computational aspects of incrementally objective algorithms for large deformation plasticity a methodology for computationally efficient formulation of the tangent stiffness matrix consistent with incrementally objective algorithms for integrating finite deformation kinematics and with closest point projection algorithms for integrating material response is developed in the context of finite deformation plasticity numerical experiments illustrate an excellent performance of the proposed formulation in comparison with other algorithms copyright 1999 john wiley sons ltd

consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm for large deformation inelasticity the present paper focuses on the consistent linearization and finite element implementation of an incrementally objective canonical form return mapping algorithm a general and modular algorithmic setting suited for almost any rate constitutive equations is presented where the finite deformation consistent tangent modulus is obtained as a by product of the integration algorithm numerical examples illustrate the good performance of the proposed formulation especially for large deformation increments with noteworthy superimposed rotation where the consistent formulation converges quadratically in a reasonable number of iteration

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

on non corotational rates of oldroyd s type and relevant issues in rate constitutive formulations the lower and upper oldroyd rates are two wellknown objective noncorotational rates a remarkable property of these rates is that they establish direct relationships between finite strain measures

finite deformation plasticity based on the additive split of the rate of deformation and hyperelasticity abstract finite deformation plasticity formulation based on additive split of rate of deformation and hyperelasticity is presented this approach is valid for finite elastic and plastic strains while rendering the choice and numerical integration of objective stress rates superfluous as the results are automatically objective for small elastic strains our method reduces to the classical hypoelastic corotational formulation provided that the dienes objective stress rate is employed while in the absence of inelastic deformation it coincides with the hyper elastic formulation the validity of the model has been examined on four test problems and the numerical results were found to be in good agreement with either the exact solution or experimental data

a mixed finite element formulation for compressible finite hyperelasticity with two fibre family reinforcement abstract a mixed finite element formulation for compressible finite hyperelasticity possibly with strongly deformation dependent stiffening fibre reinforcement is developed and verified reinforcement by two mechanically equivalent in general oblique fibre families is considered in the provided numerical experiments residual based error estimation is essential for a simple h adaptive mesh refinement strategy it is shown to be effective to resolve stress concentrations also in combination with stress oscillations checker boarding in the presence of strong anisotropy especially when mixed element constructs with enriched polynomial order are used the novel mixed generalised lagrangian metric based formulation is illustrated using a simple but fully coupled material model the ground substance is the common so called compressible neo hooke model and the standard reinforcement part is augmented by second order anisotropic invariants the theoretical formulation admits approaching the inextensible limit seamlessly by passing the use of a separate decoupled material model the hu washizu implementation used here relies on a certain extensibility for elimination of the auxiliary variables approximated in l 2 at the element level the condensation yields a purely h 1 based displacement formulation at the global level

a higher order lagrangian discontinuous galerkin hydrodynamic method for solid dynamics abstract we present a new multidimensional high order lagrangian discontinuous galerkin dg hydrodynamic method that supports hypoelastic and hyperelastic strength models for simulating solid dynamics with higher order elements we also present new one dimensional test problems that have an analytic solution corresponding to a hyperelastic plastic wave a modal dg approach is used to evolve fields relevant to conservation laws these fields are approximated high order taylor series polynomials the stress fields are represented using nodal quantities the constitutive models used to calculate the deviatoric stress are either a hypoelastic plastic infinitesimal strain hyperelastic plastic or finite strain hyperelastic plastic model these constitutive models require new methods for calculating high order polynomials for the velocity gradient and deformation gradient in an element the plasticity associated with the strength model is determined using a radial return method with a j 2 yield criterion and perfect plasticity the temporal evolution of the governing equations is achieved with the total variation diminishing runge kutta tvd rk time integration method a diverse suite of 1d and 2d test problems are calculated the new 1d piston test problems which have analytic solutions for each elastic plastic model are presented and calculated to demonstrate the stability and formal accuracy of the various models with the new lagrangian dg method 2d test problems are calculated to demonstrate the stability and robustness of the new lagrangian dg method on multidimensional problems with high order elements which have faces that can bend

large strain analysis of soft biological membranes formulation and finite element analysis abstract this paper presents a general formulation of thin incompressible membranes and investigates the behavior of soft biotissues using the finite element method in particular the underlying hyperelastic model is chosen to examine the highly non linear constitutive relation of blood vessels which are considered to be perfectly elastic homogeneous and nearly incompressible first the stress deformation relation and the elastic tangent moduli are derived in a very general material setting which is subsequently specified for blood vessels in terms of green lagrangian strains based on the principle of virtual work the finite element equations are provided and briefly discussed consistent linearization of the weak form of equilibrium and the external pressure term ensures a quadratic convergence rate of the iterative solution procedure on the computational side of this work an effort was undertaken to show a novel approach on the investigation of soft tissue biomechanics representative numerical analyses of problems in vascular mechanics are discussed that show isochoric finite deformations large rotations and large strains in particular a numerical simulation of the interaction between an inflated balloon catheter and a plaque deposit on the wall of a blood vessel is presented

a finite element formulation for finite strain elasto plastic analysis based on mixed interpolation of tensorial components abstract a quadrilateral finite element formulation for modelling finite strain elasto plastic deformation processes is presented the developed formulation is based on lee s multiplicative decomposition of the deformation gradient and on the hyperelastic expression of the associated von mises flow rule the hencky or logarithmic strain measure is used in the strain energy function the quadrilateral element is developed using displacements interpolation and interpolation of the hencky strain tensor covariant components a consistent linearization is developed obtaining a symmetric stiffness matrix

a discussion of stress rates in finite deformation problems abstract it has recently been shown that the finite elastic plastic solution of the simple shear problem exhibits an oscillatory stress response when kinematic hardening is employed while the solution for isotropic hardening gives a monotonically increasing stress this paper analyzes this response on the basis of continuum mechanical descriptions of the problem three objective stress rates are recalled and spatial descriptions of plasticity at finite deformation are reviewed for the usual generalization of the infinitesimal theory as well as a theory based on an invariant measure of true stress in light of the equations for the evolution of the yield surface the hypoelastic solution to the simple shear problem for each of the three stress rates is presented it is shown that the use of the jaumann rate in the generalization of the infinitesimal theory leads to an oscillation in the evolution of the yield surface in simple shear which is explained on the basis of the hypoelastic solution an alternative theory which makes use of the polar decomposition predicts a monotonically increasing shear stress

on the equivalence between the multiplicative hyper elasto plasticity and the additive hypo elasto plasticity based on the modified kinetic logarithmic stress rate abstract in two theorems presented herein we prove and subsequently demonstrate in several numerical examples involving homogeneous deformation that for isotropic materials hyper elasto plasticity models based on the multiplicative decomposition of the deformation gradient coincide with an additive hypo elasto plasticity model see section 3 2 that employs the spin tensor based on the modified kinetic logarithmic rate in the absence of strain induced anisotropy characterized by kinematic hardening herein this objective stress rate coincides with the kinetic logarithmic rate recently developed by jiao and fish 2017 we also show that other well known additive decomposition models such as those based on the jaumann and logarithmic rates may considerably deviate from the multiplicative model

robust numerical calculation of tangent moduli at finite strains based on complex step derivative approximation and its application to localization analysis abstract an extremely robust and efficient numerical approximation of material and spatial tangent moduli at finite strains is presented that can be easily implemented within standard fem software this method is based on the complex step derivative approximation csda approach the csda is proved to be of second order accurate and it does not suffer from roundoff errors in floating point arithmetics that limit the accuracy of other classical numerical approaches as e g finite difference approximation therefore the csda can provide approximations extremely similar to analytical solutions when perturbation values are chosen close to machine precision implementation details of the robust numerical approximation of tangent moduli from stress calculations using the csda are given and their performance is illustrated through representative examples involving finite deformations in addition to that we focus on the determination of material instabilities therefore an accompanying localization analysis is performed where the acoustic tensor is directly computed from the approximation of the moduli it is shown that classical numerical approximations are sensitive with respect to the perturbation value such that material instabilities may be artificially detected just as a result of slightly changing the perturbation on the other hand the csda approach provides high accurate and robust approximations within a wide range of perturbation values such that the material instabilities can be detected precisely

numerical study of consistency of rate constitutive equations with elasticity at finite deformation the present work is concerned with the numerical study of the elasticity consistency of the spatial rate equations using the conventional oldroyd truesdell cotter rivlin jaumann and green naghdi rates and the three novel co rotational e and l based logarithmic rates and of the rotated material rate equation describing the relationship between the material time derivatives of the rotated kirchhoff stress and material logarithmic strain to this end three integration procedures for updating stress are presented the stress responses of several typical deformation processes are simulated according to the numerical results we know that among the spatial rate equations only the logarithmic rate equation is consistent with elasticity under constant material parameters integrating the other spatial rate equations will provide path dependent stress response these numerical conclusions support the arguments in h xiao et al acta mechanica 1999 138 31 50 the reasons leading to elasticity inconsistency of spatial rate equations are analysed if the material parameters are assumed to be strain dependent the logarithmic rate equation loses also its elasticity consistent property the numerical results prove also that the spatial logarithmic and rotated material rate equations are equivalent to each other copyright 2002 john wiley sons ltd

on the numerical implementation of inelastic time dependent and time independent finite strain constitutive equations in structural mechanics abstract a number of complex issues are resolved in a way that allows the incorporation of finite strain inelastic material behavior into the piecewise numerical construction of solutions in solid mechanics without recourse to extensive continuum mechanics preliminaries an elementary time independent plasticity model an elementary time dependent creep model and a viscoelastic model are introduced as examples of constitutive equations which are routinely used in engineering calculations the constitutive equations are all suitable for problems involving large deformations and finite strains the plasticity and creep models are in rate form and use the symmetric part of the velocity gradient or the stretching to compute the co rotational time derivative of the cauchy stress the viscoelastic model computes the current value of the cauchy stress from a hereditary integral of a materially invariant form of the stretching history the current configuration is selected for evaluation of equilibrium as opposed to either the reference configuration or the last established equilibrium configuration the process of strain incrementation is examined in some depth the stretching evaluated at the mid interval and multiplied by the time step is identified as the appropriate finite strain increment to use with the selected forms of the constitutive equations discussed is the conversion of rotation rates based on the spin into incremental orthogonal rotations these rotations are used to update stresses and state variables due to rigid body rotation during the load increment comments and references to the literature are directed at numerical integration of the constitutive equations with an emphasis on doing this accurately if not exactly for any time step and stretching this material taken collectively provides an approach to numerical implementation which is marked by its simplicity

a unified stress update algorithm for explicit transient shell dynamics with combined isotropic kinematic hardening in eulerian rate type phenomenological finite elasto plasticity models abstract this paper discusses applications of the objective stress rates to a unified stress update algorithm for transient shell dynamic analysis within the context of explicit time integration we propose a unified stress update algorithm via establishing the eulerian rate type constitutive equations of hypoelastic plastic materials the constitutive equations are often given on the additive decomposition of the rate of deformation tensor under the assumption that the elastic part of the rate of deformation tensor may be characterized to be hypoelastic with grade zero i e the fourth rank constant isotropic tensor several objective stress rates are derived through the lie derivative of the cauchy or kirchhoff stress tensor those stress rates cover for instance the jaumann green naghdi truesdell oldroyd and cotter rivlin stress rates further the unified stress update algorithm is implemented along with the so called radial return method and the phenomenological plasticity theory with combined isotropic kinematic hardenings among the objective stress rates employed here we emphasize especially the stress update procedure of the green naghdi stress rate for transient shell dynamics because this algorithm developed in this work is not shown even in the representative commercial codes in the implementation of the unified stress update algorithm we applied our algorithms to the belytschko lin tsay shell theory to validate its accuracy and effectiveness moreover a slightly modified secant algorithm for the zero transverse normal stress of the shell is also developed the ultimate objectivity of this work is to provide a guideline for the best choice of an appropriate objective stress rate for phenomenological elastoplastic models several numerical examples are demonstrated including non contact transient shell dynamics examples and contact impact examples such as a ball plate impact problem by which the accuracy and effectiveness of the unified stress update algorithm developed in this work are addressed

a mixed higher order fem for fully coupled compressible transversely isotropic finite hyperelasticity a higher order mixed finite element method is presented for compressible transversely isotropic finite hyperelasticity the independent variables of the three field formulation are displacement fibre tension and fibre stretch the formulation admits the description of a fully coupled stress response which evolves from an almost isotropic one into a hyper anisotropic simply nearly inextensible response with increasing fibre tension as for example observed in soft tissue biomechanics standard displacement approximation of order p in h1 is used while the auxiliary variables are approximated element wise by square integrable functions of order p 1 in l2 for finite extensibility the auxiliary variables are statically condensed out yielding a pure displacement based method it is implemented in an hp adaptive finite element code a matching residual based error estimation capability is added and exercised numerical evidence indicating stability of approximation is supplied resolving a boundary layer caused by almost inextensible fibres coupled and uncoupled stress responses are compared a generalised compressible transversely isotropic holzapfel gasser ogden model is developed for the formulation that intentionally avoids the volumetric isochoric split solutions obtained with the mixed method compare favourably to the corresponding ones obtained with pure displacement formulation the latter fails in certain strongly anisotropic cases

a hybrid mixed model for non linear shell analysis and its applications to large rotation problems adopting an updated lagrange approach the general framework for the fully non linear analysis of curved shells is developed using a simple quadrilateral c0 model hmsh5 the governing equations are derived based on a consistent linearization of an incremental mixed variational principle of the modified hellinger reissner type with independent assumptions for displacement and strain fields emphasis is placed on devising effective solution procedures to deal with large rotations in space finite stretches and generalized rate type material models in particular a geometrically exact scheme for configuration update is developed by making use of the so called exponential mapping algorithm and the resulting element was shown to exhibit a quadratic rate of asymptotic convergence in solving practical shell problems with newton raphson type iterative schemes for the purpose of updating the spatial stress field of the element an objective generalized midpoint integration rule is utilized which relies crucially on the concept of polar decomposition for the deformation gradient and is in keeping with the underlying mixed method finally the effectiveness and practical usefulness of the hmsh5 element are demonstrated through a number of test cases involving beams plates and shells undergoing very large displacements and rotations

efficient and modular algorithms in modeling finite inelastic deformations objective integration parameter identification and sub stepping techniques this paper presents a general algorithmic formulation for constitutive equations describing inelastic material deformation the algorithmic representation focuses on the generality and modularity which is a key aspect within the development phase of constitutive equations it is shown explicitly how to solve the incremental constitutive equations by use of objective algorithms and an implicit integration scheme applying numerical as well as analytical derivatives although it is shown that using numerical gradients is significantly slower their application is easier especially in the development phase it is also demonstrated that having the constitutive equations modeled in the general context introduced the issue of parameter identification can be tackled by the same implementation using a driver algorithm besides parameter identification the general structure is also applied to two different fe simulations first the objectivity of different integration methods will be presented considering a simple shear loading superposed by a rigid rotation using a strain rate independent material model as example in the second example it is shown for a strain rate dependent material model how the simulation of the wire drawing process can be accelerated crucially by using sub stepping techniques

an effective two dimensional frictional contact model for arbitrary curved geometry a finite element model is developed on the basis of a variational formulation of the perturbed lagrange type and the classical coulomb law of friction for the analysis of frictional contact problems in two dimensions the model accounts for all geometric kinematic non linearities associated with large sliding motions as well as arbitrary contact surface curvatures explicit forms for the contact force and tangent stiffness operators and a penalty type format is utilized in the implementation an extensive number of numerical simulations are used to demonstrate the effectiveness and practical usefulness of the model

validation of an immersed thick boundary method for simulating fluid structure interactions of deformable membranes this paper constitutes an extension of the work of mendez et al 2014 36 for three dimensional simulations of deformable membranes under flow an immersed thick boundary method is used combining the immersed boundary method with a three dimensional modeling of the structural part the immersed boundary method is adapted to unstructured grids for the fluid resolution using the reproducing kernel particle method an unstructured finite volume flow solver for the incompressible navier stokes equations is coupled with a finite element solver for the structure the validation process relying on a number of test cases proves the efficiency of the method and its robustness is illustrated when computing the dynamics of a tri leaflet aortic valve the proposed immersed thick boundary method is able to tackle applications involving both thin and thick membranes closed and open membranes in significantly high reynolds number flows and highly complex geometries

on the numerical treatment of finite deformations in elastoviscoplasticity abstract this paper deals with the generalization of a geometric linear viscoplastic model to finite strains and its numerical application we owe the original formulation of the applied model to perzyna and chaboche it includes nonlinear isotropic and kinematic hardening as well as a nonlinear rate dependence the constitutive equations are integrated numerically in the context of a finite element formulation from theoretical considerations it is known that in the case of vanishing viscosity or slow processes rate independent plasticity arises as an asymptotic limit accordingly the numerical formulation includes this property in fact the stress algorithm corresponding to viscoplasticity is reduced to the asymptotic limit in a most simple way namely by setting the viscosity parameter equal to zero furthermore it is shown that the numerical integration of the constitutive model involves the solution of only one nonlinear equation for one scalar unknown this even applies to a sum of armstrong frederick terms the algorithm incorporates the inelastic incompressibility on the level of the gau points numerical computations of examples taken from metal forming technology show the physical significance of the model and the reliability of the numerical algorithm these calculations have been carried out by means of the finite element program psu

penalty function formulations for incompressible nonlinear elastostatics abstract the analysis of incompressible nonlinear elastic solids is considered by a penalty function approach in particular the necessary compatibility conditions for developing penalty forms for isotropic nonlinear elasticity are addressed and the commonly used form of strain energy functional leading to a penalty formulation of the incompressibility constraint extended the mooney rivlin model is used to show how previous developments can lead to physically meaningless situations for a certain class of incompressible materials which includes the important mooney rivlin model a new and particularly simple formulation is proposed

solution of the slump test using a finite deformation elasto plastic drucker prager model summary development of a finite deformation elasto plasticity model based on the multiplicative decomposition of the deformation gradient is presented and discussed in detail the formulation presented in this paper includes the derivation of the full set of equations for the drucker prager yield criterion the equations which are not available elsewhere are developed within a framework using a spectral decomposition approach further expressions for the consistent algorithmic tangent moduli in the finite strain regime are developed since the finite deformation framework employed to obtain the expressions presented here collapses to the classical infinitesimal plasticity framework when the finite strain assumption is no longer necessary the finite deformation consistent tangent moduli are compared to the consistent tangent moduli valid for use with infinitesimal plasticity validation of the implemented finite deformation elasto plastic drucker prager model is performed through the solution of the concrete slump test comparisons between an existing approximate analytical solution and experimental data are presented and results are discussed in detail

finite element formulations for hyperelastic transversely isotropic biphasic soft tissues abstract this paper presents a finite element model for the three dimensional 3 d nonlinear analysis of soft hydrated tissues such as articular cartilage in diarthrodial joints under physiologically relevant loading conditions a biphasic continuum description is used to represent the soft tissue as a two phase mixture of incompressible inviscid fluid and a hyperelastic transversely isotropic solid alternate mixed penalty and velocity pressure finite element formulations are used to solve the nonlinear biphasic governing equations including the effects of a strain dependent permeability and a hyperelastic solid phase under finite deformation the resulting first order nonlinear system of equations are discretized in time using an implicit finite difference scheme and solved using the newton raphson method a significant contribution of this work is the implementation and testing of a biphasic description with a transversely isotropic hyperelastic solid phase this description considers a helmholtz free energy function of five invariants of the cauchy green deformation tensor and the preferred direction of the material allowing for asymmetric behavior in tension and compression an exponential form is suggested and a set of material parameters is identified to represent the response of soft tissues in ranges of deformation and stress observed experimentally after demonstrating the behavior of this constitutive model in simple tension and compression a sample problem of unconfined compression is used to further validate the finite element implementation

formulation of implicit finite element methods for multiplicative finite deformation plasticity some constitutive and computational aspects of finite deformation plasticity are discussed attention is restricted to multiplicative theories of plasticity in which the deformation gradients are assumed to be decomposable into elastic and plastic terms it is shown by way of consistent linearization of momentum balance that geometric terms arise which are associated with the motion of the intermediate configuration and which in general render the tangent operator non symmetric even for associated plastic flow both explicit i e no equilibrium iteration and implicit finite element formulations are considered an assumed strain formulation is used to accommodate the near incompressibility associated with fully developed isochoric plastic flow as an example of explicit integration the rate tangent modulus method is reviewed in some detail an implicit scheme is derived for which the consistent tangents resulting in quadratic convergence of the equilibrium iterations can be written out in closed form for arbitrary material models all the geometrical terms associated with the motion of the intermediate configuration and the treatment of incompressibility are given explicitly examples of application to void growth and coalescence and to crack tip blunting are developed which illustrate the performance of the implicit method

application of the g g expansion method to solve coupled mkdv equations and coupled hirota satsuma coupled kdv equations abstract in the present paper we construct the travelling wave solutions involving parameters of the coupled mkdv equations and the coupled hirota satsuma coupled kdv equations by the g g expansion method the study confirms the power of the method

limitations to the large strain theory in the formulation of a large strain problem the euler cauchy stokes decomposition and the eulerian objective stress and strain rates used for the formulation of the problem are actually applicable only for some special cases and are not suitable for general large strain and rotation rates in this paper the limitations and applicabilities of the conventional large strain formulation under various deformation cases will be studied

large simple shear and torsion problems in kinematic hardening elasto plasticity with logarithmic rate abstract large simple shear and torsion problems in plasticity have been the object of a large number of papers sophisticated schemes have been developed e g j appl mech 50 1983 561 that overcome problems encountered cf e g j mech phys solids 48 2000 2231 int j solids struct 37 2000 5037 this paper substantially uses the logarithmic rate acta mechanica 124 1997a 89 which is equally based on strong mathematical and physical principles and therefore may contrast to classical approaches of cited kinds stress responses to large simple shear and torsional deformations in elastoplastic bodies are studied by applying the self consistent kinematic hardening j 2 flow model based on the logarithmic tensor rate recently established by these authors int j plasticity 15 1999 479 the application of the logarithmic stress rate in the elastic rate equation of hypoelastic type results in an exact finite hyperelastic solution in terms of hencky s logarithmic strain the plastic solution is composed of two parts the back stress and the effective stress the kirchhoff stress reduced by the back stress it is shown that the evolution equation of the back stress with the logarithmic rate is integrable to deliver a closed form relation between the back stress and hencky s logarithmic strain and the current stress moreover the effective stress is shown to be governed by a first order nonlinear ordinary differential equation with a small dimensionless material parameter multiplying the highest derivative for which the initial condition is related to the elastic plastic transition and prescribed in terms of the just mentioned small parameter a singular perturbation solution for the just mentioned equation is derived by utilizing the method of matched expansions with the analytical solution derived it is possible to make a detailed study of the coupling effect of material properties including the elastic yielding and hardening properties on elastic plastic responses for the large deformations at issue it is demonstrated that merely with three commonly known classical material constants i e the elastic shear modulus the initial tensile yield stress and the hardening modulus the simple kinematic hardening j 2 flow model with the logarithmic rate may supply satisfactory explanations for salient features of complex behaviour in experimental observation

a field boundary element formulation for material nonlinear problems at finite strains abstract a boundary element formulation for geometrical nonlinear problems is presented which is applicable for arbitrary constitutive equations based on the concept of hyperelastic response relative to an intermediate configuration this includes purely hyperelastic problems at large elastic strains in contrast to other formulations the total lagrange framework employed in this work significantly increases the numerical efficiency as the system matrices remain constant during the incremental solution and have to be computed only once a nonlinear set of equations with an identical structure for both boundary and internal unknowns is derived the basic unknowns are displacement gradients which are calculated via integral equations and not by differentiation of the shape functions the formulation can be consistently linearised to allow for the implementation of efficient iteration schemes using gradients e g the newton raphson scheme

consistent micro macro transitions at large objective strains in curvilinear convective coordinates this paper presents a computational homogenization scheme that is of particular interest for problems formulated in curvilinear coordinates the main goal of this contribution is to generalize the computational homogenization scheme to a formulation of micro macro transitions in curvilinear convective coordinates where different physical spaces are considered at the homogenized macro continuum and at the locally attached representative micro structures the deformation and the coordinate system of the micro structure are assumed to be coupled with the local deformation and the local coordinate system at a corresponding point of the macro continuum for the consistent formulation of micro macro transitions the operations scale up and scale down are introduced considering the rotated representation of tensor variables at the different physical reference frames of micro and macro structure the second goal of this paper is to use objective strain measures like the green lagrange strain tensor for the solution of boundary value problems on the micro and macro scale by providing the required transformations for the work conjugate stress strain and tangent tensors into variables admissible for the considered micro macro transitions and satisfying the averaging theorem copyright 2007 john wiley sons ltd

finite amplitude elastic waves propagating in compressible solids the paper studies the interaction of a longitudinal wave with transverse waves in general isotropic and unconstrained hyperelastic materials including the possibility of dissipation the dissipative term chosen is similar to the classical stress tensor describing a stokesian fluid and is commonly used in nonlinear acoustics the aim of this research is to derive the corresponding general equations of motion valid for any possible form of the strain energy function and to investigate the possibility of obtaining some general and exact solutions to these equations by reducing them to a set of ordinary differential equations then the reductions can lead to some exact closed form solutions for special classes of materials here the examples of the hadamard blatz ko and power law strain energy densities are considered as well as fourth order elasticity the solutions derived are in a time space separable form and may be interpreted as generalized oscillatory shearing motions and generalized sinusoidal standing waves by means of standard methods of dynamical systems theory some peculiar properties of waves propagating in compressible materials are uncovered such as for example the emergence of destabilizing effects these latter features exist for highly nonlinear strain energy functions such as the relatively simple power law strain energy but they cannot exist in the framework of fourth order elasticity

the choice of objective rates in finite elastoplasticity general results on the uniqueness of the logarithmic rate a eulerian rate formulation of finite elastoplasticity is a composite one composed of a rate equation for elastic behaviour and a flow rule for plastic behaviour as well as an evolution equation for hardening behaviour in which objective eulerian tensor rates are used among a large variety of objective rates actually infinitely many how to choose suitable ones has been one of the crucial points in finite elastoplasticity it is realized that the foregoing composite formulation of elastoplasticity should fulfil certain consistency criteria in order to avoid inconsistencies or contradictions these criteria narrow the choice of objective rates these authors bruhns and co workers and xiao and co workers have recently introduced the self consistency criterion in a composite formulation of elastoplasticity the rate equation intended for characterization of elastic behaviour should be exactly integrable to deliver an elastic relation it has been demonstrated in the work of the aforementioned authors that if a composite formulation of elastoplasticity with objective rates is required to fulfil the just stated self consistency criterion as it should be then the newly discovered logarithmic rate is the only possible choice among all objective corotational rates including the zaremba jaumann rate and the green naghdi rate etc in the afore mentioned result non corotational rates including oldroyd rates cotter rivlin rate and truesdell rate etc are not taken into consideration in general it is the main goal of this paper to further establish the uniqueness of the logarithmic rate among all corotational and non corotational objective rates essential to the attainment of this goal is the use of the yielding stationarity criterion the vanishing of the stress rate implies that the yield function is stationary it is shown that the just stated criterion is necessary for a composite eulerian rate formulation of finite elastoplasticity to be consistent and means that the stress rate must be a corotational rate the main goal of this article is thus attained by combining the just stated result and the established result stated before

on consistent stress rates in solid mechanics computational implications summary in the context of general isothermal elastic processes issues related to work conjugacy and its effect on the choice of a stress rate are discussed for processes involving large strains it is shown that a unique stress rate consistent with hyperelasticity can be defined for every conjugate pair chosen through the rate form of the associated hyperelastic constitutive equation that results in non constant and generally anisotropic spatial elasticity tensors even for a linear isotropic material the anisotropic spatial elasticity tensor connected with the green naghdi rate of kirchhoff stress is evaluated which provides a basis for the spatial formulation of processes governed by the hencky s strain energy function 8949 notions of equivalent and irreducible form of stress rate are introduced using a workless component of the deformation rate the hyperelastic constitutive models for the conjugate pairs s e and t in u which are of significance for computation are stated in their explicit form exact rate forms are derived and the order of approximation of currently employed computational algorithms discussed finally computational implications of the work presented are reviewed

is an additive decomposition of a rate of deformation and objective stress rates passe abstract the manuscript critically examines a half a century old idea of decomposing the rate of deformation into elastic and plastic parts we show that even the most recent additive variant based on the logarithmic rate is inconsistent with the notion of elasticity in so called unloading stress ratchetting obstacle test while the earlier corotational variants based on the jaumann and green naghdi rates are well known not to be integrable we then propose a stress dependent so called kinetic logarithmic stress rate that is both integrable and passes the unloading stress ratchetting obstacle test we also demonstrate that the additive hypo elasto plasticity model based on the proposed kinetic logarithmic rate is weakly invariant under isochoric reference change for simple materials in the sense of noll

incremental stresses in loaded orthotropic circular membrane tubes i theory abstract starting from the equations of general elastic nonlinear membrane theory in intrinsic form the equations governing the incremental state of stress in an orthotropic circular membrane tube are derived and discussed the tube is initially subjected to uniform internal pressure and to longitudinal extension which lead to large homogeneous deformation then some changes in loading and or geometry are considered e g an additional load is applied the shape of the boundary is changed or a slit is formed in the membrane these changes are regarded as small perturbations on the initial homogeneous state of stress the general form as well as some simplified forms of the equations are presented leading finally to a set of two equations in terms of two scalar potential functions two different variational formulations for the problem are also presented each of which may serve as a basis for numerical treatment

comparison of some stress rates abstract this paper introduces a new derivation rule by means of which two new stress rates are defined furthermore the paper compares five stress rates jaumann s truesdell s green naghdi s sowerby chu s and durban baruch s for simple shear elastic elastic perfectly plastic and elastic plastic hardening isotropic kinematic and combined material models are considered different solutions have already been published for these cases except for sowerby chu and durban baruch time derivatives using the new derivation rule the new rate form of the hyperelastic doyle ericksen formula is obtained taking advantage of the sowerby chu stress rate a new constitutive equation for hypoelastic material is given

non linear finite element implementation of micropolar hypo elastic materials in this paper updated lagrangian finite element formulations for large elastic deformation of micropolar hypo elastic materials are presented using representation theorems of tensor functions the general form of the constitutive equations for the micropolar hypo elastic materials model are presented the finite element formulations are based on the general form of the micropolar hypo elastic constitutive equations in conjunction with jaumann rate and a new rate called gyration rate gyration rate describes the deformation of the material in view of an observer attached to the micro structure an incrementally objective stress and couple stress update procedure is developed and its applicability is investigated based on the jaumann and gyration rates two planar examples based on the simplified forms of the proposed constitutive equations with jaumann and gyration rates are investigated in the first example a plate with a small circular hole is analyzed and it is shown that the results of non linear analysis converge to the linear micropolar elasticity for the case of infinitesimal deformations in another example 2d deformation of a cantilever beam which is modeled as a slender rectangle is investigated the results of jaumann and gyration rates are very close to each other but an increase in the micropolar material parameters particularly an increase in the length scale parameter results in the reduction of the deformation of the beam also it is shown that for a specimen with very small dimensions e g in a micro beam the effect of the length scale parameter is more sensible than that of a specimen with an ordinary or large dimensions this means that micropolar effects become important for deformations taking place at small scales

averaged dynamics and pole assignment for a class of stochastic systems for a linear system with markov jump parameters the moment liapunov exponents are computed explicitly in terms of averaged dynamics besides giving a necessary and sufficient condition for moment stability the computation suggests that control design for these systems can be aimed at assigning desired closed loop eigenvalues this is shown to be equivalent to a decentralized control problem to which the homotopy method applies and the control gains are obtained by continuation from the solution of the decoupled no jump problem

isogeometric collocation for large deformation elasticity and frictional contact problems abstract isogeometric collocation methods have been recently proposed as an alternative to standard galerkin approaches as they provide a significant reduction in computational cost for higher order discretizations in this work we explore the application of isogeometric collocation to large deformation elasticity and frictional contact problems we first derive the non linear governing equations for the elasticity problem with finite deformation kinematics and provide details on their consistent linearization some numerical examples demonstrate the performance of collocation in its basic and enhanced versions differing by the enforcement of neumann boundary conditions for problems with strong singularities enhanced collocation is shown to outperform basic collocation and to lead to a spatial convergence behavior very similar to galerkin whereas for weaker or no singularities enhanced and basic collocation may give very similar results a large deformation contact formulation is subsequently developed and tested in the frictional setting where collocation confirms the excellent performance already obtained for the frictionless case finally it is shown that the contact formulation in the collocation framework passes the contact patch test to machine precision in a three dimensional setting with arbitrarily inclined non matching discretizations thus outperforming most of the available contact formulations and all those with pointwise enforcement of the contact constraints

a self consistent eulerian rate type model for finite deformation elastoplasticity with isotropic damage abstract continuum models for coupled behaviour of elastoplasticity and isotropic damage at finite deformation are usually formulated by first postulating the additive decomposition of the stretching tensor d into the elastic and the plastic part and then relating each part to an objective rate of the effective stress etc it is pointed out that according to the existing models with several widely used objective stress rates none of the rate equations intended for characterizing the damaged elastic response is exactly integrable to really deliver a damaged elastic relation between the effective stress and an elastic strain measure the existing models are thus self inconsistent in the sense of formulating the damaged elastic response by consistently combining additive and multiplicative decomposition of the stretching d and the deformation gradient f and adopting the logarithmic stress rate in this article we propose a general eulerian rate type model for finite deformation elastoplasticity coupled with isotropic damage the new model is shown to be self consistent in the sense that the incorporated rate equation for the damaged elastic response is exactly integrable to yield a damaged elastic relation between the effective kirchhoff stress and the elastic logarithmic strain the rate form of the new model in a rotating frame in which the foregoing logarithmic rate is defined is derived and from it an integral form is obtained the former is found to have the same structure as the counterpart of the small deformation theory and may be appropriate for numerical integration the latter indicates in a clear and direct manner the effect of finite rotation and deformation history on the current stress and the hardening and damage behaviours further it is pointed out that in the foregoing self consistency sense of formulating the damaged elastic response the suggested model is unique among all objective eulerian rate type models of its kind with infinitely many objective stress rates to be chosen in particular it is indicated that within the context of the proposed theory a natural combination of the two widely used decompositions concerning d and f can consistently and uniquely determine the elastic and the plastic parts in the two decompositions as well as all their related kinematical quantities without recourse to any ad hoc assumption concerning a special form of the elastic part f e in the decomposition f f e f p or a related relaxed intermediate configuration as an application the proposed general model is applied to derive a self consistent eulerian rate type model for void growth and nucleation in metals experiencing finite elastic plastic deformation by incorporating a modified gurson s yield function and an associated flow rule etc two issues involved in previous relevant literature are detected and raised for consideration as a test problem the finite simple shear response of the just mentioned model is studied by means of numerical integration

stochastic evolution of microcracks in continua abstract a multifield description of continua endowed by diffused microcracks recently proposed by the authors mariano and augusti math mech sol 3 1998 183 200 within a deterministic context is extended to cover some of the stochastic aspects of microcrack distribution and evolution the microcrack state of the continuum is described by a tensor field defined as the second order approximation of the microcrack density function m p n t that represents the distribution of the number of microcracks in each direction n in a neighborhood mesodomain of the point p m p n t is considered as a submartingale process in each point p leading to a stochastic field over the continuum body generalized measures of internal actions represent the interactions voids voids and void matrix they perform work in the variation of relevant conjugated fields invariance requirements on the overall power allows to deduce both the usual balance of forces and the balance of generalized internal actions obtaining a model different from the classical internal variable models not only conceptually but also formally the introduction of a damage entropy flux whose divergence is the production of configurational entropy allows to include damage criteria within the context of clausius duhem inequality the basic features of an appropriate finite element discretization are formalized in the case of linear elastic brittle materials the stochastic distribution of microcracks is considered through the stochastic nature of the elements of the stiffness matrix among other assets the multifield approach overcomes the mesh dependence of the numerical results obtained on the basis of the more used internal variable schemes

a 3 field formulation for strongly transversely isotropic compressible finite hyperelasticity abstract a hu washizu type 3 field virtual work principle for strongly transversely isotropic compressible finite hyperelasticity is developed implemented and verified the independent variables are the stretch its energy conjugate uniaxial tension and the displacement the formulation is based on a new decomposition of the deformation gradient into a simply stretchless part and a lateral contraction free uniaxial extension in the fully constrained limit the formulation provides the constraint manifold setting of hyperelasticity with the simple internal kinematic constraint of inextensibility the finite element formulation is implemented in an hp adaptive code providing the proper flexible environment for finite elements with variable order and mixed interpolation a study using a semi inverse analytical solution corroborates the convergence characteristics for h refinements and p enrichments of the 3 field implementation the case of isostatic loading of a circular cylinder reinforced with a single family of fibres and the pressurisation of a saccular aneurysm like configuration with circumferential fibre reinforcement are used for verification and illustration a posteriori residual discretisation error estimation is used for making mesh refinements in the latter case typical applications are found in soft tissue biomechanics

the variational collocation method we propose the variational collocation method for the numerical solution of partial differential equations the conceptual basis is the establishment of a direct connection between the galerkin method and the classical collocation methods with the perspective of achieving the accuracy of the former with a computational cost of one point evaluation per degree of freedom as in the latter variational collocation requires a discrete space constructed by smooth and pointwise non negative basis functions which makes the approach immediately applicable to isogeometric analysis and some meshfree methods in this paper we concentrate on isogeometric analysis and demonstrate that there exists a set of points such that collocation of the strong form at these points produces the galerkin solution exactly we provide an estimate of these points and show that applying isogeometric collocation at the estimated points completely solves the well known odd even discrepancy in the order of spatial convergence we demonstrate the potential of variational collocation with examples of linear and non linear elasticity as well as kirchhoff plates

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic

p fem for finite deformation powder compaction the simulation of powder compaction problems die compaction and cold isostatic pressing is considered herein by an implicit high order p version finite element method in this class of problems use is made of a finite strain viscoplasticity model with evolution equations for internal variables developed for the highly compressible behavior in powder compaction processes the classical approach of implicit finite elements applies the combination of backward euler integration scheme and the multilevel newton algorithm to solve the system of differential algebraic equations resulting from the space discretized weak formulation by means of p version finite elements this approach requires on gauss point level a robust stress algorithm the challenging investigations are the incorporation of the applied highly non linear viscoplasticity model into a p version finite element formulation using follower load applications several axisymmetric numerical examples show the feasibility and good performance of this p version approach

constitutive equations for micropolar hyper elastic materials in this paper the concept of hyper elasticity in the micropolar continuum theory is investigated the restrictions on the fourth order elasticity tensors are investigated using the representation theorems a general form of constitutive equations for micropolar hyper elastic isotropic materials is presented as some special cases generalizations of the neo hookean and mooney rivlin type materials to the micropolar continuum theory are presented the generalized constitutive equations reduce to those of the microplar linear elasticity theory when the deformations are infinitesimal also updated lagrangian finite element formulations for the micropolar hyper elastic materials are presented considering two planar examples it is shown that an increase in the micropolar parameter results in the reduction of the deformation of the bodies also it is shown that for a specimen with very small dimensions e g in the micron level the micropolar effects are more sensible furthermore it is shown that the influence of the micropolar parameters is dependent not only on the size of the body but also to its geometry and loading conditions for the problems in which the deformation is very close to a homogeneous state the micropolar effects are negligible