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Finite element modelling
Published in C M Langton, C F Njeh, The Physical Measurement of Bone, 2016
In vivo, bone is saturated with fluid (blood, marrow and fat). Linear elastic FE analysis for structures and solids as described above do not explicitly account for the water content. In certain situations, however, the bone fluid can play an important role in the mechanical behaviour of bone tissue. The theory of poroelasticity has been developed to model the interaction of deformation and fluid flow in a fluid-saturated porous medium. There are several different approaches to the development of the equations for the theory of poroelasticity, but all approaches lead to the same set of equations (for a discussion of the different theories see [7]). The theory used most often for the poroelastic analysis of bone is the effective medium approach based on the work of Biot [8]. In this theory, the poroelastic field variables are the total stress, pore pressure, strain in the solid phase and variation in fluid content. The theory can also be applied to soft tissues (cartilage and fibrous-like tissues), but in these situations mixture-based theories such as the small strain incompressible poroelastic formulation by Mow et al [9] (biphasic theory) are more commonly used.
Asymptotic solutions of Biot’s equations
Published in J.-F. Thimus, Y. Abousleiman, A.H.-D. Cheng, O. Coussy, E. Detournay, Poromechanics, 2020
Biot’s equations of poroelasticity ρu¨i+ρfv¨i=σij,jρfu¨i+Nijvj=−p,iσij=cijkluk,l−Mijvk,kp=Mkluk,l−mvk,k where ρ = ϕρf + (1 − ϕ) ρs, ρf and ρs denote the density of the pore fluid and of the solid forming the skeleton, ϕ is the porosity and p is the pore pressure, are a special case of eqs (2).
Mechanics of Indentation
Published in Michelle L. Oyen, Handbook of Nanoindentation with biological applications, 2019
Poroelasticity concerns the flow of fluid through a saturated porous elastic solid. Although the components (fluid, solid) themselves are taken to be time-independent, the flow problem creates time-dependence in the response. The basic elastic problem is augmented in poroelasticity with two additional scalar variables, the pore pressure, p, and an increment of fluid content, ξ, the change in fluid volume relative to a control volume. The poroelastic framework considered here is for an isotropic elastic porous material with elastic material properties shear modulus, G, and Poisson’s ratio, v. There are two other parameters to characterize the fluid and the fluid-solid interactions, a and B, which are bounded between zero and unity. With the additional variables (p, and material constants (α, B), the stress-strain (σij, – εij) relationship for elasticity is modified by the addition of a term including the pore pressure (apδij) to become: σij+αρδij=2Gεij+(2Gv1−2v)εijδij where δij is the Kronecker delta.26 A single additional scalar expression is added to the six elastic equations to total seven constitutive equations: 2Gξ=α(1−2v1+v)(σkk+3pB)
A novel knee joint model in FEBio with inhomogeneous fibril-reinforced biphasic cartilage simulating tissue mechanical responses during gait: data from the osteoarthritis initiative
Published in Computer Methods in Biomechanics and Biomedical Engineering, 2023
Alexander Paz, Gustavo A. Orozco, Petri Tanska, José J. García, Rami K. Korhonen, Mika E. Mononen
Briefly, poroelasticity and biphasic theories consider cartilage as a material with a solid matrix saturated by fluid, both phases being intrinsically incompressible (Ateshian et al. 1997). Poroelasticity states the field equations in terms of the solid phase motion and the relative motion of the fluid phase. Biphasic theory formulates the equations based on absolute motions of the fluid and solid phases (Simon 1992). In general, the total Cauchy stress is composed of the fluid and the effective solid components, i.e., where the fluid stress is equal to the hydrostatic fluid pressure () times the second-order identity tensor (), and accounts for the stresses of all constituents in the solid phase.
Influence of gravity, magnetic field, and thermal shock on mechanically loaded rotating FGDPTM structure under Green-Naghdi theory
Published in Mechanics Based Design of Structures and Machines, 2023
Shishir Gupta, Soumik Das, Rachaita Dutta
Most of the solid bodies inherit porosity either naturally or artificially. Porous materials are usually present in volcanic rocks, mountains and basins. Eroded rocks from mountains flow with rivers and waterfalls up to a certain distance. As the altitude starts to decrease, flow velocity of the river is not strong enough to carry forward the rocks with water and deposit them on the surface. During this natural process, porous materials acquire fissures. Thus a porous material with fissures, so-called known as double porous structure, consists of two subsystems: (i) matrix porosity or macroporosity and (ii) fissure porosity or microporosity. Poroelasticity, especially multiple poroelasticity theory, has several benefits in production of gas and energy, nuclear waste treatment, oil acquisition, carbon sequestration, and tissue engineering etc. Cowin and his coworker (Cowin and Nunziato 1983; Nunziato and Cowin 1979) carried out the investigation on the theory of elastic material with voids which resulted in a generalized formation of classical theory of elasticity. The concept behind this theory is that the void volume fraction can be expressed as a ratio of the bulk density of the porous material to the density of the matrix material. Puri and Cowin (1985) considered an elastic material with voids to study plane wave propagation. Iesan (1986) modified the theory of elastic material with voids by introducing thermal characteristics. Several researchers (Iesan 2004; Sur and Kanoria 2020) have worked on the dynamics of materials and thermal wave propagation in such materials by adopting the theory of elastic material with voids. Ieşan and Quintanilla (2014) developed the theory of double porous thermoelastic solid on the basis of the Nunziato-Cowin theory of elastic material with voids. In this theory of double poro-thermoelasticity, authors illustrated the uniqueness and existence of solution along with the conditions for stability of the model. In recent years, Abdou et al. (2018, 2020) have worked on thermo-mechanical interactions in double porous material with the consideration of Iesan-Quintanilla theory.
Double poro-magneto-thermoelastic model with microtemperatures and initial stress under memory-dependent heat transfer
Published in Journal of Thermal Stresses, 2023
Shishir Gupta, Rachaita Dutta, Soumik Das, Arun Kumar Verma
Othman et al. [45] considered an initially stressed thermoelastic half-space with one type of voids and microtemperatures to investigate a plane strain problem. Kalkal et al. [37] have analyzed a functionally graded double porous thermoelastic model under TPL thermoelasticity theory. Recently, the impact of higher-order MDD on a one-dimensional double porous thermoelastic medium has been studied by Gupta et al. [47]. To the best of the authors’ knowledge, no study related to the memory response of a two-dimensional (2D) semi-infinite medium consisting of a double poro-magneto-thermoelastic material with voids and microtemperatures (DPMTMWVM) has been conducted so far. Hence, the present article intends to illuminate the memory response of an initially stressed half-space comprising DPMTMWVM. The DPL heat conduction law has been applied in the considered model. There exist several methods, such as state-space approach [48], Laplace and Fourier transform and their inversion [49], and normal mode technique [28], to solve the models corresponding to the thermo-mechanical interaction. In this study, the non-dimensional equations of motion are solved via normal mode technique to derive the theoretical expressions of the displacements, temperature, microtemperatures, void volume fractions, stresses, and heat flux moments. This method includes the harmonic formulation of functions, which is similar to the behavior of waves. The effects of magnetic field, kernel functions, and initial stress on the aforementioned field variables have been executed through several graphs. Moreover, the memory-dependent LS and DPL thermoelasticity (LSTE and DPLTE) theories are compared graphically to exhibit the variations in the prior-mentioned distributions. According to the microtemperature theory, the microelements of a thermoelastic body have different temperatures and are dependent on the microcoordinates homogeneously. This theory is widely used in nanotechnology and earthquake engineering. Chemical engineering, geophysics, biomechanics, bone mechanics, and petroleum industry are the major application areas of the poroelasticity theory. Fractional calculus has noteworthy applications in different branches of science and engineering, e.g., continuum mechanics, biophysics, electrical engineering, bioengineering, viscoelasticity, signal and image processing, and electrochemistry. In view of these myriad contributions of the double poro-magneto-thermoelasticity with microtemperatures and memory effect, authors have gained the motivation to analyze the thermoelastic deformations in the considered model.