a continuum model accounting for defect and mass densities in solids with inelastic material behaviour in this paper the analysis of structures with inelastic material behaviour is considered taking into account the evolution of defects and changes in mass density the underlying kinematical concept of an oriented continuum is general enough to describe the micro and macrobehaviour of material bodies appropriately based on the logical and consistent variational arguments for a lagrangian functional the dynamic balance laws boundary and transversality conditions all related to the evolution of defect density and mass changes are derived for macro and microstresses of deformational as well as of configurational type the adopted procedure which formally leaves the balance laws unaltered leads to the additional balance law for changes in defect density and additional boundary conditions for the changes in mass and defect densities driving forces or affinities associated with the evolution of defect and mass densities and a generalization of the j integral representing the thermodynamic forces on defects are obtained a nonlocal constitutive model accounting for changes in the defect density is presented

equivalent spatial and material descriptions of finite deformation elastoplasticity in principal axes abstract a spatial description of the theory of rate independent finite deformation elastoplasticity in which the stress tensor is defined through the strain energy function is discussed the main assumption of isotropic elastic response and invariance requirements under superposed rigid body motion restrict the acceptable forms of the strain energy function to those given in terms of principal values of the strain measure of elastic distortion the formulation is developed on a manifold and the corresponding material description is obtained simply by pull back of the derived spatial form by appealing to the notion of covariance the method of principal axes is systematically exploited to derive the explicit expression for the stress tensor computation for an arbitrary form of the strain energy function and the explicit form of the evolution equation for an arbitrary form of the yield function a model problem of volume preserving plastic flow is discussed in the closure

inverse elastostatic stress analysis in pre deformed biological structures demonstration using abdominal aortic aneurysms in stress analysis of membrane like biological structures the geometry constructed from in vivo image which often corresponds to a deformed state is routinely taken as the initial stress free geometry in this paper we show that this limitation can be completely removed using an inverse elastostatic approach namely a method for finding the initial geometry of an elastic body from a given deformed state we demonstrate the utility of the inverse approach using a patient specific abdominal aortic aneurysm model and identify the scope of error in stress estimation in the conventional approach within a realistic range of material parameter variations

a theoretical and computational framework for anisotropic continuum damage mechanics at large strains abstract the main objective of this work is the formulation and algorithmic treatment of anisotropic continuum damage mechanics at large strains based on the concept of a fictitious isotropic undamaged configuration an additional linear tangent map is introduced which allows the interpretation as a damage deformation gradient then the corresponding finger tensor denoted as damage metric constructs a second order internal variable due to the principle of strain energy equivalence with respect to the fictitious effective space and the standard reference configuration the free energy function can be computed via push forward operations within the nominal setting referring to the framework of standard dissipative materials associated evolution equations are constructed which substantially affect the anisotropic nature of the damage formulation the numerical integration of these ordinary differential equations is highlighted whereby two different schemes and higher order methods are taken into account finally some numerical examples demonstrate the applicability of the proposed framework

computational method of inverse elastostatics for anisotropic hyperelastic solids the paper presents a computational method for predicting the initial geometry of a finitely deforming anisotropic elastic body from a given deformed state the method is imperative for a class of problem in stress analysis particularly in biomechanical applications while the basic idea has been established elsewhere comput methods appl mech eng 1996 136 47 57 int j numer meth engng 1998 43 821 838 the implementation in general anisotropic solids is not a trivial exercise but comes after a systematic development of eulerian representations of constitutive equations in this paper we discuss the general representation in the context of fibrous hyperelastic solids and provide explicit stress functions for some commonly used soft tissue models including the fung model and the holzapfel model a three field mixed formulation is introduced to enforce quasi incompressibility constraints the practical utility of this method is demonstrated using an example of aneurysm stress analysis copyright 2006 john wiley sons ltd

computational methods for inverse finite elastostatics abstract in the inverse motion problem in finite hyper elasticity the classical formulation relies on conservation laws based on eshelby s energy momentum tensor this formulation is shown to be lacking in several regards for a particular class of inverse motion problems where the deformed configuration and cauchy traction are given and the undeformed configuration must be calculated it is shown that for finite element calculations a simple re examination of the equilibrium equations provides a more suitable finite element formulation this formulation is also shown to involve only minor changes to existing elements designed for forward motion calculations examples illustrating the method in simple and complex situations involving a neo hookean material are presented

inverse formulation for geometrically exact stress resultant shells the inverse elastostatics method tackles a class of problems in which the deformed geometry and the applied load are known while the initial geometry and the stress in the deformed state are to be determined the method is imperative for many biomechanics applications particularly for image based stress analysis of living organs as the geometry captured by imaging often correspond to a pre deformed configuration 3 4 the computational aspects of inverse elastostatics i e the inverse formulations have been established for general elastic solids 1 2 3 in this work the inverse formulation of thin walled structures modeled as geometrically exact stress resultant shell 5 is presented the inverse formulation builds on a fundamental assumption of the elastic media that the stress depends on local relative motion of the body if either the reference or the deformed configuration is given the other can be found from the elastostatic equilibrium the shell formulation follows the same logic albeit involving shell kinematics and kinetics regardless of the additional assumptions introduced in the shell theory the fundamental premise underlying inverse analysis remains the same in the proposed formulation the inverse problem is solved by using the standard week form of shell balance equations which are yet parameterized and linearized in terms of the inverse motion to facilitate an effective numerical computation an invariant approached is used to formulate the constitutive equations a consistent reduction from 3d energy function is also introduced numerical examples involving both isotopic and anisotropic materials are provided for validating the formulation applications to several biological thin walled structures demonstrate the practical utility of the method

on the spatial formulation of anisotropic multiplicative elasto plasticity abstract this contribution aims at a thermodynamically consistent and modular formulation of anisotropic multiplicative elasto plasticity embedded in the well established framework of non standard dissipative materials we adopt as a key ingredient the introduction of additional symmetric arguments typically structural tensors into the relevant scalar valued isotropic tensor functions on this basis the fundamental covariance relation leads us to a formulation which allows itself to be set up directly in terms of spatial arguments and thus guarantees a convenient implementation within a finite element setting

application of material forces to hyperelastostatic fracture mechanics i continuum mechanical setting the concern of this work is a consequent exploitation of the notion of material forces for the application within hyperelastostatic fracture mechanics contrary to physical forces material forces act on the material manifold thus essentially representing the tendency of defects like cracks or inclusions to move relative to the ambient material based on the formulation of the appropriate quasi static balance laws in the material space we aim at a fresh look onto classical aspects of hyperelastostatic fracture mechanics operating throughout within the geometrically nonlinear setting we emphasize on the one hand the duality of the direct and the inverse motion description and on the other hand we re establish the classical path integrals from elementary equilibrium considerations in the material space

packings of a charged line on a sphere we find equilibrium configurations of open and closed lines of charge on a sphere and track them with respect to varying sphere radius closed lines transition from a circle to a spiral like shape through two low wave number bifurcations char22 baseball seam and twist char22 which minimize coulomb energy the spiral shape is the unique stable equilibrium of the closed line other unstable equilibria arise through tip splitting events an open line transitions smoothly from an arc of a great circle to a spiral as the sphere radius decreases under repulsive potentials with faster than coulomb power law decay the spiral is tighter in initial stages of sphere shrinkage but at later stages of shrinkage the equilibria for all repulsive potentials converge on a spiral with uniform spacing between turns multiple stable equilibria of the open line are observed

a computational method to assess the in vivo stresses and unloaded configuration of patient specific blood vessels in the modelling process of cardiovascular diseases one often comes across the numerical simulation of the blood vessel wall when the vessel geometry is patient specific and is obtained in vivo via medical imaging the stress distribution throughout the vessel wall is unknown however simulating the full physiological pressure load inside the blood vessel without incorporating the in vivo stresses will result in an inaccurate stress distribution and an incorrect deformation of the vessel wall in this work a computational method is formulated to restore the zero pressure geometry of patient specific blood vessels and to recover the in vivo stress field of the loaded structures at the moment of imaging the proposed backward displacement method is able to solve the inverse problem iteratively using fixed point iterations as only an update of the mesh is required the formulation of this method allows for a straightforward implementation in combination with existing structural solvers even if the structural solver is a black box

the n vortex problem on a rotating sphere ii heterogeneous platonic solid equilibria we describe a new method of constructing point vortex equilibria on a sphere made up of n vortices with different strengths such equilibria called heterogeneous equilibria are obtained for the five platonic solid configurations hence for graphic 1 the method is based on calculating a basis set for the nullspace of a matrix obtained by enforcing the necessary and sufficient condition that the mutual distances between each pair of vortices remain constant by symmetries inherent in the platonic solid configurations this matrix is reduced for each case and we call the dimension of the nullspace the degree of heterogeneity of the structure for the tetrahedron n 4 and octahedron n 6 the degree of heterogeneity is 4 and 6 respectively hence we are free to choose each of the vortex strengths independently for the cube n 8 the degree of heterogeneity is 5 for the icosahedron n 12 it is 7 while for the dodecahedron n 20 it is 4 thus the entire set of equilibria based on the platonic solid configurations is obtained including substructures associated with each configuration constructed by taking different linear combinations of the basis elements 1 embed inline graphic 1 gif

an integrated approach to shape optimization and mesh adaptivity based on material residual forces abstract this contribution is concerned with the coupling of finite element based shape optimization to methods of mesh adaptivity therein the nodal points of a finite element mesh serve as design variables in an optimization problem that aims to minimize a cost functional with respect to different constraints in order to avoid the occurrence of oscillating boundaries in the optimal design trials we generate the respective design updates through solving a series of fictitious boundary value problems on an updated lagrangian configuration the design update process is accompanied by adaptive mesh refinement to obtain more accurate results for the structural analysis and the design sensitivity analysis for the mesh adaptation process we consider r adaptive node relocation h adaptive mesh refinement and a combination of both approaches in each case the derivations rest upon the notion of material residual forces induced by finite element discretization and the coupling of shape optimization and mesh adaptivity is of intermittent type i e mesh adaptivity is invoked once a certain number of design updates has been generated we examine the benefits of the proposed method on the basis of some numerical examples in comparison to the same shape optimization method not involving adaptive mesh refinement where we evaluate the corresponding numerical costs the gain in accuracy and the effects on the optimal shapes being obtained

construction of point vortex equilibria via brownian ratchets a theory capable of producing equilibrium configurations of point vortices in the plane along with a numerical scheme to compute them is described the theory is formulated as a problem in linear algebra where one must find solutions to the matrix equation agz 0 where a is the 1 2 n nk1 n non normal configuration matrix obtained by requiring that all intervortical distances remain fixed and g2r n are the n vortex strengths for existence of an equilibrium a must have a non trivial nullspace we consider the singular values of a when this has one or more zero singular values the nullspace of a is non empty and an equilibrium exists for some choice of g new equilibrium configurations are found numerically by randomly depositing n points in the plane which generically gives rise to a configuration matrix a with empty nullspace using the sum of squares of the k smallest singular values of a as a ratchet we thermally fluctuate the configuration allowing each point to execute a random walk in the plane retaining only those configurations which reduce this quantity at the next step the configuration is thus driven to one with nullspace a zko0 these converged states are not necessarily nearby their initial configurations typically they are asymmetric and often we can drive the same initial state to several different equilibria a reverse ratchet method is also described which can produce initial conditions that would evolve to a specified equilibrium state once a converged final state is achieved the full singular value decomposition of a is used to calculate an optimal basis set for the nullspace of a and thus all allowable g the distribution of the singular values gives important information on the size of each equilibrium state as measured by frobenius norm their distance from each other spacing and density and how far a randomly chosen system of n points in the plane is from the nearest equilibrium configuration with a specified rank as well as its shannon entropy

a numerical framework for material characterisation of inhomogeneous hyperelastic membranes by inverse analysis an inverse method for estimating the distributions of the elastic properties of hyperelastic inhomogeneous membranes is proposed the material description of the membrane is based on a versatile constitutive model in which two stiffness parameters govern the nonlinear elastic behaviour of the material the estimation procedure includes a finite element framework the two stiffness parameters in the constitutive law are assumed to vary continuously over the inhomogeneous membrane and in the finite element framework the distributions of the two parameters are approximated using standard linear shape functions experimental results are assumed to exist in terms of nodal displacements from a test with known geometry and boundary conditions the experimental membrane is modelled in the finite element framework and the deformation of it is predicted an error function quantifying the discrepancy between the experimentally obtained deformation pattern and the numerically predicted pattern is then minimised with respect to the nodal values of the two interpolated parameters in a number of numerical examples the proposed procedure is assessed by attempting to reproduce the given random reference distributions of material properties the proposed estimation method is fully able to reproduce the reference distributions of the two material parameters with excellent accuracy

a level set approach to domain wall kinetics and domain patterning in elastic ferroelectrics the level set method is a powerful tool for tracking a moving interface in ferroelectric materials the boundaries between domains of uniform polarization referred to as domain walls are viewed as interfaces in this work a variational level set method is used to formulate a model for the behavior of elastic ferroelectrics and more particularly to describe the domain formation and the domain wall kinetics a regularized version of the sharp interface theory is introduced by substitution of the singular surfaces with thin transition layers inserting so inhomogeneities into the energy density function it is proved that the forces driving domain wall motion are in fact inhomogeneity forces with this description the level set method becomes an appropriate framework to insert and exploit material forces the proposed model may interpret typical microstructures in 2 d elastic ferroelectrics with tetragonal crystal symmetry comparisons with phase field models and experiments show good agreement

the minkowski and conformal superspaces we define complex minkowski superspace in four dimensions as the big cell inside a complex flag supermanifold the complex conformal supergroup acts naturally on this superflag allowing us to interpret it as the conformal compactification of complex minkowski superspace we then consider real minkowski superspace as a suitable real form of the complex version our methods are group theoretic based on the real conformal supergroup and its lie superalgebra

vortex lattice theory a particle interaction perspective recent experiments on the formation of vortex lattices in bose einstein condensates has produced the need for a mathematical theory that is capable of predicting a broader class of lattice patterns ones that are free of discrete symmetries and can form in a random environment we give an overview of an n particle based hamiltonian theory which if formulated in terms of the o n 2 interparticle distances leads to the analysis of a nonnormal configuration matrix whose nullspace structure determines the existence or nonexistence of a lattice the singular value decomposition of this matrix leads to a method in which all lattice patterns and the associated particle strengths in principle can be classified and calculated by a random walk scheme which systematically uses the m smallest singular values as a ratchet mechanism to home in on lattices with m dimensional nullspaces where 0 m le n the resulting distribution of singular values encodes detailed geometric properties of the lattice and allows us to identify and calculate important quantitative measures associated with each lattice including its size as measured by the frobenius or 2 norms distance between the lattices hence lattice density robustness and shannon entropy as a quantitative measure of its level of disorder this article gives an overview of vortex lattice theory from 1957 to the present highlighting recent experiments in bose einstein condensate systems and formulating questions that can be addressed by understanding the singular value decomposition of the configuration matrix we discuss some of the computational challenges associated with producing large n lattices the subtleties associated with understanding and exploiting complicated hamiltonian energy surfaces in high dimensions and we highlight ten important directions for future research in this area

an augmented iterative method for identifying a stress free reference configuration in image based biomechanical modeling continued advances in computational power and methods have enabled image based biomechanical modeling to become an important tool in basic science diagnostic and therapeutic medicine and medical device design one of the many challenges of this approach however is identification of a stress free reference configuration based on in vivo images of loaded and often prestrained or residually stressed soft tissues and organs fortunately iterative methods have been proposed to solve this inverse problem among them sellier s method this method is particularly appealing because it is easy to implement convergences reasonably fast and can be coupled to nearly any finite element package by means of several practical examples however we demonstrate that in its original formulation sellier s method is not optimally fast and may not converge for problems with large deformations fortunately we can also show that a simple inexpensive augmentation of sellier s method based on aitken s delta squared process can not only ensure convergence but also significantly accelerate the method

point vortex equilibria on the sphere via brownian ratchets we describe a brownian ratchet scheme that we use to calculate relative equilibrium configurations of n point vortices of mixed strength on the surface of a unit sphere we formulate it as a proble

topology optimization utilizing inverse motion based form finding topology optimization at finite strain setting using the concept of inverse motion based form finding is introduced this novel procedure allows boundary conditions and shape of the structure in the operating deformed state to be prescribed the outcome of the optimization algorithm will be the shape of the undeformed structure i e the state in which the structure should be manufactured the objective of the optimization considered is to find the stiffest structure for a given amount of material the problem is regularized using a helmholtz filter which is formulated in the deformed configuration both the elastic boundary value problem and the partial differential equation associated with the helmholtz filter are solved using the finite element method the optimization problem is solved using a sequence of convex separable approximations the paper is closed by 2d as well as 3d numerical examples that clearly illustrates that the method is able to find optimal solutions for inverse motion finite strain topology optimization problems

green s function for the laplace beltrami operator on a toroidal surface green s function for the laplacebeltrami operator on the surface of a three dimensional ring torus is constructed an integral ingredient of our approach is the stereographic projection of the toru

vector fields on glm n c flag supermanifolds abstract the main result of this paper is the computation of the lie superalgebras of holomorphic vector fields on complex flag supermanifolds introduced by yu i manin we prove that with several exceptions any holomorphic vector field is fundamental with respect to the natural action of the lie superalgebra gl m n c

strain measures integrability condition and frame indifference in the theory of oriented media this paper is concerned with the appropriate choice of state variables within the continuum model or oriented media it is shown that residual deformation strain and wryness can be considered as such quantities the compatibility conditions for them are derived which make the inverse problem of determining the displacement and director triad fields well posed the principle of frame indifference justifies the use of these quantities as state variables in the free energy density governing and constitutive equations are studied in detail a comparison with the continuum model of crystals with continuously distributed dislocations is provided

on the numerical treatment and analysis of finite deformation ductile single crystal plasticity abstract this contribution is concerned with the numerical analysis and simulation of instability phenomena such as shearband localization in plastic flow processes within the framework of ductile single fcc crystal viscoplasticity to this end a family of implicit constitutive algorithms and their implementation are considered in detail issues of accuracy and conservation of plastic incompressibility are examined moreover the implementation within an overall newton raphson finite element solution strategy relies on the correct evaluation of the algorithmic tangent moduli which result from linearizing the stress update integrator likewise these algorithmic tangent moduli are employed within the numerical analysis of the spatial localization tensor examples demonstrating the performance of the proposed strategy at the local constitutive level as well as at the global structural level are given

numerical computation of algorithmic consistent tangent moduli in large strain computational inelasticity abstract an algorithm for the numerical computation of so called algorithmic or consistent tangent moduli in finite inelasticity is presented these moduli determine the sensitivity of algorithmic expressions for stresses with respect to the change in total deformation they serve as iteration operators by application of newton type solvers for the iterative solution of non linear initial boundary value problems in finite inelasticity the underlying concept of the numerical computation is a perturbation technique based on a forward difference approximation which reduces the computation of the tangent moduli to a multiple stress computation the algorithmic procedure is material independent and surprisingly simple it is outlined for a lagrangian as well as an eulerian geometric setting and applied to model problems of finite elasticity and finite elastoplasticity

computational methods for inverse deformations in quasi incompressible finite elasticity this paper presents a formulation for incorporating quasi incompressibility in inverse design problems for finite elastostatics where deformed configurations and cauchy tractions are known in the recent paper of govindjee and mihalic 1996 comput methods appl mech engng 136 47 57 a method for solving this class of inverse problems was presented for compressible materials here we extend this work to the important case of nearly incompressible materials a displacement pressure mixed formulation is combined with a penalty method to enforce the quasi incompressible constraint without locking numerical examples are presented and compared to known solutions further examples present practical applications of this research to active problems in elastomeric component design 1998 john wiley sons ltd

point vortex equilibria and optimal packings of circles on a sphere we answer the question of whether optimal packings of circles on a sphere are equilibrium solutions to the logarithmic particle interaction problem for values of n 3 12 and 24 the only values of n for which the optimal packing problem also known as the tammes problem has rigorously known solutions we also address the cases n 13 23 where optimal packing solutions have been computed but not proven analytically as in jamaloodeen newton jamaloodeen newton 2006 proc r soc lond ser a 462 3277 3299 doi 10 1098 rspa 2006 1731 1 a logarithmic or point vortex equilibrium is determined by formulating the problem as the one in linear algebra graphic 2 where a is a n n 1 2 n non normal configuration matrix obtained by requiring that all interparticle distances remain constant if a has a kernel the strength vector graphic 3 is then determined as a right singular vector associated with the zero singular value or a vector that lies in the nullspace of a where the kernel is multi dimensional first we determine if the known optimal packing solution for a given value of n has a configuration matrix a with a non empty nullspace the answer is yes for n 3 9 11 14 16 and no for n 10 15 17 24 we then determine a basis set for the nullspace of a associated with the optimally packed state answer the question of whether n equal strength particles graphic 4 form an equilibrium for this configuration and describe what is special about the icosahedral configuration from this point of view we also find new equilibria by implementing two versions of a random walk algorithm first we cluster sub groups of particles into patterns during the packing process and grow a packed state using a version of the yin yang algorithm of longuet higgins longuet higgins 2008 proc r soc a doi 10 1098 rspa 2008 0219 we also implement a version of our brownian ratchet algorithm to find new equilibria near the optimally packed state for n 10 15 17 24 1 http dx doi org 10 1098 rspa 2006 1731 2 embed inline graphic 1 gif 3 embed inline graphic 2 gif 4 embed inline graphic 3 gif

snub polyhedra and organic growth this paper describes a new application of polyhedral theory to the growth of the outer sheath of certain viruses such structures are often modular consisting of one or two types of units arranged in a symmetric pattern in particular the polyoma virus has a structure apparently related to the snub dodecahedron here we consider the problem of how such patterns might grow in time starting from a given number n of randomly placed circles on the surface of a sphere the circles are first jostled by random perturbations then their radii are enlarged then they are jostled again and so on this yin yang method of growth can result in some very close packings when n 12 the closest packing corresponds to the snub tetrahedron and when n 24 the closest packing corresponds to the snub cube however when n 60 the closest packing does not correspond to the snub dodecahedron but to a less symmetric arrangement special attention is given to the structure of the human polyoma virus for which n 72 it is shown that the yin yang procedure successfully assembles the observed structure provided that the 72 circles are pre assembled in clusters of six each cluster consists of five circles arranged symmetrically around a sixth at the centre as in a flower with five petals this has implications for the assembly of the capsomeres in a polyoma virus

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

on configurational forces in multiplicative elastoplasticity abstract the main goal of this work consists in the elaboration of the material or rather configurational mechanics in the context of multiplicative elastoplasticity this nowadays well established approach which is inherently related to the concept of a material isomorphism or in other words to a local rearrangement is adopted as a paradigm for the general modelling of finite inelasticity the overall motion in space is throughout assumed to be compatible and sufficiently smooth according to the underlying configurations namely the material and the spatial configuration as well as what we call the intermediate configuration different representations of balance of linear momentum are set up for the static case the underlying flux terms are thereby identified as stress tensors of piola and cauchy type and are assumed to derive from a free energy density function thus taking hyperelastic formats moreover the incorporated source terms namely the configurational volume forces are identified by comparison arguments these quantities include gradients of distortions as well as dislocation density tensors in particular those dislocation density tensors related to the elastic or plastic distortion do not vanish due to the general incompatibility of the intermediate configuration as a result configurational volume forces which are settled in the intermediate configuration embody non vanishing dislocation density tensors while their material counterparts directly incorporate non vanishing gradients of distortions this fundamental property enables us to recover the celebrated peach koehler force for finite inelasticity acting on a single dislocation from the intermediate configuration volume forces

on the determination of the crystal reference in nonlinear continuum theory of dislocations the paper is concerned with the determination of the anholonomic crystal reference from the given elastic strain metric and dislocation density cartan torsion the incompatible inverse elastic deformation is shown to satisfy a system of linear first order differential equations the integrability condition requires that the curvature of the crystal connection should vanish the equivalent system of equations and integrability condition for the inverse elastic rotation are also obtained formulae for the inverse elastic deformation and the inverse elastic rotation in form of tensor series are derived the dual problem of determining the plastic deformation and the plastic rotation from the plastic strain and the dislocation density is solved in a similar manner a linearization procedure is carried out and a comparison with kroner s theory is provided

finiteness of relative equilibria of the four body problem we show that the number of relative equilibria of the newtonian four body problem is finite up to symmetry in fact we show that this number is always between 32 and 8472 the proof is based on symbolic and exact integer computations which are carried out by computer

exponential map algorithm for stress updates in anisotropic multiplicative elastoplasticity for single crystals this paper presents a new stress update algorithm for large strain rate independent single crystal plasticity the theoretical frame is the well established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts a distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts resulting in a very convenient representation of the schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of eulerian vectors the key contribution of this paper is an algorithmic formulation of the exponential map exp sl 3 sl 3 for updating the special linear group sl 3 of unimodular plastic deformation maps this update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration the resulting fully implicit stress update algorithm treats the possibly redundant constraints of single crystal plasticity by means of an active set search it exploits intrinsically the simple representation of the schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of eulerian vectors updates the performance of the proposed algorithm is demonstrated by means of a representative numerical example

on natural strain measures of the non linear micropolar continuum we discuss three different ways of defining the strain measures in the non linear micropolar continuum a by a direct geometric approach b considering the strain measures as the fields required by the structure of local equilibrium conditions and c requiring the strain energy density of the polar elastic body to satisfy the principle of invariance under superposed rigid body deformations the geometric approach a generates several two point deformation measures as well as some lagrangian and eulerian strain measures the ways b and c allow one to choose those lagrangian strain measures which satisfy the additional mechanical requirements these uniquely selected relative strain measures are called the natural ones all the strain measures discussed here are formulated in the general coordinate free form they are valid for unrestricted translations stretches and changes of orientations of the micropolar body and are required to identically vanish in the absence of deformation the relation of the lagrangian stretch and wryness tensors derived here to the ones proposed in the literature is thoroughly discussed

vector fields on osp2m 2n c and πspn c flag supermanifolds abstract the paper is devoted to a computation of the lie superalgebras of holomorphic vector fields on isotropic flag supermanifolds of maximal type corresponding to the lie superalgebras osp 2 m 2 n c and sp n c the result is that under some restrictions on the flag type any holomorphic vector field is fundamental with respect to the natural action of the lie superalgebras osp 2 m 2 n c or sp n c

some remarks on the continuum theory of defects in solids a succinct account of the classical continuum theory of defects is given and some critical remarks on its relevance for the mechanics of the model are discussed

coupled fluid structure interaction hemodynamics in a zero pressure state corrected arterial geometry abstract hemodynamic conditions in large arteries are significantly affected by the interaction of the pulsatile blood flow with the distensible arterial wall a numerical procedure for solving the fluid structure interaction problem encountered in cardiovascular flows is presented we consider a patient specific carotid bifurcation geometry obtained from 3d reconstruction of in vivo acquired tomography images which yields a geometrical representation of the artery corresponding to its pressurized state to recover the geometry of the artery in its zero pressure state which is required for a fluid structure interaction simulation we utilize inverse finite elastostatics time dependent flow simulations with in vivo measured inflow volume flow rate in the 3d undeformed artery are performed through the finite element method the coupled momentum method for fluid structure interaction is adopted to incorporate the influence of wall compliance in the numerical computation of the time varying flow domain to demonstrate the importance in recovering the zero pressure state of the artery in hemodynamic simulations we compute the time varying flow field with compliant walls for the original and the zero pressure state corrected geometric configurations of the carotid bifurcation the most important resulting effects in the hemodynamic environment are evaluated our results show a significant change in the wall shear stress distribution and the spatiotemporal extent of the recirculation regions

the helicoidal modeling in computational finite elasticity part i variational formulation the finite elasticity mechanics of continua capable of a polar description is formulated by an alternative modeling to keeping position and orientation as uncoupled fields the rototranslation between two material particles can be described by a single complex tensorial quantity which is recognized to be orthogonal its linearization gives the characteristic curvature and differential vectors underlying the helicoidal modeling in both the sense of the body geometric description and the evolution of a deforming body after due introduction to dual tensors and rototranslations the polar description of the continuum is addressed with particular care to mixed differentiations of the rototranslation field then a thorough variational framework is established for the most general polar continuum under hyperelasticity hypothesis and the three field two field and one field principles are drawn and linearized the proposed modeling is expected to be profitably exploited in non linear finite element analyses of solids undergoing finite displacements rotations and strains

energies and spacings of point charges on a sphere numerical methods are in general required for the determination of the stable configurations of n point charges on a sphere the stable configurations for n up to 50 have previously been ascertained and the authors extend the calculations for values up to 101 they report for the first time some remarkable global features of these configurations they show that the minimum energy accurately follows a simple half integral power law in 1 n over the full range they have investigated this power law is explicable in terms of the idealization of mapping a planar wigner lattice onto the surface of the unit sphere the pair distribution functions of the larger n configurations indicate predominant hexagonal coordination the coefficients of the observed power law are closely straddled by values calculated on the basis of hexagonal and square wigner lattices this highly accurate description of the energy permits them to remark on the detailed deviations of the individual structures from the general trend for n 30 they note that structures with n prime are relatively less stable while structures with n equal to 6 12 32 44 48 and 60 seem more stable

approximate solution of the trust region problem by minimization over two dimensional subspaces the trust region problem minimization of a quadratic function subject to a spherical trust region constraint occurs in many optimization algorithms in a previous paper the authors introduced an inexpensive approximate solution technique for this problem that involves the solution of a two dimensional trust region problem they showed that using this approximation in an unconstrained optimization algorithm leads to the same theoretical global and local convergence properties as are obtained using the exact solution to the trust region problem this paper reports computational results showing that the two dimensional minimization approach gives nearly optimal reductions in then dimension quadratic model over a wide range of test cases we also show that there is very little difference in efficiency and reliability between using the approximate or exact trust region step in solving standard test problems for unconstrained optimization these results may encourage the application of similar approximate trust region techniques in other contexts

determining the finite elasticity reference state from a loaded configuration there are a number of situations where the deformed configuration of a body is known and it is necessary to determine the reference state previous methods developed to calculate the reference state involve the formulation of the finite elasticity equations in terms of the deformed configuration this paper demonstrates that the undeformed reference state can be accurately determined from a deformed configuration and the associated loading conditions by using conventional finite elasticity balance equations together with a solution procedure that treats the reference configuration as the unknowns the mathematical theory behind the solution method is described validated with an analytical solution and verified using experimental studies on gel phantoms the practical utility of this method is then demonstrated in the field of breast biomechanics copyright 2007 john wiley sons ltd

finite element modelling of inverse design problems in large deformations anisotropic hyperelasticity this paper introduces a finite element model for the inverse design of pieces with large displacements in the elastic range the problem consists in determining the initial shape of the piece such that it attains the designed shape under the effect of service loads the model is particularly focused on the design of parts with a markedly anisotropic behavior like laminated turbine blades although the formulation expresses equilibrium on the distorted configuration it uses a standard constitutive equations library that is expressed as usual for measures attached to the undistorted configuration in this way the modifications to a standard finite elements code are limited to the routines for the computation of the finite element internal forces and tangent matrix two examples are given the first one for validation purposes while the second is an application which has industrial interest for the design of turbine blades copyright 2007 john wiley sons ltd

on a discrete variational problem involving interacting particles we consider the minimization problem quad min cal f bf z equiv sum k j log z k z j qquad k j 1 2 dots n for bf z z 1 z 2 dots z n in mathbb c n satisfying sum k 1 n z k 2 1 and z k ne z j for 1 le k j le n this problem arises in different contexts in several physical models such as incompressible euler equations and the ginzburg landau model in superconductivity we study the stability properties of some symmetric critical points such as regular polygons and configurations that consist of a regular polygon plus the origin we also establish the existence of an infinite number of critical points enjoying different kinds of symmetry we show that when the number of particles n exceeds a critical value the global minimizer cannot be a regular polygon n ge6 and if n ge 11 it cannot be a star configuration i e an n 1 sides regular polygon plus the origin

multisurface thermoplasticity for single crystals at large strains in terms of eulerian vector updates this paper presents a formulation of large strain rate independent multisurface thermoplasticity for single crystals and addresses aspects of its numerical implementation the theoretical frame is the well established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts a key feature of the present paper is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts resulting in a very convenient representation of the schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of eulerian vectors this observation is intrinsically exploited on the numerical side by formulating a new fully implicit stress update algorithm and the associated consistent elastoplastic moduli in terms of these eulerian vectors the proposed return mapping algorithm treats the possibly redundant constraints of large strain multisurface plasticity for single crystals by means of an active set search furthermore it satisfies in an algorithmically exact way the plastic incompressibility condition in situations of multislip the performance of the proposed formulation is demonstrated for two representative numerical simulations of thermoplastic deformation processes in single crystals with isotropic taylor type hardening

the closest packing of equal circles on a sphere the closest packing of x circles on the surface of a sphere is examined with the use of techniques that have been developed to determine the stereochemical arrangement of atoms packed around a central atom the technique is based on the concept that the centres of the circles repel one another and minimizing the total repulsion energy leads to an approximate structure from which exact solutions can be determined a number of improved packings have been discovered for values of x in the range 20 40 many different types of structure are found that are of lower symmetries than those previously described the packing density p defined as the fraction of the spherical surface that is enclosed by the circles is found to increase as the number of circles increases in contrast to the conclusion from previous studies however this increase in p is very slight and the values remain substantially below that for an infinite number of circles or a close packed plane

on the comparison of two approaches to compute material forces for inelastic materials application to single slip crystal plasticity abstract the main goal of this contribution is the application of the material force method to dissipative materials whereby special emphasis is placed on the application to crystal plasticity balance equations on the material manifold as e g the material motion balance of momentum generally necessitate the computation of the spatial gradient of additional internal degrees of freedom or rather internal variables we therefore focus in this work on the comparison of two different numerical approaches to compute these gradient fields within a finite element setting namely a node point based and an integration point based formulation several numerical examples underline that the latter framework which is the convenient numerically comparatively cheap and well established approach in computational inelasticity turns out to be sufficiently accurate for the problem at hand

inverse motion based form finding for quasi incompressible finite electroelasticity this work deals with inverse motion based form finding for electroelasticity the inverse motion problem is formulated for the electroelastic case and the resulting equations are implemented within a finite element framework a four field variational approach is adopted taking into consideration the typically incompressible behavior of the elastomer materials commonly used in electromechanical applications by means of numerical simulations the inverse motion based form finding makes it possible to design the referential configuration so that a given set of loads and boundary conditions results in a prespecified deformed configuration the computational finite element framework established in this work allows for such numerical simulations and testing and thereby the possibility to improve the design and accuracy in electroelastic applications such as grippers sensors and seals less

on referential and spatial formulations of inverse elastostatic analysis abstract there are two families of element formulation for inverse elastostatic analysis in the literature they employ different computation procedures at the element and material levels it has been suggested that one of them can preserve material library while the other requires redeveloping the material functions in this article we show that these two formulations are completely equivalent under certain conditions both can preserve the material library while under other conditions neither can we show that the modification of material functions is caused by the need of updating material symmetry in the inverse solution process several theoretical results were obtained while establishing the equivalence including an identity relating the inverse and forward tangent tensors for isotropic materials a comprehensive documentation of these two formulations is provided

on holomorphic functions on a compact complex homogeneous supermanifold abstract it is well known that non constant holomorphic functions do not exist on a connected compact complex manifold this statement is false for a supermanifold with a connected compact reduction in this paper we study the question under what conditions non constant holomorphic functions do not exist on a connected compact homogeneous complex supermanifold we describe also the vector bundles determined by split homogeneous complex supermanifolds as an application we compute the algebra of holomorphic functions on the classical flag supermanifolds introduced in manin 1997 11

a kinematically and thermodynamically consistent volumetric growth model based on the stress free configuration a continuum volumetric growth theory is presently elaborated based on the stress free configuration the total deformation gradient is the composition of a growth mapping followed by an elastic mapping ensuring the compatibility of the body a measure of the growth rate fully lying in the stress free configuration is elaborated and the balance of momentum accounting for mass changes due to growth is written in the same configuration highlighting measures of kinematic incompatibilities an eshelby stress is identified as the driving force for growth in the sense that it appears as the true dissipative thermodynamic force conjugated to the growth rate examples of incompatible cylindrical growth with residual stress generation and tumor growth with mechanics coupled to diffusion of nutrients illustrate the proposed theory

material forces in open system mechanics the basic concern of the present work is the exploitation of the notion of material forces in the theoretical and computational analysis of open systems with particular application to biomechanical problems based on a completely dual framework for the spatial and the material motion problem we introduce the balance equations for open system thermodynamics in combination with the appropriate constitutive equations they constitute the basis of the finite element formulation derived thereafter for the spatial motion problem the solution of the governing equations basically the balance of mass and momentum renders the discrete nodal values of the density and the deformation as primary unknowns for the material motion problem the computational analysis of the balance of momentum yields the discrete material node point forces which can be interpreted as driving forces for the local rearrangement of material inhomogeneities once the spatial motion problem has been solved the material force method is nothing but a mere post processing step from an algorithmic point of view as a convincing benefit of the proposed strategy the computation of the material forces is extremely cheap and requires no additional finite element data structure in the context of biomechanics material forces give further insight into complex biomechanically induced processes such as functional adaption morphogenesis healing or growth for example material forces on the boundary can be considered as a measure of shape sensitivity and thus indicate morphological changes while material forces in the interior indicate the tendency to create new mass locally or to equilibrate local density inhomogeneities

a subspace interior and conjugate gradient method for large scale bound constrained minimization problems a subspace adaptation of the coleman li trust region and interior method is proposed for solving large scale bound constrained minimization problems this method can be implemented with either sparse cholesky factorization or conjugate gradient computation under reasonable conditions the convergence properties of this subspace trust region method are as strong as those of its full space version computational performance on various large test problems is reported advantages of our approach are demonstrated our experience indicates that our proposed method represents an efficient way to solve large bound constrained minimization problems

elastic properties of anisotropic vascular membranes examined by inverse analysis an inverse method for estimating the distributions of the nonlinear elastic properties of inhomogeneous and anisotropic vascular membranes such as cerebral aneurysms is proposed the material description of the membrane is based on a versatile structural model able to represent multiple collagen layers and the passive response of the vascular wall each individual layer is assumed to behave transversely isotropic following exponential stiffening with increasing loading the model includes four parameters to be explainable physically two initial stiffnesses of the collagen fabric a parameter related to the nonlinearity of the collagen fabric angle between the principal directions of the collagen fabric and a reference coordinate system for this finite deformation problem a finite element framework for membranous structures considering pressure boundary loading is outlined i e the principle of virtual work its linearisation and the related spatial discretisation the estimation procedure consists of the following three steps i in vivo or in vitro approaches record the mechanical responses of membranous structures whose properties are to be determined ii define a corresponding finite element model iii minimise an error function regarding the unknown parameters that quantifies the deviation of the numerical prediction from the recorded data to achieve a robust parameter estimation an element partition method is employed the outcome of the procedure is affected by the number of nodes defined on the membrane surface and the number of load steps in a numerical example the proposed procedure is assessed by reestablishing given reference distributions in a reference membrane the deviations of the estimated material parameter distributions from the related reference fields are within just a few percent in most of the investigated cases the standard deviation for the resulting maximum principal stress was even below 1 which is accurate enough for rupture risk assessment of vascular membranes

geometrically non linear anisotropic inelasticity based on fictitious configurations application to the coupling of continuum damage and multiplicative elasto plasticity the objective of this contribution is the formulation and algorithmic treatment of a phenomenological framework to capture anisotropic geometrically non linear inelasticity in addition to the intermediate configuration of multiplicative elasto plasticity we further introduce two microscopic configurations of lagrangian and eulerian type which characterize the so called fictitious undamaged material this kinematical framework enables us to apply two well established postulates based on standard terminology in non linear continuum mechanics concerning the free energy function the postulate of strain energy equivalence is adopted and in view of the plastic dissipation potential the concept of effective stress is a natural outcome of the underlying kinematical assumptions finally we focus on the integration technique for the class of obtained evolution equations and present numerical examples for a prototype model to underline the applicability of the proposed framework copyright 2003 john wiley sons ltd

an arc length method including line searches and accelerations this paper describes a method for introducing line searches into the arc length solution procedure such line searches may be used at each iteration to calculate an optimum scalar step length which scales the normal iterative vector in practice a loose tolerance is provided so that on many iterations the line searches are avoided however on difficult iterations the line searches are shown to lead to a substantial improvement in the convergence characteristics a simple single parameter acceleration is also developed using line search concepts the new arc length method is applied to both the geometrically nonlinear analysis of shallow shells and the materially nonlinear analysis of reinforced concrete beams and slabs significant improvements are demonstrated in relation to the standard arc length method

completely integrable gradient flows in this paper we exhibit the toda lattice equations in a double bracket form which shows they are gradient flow equations on their isospectral set on an adjoint orbit of a compact lie group representations for the flows are given and a convexity result associated with a momentum map is proved some general properties of the double bracket equations are demonstrated including a discussion of their invariant subspaces and their function as a lie algebraic sorter

spectral gradient flow and equilibrium configurations of point vortices we formulate the problem of finding equilibrium configurations of n point vortices in the plane in terms of a gradient flow on the smallest singular value of a skew symmetric matrix m whose nullspa