Fully non-linear wave models in fiber-reinforced anisotropic incompressible hyperelastic solids

Many composite materials, including biological tissues, are modeled as non-linear elastic materials reinforced with elastic fibers. In the current paper, the full set of dynamic equations for finite deformations of incompressible hyperelastic solids containing a single fiber family are considered. Finite-amplitude wave propagation ansätze compatible with the incompressibility condition are employed for a generic fiber family orientation. Corresponding non-linear and linear wave equations are derived. It is shown that for a certain class of constitutive relations, the fiber contribution vanishes when the displacement is independent of the fiber direction.Point symmetries of the derived wave models are classified with respect to the material parameters and the angle between the fibers and the wave propagation direction. For planar shear waves in materials with a strong fiber contribution, a special wave propagation direction is found for which the non-linear wave equations admit an additional symmetry group. Examples of exact time-dependent solutions are provided in several physical situations, including the evolution of pre-strained configurations and traveling waves.     

Compressible and incompressible constituents in anisotropic poroelasticity: The problem of unconfined compression of a disk

The governing equations for the theory of anisotropic poroelastic materials with incompressible constituents undergoing small deformations are developed from the theory of anisotropic poroelastic materials without the constituent incompressibility constraint. The development of the constituent specific incompressibility constraint is accomplished by restricting the elastic constants rather than by introducing Lagrange multipliers, the conventional method. The advantage of the approach is insight into the nature of the elastic response that characterizes incompressible poroelasticity. An application of the theory to the unconfined compression of a circular porous disk is presented to illustrate the effects of compressibility vs. incompressibility and transverse isotropy vs. isotropy.     

Linearization of response mappings in constrained elasticity

The linearization of response mappings for elastic materials with internal constraints is discussed with a geometric perspective. It is shown that, when the active residual stress is not zero, the Weingarten map of the constraint manifold plays an important role in the deduction of linearized constitutive equations. As an example, the Weingarten map is computed for a material which is incompressible and inextensible in a given direction.     

The Boussinesq problem in dipolar gradient elasticity

The three-dimensional axisymmetric Boussinesq problem of an isotropic half-space subjected to a concentrated normal quasi-static load is studied within the framework of dipolar gradient elasticity involving linear constitutive relations and small strains. Our main concern is to determine possible deviations from the predictions of classical linear elastostatics when a more refined theory is employed to attack the problem. Of special importance is the behavior of the new solution near to the point of application of the load where pathological singularities exist in the classical solution. The use of the theory of gradient elasticity is intended here to model the response of materials with microstructure in a manner that the classical theory cannot afford. A linear version of this theory (as regards both kinematics and constitutive response) results by considering a linear isotropic expression for the strain-energy density that depends on strain gradient terms, in addition to the standard strain terms appearing in classical elasticity and by considering small strains. Through this formulation, a microstructural material constant is introduced, in addition to the standard Lamé constants. The solution method is based on integral transforms and is exact. The present results show significant departure from the predictions of classical elasticity. Indeed, continuous and bounded displacements are predicted at the points of application of the concentrated load. Such a behavior of the displacement field is, of course, more natural than the singular behavior exhibited in the classical solution.     

Stroh formulation for a generally constrained and pre-stressed elastic material

The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.     

Variational formulation of a simplified strain gradient elasticity theory and its application to a pressurized thick-walled cylinder problem

A detailed variational formulation is provided for a simplified strain gradient elasticity theory by using the principle of minimum total potential energy. This leads to the simultaneous determination of the equilibrium equations and the complete boundary conditions of the theory for the first time. To supplement the stress-based formulation, the coordinate-invariant displacement form of the simplified strain gradient elasticity theory is also derived anew. In view of the lack of a consistent and complete formulation, derivation details are included for the tutorial purpose. It is shown that both the stress and displacement forms of the simplified strain gradient elasticity theory obtained reduce to their counterparts in classical elasticity when the strain gradient effect (a measure of the underlying material microstructure) is not considered. As a direct application of the newly obtained displacement form of the theory, the problem of a pressurized thick-walled cylinder is analytically solved. The solution contains a material length scale parameter and can account for microstructural effects, which is qualitatively different from Lamé’s solution in classical elasticity. In the absence of the strain gradient effect, this strain gradient elasticity solution reduces to Lamé’s solution. The numerical results reveal that microstructural effects can be large and Lamé’s solution may not be accurate for materials exhibiting significant microstructure dependence.     

Propagation of acceleration waves in incompressible saturated porous solids

Within the framework of the incompressible porous media model, the propagation properties of acceleration waves in liquid-filled porous solids is discussed. The incompressibility of the two constituents in the model forces the amplitudes of the longitudinal waves in the skeleton and in the liquid to satisfy a certain relation. The two propagation speeds are presented by examination for the existence of acceleration waves and only longitudinal and transverse waves are realizable in the incompressible two-phase porous materials.     

High‐order implicit time integration for unsteady incompressible flows

The spatial discretization of unsteady incompressible Navier–Stokes equations is stated as a system of differential algebraic equations, corresponding to the conservation of momentum equation plus the constraint due to the incompressibility condition. Asymptotic stability of Runge–Kutta and Rosenbrock methods applied to the solution of the resulting index‐2 differential algebraic equations system is analyzed. A critical comparison of Rosenbrock, semi‐implicit, and fully implicit Runge–Kutta methods is performed in terms of order of convergence and stability. Numerical examples, considering a discontinuous Galerkin formulation with piecewise solenoidal approximation, demonstrate the applicability of the approaches and compare their performance with classical methods for incompressible flows. Copyright © 2011 John Wiley & Sons, Ltd.     

Hamiltonian dynamics of an elastica and the stability of solitary waves

A method is presented for deriving unconstrained Hamiltonian systems of partial differential equations equivalent to given constrained Lagrangian systems. The method is applied to the theory of planar, finite-amplitude motions of inextensible and unshearable elastic rods. The constraints of inextensibility and unshearability become integrals of motion in the Hamiltonian formulation.It is known that in the theory of uniform, inextensible, unshearable rods of infinite length there arise solitary-wave solutions with the property that each profile can move at arbitrary speed. The Hamiltonian formulation is exploited to analyze the stability properties of these solitary waves. The wave profiles are first characterized as critical points of an appropriate time-invariant functional. It is then shown that for a certain range of wave speeds the solitary-wave profiles are actually nonisolatedminimizers of the functional, a fact with implications for nonlinear stability.     

A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation

In recent years there have been many papers that considered the effects of material length scales in the study of mechanics of solids at micro- and/or nano-scales. There are a number of approaches and, among them, one set of papers deals with Eringen's differential nonlocal model and another deals with the strain gradient theories. The modified couple stress theory, which also accounts for a material length scale, is a form of a strain gradient theory. The large body of literature that has come into existence in the last several years has created significant confusion among researchers about the length scales that these various theories contain. The present paper has the objective of establishing the fact that the length scales present in nonlocal elasticity and strain gradient theory describe two entirely different physical characteristics of materials and structures at nanoscale. By using two principle kernel functions, the paper further presents a theory with application examples which relates the classical nonlocal elasticity and strain gradient theory and it results in a higher-order nonlocal strain gradient theory. In this theory, a higher-order nonlocal strain gradient elasticity system which considers higher-order stress gradients and strain gradient nonlocality is proposed. It is based on the nonlocal effects of the strain field and first gradient strain field. This theory intends to generalize the classical nonlocal elasticity theory by introducing a higher-order strain tensor with nonlocality into the stored energy function. The theory is distinctive because the classical nonlocal stress theory does not include nonlocality of higher-order stresses while the common strain gradient theory only considers local higher-order strain gradients without nonlocal effects in a global sense. By establishing the constitutive relation within the thermodynamic framework, the governing equations of equilibrium and all boundary conditions are derived via the variational approach. Two additional kinds of parameters, the higher-order nonlocal parameters and the nonlocal gradient length coefficients are introduced to account for the size-dependent characteristics of nonlocal gradient materials at nanoscale. To illustrate its application values, the theory is applied for wave propagation in a nonlocal strain gradient system and the new dispersion relations derived are presented through examples for wave propagating in Euler–Bernoulli and Timoshenko nanobeams. The numerical results based on the new nonlocal strain gradient theory reveal some new findings with respect to lattice dynamics and wave propagation experiment that could not be matched by both the classical nonlocal stress model and the contemporary strain gradient theory. Thus, this higher-order nonlocal strain gradient model provides an explanation to some observations in the classical and nonlocal stress theories as well as the strain gradient theory in these aspects.     

Finite Elastic Deformations in Liquid-Saturated and Empty Porous Solids

Based on the Theory of Porous Media (TPM), a formulation of a fluid-saturated porous solid is presented where both constituents, the solid and the fluid, are assumed to be materially incompressible. Therefore, the so-called point of compaction exists. This deformation state is reached when all pores are closed and any further volume compression is impossible due to the incompressibility constraint of the solid skeleton material. To describe this effect, a new finite elasticity law is developed on the basis of a hyperelastic strain energy function, thus governing the constraint of material incompressibility for the solid material. Furthermore, a power function to describe deformation dependent permeability effects is introduced.After the spatial discretization of the governing field equations within the finite element method, a differential algebraic system in time arises due to the incompressibility constraint of both constituents. For the efficient numerical treatment of the strongly coupled nonlinear solid-fluid problem, a consistent linearization of the weak forms of the governing equations with respect to the unknowns must be used.     

The initial-value problem for elastodynamics of incompressible bodies

We prove short-time well-posedness of the Cauchy problem for incompressible strongly elliptic hyperelastic materials. Our method consists in:

1. Reformulating the classical equations in order to solve for the pressure gradient (The pressure is the Lagrange multiplier corresponding to the constraint of incompressibility.) This formulation uses both spatial and material variables.
2. Solving the reformulated equations by using techniques which are common for symmetric hyperbolic systems. These are:

1. Using energy estimates to bound the growth of various Sobolev norms of solutions.
2. Finding the solution as the limit of a sequence of solutions of linearized problems.

Our equations differ from hyperbolic systems, however, in that the pressure gradient is a spatially non-local function of the position and velocity variables.     

Version of the linear elasticity theory with a structural parameter

It is shown that, in the planar case, the system of constitutive equations of the linear elasticity theory should contain five independent equations. In the classical theory, only three equations are formulated, while the other two are contained in implicit form in the postulate of diffeomorphism, which is the assumption of smoothness of the displacement field. A closed elasticity model is constructed without the assumption of diffeomorphism, and it contains a structural parameter having a dimension of length. It is shown that, in a static version, macrodeformations depend on stresses and second derivatives of stresses with respect to the coordinates, while there is dispersion of longitudinal and transverse waves in the dynamics.     

WAVE PROPAGATION IN A SYSTEM OF COAXIAL TUBES FILLED WITH INCOMPRESSIBLE MEDIA: A MODEL OF PULSE TRANSMISSION IN THE INTRACRANIAL ARTERIES

The propagation of harmonic waves through a system formed of coaxial tubes filled with incompressible continua is considered as a model of arterial pulse propagation in the craniospinal cavity. The inner tube represents a blood vessel and is modelled as a thin-walled membrane shell. The outer tube is assumed to be rigid to account for the constraint imposed on the vessels by the skull and the vertebrae. We consider two models: in the first model the annulus between the tubes is filled with fluid; in the second model the annulus is filled with a viscoelastic solid separated from the tubes by thin layers of fluid. In both models, the elastic tube is filled with fluid. The motion of the fluid is described by the linearized form of the Navier–Stokes equations, and the motion of the solid by classical elasticity theory. The results show that the wave speed in the system is lower than that for a fluid-filled elastic tube free of any constraint. This is due to the stresses generated to satisfy the condition that the volume in the system has to be conserved. However, the effect of the constraint weakens as the radius of the outer tube is increased, and it should be insignificant for the typical physiological parameter range.     

A second strain gradient elasticity theory with second velocity gradient inertia – Part II: Dynamic behavior

This paper is the sequel of a companion Part I paper devoted to the constitutive equations and to the quasi-static behavior of a second strain gradient material model with second velocity gradient inertia. In the present Part II paper, a multi-cell homogenization procedure (developed in the Part I paper) is applied to a nonhomogeneous body modelled as a simple material cell system, in conjunction with the principle of virtual work (PVW) for inertial actions (i.e. momenta and inertia forces), which at the macro-scale level takes on the typical format as for a second velocity gradient inertia material model. The latter (macro-scale) PVW is used to determine the equilibrium equations relating the (ordinary, double and triple) generalized momenta to the inertia forces. As a consequence of the surface effects, the latter inertia forces include (ordinary) inertia body forces within the bulk material, as well as (ordinary and double) inertia surface tractions on the boundary layer and (ordinary) inertia line tractions on the edge line rod; they all depend on the acceleration in a nonstandard way, but the classical laws are recovered in the case of no higher order inertia. The classical linear and angular momentum theorems are extended to the present context of second velocity gradient inertia, showing that the extended theorems—used in conjunction with the Cauchy traction theorem—lead to the local force and moment (stress symmetry) motion equations, just like for a classical continuum. A gradient elasticity theory is proposed, whereby the dynamic evolution problem for assigned initial and boundary conditions is shown to admit a Hamilton-type variational principle; the uniqueness of the solution is also discussed. A few simple applications to wave propagation and dispersion problems are presented. The paper indicates the correct way to describe the inertia forces in the presence of higher order inertia; it extends and improves previous findings by the author [Polizzotto, C., 2012. A gradient elasticity theory for second-grade materials and higher order inertia. Int. J. Solids Struct. 49, 2121–2137]. Overall conclusions are drawn at the end of the paper.     

Generating functions for volume-preserving transformations

A general implicit solution for determining volume-preserving transformations in the n-dimensional Euclidean space is obtained in terms of a set of 2n generating functions in mixed coordinates. For n=2, the proposed representation corresponds to the classical definition of a potential stream function in a canonical transformation. For n=3, the given solution defines a more general class of isochoric transformations, when compared to existing methods based on multiple potentials. Illustrative examples are discussed both in rectangular and in cylindrical coordinates for applications in mechanical problems of incompressible continua. Solving exactly the incompressibility constraint, the proposed representation method is suitable for determining three-dimensional isochoric perturbations to be used in bifurcation theory. Applications in non-linear elasticity are envisaged for determining the occurrence of complex instability patterns for soft elastic materials.     

Weak shock waves and shear bands in compressible,inextensible thermoelastic solids

A study of weak shock waves propagating into a solid, which is compressible but temperature-dependent extensible in a specified direction is presented. The inextensible solid is also considered. The constitutive equations of constrained thermoelastic material are written as the summation of constrained and unconstrained counterparts of the relevant quantities. The equation of motion of weak shock waves, which is recovered by the theory of singular surfaces, reduces to an eigenvalue problem. The solution of this eigenvalue problem yields the speeds of propagation of weak shock waves. In the case of an undeformed solid, the speeds of these waves are explicitly expressed. Additionally, a discussion on the ductility limits of constrained thermoelastic material subjected to the uniaxial and biaxial extensions is presented.     

A stabilized formulation for incompressible plasticity using linear triangles and tetrahedra

In this paper, a stabilized finite element method to deal with incompressibility in solid mechanics is presented. Both elastic and J2-plastic constitutive behavior have been considered. A mixed formulation involving pressure and displacement fields is used and a continuous linear interpolation is considered for both fields. To circumvent the Babuška–Brezzi condition a stabilization technique based on the orthogonal sub-scale method is introduced. The main advantage of the method is the possibility of using linear triangular or tetrahedral finite elements, which are easy to generate for real industrial applications. Results are compared with standard Galerkin and Q1P0 mixed formulations in either elastic or elasto-plastic incompressible problems.     

On wave propagation in inextensible elastic bodies

In this paper we study the propagation of acceleration waves in inextensible elastic bodies. ¹ While the computations are but an exercise, the results are interesting and quite unlike the corresponding results for unconstrained bodies. Indeed, a wave travelling in the direction of inextensibility must necessarily be transverse, and, when the reaction stress is compressive and sufficiently large, the corresponding speed of propagation becomes non-real, so that even transverse waves fail to exist.We also study (infinitesimal) progressive waves and we find the corresponding propagation condition to be the same as that for acceleration waves. Here, however, non-real speeds of propagation have a definite physical meaning: they imply exponential growth of the wave. Thus, in particular, when the reaction stress is compressive and sufficiently large, a transverse progressive wave travelling in the direction of inextensibility grows without bound. We conjecture that this indicates the presence of local buckling. ²