𝒞1 continuous discretization of gradient elasticity

For the modeling of size effects, gradient continua can be applied. In this contribution, strain gradients are used in a nonlinear hyperelastic material model. The resulting partial differential equations are solved numerically in terms of a Galerkin scheme requiring 𝒞¹ continuity of the shape functions. The performance of three 𝒞¹ continuous finite elements and the 𝒞¹ Natural Element method is compared. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Diagonalization of the system of static Lamé equations of isotropic linear elasticity

We find a simplest representation for the general solution to the system of the static Lamé equations of isotropic linear elasticity in the form of a linear combination of the first derivatives of three functions that satisfy three independent harmonic equations. The representation depends on 12 free parameters choosing which it is possible to obtain various representations of the general solution and simplify the boundary value conditions for the solution of boundary value problems as well as the representation of the general solution for dynamic Lamé equations. The system of Lamé equations diagonalizes; i.e., it is reduced to the solution of three independent harmonic equations. The representation implies three conservation laws and some formula for producing new solutions which makes it possible, given a solution, to find new solutions to the static Lamé equations by derivations. In the two-dimensional case of a plane deformation, the so-found solution immediately implies the Kolosov-Muskhelishvili representation for shifts by means of two analytic functions of complex variable. Two examples are given of applications of the proposed method of diagonalization of the two-dimensional elliptic systems.     

The Reissner-Mindlin plate is the Γ-limit of Cosserat elasticity

The linear Reissner-Mindlin membrane-bending plate model is the rigorous Γ-limit for zero thickness of a linear isotropic Cosserat bulk model with symmetric curvature. For this result we use the natural nonlinear scaling for the displacements u and the linear scaling for the infinitesimal microrotations Ā ∈ so(3) affecting the in-plane drill rotations. No boundary conditions on the microrotations are prescribed. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lamé system of elasticity theory in a plane orthotropic medium

The authors develop a functional-theoretic approach to solving boundary-value problems for the Lamé system of elasticity theory. Special attention is paid to the case of a plane orthotropic medium. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 53, Suzdal Conference-2006, Part 1, 2008.     

Mean–variance portfolio selection under a constant elasticity of variance model

This paper discusses a mean–variance portfolio selection problem under a constant elasticity of variance model. A backward stochastic Riccati equation is first considered. Then we relate the solution of the associated stochastic control problem to that of the backward stochastic Riccati equation. Finally, explicit expressions of the optimal portfolio strategy, the value function and the efficient frontier of the mean–variance problem are expressed in terms of the solution of the backward stochastic Riccati equation.     

Spectrum and analyticity of semigroups arising in elasticity theory and hydromechanics

Cauchy problems for a second order linear differential operator equation

Legendre–Fenchel duality in elasticity

We show that the displacement and strain formulations of the displacement–traction problem of three-dimensional linearized elasticity can be viewed as Legendre–Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre–Fenchel duality.     

Linear elasticity obtained from finite elasticity by Γ-convergence under weak coerciveness conditions

The energy functional of linear elasticity is obtained as Γ  -limit of suitable rescalings of the energies of finite elasticity. The quadratic control from below of the energy density W(∇v)$W(\mathrm{\nabla }v)$ for large values of the deformation gradient ∇v   is replaced here by the weaker condition W(∇v)?|∇v|^p$W(\mathrm{\nabla }v)?{|\mathrm{\nabla }v|}^p$, for some p>1$p>1$. Energies of this type are commonly used in the study of a large class of compressible rubber-like materials.     

Saint–Venant torsion of a circular bar with radial cracks incorporating surface elasticity

The Saint–Venant torsion problem of a circular cylinder containing a radial crack with surface elasticity is studied. The surface elasticity is incorporated into the crack faces by using the continuum-based surface/interface model of Gurtin and Murdoch. Both an internal crack and an edge crack are considered. By using the Green’s function method, the boundary value problem is reduced to a Cauchy singular integro-differential equation of the first order, which can be numerically solved by using the Gauss–Chebyshev integration formula, the Chebyshev polynomials and the collocation method. Due to the incorporation of surface elasticity, the stresses exhibit the logarithmic singularity at the crack tips. The torsion problem of a circular cylinder containing two symmetric collinear radial cracks of equal length with surface elasticity is also solved by using a similar method. The strengths of the logarithmic singularity and the size-dependent torsional rigidity are calculated.     

The relative importance of surface compressional modulus of elasticity and reynolds number in interfacial turbulence — mass transfer

Surface renewal theories have gained recognition, prominence and widespread acceptance in explaining mass transfer processes at interfaces that become turbulent.Eddies are envisaged as being impelled into the interface from the bulk phase in mechanically stirred systems where they renew the surface with fresh liquid or fluid in a dynamic process.The effect of surface active agents has been known to damp mass transfer by inhibiting tangential eddy movements, hence surface renewal, and confirmed experimentally.Since the problem is dynamic, the mode of eddy propagation into the interface and the amount of area renewed are visualized as important factors in determining mass transfer across turbulent interfaces.The purpose of this paper was to construct a mathematical model which reveals the relative importance of the surface compressional modulus of elasticity Cs⁻¹, as opposed to the degree of agitation, which is synonymous with turbulence, hence Reynolds number, as each one tends to offset the other in mass transfer processes.The results are extremely important in industrial processes as they affect design and efficiency, as surface active agents become an undesirable nuisance in retarding mass transfer in both mixing and extraction processes and could prove to be well nigh indispensable.     

Homogenization of the Neumann problem for the Lamé equations of linear elasticity in domains with a periodic system of channels of small length

The problem of homogenization is considered for the solutions of the Neumann problem for the Lamé system of plane elasticity in two-dimensional domains with channels that have the form of rectilinear cylinders of length ε ^q (ε is a small positive parameter, q = const > 0) and radius a _ɛ. The bases of the channels form an ε-periodic structure on the hyperplane {x ∈ ℝ²: x ₁ = 0} and their number is equal to N _ɛ= O(ɛ⁻¹) as ε → 0. Under the limit condition lim on the parameters characterizing the geometry of the domain, the weak H ¹-limit of the generalized solution of this problem is found. __________ Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 25, pp. 310–322, 2005.     

Agmon’s condition for incompressible elasticity: A formulation of Mielke–Sprenger type

Agmon’s Condition for incompressible elasticity is recast in a tractable variational form analogous to that developed by Mielke and Sprenger for the compressible case.

“A teacher affects eternity; he can never tell where his influence will stop.” – Henry Adams, The Education of Henry Adams, 1918.

To Professor Michael Hayes on the Occasion of his Seventieth Birthday     

Homogenization in fractional elasticity – One spatial dimension

In this note we treat the equations of fractional elasticity in one spatial dimension. After establishing well-posedness, we use an abstract result in the theory of homogenization to derive effective equations in fractional elasticity with highly oscillating coefficients. The approach also permits the consideration of non-local operators (in time and space). (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

The time-dependent von K��rm��n plate equation as a limit of 3d nonlinear elasticity

The asymptotic behaviour of solutions of three-dimensional nonlinear elastodynamics in a thin plate is studied, as the thickness h of the plate tends to zero. Under appropriate scalings of the applied force and of the initial values in terms of h, it is shown that three-dimensional solutions of the nonlinear elastodynamic equation converge to solutions of the time-dependent von Kármán plate equation.     

Boundary stabilization of elastodynamic systems,part I: Rellich-type relations for a problem in elasticity involving singularities

For the Lamé’s system, mixed boundary conditions generate singularities in the solution, mainly when the boundary of the domain is connected. We here prove Rellich relations involving these singularities.     

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the varia-tional inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.     

A locking-free anisotropic nonconforming rectangular finite element approximation for the planar elasticity problem

This paper deals with a new nonconforming anisotropic rectangular finite element approximation for the planar elasticity problem with pure displacement boundary condition. By use of the special properties of this element, and by introducing the complementary space and a series of novel techniques, the optimal error estimates of the energy norm and the L^2-norm are obtained. The restrictions of regularity assumption and quasi-uniform assumption or the inverse assumption on the meshes required in the conventional finite element methods analysis are to be got rid of and the applicable scope of the nonconforming finite elements is extended.