mixed variational principles and robust finite element implementations of gradient plasticity at small strains summary this work outlines a theoretical and computational framework of gradient plasticity based on a rigorous exploitation of mixed variational principles in contrast to classical local approaches to plasticity based on locally evolving internal variables order parameter fields are taken into account governed by additional balance type pdes including micro structural boundary conditions this incorporates non local plastic effects based on length scales which reflect properties of the material micro structure we develop a unified variational framework based on mixed saddle point principles for the evolution problem of gradient plasticity which is outlined for the simple model problem of von mises plasticity with gradient extended hardening softening response the mixed variational structure includes the hardening softening variable itself as well as its dual driving force the numerical implementation exploits the underlying variational structure yielding a canonical symmetric structure of the monolithic problem it results in a novel finite element fe design of the coupled problem incorporating a long range hardening softening parameter and its dual driving force this allows a straightforward local definition of plastic loading unloading driven by the long range fields providing very robust fe implementations of gradient plasticity this includes a rational method for the definition of elastic plastic boundaries in gradient plasticity along with a post processor that defines the plastic variables in the elastic range we discuss alternative mixed fe designs of the coupled problem including a local global solution strategy of short range and long range fields this includes several new aspects such as extended q1p0 type and mini type finite elements for gradient plasticity all methods are derived in a rigorous format from variational principles numerical benchmarks address advantages and disadvantages of alternative fe designs and provide a guide for the evaluation of simple and robust schemes for variational gradient plasticity copyright 2013 john wiley sons ltd

sharp bounds for generalized elliptic integrals of the first kind abstract in this paper we prove that the double inequality 1 r 2 k a r sin a log e r a 2 r 1 r 2 holds for all a 0 1 2 and r 0 1 if and only if r a sin a 1 and a 1 a where r 1 r 2 k a r is the generalized elliptic integral of the first kind and r x is the ramanujan constant function besides as the key tool the series expression for the ramanujan constant function r x is given

a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis a discontinuous galerkin formulation is developed and analyzed for the cases of classical and gradient plasticity the model of gradient plasticity is based on the von mises yield function in which dependence is on the isotropic hardening parameter and its laplacian the problem takes the form of a variational inequality of the second kind the discontinuous galerkin formulation is shown to be consistent and convergent error estimates are obtained for the cases of semi and fully discrete formulations these mimic the error estimates obtained for classical plasticity with the conventional galerkin formulation

discontinuous galerkin method based on peridynamic theory for linear elasticity modeling of discontinuities shock waves crack surfaces etc in solid mechanics is one of the major research areas in modeling the mechanical behavior of materials among the numerical methods the discontinuous galerkin method dgm poses some advantages in solving these problems in this study a novel formulation for dgm is derived for elastostatics based on the peridynamic theory derivation of the proposed formulation is presented numerical analyses are performed for different problems and the numerical results are compared to that of the known exact solutions of the problems the proposed weak formulation is stable and coercive peridynamic discontinuous galerkin formulation is found to be robust and successful in modeling elastostatic problems copyright 2011 john wiley sons ltd

inequalities and bounds for generalized complete elliptic integrals computable bounds for the generalized complete elliptic integrals of the first and second kind are obtained also bounds for some combinations and products for integrals under discussion are established it has been proven that both families of integrals are logarithmically convex as functions of the first parameter this property has been employed to obtain several inequalities involving integrals in question

variational projection methods for gradient crystal plasticity using lie algebras summary a computational method is developed for evaluating the plastic strain gradient hardening term within a crystal plasticity formulation while such gradient terms reproduce the size effects exhibited in experiments incorporating derivatives of the plastic strain yields a nonlocal constitutive model rather than applying mixed methods we propose an alternative method whereby the plastic deformation gradient is variationally projected from the elemental integration points onto a smoothed nodal field crucially the projection utilizes the mapping between lie groups and algebras in order to preserve essential physical properties such as orthogonality of the plastic rotation tensor following the projection the plastic strain field is directly differentiated to yield the nye tensor additionally an augmentation scheme is introduced within the global newton iteration loop such that the computed nye tensor field is fed back into the stress update procedure effectively this method results in a fully implicit evolution of the constitutive model within a traditional displacement based formulation an elemental projection method with explicit time integration of the plastic rotation tensor is compared as a reference a series of numerical tests are performed for several element types in order to assess the robustness of the method with emphasis placed upon polycrystalline domains and multi axis loading this article is protected by copyright all rights reserved

adaptive stabilization of discontinuous galerkin methods for nonlinear elasticity motivation formulation and numerical examples the goal of this paper is to motivate introduce and demonstrate a novel approach to stabilizing discontinuous galerkin dg methods in nonlinear elasticity problems the stabilization term adapts to the solution of the problem by locally changing the size of a penalty term on the appearance of discontinuities with the goal of better approximating the solution consequently it is called an adaptive stabilization strategy the need for such a strategy is motivated through two and three dimensional examples in nonlinear elasticity the proposed scheme is simple to implement and compute and its performance is demonstrated with two and three dimensional numerical examples the accuracy of the proposed method is compared against a conforming method of the same order and a dg method with a traditional form of stabilization results for trilinear hexahedral elements indicate that the new stabilization strategy is more robust and more accurate when compared to a traditional form of stabilization however conforming trilinear hexahedral elements proved to be more computationally efficient for the examples shown here a two dimensional example with linear triangular elements showed comparable performances between the proposed method and a conforming one

monotonicity properties of generalized elliptic integrals with respect to the parameter abstract for a 0 1 2 and r 0 1 let k a r and e a r k r and e r denote the generalized elliptic integrals the complete elliptic integrals respectively of the first and second kinds respectively in this paper the authors present the necessary and sufficient conditions under which certain familiar combinations defined in terms of the generalized elliptic integrals and elementary functions are monotone in a 0 1 2 by these results sharp bounds expressed in terms of k r e r and elementary functions are obtained for k a r and e a r thus showing the dependence on the parameter a of k a r and e a r and substantially improving the known related results

some mean values related to the quasi arithmetic mean in this article we study new mean values and their properties related to the quasi arithmetic mean and their properties in the spirit of h haruki and th m rassias

a discontinuous galerkin formulation of a model of gradient plasticity at finite strains abstract the work presented here constitutes an extension to the finite strain regime of a discontinuous galerkin based strain gradient plasticity formulation presented in djoko et al j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis comput methods appl mech engrg 197 2007 1 21 the focus here is on algorithmic and computational aspects of the formulation at finite strains the adoption of a logarithmic hyperelastic plastic formulation preserves the essential features of the infinitesimal formulation this key ingredient allows the predictor corrector solution algorithms developed for the infinitesimal gradient formulation to be extended readily to the finite strain regime the use of low order elements is essential to contain the computational expense of the formulation but these elements are prone to locking the method of enhanced assumed strains for geometrically nonlinear problems is utilised to circumvent this limitation the form of the consistent tangent modulus is derived for the case of gradient plasticity two numerical examples are presented to illustrate aspects of the approximation scheme and the algorithm as well as features of the model of gradient plasticity

on a mathematical model of the productivity index of a well from reservoir engineering motivated by the reservoir engineering concept of the productivity index of a produc ing oil well in an isolated reservoir we analyze a time dependent functional diffusive capacity on the solutions to initial boundary value problems for a parabolic equation sufficient conditions providing for time independent diffusive capacity are given for different boundary conditions the dependence of the constant diffusive capacity on the type of the boundary condition dirichlet neumann or third boundary condition is investigated using a known variational principle and confirmed numer ically for various geometrical settings an important comparison between two principal constant values of a diffusive capacity is made leading to the establishment of criteria when the so called pseudo steady state and boundary dominated productivity indices of a well significantly differ from each other the third boundary condition is shown to model the thin skin effect for the constant wellbore pressure production regime for a damaged well the questions of stabilization and unique ness of the time independent values of the diffusive capacity are addressed the derived formulas are used in numerical study of evaluating the productivity index of a well in a general three dimensional reservoir for a variety of well configurations

some properties of the difference between the ramanujan constant and beta function abstract the authors present the power series expansions of the function r a b a at a 0 and at a 1 2 show the monotonicity and convexity properties of certain familiar combinations defined in terms of polynomials and the difference between the so called ramanujan constant r a and the beta function b a b a 1 a and obtain asymptotically sharp lower and upper bounds for r a in terms of b a and polynomials in addition some properties of the riemann zeta function n n n and its related sums are derived

monotonicity properties and inequalities for the generalized elliptic integral of the first kind abstract the authors obtain some monotonicity and concavity properties and asymptotically sharp bounds for the generalized elliptic integral k a r of the first kind and its special case k r the complete elliptic integral of the first kind by studying the monotonicity and concavity properties of certain combinations defined in terms of the gaussian hypergeometric function f a 1 a 1 x and elementary functions thus improving some well known results

discontinuous galerkin methods for solving a quasistatic contact problem we consider the numerical solution of a nonlinear evolutionary variational inequality arising in the study of quasistatic contact problems we study spatially semi discrete and fully discrete schemes for the problem with several discontinuous galerkin discretizations in space and finite difference discretization in time under appropriate regularity assumptions on the solution a unified error analysis is established for the schemes reaching the optimal convergence order for linear elements numerical results are presented on a two dimensional test problem to illustrate numerical convergence orders

generalized elliptic integrals and the legendre m function abstract we study monotonicity and convexity properties of functions arising in the theory of elliptic integrals and in particular in the case of a schwarz christoffel conformal mapping from a half plane to a trapezoid we obtain sharp monotonicity and convexity results for combinations of these functions as well as functional inequalities and a linearization property

analysis of generalized forchheimer flows of compressible fluids in porous media this work is focused on the analysis of nonlinear flows of slightly compressible fluids in porous media not adequately described by darcy s law we study a class of generalized nonlinear momentum equations which covers all three well known forchheimer equations the so called two term power and three term laws the generalized forchheimer equation is inverted to a nonlinear darcy equation with implicit permeability tensor depending on the pressure gradient this results in a degenerate parabolic equation for the pressure two classes of boundary conditions are considered given pressure and given total flux in both cases they are allowed to be unbounded in time the uniqueness lyapunov and asymptotic stabilities and other long time dynamical features of the corresponding initial boundary value problems are analyzed the results obtained in this paper have clear hydrodynamic interpretations and can be used for quantitative evaluation of engineering parameters some numerical simulations are also included

multiscale computational homogenization methods with a gradient enhanced scheme based on the discontinuous galerkin formulation abstract when considering problems of dimensions close to the characteristic length of the material the size effects can not be neglected and the classical so called first order multiscale computational homogenization scheme fmch looses accuracy motivating the use of a second order multiscale computational homogenization smch scheme this second order scheme uses the classical continuum at the micro scale while considering a second order continuum at the macro scale although the theoretical background of the second order continuum is increasing the implementation into a finite element code is not straightforward because of the lack of high order continuity of the shape functions in this work we propose a smch scheme relying on the discontinuous galerkin dg method at the macro scale which simplifies the implementation of the method indeed the dg method is a generalization of weak formulations allowing for inter element discontinuities either at the c 0 level or at the c 1 level and it can thus be used to constrain weakly the c 1 continuity at the macro scale the c 0 continuity can be either weakly constrained by using the dg method or strongly constrained by using usual c 0 displacement based finite elements therefore two formulations can be used at the macro scale i the full discontinuous galerkin formulation fdg with weak c 0 and c 1 continuity enforcements and ii the enriched discontinuous galerkin formulation edg with high order term enrichment into the conventional c 0 finite element framework the micro problem is formulated in terms of standard equilibrium and periodic boundary conditions a parallel implementation in three dimensions for non linear finite deformation problems is developed showing that the proposed method can be integrated into conventional finite element codes in a straightforward and efficient way

convexity with respect to holder mean involving zero balanced hypergeometric functions in this note we investigate the convexity of zero balanced hypergeometric functions with respect to holder mean

a fast convergent rate preserving discontinuous galerkin framework for rate independent plasticity problems abstract in this paper a finite element framework based on the incomplete interior penalty galerkin formulation a non symmetric discontinuous galerkin method is consistently formulated for modeling plasticity problems with small deformation because of its pure displacement based framework this proposed discontinuous galerkin method is possibly able to completely preserve numerical integration algorithms efficiently developed in the traditional continuous galerkin framework besides stresses on element interior quadrature points stresses on element surface quadrature points are also required to return on yielding surfaces in this discontinuous galerkin framework which is able to provide more accurate material yielding profiles than the continuous galerkin framework the performance of the proposed discontinuous galerkin framework has been evaluated in detail for j 2 and pressure dependent plasticities using perfect plasticity plasticity with hardening and associative and non associative material models quadratic convergent rates compatible to the tradition continuous galerkin method for modeling plasticity problems have been achieved within a large penalty range in a nodal based discontinuous galerkin implementation

accurate approximations for the complete elliptic integral of the second kind abstract in this paper we prove that the double inequality s 11 4 7 4 1 r e r s 11 4 7 4 1 r holds for all r 0 1 if and only if 2 1 570796 and 11 7 1 571428 where r 1 r 2 1 2 e r 0 2 1 r 2 sin 2 t d t is the complete elliptic integral of the second kind and s p q a b q a p b p p a q b q 1 p q is the stolarsky mean of a and b

on the spatial formulation of discontinuous galerkin methods for finite elastoplasticity abstract in this paper we present a consistent spatial formulation for discontinuous galerkin dg methods applied to solid mechanics problems with finite deformation this spatial formulation provides a general accurate and efficient dg finite element computational framework for modeling nonlinear solid mechanics problems to obtain a consistent formulation we employ the incomplete interior penalty galerkin iipg method another requirement for achieving a fast convergence rate for newton s iterations is the consistent formulation of material integrators we show that material integrators that are well developed and tested in continuous galerkin cg methods can be fully exploited for dg methods by additionally performing stress returning on element interfaces finally for problems with pressure or follower loading stiffness contributed from loaded surfaces must also be consistently incorporated in this work we propose the truesdell objective stress rate for both hypoelastoplastic and hyperelastoplastic problems two formulations based on the co rotational and multiplicative decomposition based frameworks are implemented for hypoelastoplasticity and hyperelastoplasticity respectively two new terminologies the so called standard surface geometry stiffness and the penalty surface geometry stiffness are introduced and derived through consistently linearizing the virtual work contributed from interior surface integrals the performance of our dg formulation has been demonstrated through solving a cantilever beam problem undergoing large rotations as well as a bipolar void coalescence problem where the voids grow up to several hundred times of their original volumes fast convergence rates for newton s iterations have been achieved in our iipg implementation

domain decomposition for linear elasticity with dg jumps and mortars in this article we present a domain decomposition method for solving a linear elasticity system data are transmitted by jumps as in a discontinuous galerkin method and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains a decoupling algorithm condenses the unknowns on the interface the matrix of the system is constructed by parallel computation numerical experiments confirming the theory are presented

a new weighted metric the relative metric ii in the first part of this investigation we generalized a weighted distance function of r c li s and found necessary and sufficient conditions for it being a metric in this paper some properties of this so called m relative metric are established specifically isometries and quasiconvexity results are derived we also illustrate connections between our approach and generalizations of the hyperbolic metric

turan type inequalities for generalized complete elliptic integrals in this paper our aim is to establish some turan type inequalities for gaussian hypergeometric functions and for generalized complete elliptic integrals these results complete the earlier result of p turan proved for legendre polynomials moreover we show that there is a close connection between a turan type inequality and a sharp lower bound for the generalized complete elliptic integral of the first kind at the end of this paper we prove a recent conjecture of t sugawa and m vuorinen related to estimates of the hyperbolic distance of the twice punctured plane

on applications of hypergeometric functions abstract this is a survey article on the author s involvement over the years with hypergeometric functions we discuss our counter example to one of m robertson s conjectures our results on the omitted values problems brannan s conjecture on the coefficients of a certain power series generalizations of ramanujan s asymptotic formulas for complete elliptic integrals and muir s 1883 approximation to the arc length of an ellipse involving an inequality for some 3f2 s

functional inequalities for the ratio of complete p elliptic integrals abstract in this paper we show monotonicity and convexity properties of a function involving the ratio of complete p elliptic integrals which reduces to the modulus of the grotzsch ring as applications we obtain several sharp functional inequalities

long term dynamics for well productivity index for nonlinear flows in porous media motivated by the reservoir engineering concept of the well productivity index pi we study a time dependent functional for general nonlinear forchheimer equation the pi of the well characterizes the well capacity with respect to drainage area of the well unlike the linear case for which this concept is well developed there are only a few recent publications dedicated to the pi for nonlinear case in this paper the pi is comprehensively studied both theoretically and numerically the impact of the nonlinearity of the flow filtration on the value of the pi is analyzed exact formula for the so called skin factor in radial case is derived depending on the rate of the flow the order of nonlinearity and the geometric parameters dynamics of the pi for the class of boundary conditions is studied and its convergence to the specific value of steady state pi was justified developed framework is applied to obtain nonlinear analog of peaceman formula for the well block pressure in unstructured grid numerica

fluid flows of mixed regimes in porous media in porous media there are three known regimes of fluid flows namely pre darcy darcy and post darcy because of their different natures these are usually treated separately in the literature to study complex flows when all three regimes may be present in different portions of a same domain we use a single equation of motion to unify them several scenarios and models are then considered for slightly compressible fluids a nonlinear parabolic equation for the pressure is derived which is degenerate when the pressure gradient is either small or large we estimate the pressure and its gradient for all time in terms of initial and boundary data we also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one moreover the continuous dependence of the solutions on initial and boundary data and the structural stability for the equation are established in porous media there are three known regimes of fluid flows namely pre darcy darcy and post darcy because of their different natures these are usually treated separately in the literature to study complex flows when all three regimes may be present in different portions of a same domain we use a single equation of motion to unify them several scenarios and models are then considered for slightly compressible fluids a nonlinear parabolic equation for the pressure is derived which is degenerate when the pressure gradient is either small or large we estimate the pressure and its gradient for all time in terms of initial and boundary data we also obtain their particular bounds for large time which depend on the asymptotic behavior of the boundary data but not on the initial one moreover the continuous dependence of the solutions on initial and boundary data and the structural stability for the equation are established

asymptotical bounds for complete elliptic integrals of the second kind abstract in this paper we establish several asymptotical bounds for the complete elliptic integrals of the second kind e r and improve the well known conjecture e r 1 1 r 2 3 4 2 2 3 2 for all r 0 1 proposed by m vuorinen

some gamma function inequalities a class of completely monotonic functions are presented involving the gamma function as well as the derivative of the psi function as a consequence new upper and lower bounds for the ratio f x 1 f x s are obtained and compared with related bounds given in part by j d keckic and p m vasic our results are further applied to obtain functions which are laplace transforms of infinitely divisible probability measures

exponential gamma and polygamma functions simple proofs of classical and new inequalities abstract some inequalities for the exponential gamma and polygamma functions are established by means of the mean value theorem of differential calculus in particular the cases of the ratio of gamma and polygamma functions are considered in some cases the results are new in some others classical results are recovered or extended

sharp inequalities between one parameter and power means for all a b 0 with a b we prove that j p a b m 1 2p 3 a b for p 2 1 2 u 1 and j p a b m 1 2p 3 a b for p 00 2 1 2 1 and the parameter 1 2p 3 in either case is the best possible here j p a b and m p a b are the one parameter and power means of order p of two positive numbers a and b respectively

a low order locking free hybrid discontinuous galerkin element formulation for large deformations abstract in this work a hybrid discontinuous galerkin dg quadrilateral element formulation is presented as usual in hybrid formulations the interior of the elements is treated separately from the skeleton which represents the element boundaries these are kinematically decoupled i e displacement jumps can occur between the skeleton and the interior of the elements the global degrees of freedom dofs are defined on the skeleton as the displacements at the corners which allows the implementation into existing finite element codes as usual in hybrid dg formulations the degrees of freedom in the interior are condensed out on the element level leading to the same number of global degrees of freedom as for continuous bilinear elements instead of using conventional shape functions in the interior the deformation gradient f is assumed constant within the element furthermore f is connected to the skeleton degrees of freedom via the weak form this leads to a very simple formulation and implementation the element is tested for several computational examples from the literature special choices of the penalty parameter are investigated which are partially derived analytically it is found that the element is free of volumetric and shear locking moreover the convergence is similar to that of other well known locking free finite element formulations

concavity of the complete elliptic integrals of the second kind with respect to holder means abstract in this paper we establish a necessary and sufficient condition for the concavity of the complete elliptic integrals of the second kind with respect to holder means

geometrically concave univariate distributions in this paper our aim is to show that if a probability density function is geometrically concave convex then the corresponding cumulative distribution function and the survival function are geometrically concave convex too under some assumptions the proofs are based on the so called monotone form of l hospital s rule and permit us to extend our results to the case of the concavity convexity with respect to holder means to illustrate the applications of the main results we discuss in details the geometrical concavity of the probability density function cumulative distribution function and survival function of some common continuous univariate distributions moreover at the end of the paper we present a simple alternative proof to schweizer s problem related to the mulholland s generalization of minkowski s inequality

generalized convexity and inequalities involving special functions abstract in this paper the authors show a relation between the generalized convexity and super sub multiplicative property and discuss some generalized convexity and inequalities involving the gaussian hypergeometric function the generalized distortion function k a x and the generalized grotzsch function a r

a hybrid high order method for finite elastoplastic deformations within a logarithmic strain framework we devise and evaluate numerically a hybrid high order hho method for finite plasticity within a logarithmic strain framework the hho method uses as discrete unknowns piecewise polynomials of order k 1 on the mesh skeleton together with cell based polynomi als that can be eliminated locally by static condensation the hho method leads to a primal formulation supports polyhedral meshes with non matching interfaces is free of volumetric locking the integration of the behavior law is performed only at cell based quadrature nodes and the tangent matrix in newton s method is symmetric moreover the principle of virtual work is satisfied locally with equilibrated tractions various two and three dimensional benchmarks are presented as well as comparison against known solutions with an industrial software using conforming and mixed finite elements

theory and numerics of a thermodynamically consistent framework for geometrically linear gradient plasticity the paper presents the theory and the numerics of a thermodynamically consistent formulation of gradient plasticity at small strains starting from the classical local continuum formulation which fails to produce physically meaningful and numerically converging results within localization computations a thermodynamically motivated gradient plasticity formulation is envisioned the model is based on an assumption for the helmholtz free energy incorporating the gradient of the internal history variable a yield condition and the postulate of maximum dissipation resulting in an associated structure as a result the driving force conjugated to the hardening evolution is identified as the quasi non local drag stress which incorporates besides the strictly local drag stress essentially the divergence of a vectorial hardening flux at the numerical side besides the balance of linear momentum the algorithmic consistency condition has to be solved in weak form thereby the crucial issue is the determination of the active constraints exhibiting plastic loading which is solved by an active set search algorithm borrowed from convex non linear programming moreover different discretization techniques are proposed in order to compare the fe performance in local plasticity with the advocated gradient formulation both for hardening and softening copyright 2001 john wiley sons ltd

sharp landen transformation inequalities for hypergeometric functions with applications abstract the authors present sharp landen transformation inequalities for the hypergeometric functions f 1 2 a b a b x and f 1 2 a b a b 1 2 x by showing the monotonicity properties of certain combinations defined in terms of one of these two hypergeometric functions and linear or rational functions thus giving complete solutions of the problem on extending the well known landen transformation identities for the complete elliptic integrals of the first kind to these two hypergeometric functions and substantially improving the related known results as applications of these results sharp landen transformation inequalities are obtained for the generalized grotzsch ring functions and the modular functions which appear in ramanujan s modular equations some other properties of hypergeometric functions and several properties of the ramanujan constant are obtained too

a general refinement of jordan type inequality in this work a general form of jordan s inequality p 2 n x a n 1 p 2 4x 2 n email protected email protected p 2 n x email protected n 0na email protected 2 email protected 2 n 1 p 2 4x 2 n 1 is established where email protected 0 p 2 p 2 n x email protected n 0 na n p 2 4x 2 n a email protected a email protected 3 a n 1 2n 12 n 1 p 2a n 116n n 1 p 2a n 1 and n 0 is a natural number the applications of the above result give the general improvement of the yang le inequality and a new infinite series sinx email protected n 0 a n p 2 4x 2 n for 0 x email protected 2

concavity with respect to holder means involving the generalized grotzsch function the generalized grotzsch function has numerous applications in geometric function theory and analytic number theory and its properties have been investigated by many authors in this paper we study the concavity of the generalized grotzsch function with respect to holder means

discontinuous galerkin methods for solving elliptic variational inequalities we study discontinuous galerkin methods for solving elliptic variational inequalities of both the first and second kinds analysis of numerous discontinuous galerkin schemes for elliptic boundary value problems is extended to the variational inequalities we establish a priori error estimates for the discontinuous galerkin methods which reach optimal order for linear elements results from some numerical examples are reported

two optimal double inequalities between power mean and logarithmic mean for email protected r the power mean m p a b of order p the logarithmic mean l a b and the arithmetic mean a a b of two positive real values a and b are defined by m p a b a p b p2 1 p p 0 ab p 0 l a b b alogb loga a b a a b and a a b a b2 respectively in this article we answer the questions what are the greatest values p and r and the least values q and s such that the inequalities m p a b 13a a b 23l a b m q a b and m r a b 23a a b 13l a b m s a b hold for all a b 0

inequalities for means abstract a monotone form of l hospital s rule is obtained and applied to derive inequalities between the arithmetic geometric mean of gauss the logarithmic mean and stolarsky s identric mean some related inequalities are given for complete elliptic integrals

adaptive discontinuous galerkin methods for solving an incompressible stokes flow problem with slip boundary condition of frictional type abstract we consider the numerical solution of a variational inequality arising in the study of an incompressible stokes flow with a nonlinear slip boundary condition of frictional type we study several discontinuous galerkin dg schemes for solving the problem and derive reliable and efficient residual type a posteriori error estimators an adaptive dg method is constructed based on the error estimators to solve the problem some numerical examples are presented to illustrate the efficiency of the adaptive dg method

generalized convexity and inequalities abstract let r 0 and let m be the family of all mean values of two numbers in r some examples are the arithmetic geometric and harmonic means given m 1 m 2 m we say that a function f r r is m 1 m 2 convex if f m 1 x y m 2 f x f y for all x y r the usual convexity is the special case when both mean values are arithmetic means we study the dependence of m 1 m 2 convexity on m 1 and m 2 and give sufficient conditions for m 1 m 2 convexity of functions defined by maclaurin series the criteria involve the maclaurin coefficients our results yield a class of new inequalities for several special functions such as the gaussian hypergeometric function and a generalized bessel function

sharp cusa type inequalities with two parameters and their applications in the article we present several sharp cusa type inequalities with two parameters as applications some new shafer fink type and carlson type inequalities for inverse sine and cosine functions and new inequalities for bivariate means are found

on approximating the arithmetic geometric mean and complete elliptic integral of the first kind abstract in the article we prove that the double inequalities 1 6 p 7 r p 5 p 6 r tanh 1 r 2 r k r 1 6 q 7 r q 5 q 6 r tanh 1 r 2 r q a 1 r 5 q 6 g 1 r a 1 r 6 q 7 g 1 r l 1 r a g m 1 r p a 1 r 5 p 6 g 1 r a 1 r 6 p 7 g 1 r l 1 r hold for all r 0 1 if and only if p 2 1 570796 and q 89 69 1 289855 where k r 0 2 1 r 2 sin 2 t 1 2 d t is the complete elliptic integral of the first kind tanh 1 r log 1 r 1 r 2 is the inverse hyperbolic tangent function r 1 r 2 and a 1 r 1 r 2 g 1 r r l 1 r r 1 log r and a g m 1 r are the arithmetic geometric logarithmic and gaussian arithmetic geometric means of 1 and r respectively

continuous discontinuous finite element approximations of fourth order elliptic problems in structural and continuum mechanics with applications to thin beams and plates and strain gradient elasticity abstract a new finite element method for fourth order elliptic partial differential equations is presented and applied to thin bending theory problems in structural mechanics and to a strain gradient theory problem the method combines concepts from the continuous galerkin cg method the discontinuous galerkin dg method and stabilization techniques a brief review of the cg method the dg method and stabilization techniques highlights the advantages and disadvantages of these methods and suggests a new approach for the solution of fourth order elliptic problems a continuous discontinuous galerkin c dg method is proposed which uses c 0 continuous interpolation functions and is formulated in the primary variable only the advantage of this formulation over a more traditional mixed approach is that the introduction of additional unknowns and related difficulties can be avoided in the context of thin bending theory the c dg method leads to a formulation where displacements are the only degrees of freedom and no rotational degrees of freedom need to be considered the main feature of the c dg method is the weak enforcement of continuity of first and higher order derivatives through stabilizing terms on interior boundaries consistency stability and convergence of the method are shown analytically numerical experiments verify the theoretical results and applications are presented for bernoulli euler beam bending poisson kirchhoff plate bending and a shear layer problem using toupin mindlin strain gradient theory

a primal dual finite element approximation for a nonlocal model in plasticity we study the numerical approximation of a static infinitesimal plasticity model of kinematic hardening with a nonlocal extension here the free energy to be minimized is a combination of the elastic energy and an additional term depending on the curl of the plastic variable first we introduce the stress as dual variable and provide an equivalent primal dual formulation resulting in a local flow rule the discretization is based on curl conforming nedelec elements to obtain optimal a priori estimates the finite element spaces have to satisfy a uniform inf sup condition this can be guaranteed by adding locally defined face and element bubbles second the discrete variational inequality system is reformulated as a nonlinear equality we show that the classical radial return algorithm applied to the mixed inequality formulation is equivalent to a semismooth newton method for the nonlinear system of equations numerical results illustrate the convergence of the applied discretization and the solver

gradient dependent plasticity formulation and algorithmic aspects a plasticity theory is proposed in which the yield strength not only depends on an equivalent plastic strain measure hardening parameter but also on the laplacian thereof the consistency condition now results in a differential equation instead of an algebraic equation as in conventional plasticity to properly solve the set of non linear differential equations the plastic multiplier is discretized in addition to the usual discretization of the displacements for appropriate boundary conditions this formulation can also be derived from a variational principle accordingly the theory is complete

precise estimates for differences of the gaussian hypergeometric function abstract by showing certain combinations of the gaussian hypergeometric functions f a b a b 1 x c and f a b a b 1 x d to be monotone on 0 1 for given a b c d 0 a b and c d the authors study the problem of comparing these two functions they find sup 0 a f a b a b 1 x c f a b a b 1 x d for x 0 1 thus solving an open problem and a recent conjecture

a variational principle for gradient plasticity abstract we elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by aifantis and co workers the generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher order spatial gradients of the equivalent plastic strain in the yield condition first it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands due to the presence of higher order gradients additional boundary conditions for the equivalent plastic strain are required this question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work we demonstrate here that upon assuming a certain type of additional boundary conditions the structural symmetries of the gradient dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain the variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given

inequalities for compound mean iterations with logarithmic asymptotes abstract we consider the compound means arising as limits from the arithmetic geometric mean iteration and related iterations each of these iterations possesses a logarithmic asymptote we show that these limit means satisfy very precise inequalities these can be deduced in a quite uniform fashion from a comparison lemma for compound means

numerical approximation of incremental infinitesimal gradient plasticity we investigate a representative model of continuum infinitesimal gradient plasticity the formulation is an extension of classical rate independent infinitesimal plasticity based on the additive decomposition of the symmetric strain tensor into elastic and plastic parts it is assumed that dislocation processes contribute to the storage of energy in the material whereby the curl of the plastic distortion appears in the thermodynamic potential and leads to an additional nonlocal backstress tensor the formulation is cast into a numerical framework by a saddle point approximation of the corresponding minimization problem in each incremental loading step this allows one to reformulate the nonlocal dissipation inequality to a point wise flow rule and yields a solution scheme which is a direct extension of the standard approach in classical plasticity our numerical results show the regularizing effects of the additional physically motivated terms copyright 2008 john wiley sons ltd

a discontinuous galerkin formulation for a strain gradient dependent damage model abstract the numerical solution of strain gradient dependent continuum problems has been dogged by continuity demands on the basis functions for most commonly accepted models solutions using the finite element method demand c 1 continuity of the shape functions here recent developments in discontinuous galerkin methods are explored and exploited for the solution of a prototype non linear strain gradient dependent continuum model a formulation is developed that allows the rigorous solution of a strain gradient damage model using standard c 0 shape functions the formulation is tested in one dimension for the simplest possible finite element formulation continuous piecewise linear displacement and constant on elements internal variable numerical results are shown to compare excellently with a benchmark solution the results are remarkable given the simplicity of the proposed formulation

monotonicity theorems and inequalities for the complete elliptic integrals we prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind k and e these results lead to sharp symmetrical bounds for k and e which improve recently discovered inequalities

sharp power mean bounds for the gaussian hypergeometric function sharp inequalities are established between the gaussian hypergeometric function and the power mean these results extend known inequalities involving the complete elliptic integral and the hypergeometric mean

inequalities for the generalized elliptic integrals with respect to holder means abstract in this paper the authors prove some inequalities for the generalized elliptic integrals with respect to holder means

inequalities for zero balanced hypergeometric functions the authors study certain monotoneity and convexity properties of the gaussian hypergeometric function and those of the euler gamma function

a discontinuous galerkin formulation for classical and gradient plasticity part 2 algorithms and numerical analysis this work is the second of a two part investigation into the use of discontinuous galerkin methods for obtaining approximate solutions to problems of classical and gradient plasticity part i j k djoko f ebobisse a t mcbride b d reddy a discontinuous galerkin formulation for classical and gradient plasticity part 1 formulation and analysis comput methods appl mech engrg 196 2007 3881 3897 presented the formulation and analysis of such problems this part focusses on algorithmic and computational aspects of the problem in particular it is shown that the predictor corrector algorithms of classical plasticity are readily extended to the case of gradient plasticity and to discontinuous galerkin formulations conditions for convergence of the algorithms are presented for the elastic secant and consistent tangent predictors the form of the consistent tangent modulus is established for the case of gradient plasticity a selection of numerical examples is presented and discussed with a view to illustrating aspects of the approximation scheme and algorithms as well as features of the model of gradient plasticity adopted here

an algorithm for gradient regularized plasticity coupled to damage based on a dual mixed fe formulation abstract a thermodynamically consistent theory of gradient regularized plasticity with coupling to isotropic damage was presented by svedberg and runesson 23 its main purpose was to describe the successive development of the localization zone that is pertinent to the softening regime due to excessive damage close to failure explicit constitutive relations were established for a model problem based on von mises plasticity with isotropic hardening of two types local and gradient enhanced the main purpose of this paper is to describe a new numerical algorithm for the gradient enhanced formulation using the backward euler rule leads to a gradient enhanced format of the classical closest point projection method the plastic multiplier increment is solved via a dual mixed method which is a convenient feature in the present context this leads to an unambiguous representation of the constitutive relations in the equilibrium equation moreover the classical local algebraic kuhn tucker conditions are obtained in a straightforward fashion in the limit case of local theory the developed algorithm was implemented in the educational finite element package 5 numerical results are shown for the extension of a plate in plane stress these results demonstrate the convergence characteristics with mesh refinement as well as the salient features of the adopted damage based constitutive model

remarks on generalized elliptic integrals we study the monotonicity for certain combinations of generalized elliptic integrals thus generalizing analogous well known results for classical complete elliptic integrals and prove a conjecture put forward by heikkala vamanamurthy and vuorinen

a hybridizable discontinuous galerkin formulation for non linear elasticity abstract we revisit the hybridizable discontinuous galerkin method for non linear elasticity introduced by s c soon 2008 we show that it can be recast as a minimization problem of a non linear functional over a space of discontinuous approximations to the displacement the functional can be written as the sum over the elements of the classic potential energy plus a new energy associated to the inter element jumps of the displacement we then show that if this new energy is not properly weighted the minimizers might not converge to the exact solution we construct an example illustrating this phenomenon and show how to overcome it by suitably increasing the weight of the energy of the inter element jumps finally we explore the performance of the method for the case of piecewise linear approximations in rather demanding situations in both two dimensional and for the first time three dimensional situations they include almost incompressible materials large deformations with large shear layers and cavitation we also compare the method with the continuous galerkin method and a previously explored discontinuous galerkin method and show that when using piecewise linear approximations and a moderate number of degrees of freedom the current method turns out to be more efficient for the computation of the gradient

a discontinuous galerkin method for strain gradient dependent damage study of interpolations and convergence abstract a discontinuous galerkin method has been developed for strain gradient dependent damage the strength of this method lies in the fact that it allows the use of c 0 interpolation functions for continuum theories involving higher order derivatives while in a conventional framework at least c 1 interpolations are required the discontinuous galerkin formulation thereby offers significant potential for engineering computations with strain gradient dependent models when using basis functions with a low degree of continuity jump conditions arise at element edges which are incorporated in the weak form in addition to the formulation itself a detailed study of the convergence properties of the method for various element types is presented and an error analysis is undertaken numerical results of some one and two dimensional problems are presented and discussed