model setups. We then perform a 2D reduction of the cell problems, again presenting the 2D boundary loads that are required to solve the problems numerically. The results of our numerical simulations show that whenever investigating a poroelastic composite material with porosity exceeding 5% then the LMRP model should be considered more appropriate in incorporating the structural details in the Young’s moduli E1 and E3 and the shear C44. Whenever the porosity exceeds 20% it should also be used to investigate the shear C66. We find that for materials with less than 5% porosity that the voids are so small that a standard poroelastic approach or the LMRP model produce the same results.

Semi-analytical IGA-based computation of wave dispersion in fluid-coupled anisotropic poroelastic plates

Understanding dispersion relations and wave mode shapes is vital in nondestructive control of dynamic behaviors of poroelastic composites. In the framework of semi-analytical finite element (SAFE) method, this paper presents two numerical approaches so-called semi-analytical isogeometric Galerkin (SAIGA-G) and semi-analytical isogeometric collocation (SAIGA-C) for computing dispersion of guided-waves in anisotropic poroelastic plates immersed in acoustic fluids. Biot’s theory was used for describing the dynamic behavior of anisotropic poroelastic material. Assuming the structures is homogeneous along its axial direction, the Non-Uniform Rational B-splines (NURBS) was successfully employed in procedures using isogeometric Galerkin and collocation methods. The numeral studies showed that the SAIGA-G method using high continuity NURBS basis allowed to significantly improve the accuracy as well as the convergence rate of the wave dispersion solutions in compared with the conventional SAFE method, which used Lagrange basis functions. Otherwise, the SAIGA-C method was shown to have similar performance in terms of accuracy to the SAFE method.

Homogenized Balance Equations for Nonlinear Poroelastic 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.

Effective balance equations for poroelastic composites

We derive the quasi-static governing equations for the macroscale behaviour of a linear elastic porous composite comprising a matrix interacting with inclusions and/or fibres, and an incompressible Newtonian fluid flowing in the pores. We assume that the size of the pores (the microscale) is comparable with the distance between adjacent subphases and is much smaller than the size of the whole domain (the macroscale). We then decouple spatial scales embracing the asymptotic (periodic) homogenization technique to derive the new macroscale model by upscaling the fluid–structure interaction problem between the elastic constituents and the fluid phase. The resulting system of partial differential equations is of poroelastic type and encodes the properties of the microstructure in the coefficients of the model, which are to be computed by solving appropriate cell problems which reflect the complexity of the given microstructure. The model reduces to the limit case of simple composites when there are no pores, and standard Biot’s poroelasticity whenever only the matrix–fluid interaction is considered. We further prove rigorous properties of the coefficients, namely (a) major and minor symmetries of the effective elasticity tensor, (b) positive definiteness of the resulting Biot’s modulus, and (c) analytical identities which allow us to define an effective Biot’s coefficient. This model is applicable when the interactions between multiple solid phases occur at the porescale, as in the case of various systems such as biological aggregates, constructs, bone, tendons, as well as rocks and soil.

Concurrent factors determine toughening in the hydraulic fracture of poroelastic composites

Brittle materials fail catastrophically. In consequence of their limited flaw-tolerance, failure occurs by localized fracture and is typically a dynamic process. Recently, experiments on epithelial cell monolayers have revealed that this scenario can be significantly modified when the material susceptible to cracking is adhered to a hydrogel substrate. Thanks to the hydraulic coupling between the brittle layer and the poroelastic substrate, such a composite can develop a toughening mechanism that relies on the simultaneous growth of multiple cracks. Here, we study this remarkable behaviour by means of a detailed model, and explore how the material and loading parameters concur in determining the macroscopic toughness of the system. By extending a previous study, our results show that rapid loading conveys material toughness by promoting distributed cracking. Moreover, our theoretical findings may suggest innovative architectures of flaw-insensitive materials with higher toughness.

On reconstruction of dynamic permeability and tortuosity from data at distinct frequencies

This article focuses on the mathematical problem of reconstructing the dynamic permeability and dynamic tortuosity of poroelastic composites from permeability data at different frequencies, utilizing the analytic structure of the Stieltjes function representation of derived by Avellaneda and Tortquato (1991 Phys. Fluids A 3 2529), which is valid for all pore space geometry. The integral representation formula (IRF) for dynamic tortuosity is derived and its analytic structure exploited for reconstructing the function from a finite data set. All information of pore-space microstructure is contained in the measure of the IRF. The theory of multipoint Padé approximates for Stieltjes functions guarantees the existence of relaxation kernels that can approximate the dynamic permeability function and the dynamic tortuosity function with high accuracy. In this paper, a numerical algorithm is proposed for computing the relaxation time and the corresponding strength for each element in the relaxation kernels. In the frequency domain, this approximation can be regarded as approximating the Stieltjes function by rational functions with simple poles and positive residues. The main difference between this approach and the curve fitting approach is that the relaxation times and the strengths are computed from the partial fraction decomposition of the multipoint Padé approximates, which is the main subject of the proposed approximation scheme. With the idea from dehomogenization, we also established the exact relations between the moments of the positive measures in the IRFs of permeability and tortuosity with two important parameters in the theory of poroelasticity: the infinite-frequency tortuosity for the general case and the weighted volume-to-surface ratio Λ for the JKD model, which is regarded as a special case of the general model. From these relations, we suggest a new way for evaluating these two microstructure-dependent parameters from a finite data set of permeability at different frequencies, without assuming any specific forms of the functions except the fact that they satisfies the IRFs. Numerical results for JKD permeability and tortuosity are presented.

An XFEM-based numerical procedure for the analysis of poroelastic composites with coherent imperfect interface

Composites under consideration consist of a poroelastic matrix in which poroelastic inclusions are embedded via a poroelastic transition layer of small uniform thickness. A prototype of such poroelastic composites is concrete in which aggregates are often bonded to a cement paste through an interfacial transition zone (ITZ). In this work, the poroelastic transition layer is replaced by an interface of zero thickness described by the poroelastic coherent imperfect interface model. According to this model, the displacement and pore pressure are continuous across the interface but the traction and normal fluid flux suffer interfacial jumps. The main objective of this work is to propose an efficient numerical solution to the problem of poroelastic composites with coherent imperfect interface under the regime of undrained saturation. To this end, the weak formulations are first established for the relevant boundary value problems of linear poroelasticity and fluid transport. The extended finite element method (XFEM) and level-set method (LSM) are then used to elaborate a numerical procedure. The accuracy and convergence rate of this procedure are analyzed with the aid of a benchmark problem whose analytical exact solution is derived. The numerical procedure proposed turns out to be efficient in capturing the interfacial discontinuities and computing the elastic and transport fields, and can be applied, for example, to solving the important problem of homogenizing poroelastic composites with coherent imperfect interface.

Adaptive energy-absorbing materials using field-responsive fluid-impregnated cellularsolids

Adaptive materials with rapidly controllable and switchable energy-absorption andstiffness properties have a number of potential applications. We have developed,characterized and modeled a class of adaptive energy-absorbing systems consistingof nonlinear poroelastic composites wherein a field-responsive fluid, such as amagnetorheological fluid or a shear-thickening fluid, has been used to modulate themechanical properties of a cellular solid. The mechanical properties and energy-absorbingcapabilities of the composite are studied for variations in design parameters includingimposed field strength, volume fraction of the field-responsive fluid within thecomposite and impact strain rates. The total energy absorbed by these materials canbe modulated by a factor of 1- to 50-fold for small volume fractions of the fluid(∼15%) using moderate magnetic fields varying from 0 to 0.2 T. A scaling model is also proposedfor the fluid–solid composite mechanical behavior that collapses experimental data onto asingle master curve. The model allows optimization of the composite properties in tunewith the application requirements. Potential application areas are discussed with emphasison applicability in impact-absorbing headrests and cushioned assemblies for energymanagement.

Effective Medium Theories for Multicomponent Poroelastic Composites

It is demonstrated that effective medium theories for poroelastic composites such as rocks can be formulated easily by analogy to well-established methods used for elastic composites. An identity analogous to Eshelby’s classic result has been derived previously for use in composites containing arbitrary ellipsoidal-shaped inclusions. This result is the starting point for new methods of estimation, including generalizations of the coherent potential approximation, differential effective medium theory, and two explicit schemes. Results are presented for estimating drained shear and bulk modulus, the Biot–Willis parameter, and Skempton’s coefficient. Three of the methods considered appear to be quite reliable estimators, while one of the explicit schemes is found to have some undesirable characteristics. Furthermore, the results obtained show that the actual microstructure should be taken carefully into account when trying to decide which of these methods to apply in a given situation.

Finite element analysis of poroelastic composites undergoing thermal and gas diffusion

A theory for time-dependent thermal and gas diffusion in mechanically time-rate-independent anisotropic poroelastic composites has been developed. This theory advances previous work by the latter two authors by providing for critical transverse shear through a three-dimensional axisymmetric formulation and using it in a new hypothesis for determining the Biot fluid pressure-solid stress coupling factor. The derived governing equations couple material deformation with temperature and internal pore pressure and more strongly couple gas diffusion and heat transfer than the previous theory. Hence the theory accounts for the interactions between conductive heat transfer in the porous body and convective heat carried by the mass flux through the pores. The Bubnov Galerkin finite element method is applied to the governing equations to transform them into a semidiscrete finite element system. A numerical procedure is developed to solve the coupled equations in the space and time domains. The method is used to simulate two high temperature tests involving thermal-chemical decomposition of carbon-phenolic composites. In comparison with measured data, the results are accurate. Moreover unlike previous work, for a single set of poroelastic parameters, they are consistent with two measurements in a restrained thermal growth test.

A constitutive model for thermomechanical response of decomposing composites under high heating rates

The thermomechanical response model of chemically decomposing composites is established. Mass loss due to thermochemical decomposition is modeled by the Arrhenius equation. Phenomenological shrinkage models are proposed and the pore formation due to mass loss is examined from a micro-mechanical point of view. The effect of gas pore pressure caused by decomposition on the deformation of the composite is considered based on the thermal effective stress laws for linearly poroelastic composites. The coupling between the thermochemical decomposition and thermomechanical deformation is made clear throughout the formulation. The results of laboratory tests known as free thermal expansion tests and restrained thermal growth tests are theoretically predicted by the obtained model.

Realizability of Negative Pore Compressibility in Poroelastic Composites

For elastic materials containing fluid-saturated porosity, the pore compressibility is a measure of the deformation of a unit pore volume in response to a change in fluid pressure. Rather than being measured, this quantity has been routinely set equal to an effective solid compressibility, since this equality is exact whenever a single solid component is present. However, we show that the pore compressibility and solid compressibility may be uncorrelated in general. In certain special circumstances they do not even share the same sign. Although thermodynamic and mechanical stability constraints cause solid and drained-frame bulk moduli of a porous composite to be positive and bounded by component properties, the pore compressibility is unconstrained and, therefore, can have negative values. For special realizable model materials, the value of the pore compressibility can be found using an exact expression valid for a composite made up of one fluid and two solid components, i.e., two porous components. In order to quantify how various factors affect the sign and magnitude of the pore compressibility, pore compressibilities were calculated for models that used two porous components having the microgeometry of an assemblage of concentric spheres. This model implicitly assumes the pores are on a much smaller length scale than the concentric spheres. Modeling results show that with the stiffer porous material forming the outer shells of the concentric spheres, the pore compressibility of such materials is negative when solid component bulk moduli differ by at least a factor of 5, if, in addition, the porosities and drained frame moduli of the two porous components are relatively low. Negative pore compressibilities were found for realizable models whose two porous constituents had the properties of silicon nitride and either sandstone or clay. For models using combinations of alumina and glass foam properties, pore compressibilities were non-negative but smaller than the compressibilities of the solid components.