singular values of a real rectangular tensor real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics in this paper we systematically study properties of singular values of a real rectangular tensor and give an algorithm to find the largest singular value of a nonnegative rectangular tensor numerical results show that the algorithm is efficient

periodic boundary value problems for a class of functional differential equations in this paper we show that the method of upper and lower solutions coupled with the monotone iterative technique is valid to obtain constructive proofs of existence of solutions for nonlinear periodic boundary value problems of functional differential equations without assuming properties of monotonicity in the nonlinear part

singular value inclusion sets of rectangular tensors abstract in this paper two singular value inclusion sets for rectangular tensors via a digraph and a lifting square tensor are given respectively furthermore it is shown that these two singular value inclusion sets are both tighter than the known result given by zhao and li in theorem 2 1 of 25 and singular value inclusion sets via a digraph in special case are tighter than the inclusion sets via a lifting square tensor numerical examples are still given to show the relation among the inclusion sets via a digraph the inclusion sets via a lifting square tensor and the known result given by zhao and li 25 finally by using the lifting square tensors the result of tensors provided by bu et al 24 is extended to rectangular tensors

monotone method for second order delayed differential equations with boundary value conditions in this paper we investigate the existence of extremal solutions of boundary value problems for second order differential equations with delayed arguments we show that a lower and upper solution method has been applied successfully to such problems an one sided lipschitz condition with corresponding constants is important assumption in our research

ordinary and strong ellipticity in the equilibrium theory of incompressible hyperelastic solids in this paper explicit necessary and sufficient conditions are established for the ordinary and strong ellipticity of the three dimensional field equations in the nonlinear equilibrium theory of incompressible homogeneous and isotropic hyperelastic solids the resulting system of inequalities involves the local principal stretches directly and in addition restricts the first and second partial derivatives of the strain energy density with respect to the deformation invariants or the principal stretches the conditions of ordinary and strong ellipticity are found to coalesce for materials that obey the baker ericksen inequalities and possess a positive shear modulus at infinitesimal deformations various implications of these ellipticity conditions for special classes of materials and deformations are explored

ellipticity and deformations with discontinuous gradients in finite elastostatics loss of ellipticity of the equilibrium equations of finite elastostatics is closely related to the possible emergence of elastostatic shocks i e deformations with discontinuous gradients in certain situations where constitutive response functions are essentially one dimentional such as anti plane shear or bar theories strong ellipticity is closely related to convexity of the elastic potential and invertibility of certain constitutive response functions the present work addresses the analogous issues within the context of three dimensional elastostatics of compressible but not necessarily isotropic hyperelastic materials a certain direction dependent resolution of the deformation gradient is introduced and its existence and uniqueness for a given direction are established the elastic potential is expressed as a function of kinematic variables arising from this resolution strong ellipticity is shown to be equivalent to the positive definiteness of the hessian matrix of this function thus sufficing for its strict convexity the underlying variables are interpretable physically as simple shears and extensions their work conjugates define a traction response mapping it is shown that discontinuous deformation gradients are sustainable if and only if this mapping fails to be invertible this result is explicit in the sense that it characterizes the set of all possible piecewise homogeneous deformations given the elastic potential function

on nonlinear two point boundary value problems for impulsive differential algebraic problems in this paper we investigate boundary value problems for first order impulsive differential algebraic problems with causal operators note that a corresponding boundary condition is given by a nonlinear function using a monotone iterative method we formulate general sufficient conditions under which such problems have solutions extremal or a unique an example shows that corresponding assumptions are satisfied the results are new

spherically symmetric two phase deformations and phase transition zones this paper is concerned with characterization and stability assessment of two phase spherically symmetric deformations that can be supported by a nonlinear elastic isotropic material we study general properties of equilibrium two phase spherically symmetric deformations then we specialize to phase transformations of a solid sphere that is subjected to an all round tension pressure two material models are used to demonstrate a variety of transformation behaviours and some common features for both materials we construct phase transition zones ptzs formed in the space of principal stretches by those which can exist adjacently to an equilibrium interface then we demonstrate how the ptz can be used for the prediction of the number of two phase spherically symmetric solutions and study how the deformation field associated with each solution is related to the ptz we show that even in the simplest case of one interface the solution is not unique two equilibrium two phase solutions as well as one uniform one phase solution are found under the same boundary conditions for the three solutions we construct their load deformation diagrams and compare the associated total energies the stability of the two phase states with respect to radial and small wavelength perturbations is also examined we observe how unstable solutions are related with the ptz

elasticity m tensors and the strong ellipticity condition abstract in this paper we establish two sufficient conditions for the strong ellipticity of any fourth order elasticity tensor and investigate a class of tensors satisfying the strong ellipticity condition the elasticity m tensor the first sufficient condition is that the strong ellipticity holds if the unfolding matrix of this fourth order elasticity tensor can be modified into a positive definite one by preserving the summations of some corresponding entries second an alternating projection algorithm is proposed to verify whether an elasticity tensor satisfies the first condition or not besides the elasticity m tensor is defined with respect to the m eigenvalues of elasticity tensors we prove that any nonsingular elasticity m tensor satisfies the strong ellipticity condition by employing a perron frobenius type theorem for m spectral radii of nonnegative elasticity tensors other equivalent definitions of nonsingular elasticity m tensors are also established

monotone iterative technique for periodic boundary value problems with causal operators abstract in this paper we develop the monotone iterative technique for periodic boundary value problems with causal operators

solvability of three point boundary value problems for second order differential equations with deviating arguments the monotone iterative technique is used to boundary problems for second order ordinary differential equations with deviating arguments corresponding results are formulated when the problem has extremal solutions or weakly coupled extremal quasi solutions

boundary value problems with causal operators abstract in this paper we apply the monotone iterative method for nonlinear two point boundary value problems with causal operators we formulate sufficient conditions under which such problems have extremal or quasisolutions in a corresponding sector we also investigate differential inequalities

on energy based coupled elastoplastic damage theories constitutive modeling and computational aspects abstract novel energy based coupled elastoplastic damage theories are presented in this paper the proposed formulation employs irreversible thermodynamics and internal state variable theory for ductile and brittle materials at variance with lemaitre s work on damage elastoplasticity the present formulation renders rational thermodynamic potential and damage energy release rate in contrast to previous work by simo and ju featuring an additive split of the stress tensor current formulation assumes an additive split of the strain tensor it is shown that the strain split damage elastoplasticity formulation leads to more robust tangent moduli than the stress split formulation the plastic flow rule and hardening law are characterized in terms of the effective quantities viz the effective stress space plasticity this mechanism is both physically well motivated and computationally efficient further a fourth order anisotropic damage mechanism is proposed for brittle materials rational mechanisms are also presented to account for the microcrack opening and closing operations as well as the strain rate dependency of microcrack growth efficient computational algorithms for proposed elasloplaslic damage models are subsequently explored by making use of the operator splitting methodology in particular new three step operator split algorithms are presented application is made to a class of inviseid and rate dependent cap damage models for concrete and mortar experimental validations are also given to illustrate the applicability of the proposed damage models

review paper characterization and stability of two phase piecewise homogeneous deformations many solid materials exhibit stress induced phase transformations such phenomena can be modelled with the aid of the nonlinear elasticity theory with appropriate choices of the strain energy function it is known that if a two phase deformation with gradient f in a finite elastic body is a local energy minimizer then given any point p of the surface of discontinuity the piecewise homogeneous deformation corresponding to the two values f p of f p is a global energy minimizer thus instability of the latter state would imply instability of the former state in this paper we investigate the stability properties of such piecewise homogeneous deformations more precisely we are concerned with two joined half spaces that correspond to two different phases of the same material we first show how such a two phase deformation can be constructed then the stability of the piecewise homogeneous deformation is investigated with the aid of two test criteria one is a kinetic stability criterion based on a quasi static approach and on the growth decay behaviour of the interface in the undeformed configuration when it is perturbed the other referred to as the energy criterion is used to determine whether the deformation is a minimizer of the total energy with respect to perturbations of the interface in both the current and undeformed configurations we clarify the differences between the two criteria and provide a compact formula which can be used to establish the stability instability of any two phase piecewise homogeneous deformations

on discontinuous strain fields in finite elastostatics abstract a general method for the study of piece wise homogeneous strain fields in finite elasticity is proposed critical homogeneous deformations supporting strain jumping are defined for any anisotropic elastic material under constant piola kirchhoff stress field in three dimensional elasticity since maxwell s sets appear in the neighborhood of singularities higher than the fold the existence of a cusp singularity is a sufficient condition for the emergence of piece wise constant strain fields general formulae are derived for the study of any problem without restrictions or fictitious stress strain laws the theory is implemented in a simple shearing plane strain problem nevertheless the procedure is valid for any anisotropic material and three dimensional problems

existence and approximation of solutions for nonlinear functional differential equations with periodic boundary value conditions abstract this paper is concerned with the existence and approximation of solutions for a class of first order functional differential equations with periodic boundary conditions we present a new comparison result and extend previous results

remarks on periodic boundary value problems for functional differential equations we extend some results on existence and approximation of solution for a class of first order functional differential equations with periodic boundary conditions we show the validity of the monotone iterative technique under weaker hypotheses and present some examples

a monotone method for constructing extremal solutions to second order periodic boundary value problems abstract in this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the periodic boundary value problem u t f t u u t t 0 2 u 0 u 2 u 0 u 2 in the presence of an upper solution and a lower solution with

coupled plastic damaged model abstract a constitutive model that couples plasticity and damage is presented the model is thermodynamically consistent and comes from a generalization of classical plasticity theory and isotropic damage theory of kachanov coupling between plasticity and damage is achieved through a simultaneous solution of the plastic and the damage problem after a description of the model a numerical algorithm for the integration of the resulting constitutive equations is presented it is an euler backward type of algorithm that is particularly suitable to solve plain stress non linear problems with a 2d finite element program the consistent stiffness matrix is also derived the paper is completed with some application examples that show that the model presented accurately reproduces the behaviour of elastic plastic damaged materials

coupled damage and plasticity models derived from energy and dissipation potentials a theoretical framework is defined that allows plasticity and damage models of inelastic behaviour to be combined within a consistent approach much emphasis is placed on the fact that within this framework the entire constitutive response is specified through two potential functions with no additional assumptions or evolution equations being necessary both plastic strain and damage parameter have roles as internal variables within the theory two classes of models are derived involving respectively uncoupled and coupled plasticity and damage examples of application of the theory are presented

convergence of an algorithm for the largest singular value of a nonnegative rectangular tensor abstract in this paper we present an iterative algorithm for computing the largest singular value of a nonnegative rectangular tensor we establish the convergence of this algorithm for any irreducible nonnegative rectangular tensor

positive definiteness of paired symmetric tensors and elasticity tensors abstract in this paper we consider higher order paired symmetric tensors and strongly paired symmetric tensors elasticity tensors and higher order elasticity tensors in solid mechanics are strongly paired symmetric tensors a strongly paired symmetric tensor is said to be positive definite if the homogeneous polynomial defined by it is positive definite positive definiteness of elasticity and higher order elasticity tensors is strong ellipticity in solid mechanics which plays an important role in nonlinear elasticity theory we mainly investigate positive definiteness of fourth order three dimensional and sixth order three dimensional strongly paired symmetric tensors we first show that the concerned strongly paired symmetric tensor is positive definite if and only if its smallest m eigenvalue is positive second we propose several necessary and sufficient conditions under which the concerned strongly paired symmetric tensor is positive definite third we study the conditions under which the homogeneous polynomial defined by a fourth order three dimensional or sixth order three dimensional strongly paired symmetric tensor can be written as a sum of squares of polynomials and further propose several necessary and or sufficient conditions to judge whether the concerned strongly paired symmetric tensors are positive definite or not fourth by using semidefinite relaxation we propose a sequential semidefinite programming method to compute the smallest m eigenvalue of a fourth order three dimensional strongly paired symmetric tensor by which we can check positive definiteness of the concerned tensor the preliminary numerical results confirm our theoretical findings

nonlinear boundary value problems for second order differential equations with causal operators in this paper we deal with second order differential equations with causal operators to obtain sufficient conditions for existence of solutions we use a monotone iterative method we investigate both differential equations and differential inequalities an example illustrates the results obtained

biquadratic optimization over unit spheres and semidefinite programming relaxations this paper studies the so called biquadratic optimization over unit spheres min x in mathbb r n y in mathbb r m sum 1 leq i k leq n 1 leq j l leq m b ijkl x i y j x k y l subject to x 1 y 1 we show that this problem is np hard and there is no polynomial time algorithm returning a positive relative approximation bound then we present various approximation methods based on semidefinite programming sdp relaxations our theoretical results are as follows for general biquadratic forms we develop a frac 1 2 max m n 2 approximation algorithm under a slightly weaker approximation notion for biquadratic forms that are square free we give a relative approximation bound frac 1 nm when min n m is a constant we present two polynomial time approximation schemes ptass which are based on sum of squares sos relaxation hierarchy and grid sampling of the standard simplex for practical computational purposes we propose the first order sos relaxation a convex quadratic sdp relaxation and a simple minimum eigenvalue method and show their error bounds some illustrative numerical examples are also included

on monotone method for first and second order periodic boundary value problems and periodic solutions of functional differential equations abstract in this paper we show that the method of monotone iterative technique is valid to obtain two monotone sequences that converge uniformly to extremal solutions of first and second order periodic boundary value problems and periodic solutions of functional differential equations we obtain some new results relative to the lower solution and upper solution with

a strain based plastic viscous damage model for massive concrete structures abstract within the framework of continuum damage mechanics a new constitutive damage model for massive concrete is presented mainly intended for the seismic analysis of gravity and arch dams consistent with thermodynamic requirements a straindriven formalism is adopted improving the algorithmic efficiency as much as required for the analysis of large scale problems to become feasible two scalar damage variables are introduced as internal variables as well as a plastic strain tensor an extension to account for the concrete strain rate dependency suitable for seismic analysis is presented at the end the efficiency of numerical predictions from the constitutive model is illustrated through numerical applications and algorithmic implementation is also detailed

on copositive matrices and strong ellipticity for isotropic elastic materials on etablit les conditions necessaires et suffisantes pour une forte ellipticite des equations d un materiau elastique non lineaire isotrope compressible a l equilibre on utilise un theoreme de representation de matrices copositives du a cottle habetler et lemke 1970

monotone iterative method for first order functional difference equations with nonlinear boundary value conditions in this paper we show that the method of monotone iterative coupled with the upper and lower solutions is valid to obtain constructive proofs to the existence of solutions for first order functional difference equations with nonlinear boundary value conditions an example is given to illustrate the results obtained

on differential algebraic problems in this paper the existence of extremal or quasi solutions for differential algebraic problems with causal operators is investigated the case when such problems have a unique solution is also discussed to obtain existence results a monotone iterative technique is applied

constitutive inequalities for an isotropic elastic strain energy function based on hencky s logarithmic strain tensor hencky s strain energy function for finite isotropic elasticity is obtained by the replacement of the infinitesimal strain measure occurring in the classical strain energy function of infinitesimal

advanced differential equations with nonlinear boundary conditions we investigate the existence of solutions for advanced differential equations with nonlinear boundary conditions sufficient conditions when the problem has extremal solutions or a unique solution are formulated linear advanced differential inequalities are also discussed

on copositiveness identification of partially symmetric rectangular tensors abstract this paper studies the copositiveness of a partially symmetric rectangular tensor some new criteria for identifying copositive rectangular tensors are established via the representation of the multivariate form in barycentric coordinates based on this we propose a numerical method for identifying the copositiveness of a partially symmetric rectangular tensor preliminary numerical results are reported to show the efficiency of the proposed method

on isotropic rank 1 convex functions let f be a rotationally invariant function defined on the set lin of all tensors with positive determinant on a vector space of arbitrary dimension necessary and sufficient conditions are given for the rank 1 convexity of f in terms of its representation through the singular values for the global rank 1 convexity on lin the result is a generalization of a two dimensional result of aubert generally the inequality on contains products of singular values of the type encountered in the definition of polyconvexity but is weaker it is also shown that the rank 1 convexity is equivalent to a restricted ordinary convexity when f is expressed in terms of signed invariants of the deformation

the method of lower and upper solutions for second third fourth and higher order boundary value problems abstract in this paper we develop the monotone method in the presence of lower and upper solutions for the problem u n t t u t u i a u i b i r i 0 n 1 where is a caratheodory function we obtain necessary and sufficient conditions in to guarantee the existence of solutions between a lower solution and an upper solution for n 2 if n 3 either or and n 4 if furthermore we obtain sufficient conditions in for n 2 k 6 when

rank 1 perturbations of deformation gradients rank 1 connections i e pairs f g of tensors matrices differing by a rank 1 tensor g f f n play an important role in the nonlinear elasticity and in particular in the theory of solid to solid phase transformations if f g are rank 1 connected g is said to be a rank 1 perturbation of f this paper describes the set of all rank 1 perturbations of f with prescribed singular values in an n dimensional space this set is shown to consist of 2n families of dimension n 1 and within each family g and f may be expressed as functions of the unit vector n given by explicit formulas these are then specialized to dimensions n 2 and 3 some illustrative examples are given