Experimental data for porous media exhibit nonlinear pressure-volumetric strain relations and a strong dependence on the Terzaghi pressure defined as confining pressure minus pore pressure. However, ...a clear explanation of why this pressure plays such a dominant role appears to be missing. Several authors have suggested that shear must be a significant factor in predicting change in porosity even for purely hydrostatic loading. Here this idea is explored in detail by analyzing a representative volume element consisting of a hollow sphere within a unit cube subjected only to hydrostatic compression. The results are presented independent of this particular geometry with the use of volume fractions. The analysis shows that the stress field within a relatively small region around a pore contains a measure of shear stress, called the Terzaghi shear, which is similar, but not equal, to the Terzaghi pressure. Shear strain in the hollow sphere does not affect the volumetric strain of the hollow sphere itself but is a major contribution to the volumetric strain of the pore, and hence to the bulk volumetric strain. The constitutive equation between Terzaghi shear and shear strain within a conditioned state is nonlinearly elastic, whereas transitioning from one conditioned state to another is governed by strain-hardening plasticity. The plastic strain that develops introduces a residual Terzaghi shear, which provides an explanation as to why the elastic response in a second conditioned state is different from that in the first. The formulation also explains the interesting phenomenon of shear-enhanced compaction under hydrostatic loading. Explicit expressions are given for the compressibilities relating increments of confining and pore pressures to increments of bulk and pore volumetric strains. Relations between the compressibilities are similar to classical equations with the exception that the new formulation extends beyond a single conditioned state.
We study the three-dimensional problem associated with an incompressible nonlinear elastic spherical inhomogeneity embedded in an infinite linear isotropic elastic matrix subjected to a uniform ...deviatoric load at infinity. The nonlinear elastic material can incorporate both power-law hardening and softening materials. The inhomogeneity-matrix interface is a spring-type imperfect interface characterized by a common interface parameter for both the normal and tangential directions. It is proved that the internal stresses and strains within the spherical inhomogeneity are unconditionally uniform. The original boundary value problem is reduced to a single non-linear equation which is proved rigorously to have a unique solution which can be found numerically. Furthermore, the neutrality of the imperfectly bonded nonlinear elastic spherical inhomogeneity is accomplished in an analytical manner. Finally, we prove the uniformity of the internal elastic field of stresses and strains inside an incompressible power-law hardening or softening nonlinear elastic ellipsoidal inhomogeneity perfectly bonded to an infinite linear isotropic elastic matrix subjected to uniform remote shear stresses and strains.
Nonlinear ultrasonic diffuse energy imaging is a highly sensitive method for the measurement of elastic nonlinearity. While the underlying principles that govern the technique are understood, the ...precise behavior and sensitivity have not previously been quantified. This paper presents experimental, theoretical, and numerical modeling studies undertaken to characterize nonlinear diffuse energy imaging. The influence of incoherent noise, elastic nonlinearity, and instrumentation error are quantified. This paper enables the prediction of spatial sensitivity, aperture, and amplitude dependence of the measurement, all of which moves the technique toward industrial viability. Furthermore, while previous studies have focused on the detection of closed cracks, the ultimate aim for nonlinear ultrasonic imaging in application to material testing is the detection of damage precursors, which requires a sensitivity to weak classical nonlinearity. This paper identifies the experimental requirements necessary for this to be achieved, greatly expanding the potential applicability of nonlinear ultrasonic array imaging.
A central question in developmental biology is how size and position are determined. The genetic code carries instructions on how to control these properties in order to regulate the pattern and ...morphology of structures in the developing organism. Transcription and protein translation mechanisms implement these instructions. However, this cannot happen without some manner of sampling of epigenetic information on the current patterns and morphological forms of structures in the organism. Any rigorous description of space- and time-varying patterns and morphological forms reduces to one among various classes of spatio-temporal partial differential equations. Reaction-transport equations represent one such class. Starting from simple Fickian diffusion, the incorporation of reaction, phase segregation and advection terms can represent many of the patterns seen in the animal and plant kingdoms. Morphological form, requiring the development of three-dimensional structure, also can be represented by these equations of mass transport, albeit to a limited degree. The recognition that physical forces play controlling roles in shaping tissues leads to the conclusion that (nonlinear) elasticity governs the development of morphological form. In this setting, inhomogeneous growth drives the elasticity problem. The combination of reaction-transport equations with those of elasto-growth makes accessible a potentially unlimited spectrum of patterning and morphogenetic phenomena in developmental biology. This perspective communication is a survey of the partial differential equations of mathematical physics that have been proposed to govern patterning and morphogenesis in developmental biology. Several numerical examples are included to illustrate these equations and the corresponding physics, with the intention of providing physical insight wherever possible.
•I have identified the dual roles of patterning and morphogenesis, and their coupling in determining the form of organisms.•The manuscript draws a rigorous connection between organization of multicellular tissues and phase segregation phenomena.•It goes on to highlight several aspects of phase field models and their bearing on diverse features of biological patterning.•Throughout the manuscript the central questions of developmental biology are placed within context of the considered models: How are size and position controlled by the physics that the mathematical models describe?
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Different formulations of the constitutive laws and governing equations for nonlinear electroelastic solids are reviewed and two new variational principles for electroelastostatics are introduced. ...One is based on use of the electrostatic scalar potential and one on the vector potential, combined with the deformation function. In each case Lagrangian forms of the electric variables are used. Their connections with several formulations of nonlinear electroelasticity in the literature are established and some differences highlighted.
The active response of cells to mechanical cues due to their interaction with the environment has been of increasing interest, since it is involved in many physiological phenomena, pathologies, and ...in tissue engineering. In particular, several experiments have shown that, if a substrate with overlying cells is cyclically stretched, they will reorient to reach a well-defined angle between their major axis and the main stretching direction. Recent experimental findings, also supported by a linear elastic model, indicated that the minimization of an elastic energy might drive this reorientation process. Motivated by the fact that a similar behaviour is observed even for high strains, in this paper we address the problem in the framework of finite elasticity, in order to study the presence of nonlinear effects. We find that, for a very large class of constitutive orthotropic models and with very general assumptions, there is a single linear relationship between a parameter describing the biaxial deformation and
cos
2
θ
eq
, where
θ
eq
is the orientation angle of the cell, with the slope of the line depending on a specific combination of four parameters that characterize the nonlinear constitutive equation. We also study the effect of introducing a further dependence of the energy on the anisotropic invariants related to the square of the Cauchy–Green strain tensor. This leads to departures from the linear relationship mentioned above, that are again critically compared with experimental data.
A local universal relation is an equation between the stress components and the position vector components which holds for any material in an assigned constitutive family. Although universal ...relations may be of great help to modellers in characterizing the material behaviour, they are often ignored. In this paper we briefly discuss the valuable insights that universal relations may offer in solid mechanics and determine novel universal relations associated with shearing motions in nonlinearly elastic and nonlinearly viscoelastic materials of differential type.
We consider a nonlinear elasticity problem in a bounded domain, its boundary is decomposed in three parts: lower, upper, and lateral. The displacement of the substance, which is the unknown of the ...problem, is assumed to satisfy the homogeneous Dirichlet boundary conditions on the upper part, and not homogeneous one on the lateral part, while on the lower part, friction conditions are considered. In addition, the problem is governed by a particular constitutive law of elasticity system with a strongly nonlinear strain tensor. The functional framework leads to using Sobolev spaces with variable exponents. The formulation of the problem leads to a variational inequality, for which we prove the existence and uniqueness of the solution of the associated variational problem.
This article considers the influence of incompressibility on the compliance and stiffness constants that appear in the weakly nonlinear theory of elasticity. The formulation first considers the ...incompressibility constraint applied to compliances, which gives explicit finite limits for the second-, third-, and fourth-order compliance constants. The stiffness/compliance relationships for each order are derived and used to determine the incompressible behavior of the second-, third-, and fourth-order stiffness constants. Unlike the compressible case, the fourth-order compliances are not found to be dependent on the fourth-order stiffnesses.
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