on the physical significance of higher order kinematic and static variables in a three dimensional shell formulation abstract in recent years considerable attention has been given to the development of higher order plate and shell models these models are able to approximately represent three dimensional effects while pertaining the efficiency of a two dimensional formulation due to pre integration of the structural stiffness matrix across the thickness especially the possibility to use unmodified complete three dimensional material laws within shell analysis has been a major motivation for the development of such models while the theoretical and numerical formulation of so called 7 parameter shell models including a thickness stretch of the shell has been discussed in numerous papers no thorough investigation of the physical significance of the additional kinematic and static variables coming along with the extension into three dimensions is known to the authors however realization of the mechanical meaning of these quantities is decisive for both a proper modeling of shell structures e g concerning loading and kinematic boundary conditions and a correct interpretation of the results in the present paper the significance of kinematic and static variables appearing in a 7 parameter model proposed by buchter and ramm 1992a are discussed it is shown how these quantities refine the model behavior and how they can be related to the classical variables such as curvatures and stress resultants furthermore the special role of the material law within such a formulation is addressed it is pointed out that certain requirements must hold for the variation of kinematic and static variables across the thickness to ensure correct results in this context it is found that the considered 7 parameter model can be regarded as optimal with respect to the number of degrees of freedom involved

a 4 node finite shell element for the implementation of general hyperelastic 3d elasticity at finite strains abstract in this paper a finite shell element for large deformations is presented based on extensible director kinematics the essential feature is an interface to arbitrary three dimensional material laws the non linear lagrangian formulation is based on the three field variational principle parametrized with the displacement vector enhanced green lagrangian strain tensor and second piola kirchhoff stress tensor the developed quadrilateral shell element is characterized by a course mesh accuracy and distortion insensitivity compared with bilinear displacement approaches furthermore plane stress response is approximately recovered in the asymptotic case of vanishing thickness a number of example problems investigating large deformation as well as finite strain applications are presented compressible and incompressible hyperelastic materials of the st venant kirchhoff neo hookean and mooney rivlin type are particularly used

mesh adaptive direct search algorithms for constrained optimization this paper addresses the problem of minimization of a nonsmooth function under general nonsmooth constraints when no derivatives of the objective or constraint functions are available we introduce the mesh adaptive direct search mads class of algorithms which extends the generalized pattern search gps class by allowing local exploration called polling in an asymptotically dense set of directions in the space of optimization variables this means that under certain hypotheses including a weak constraint qualification due to rockafellar mads can treat constraints by the extreme barrier approach of setting the objective to infinity for infeasible points and treating the problem as unconstrained the main gps convergence result is to identify limit points hat x where the clarke generalized derivatives are nonnegative in a finite set of directions called refining directions although in the unconstrained case nonnegative combinations of these directions span the whole space the fact that there can only be finitely many gps refining directions limits rigorous justification of the barrier approach to finitely many linear constraints for gps the main result of this paper is that the general mads framework is flexible enough to allow the generation of an asymptotically dense set of refining directions along which the clarke derivatives are nonnegative we propose an instance of mads for which the refining directions are dense in the hypertangent cone at hat x with probability 1 whenever the iterates associated with the refining directions converge to a single hat x the instance of mads is compared to versions of gps on some test problems we also illustrate the limitation of our results with examples

classic and inverse compositional gauss newton in global dic today effective implementations of digital image correlation dic are based on iterative algorithms with constant linear operators a relevant idea of the classic finite element or more generally global dic fe dic solver consists in replacing the gradient of the deformed state image with that of the reference image so as to obtain a constant operator different arguments small strains small deformations equality of the two gradients close to the solution have been given in the literature to justify this approximation but none of them are fully accurate indeed the convergence of the optimization algorithm has to be investigated from its ability to produce descent directions through such a study this paper attempts to explain why this approximation works and what is its domain of validity then an inverse compositional gauss newton icgn implementation of fe dic is proposed as a cost effective and mathematically sound alternative to this approximation

a domain decomposition approach for full field measurements based identification of local elastic parameters abstract we propose a domain decomposition formalism specifically designed for the identification of local elastic parameters based on full field measurements this technique is made possible by a multi scale implementation of the constitutive compatibility method contrary to classical approaches the constitutive compatibility method resolves first some eigenmodes of the stress field over the structure rather than directly trying to recover the material properties a two steps micro macro reconstruction of the stress field is performed a dirichlet identification problem is solved first over every subdomain the macroscopic equilibrium is then ensured between the subdomains in a second step we apply the method to large linear elastic 2d identification problems to efficiently produce estimates of the material properties at a much lower computational cost than classical approaches

stochastic identification of elastic constants for anisotropic materials this paper presents an energy based characterization technique that stochastically identifies the elastic constants of anisotropic materials by modeling the measurement noise and removing its effect unlike conventional deterministic techniques which deterministically identify the elastic constants directly from noisy measurements the technique recursively estimates the elastic constants at every acquisition of measurements using kalman filter owing to the non linear expression of the measurement model a kalman gain has been newly derived and achieves optimal estimation since the variances in addition to the means are computed the proposed technique can not only identify the elastic constants but also describe their certainty as an additional advantage the validity of the proposed technique and its superiority to the conventional technique were first demonstrated via parametric studies of low dimensional problems the proposed technique was then successfully applied to the identification of elastic constants of an anisotropic material copyright 2009 john wiley sons ltd

geometrically non linear enhanced strain mixed methods and the method of incompatible modes a class of assumed strain mixed finite element methods for fully non linear problems in solid mechanics is presented which when restricted to geometrically linear problems encompasses the classical method of incompatible modes as a particular case the method relies crucially on a local multiplicative decomposition of the deformation gradient into a conforming and an enhanced part formulated in the context of a three field variational formulation the resulting class of mixed methods provides a possible extension to the non linear regime of well known incompatible mode formulations in addition this class of methods includes non linear generalizations of recently proposed enhanced strain interpolations for axisymmetric problems which cannot be interpreted as incompatible modes elements the good performance of the proposed methodology is illustrated in a number of simulations including 2 d 3 d and axisymmetric finite deformation problems in elasticity and elastoplasticity remarkably these methods appear to be specially well suited for problems involving localization of the deformation as illustrated in several numerical examples

adjoint weighted equation for inverse problems of incompressible plane stress elasticity the adjoint weighted equation awe formulation is applied to incompressible isotropic plane stress elasticity for the solution of inverse problems for the strong form of the problem to have a solution a restrictive strain compatibility condition must hold in the awe variational formulation on the other hand a much milder sufficient condition is that uniaxial stress states be precluded making the method more robust to noisy data we prove that the formulation leads to a stable converging numerical method and show through computations that the method performs well

general framework for the identification of constitutive parameters from full field measurements in linear elasticity in this paper several approaches available in the literature for identifying the constitutive parameters of linear elastic materials from full field measurements are presented and their sensitivity to a white noise added to the data is compared the first investigated approach is the virtual fields method vfm it is shown that the uncertainty of the parameters identified with the vfm when a white noise is added to the data depends on the choice of a relevant set of virtual fields optimal virtual fields exist thus minimizing the uncertainty and providing the maximum likelihood solution the other approaches investigated in this paper are based on finite element model updating femu it is proved that femu approaches actually yield equations similar to the ones derived from the vfm but with nonoptimal sets of virtual fields therefore the femu approaches do not provide the maximum likelihood solution however the uncertainty of femu approaches varies dramatically with the cost function to minimize on one hand the femu approach based on the displacement gap minimization yields equations which are very close to the ones of the vfm approach and therefore its uncertainty is almost the same as the vfm one on the other hand it is shown that other approaches based on the constitutive equation gap minimization or the equilibrium gap minimization provide biased solutions for all the approaches very fast algorithms converging in only two iterations have been devised they are finally applied to real experimental data obtained on an orthotropic composite material results confirm the success of two methods the vfm approach which provides the maximum likelihood solution and the femu approach based on the displacement gap minimization

gradient enhanced damage for quasi brittle materials summary conventional continuum damage descriptions of material degeneration suffer from loss of well posedness beyond a certain level of accumulated damage as a consequence numerical solutions are obtained which are unacceptable from a physical point of view the introduction of higher order deformation gradients in the constitutive model is demonstrated to be an adequate remedy to this deficiency of standard damage models a consistent numerical solution procedure of the governing partial differential equations is presented which is shown to be capable of properly simulating localization phenomena

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

geometrical ray theory rays and traveltimes in seismic systems second order approximations of the traveltimes in this paper i establish general laws that represent global properties for the reflected and transmitted rays in seismic systems the method used for establishing these laws is based on hamilton s original ideas the laws do not depend on the specific seismic system being used the reflected and transmitted rays as well as their traveltimes are described by the same four 2 x 2 matrices these matrices are interrelated by three equations that provide the basis for the general laws and also provide relations between reflected and transmitted rays that eventually will be exploitable for downward continuation and migration the traveltimes of reflected and transmitted rays are summed up in single functions giving the traveltime for any pair of source and receiver positions in a seismic system and corresponding to hamilton s point characteristics in optics the traveltimes of the gathers commonly used in seismic exploration are special cases of the general point characteristics the consequences from the derived laws are directly available as a priori knowledge before any actual ray tracing and computation of traveltimes are done in any particular seismic system the laws are also available as a general foundation for future treatments of 1 downward continuation 2 amplitudes of reflected and transmitted events 3 true amplitude migration 4 examples of nonuniqueness of seismic interpretations 5 focus phenomena and 6 higher order approximations

quadratic wavefront and traveltime approximations in inhomogeneous layered media with curved interfaces a quadratic approximation for the square of the traveltime from a source region to a receiver region is given for a three dimensional 3 d medium consisting of inhomogeneous layers with curved interfaces the square of the traveltime being a function of source and receiver coordinates is developed in a taylor series around a reference source and receiver point the relationships of the traveltime parameters to the ray parameters and the wavefront curvature matrices are shown using midpoint half offset coordinates gives a simplified traveltime function compared to using source receiver coordinates only in the case that the reference source point and the reference receiver point coincide zero offset approximation for a medium consisting of homogeneous layers with plane dipping interfaces the traveltime approximation is further simplified the derived traveltime approximation is shown to be exact for a reflection from a plane dipping interface in a homogeneous medium explicit expressions for the tr

normal moveout revisited inhomogeneous media and curved interfaces the equation of normal moveout t2 t02 x2 va2 is valid for a reflection from the base of a single homogeneous and isotropic bed but is only an approximation in the real world of multilayered inhomogeneous media and curved interfaces using the theory of geometrical optics we can find another second order equation which represents hyperbolas that are also symmetrical about the time axis however the centers of these hyperbolas do not coincide with the center of coordinates but are shifted along the time axis the equation describing this second type of hyperbola is t tp t0 2 tp2 x2 v12 where tp is the time of focusing depth and v1 the velocity of the input medium this equation is not only more accurate than the usual normal moveout but its use is more economical on a vector computer because the traditional dynamic correction is a static correction in the tp analysis this procedure makes it possible to compute velocities for all the samples of all the stacked traces and produces a velocity secti

large strain deformations of elastic shells constitutive modelling and finite element analysis this paper is concerned with the constitutive modelling of large elastic strains with applications to shells the shell theory itself is characterized by seven degrees of freedom and enjoys the following features 1 it circumvents the use of a rotation tensor preserving the euclidean structure of the configuration space and 2 it is capable to the application of a three dimensional constitutive law in addition an elastic finite strain constitutive law is developed which 1 covers the whole range of deformation experimentally given by treloar and 2 is formulated in terms of invariants avoiding the use of principal stretches the second feature means that the model is generalizable to anisotropic material behaviour and that it allows for straightforward computations of the tangent operator making the whole theory well suited for large strain computations in addition to the four node and nine node displacement based finite elements an enhanced strain finite element is examined its efficiency is compared with that of displacement based element formulations different examples of large strain shell deformations are presented it is shown that the application of the enhanced strain concept in the nonlinear constitutive range limits severely the stability of computation

identification of orthotropic elastic constants using the eigenfunction virtual fields method abstract the virtual fields method vfm pierron and grediac 2012 an inverse method based on the principle of virtual work pvw is being increasingly used to estimate mechanical properties of materials from full field deformations obtained from techniques such as digital image correlation moire and speckle interferometry and grid methods by making specific choices for virtual fields vfs in pvw one obtains a system of algebraic equations which is then solved for the unknown material constants recently a new variant of vfm known as the eigenfunction virtual fields method evfm has been proposed subramanian 2013 in evfm principal components of the measured i e true strain fields are used to systematically generate vfs we extend evfm to orthotropic elastic materials in this work and estimate the relevant material parameters from full field strain data generated from a finite element model of an unnotched iosipescu test varying levels of gaussian white noise are added to the synthetic strain data to evaluate the sensitivity of evfm to input noise it is observed that for low to moderate noise the material properties estimated by the proposed method are relatively insensitive to noise however when noise levels are high the proposed method yields large variance in some of the computed properties when compared to the state of the art optimized piecewise continuous vfm toussaint et al 2006 pierron and grediac 2012 some of the large variance in properties estimated from noisy data using evfm is traced to the sensitivity of the third dominant eigenfunction and modifications to the proposed method to address this issue are suggested

signal and traveltime parameter estimation using singular value decomposition abstractsignal detection and traveltime parameter estimation can be performed by computing a coherence function in a data window centered around a traveltime function defined by its parameters we used singular value decomposition of the data matrix not eigendecomposition of a covariance matrix to review the most commonly used coherence measures this resulted in a new reduced semblance coefficient defined from the first eigenimage assuming that the signal amplitude was the same on all data channels as in classical semblance in a second signal model the time signal was constant on each channel but the amplitude changed then the semblance coefficient is the square of the first singular value divided by the energy of the data two normalized crosscorrelation coefficients derived from the first eigenimage can also be used as a coherence measure the normalized crosscorrelation of the spatial singular vector with a vector with all elements equal to one and the normalized crosscorrelation of the tem

a fully integrated noise robust strategy for the identification of constitutive laws from digital images constitutive parameter identification has been greatly improved by the achievement of full field measurements in this context noise sensitivity has been shown to be of great importance it is crucial to incorporate noise sensitivity minimization in the design of robust identification procedures in this paper we investigate noise sensitivity reduction techniques for constitutive parameter identification based on finite element model updating after examining the existing strategies we propose a single step algorithm based on a mixed optical mechanical cost function the key point of this novel procedure is that no boundary conditions are needed a first example on a real case illustrates the advantages of the proposed methodology in terms of noise sensitivity a second example shows its capabilities to identify a non linear consitutive law copyright 2010 john wiley sons ltd

traveltime and wavefront curvature calculations in three dimensional inhomogeneous layered media with curved interfaces the seismic rays and wavefront curvatures are determined by solving a system of nonlinear ordinary differential equations for media with constant velocity and for media with constant velocity gradient simplified solutions exist in a general inhomogeneous medium these equations must be solved by numerical approximations the integration of the ray tracing and wavefront curvature equations is then performed by a modified divided difference form of the adams pece predict evaluate correct evaluate formulas and local extrapolation the interfaces between the layers are represented by bicubic splines the changes in ray direction and wavefront curvature at the interfaces are computed using standard formulas for three dimensional media two quadratic traveltime approximations have been proposed both are based on a taylor series expansion with reference to a ray from a reference source point to a reference receiver point the first approximation corresponds to expanding the square of the traveltime in a ta

determination of the size of the representative volume element for random composites statistical and numerical approach the representative volume element rve plays a central role in the mechanics and physics of random heterogeneous materials with a view to predicting their effective properties a quantitative definition of its size is proposed in this work a rve size can be associated with a given precision of the estimation of the wanted overall property and the number of realizations of a given volume v of microstructure that one is able to consider it is shown to depend on the investigated morphological or physical property the contrast in the properties of the constituents and their volume fractions the methodology is applied to a specific random microstructure namely a two phase three dimensional voronoi mosaic finite element simulations of volumes of different sizes are performed in the case of linear elasticity and thermal conductivity the volumes are subjected to homogeneous strain stress or periodic boundary conditions the effective properties can be determined for large volumes and a small number of realizations conversely smaller volumes can be used providing that a sufficient number of realizations are considered a bias in the estimation of the effective properties is observed for too small volumes for all types of boundary conditions the variance of computed apparent properties for each volume size is used to define the precision of the estimation the key notion of integral range is introduced to relate this error estimation and the definition of the rve size for given wanted precision and number of realizations one is able to provide a minimal volume size for the computation of effective properties the results can also be used to predict the minimal number of realizations that must be considered for a given volume size in order to estimate the effective property for a given precision the rve sizes found for elastic and thermal properties but also for a geometrical property like volume fraction are compared

special virtual fields for the direct determination of material parameters with the virtual fields method 2 application to in plane properties this paper deals with the direct identification of mechanical parameters that govern the in plane constitutive law of orthotropic materials those parameters are extracted from heterogeneous strain fields that occur in a short beam specimen tested in a iosipescu fixture the procedure used is the virtual fields method with special virtual fields the case of linear elasticity is first addressed it is shown that the parameters are directly extracted with this method no iterative calculations are required the stability is also discussed in different cases a non linear shear response is then considered the parameter that governs this non linearity is also directly identified with the special virtual fields

a new stable inverse method for identification of the elastic constants of a three dimensional generally anisotropic solid abstract this article presents a new approach for inverse identification of all elastic constants of a 3d generally anisotropic solid with arbitrary geometry via measured strain data to eradicate the nonlinear inequality constraints posed on the elastic constants the problem is first transformed to an unconstrained one by the cholesky factorization theorem the cost function is defined by the tikhonov regularization method and the inverse problem is solved using the damped gauss newton technique where a meshless method is employed for the direct and sensitivity analyses to demonstrate the effectiveness of the proposed approach several examples are presented in the end where all experimental data are numerically simulated analyses of these examples show that all twenty one elastic constants of an example material can be correctly identified even when measurement errors are relatively large and initial guesses are far from exact values

properties of the zeroth first and higher order approximations of attributes of elastic waves in weakly anisotropic media use of the perturbation theory in the study of attributes of elastic waves propagating in weakly anisotropic media leads to approximate but transparent and simple formulas which have many applications in forward and inverse wave modeling we present and study such formulas we show that all studied attributes depend on elements of a matrix linearly dependent on parameters of a medium we study this dependence with the goal to understand which parameters of the medium and in which combinations affect individual wave attributes alternative auxiliar vector bases in which the matrix can be specified are proposed and studied the vector bases offer alternative specifications of polarization vectors of qs waves one of the important observations is that the higher order n 2 perturbation formulas for qs waves are obtained separately for qs1 and qs2 waves we also study effects of the use of the perturbation theory on the accuracy of the determination of the acoustical axes in weakly anisotropic media we show that longitudinal directions in the first order approximation are identical with actual ones in singular directions however the first order formulas provide directions which may deviate from the exact ones or they may even indicate false singular directions again the above mentioned matrix depending linearly on the parameters of the medium plays a central role in this study

the two faces of nmo abstract in many present day applications in seismic processing the assumption of a homogeneous model leads to simple yet powerful approximations which also work well when heterogeneity is not negligible while the classical normal moveout nmo hyperbola assumes an effective constant velocity medium de bazelaire based on optical projections introduced an alternative way to account for heterogeneity by shifting the reference time rather than the velocity although modern multiparameter nmo descriptions usually are based on the same set of parameters we show through reparameterization that they can be divided into the same two types of approximations one assuming an effective medium the other describing the optical analogue in a medium of constant near surface velocity we provide new insights into the auxiliary medium concept and introduce a generalized osculating equation which allows for the forward and inverse transformation between the effective and the optical domain thereby providing a uni

shape features and finite element model updating from full field strain data finite element model updating is an inverse problem based on measured structural outputs in this case maximum principal strain measured using digital image correlation full field responses in the form of strain maps contain valuable information for model updating but within large volumes of highly redundant data in this paper shape descriptors based on zernike polynomials having the properties of orthogonality and rotational invariance are shown to be powerful decomposition kernels for defining the shape or map of the strain distribution a square plate with a circular hole subject to a uniaxial tensile load is considered and effective shape features are constructed using a set of modified zernike polynomials the modification includes the application of a decaying weighting function to the zernike polynomials so that high strain magnitudes around the hole are well represented the gram schmidt process is then used to ensure orthogonality for the obtained decomposition kernels over the domain of the specimen i e excluding the hole results show that only a very small number of zernike moment descriptors are necessary and sufficient to represent the full field data the onset of yielding may be quantified using the descriptors furthermore model updating of nonlinear elasto plastic material properties is carried out using the zernike moment descriptors derived from full field strain measurements

a continuum based shell theory for non linear applications the paper introduces a non linear shell theory which provides a complete three dimensional state of stress since the theory is derived from simple three dimensional continuum mechanics it is very easy to understand as an example the theory is applied to quadrilateral shell elements which provide only displacement degrees of freedom located at nodes on the outer surfaces and one degree of freedom at the middle surface it is proposed to eliminate this degree of freedom on element level so that the elements have the same layout as the equivalent brick elements but have a better behaviour in bending have stress resultants and are cheaper with respect to computational effort the advantages with respect to implementation in a finite element program as well as in special applications are obvious however well known conditioning problems in thin shell applications must be expected therefore emphasis is put on this issue in the example problems it is shown that the elements can give acceptable answers in engineering applications and offer a potential for material non linear applications which will be considered in a forthcoming paper

a domain coupling method for finite element digital image correlation with mechanical regularization application to multiscale measurements and parallel computing a general method is proposed to couple two sub regions analysed with finite element digital image correlation fe dic even when using a mechanical regularization r dic a lagrange multiplier is introduced to stitch both displacements fields in order to recover continuity over the full region of interest another interface unknown is introduced to ensure additionally the equilibrium of the mechanical models used for regularization as a first application the method is used to perform a single measurement from images at two different resolutions secondly the method is also extended to parallel computing in r dic the problem is formulated at the interface and solved with gmres a dedicated preconditioner is proposed to significantly accelerate convergence the resulting method is a good candidate for the analysis of large data sets copyright

a second order homogenization procedure for multi scale analysis based on micropolar kinematics the paper presents a higher order homogenization scheme based on non linear micropolar kinematics representing the macroscopic variation within a representative volume element rve of the material on the microstructural level the micro macro kinematical coupling is introduced as a second order taylor series expansion of the macro displacement field and the microstructural displacement variation is gathered in a fluctuation term this approach relates strongly to second gradient continuum formulations presented by e g kouznetsova et al int j numer meth engng 2002 54 1235 1260 thus establishing a link between second gradient and micropolar theories the major difference of the present approach as compared to second gradient formulations is that an additional constraint is placed on the higher order deformation gradient in terms of the micropolar stretch the driving vehicle for the derivation of the homogenized macroscopic stress measures is the hill mandel condition postulating the equivalence of microscopic and macroscopic homogenized virtual work thereby the resulting homogenization procedure yields not only a stress tensor conjugated to the micropolar stretch tensor but also the couple stress tensor conjugated to the micropolar curvature tensor the paper is concluded by a couple of numerical examples demonstrating the size effects imposed by the homogenization of stresses based on the micropolar kinematics copyright 2006 john wiley sons ltd

shear band capturing using a multiscale extended digital image correlation technique abstract finite elements have been used recently to solve the optical flow conservation principle invoked to determine displacement fields by digital image correlation inspired by these recent advances and by the computational effort that has been accomplished during the past 10 years for the simulation of discontinuities by the extended finite element method xfem an extended correlation technique is introduced for capturing shear band like discontinuities from images of real mechanical tests because of the specific information at hand i e gray level images this extended finite element approach is included in a non linear multi grid solver the performances of the proposed approach are discussed on two examples namely the analysis of a bolted assembly and the formation of piobert luders bands

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

amplitude fresnel zone and nmo velocity for pp and ss normal incidence reflections inspired by recent ray theoretical developments the theory of normal incidence rays is generalized to accommodate p and s waves in layered isotropic and anisotropic media the calculation of the three main factors contributing to the two way amplitude i e geometric spreading phase shift from caustics and accumulated reflection transmission coefficients is formulated as a recursive process in the upward direction of the normal incidence rays this step by step approach makes it possible to implement zero offset amplitude modeling as an efficient one way wavefront construction process for the purpose of upward dynamic ray tracing the one way eigensolution matrix is introduced having as minors the paraxial ray tracing matrices for the wavefronts of two hypothetical waves referred to by hubral as the normal incidence point nip wave and the normal wave dynamic ray tracing expressed in terms of the one way eigensolution matrix has two advantages the formulas for geometric spreading phase shif

the constitutive compatibility method for identification of material parameters based on full field measurements abstract we revisit here the concept of the constitutive relation error for the identification of elastic material parameters based on image correlation an additional concept so called constitutive compatibility of stress is introduced defining a subspace of the classical space of statically admissible stresses the key idea is to define stresses as compatible with the observed deformation field through the chosen class of constitutive equation this makes possible the uncoupling of the identification of stress from the identification of the material parameters as a result the global cost of the identification is strongly reduced this uncoupling also leads to parametrized solutions in cases where the solution is non unique as demonstrated on 2d numerical examples

anisotropic linear elastic parameter estimation using error in the constitutive equation functional a modified error in the constitutive equation based approach for identification of heterogeneous and linear anisotropic elastic parameters involving static measurements is proposed and explored following an alternating minimization procedure associated with the underlying optimization problem the new strategy results in an explicit material parameter update formula for general anisotropic material this immediately allows us to derive the necessary constraints on measured data and thus restrictions on physical experimentation to achieve the desired reconstruction we consider a few common materials to derive such conditions then we exploit the invariant relationships of the anisotropic constitutive tensor to propose an identification procedure for space dependent material orientations finally we assess the numerical efficacy of the developed tools against a few parameter identification problems of engineering interest

on the existence of a global minimum in inverse parameters identification by self optimizing inverse analysis method abstract in this paper a mathematical proof of the existence of a global minimum of self optim self optimizing inverse analysis method cost functional is presented based upon weak solution theory of partial differential equations the self optim provides single global minimum rather than having multiple global minima corresponding to unrealistic solutions of the inverse problem furthermore discrete approximation of the inverse problem and computational methods for the cost functional are proposed and the proof is numerically verified this paper provides a rigorous mathematical foundation for applications of the self optim method to various inverse problems in mechanics

constitutive error based parameter estimation technique for plate structures using free vibration signatures abstract in this paper a variant of constitutive equation error based material parameter estimation procedure for linear elastic plates is developed from partially measured free vibration sig natures it has been reported in many research articles that the mode shape curvatures are much more sensitive compared to mode shape themselves to localize inhomogeneity complying with this idea an identification procedure is framed as an optimization problem where the proposed cost function measures the error in constitutive relation due to incompatible curvature strain and moment stress fields unlike standard constitutive equation error based procedure wherein a solution of a couple system is unavoidable in each iteration we generate these incompatible fields via two linear solves a simple yet effective penalty based approach is followed to incorporate measured data the penalization parameter not only helps in incorporating corrupted measurement data weakly but also acts as a regularizer against the ill posedness of the inverse problem explicit linear update formulas are then developed for anisotropic linear elastic material numerical examples are provided to show the applicability of the proposed technique finally an experimental validation is also provided

first order ray tracing for qp waves in inhomogeneous weakly anisotropic media abstractwe propose approximate ray tracing equations for qp waves propagating in smooth inhomogeneous weakly anisotropic media for their derivation we use perturbation theory in which deviations of anisotropy from isotropy are considered to be the first order quantities the proposed ray tracing equations and corresponding traveltimes are of the first order accuracy of the traveltimes can be increased by calculating a secondorder correction along first order rays the first order ray tracing equations for qp waves propagating in a general weakly anisotropic medium depend on only 15 weak anisotropy parameters generalization of thomsen s parameters the equations are thus considerably simpler than the exact ray tracing equations for higher symmetry anisotropic media the equations differ only slightly from equations for isotropic media they can thus substitute for the traditional isotropic ray tracers used in seismic processing for vanishing anisotropy the first order ray tracing equations reduce

computational homogenization based on a weak format of micro periodicity for rve problems computational homogenization with a priori assumed scale separation is considered whereby the macroscale stress is obtained via averaging on representative volume elements rve s a novel variational formulation of the rve problem based on the assumption of weak micro periodicity of the displacement fluctuation field is proposed notably independent fe discretization of boundary tractions lagrange multipliers allows for a parameterized transition between the conventional strong periodicity and neumann boundary conditions in this paper the standard situation of macroscale strain control is considered numerical results demonstrate the convergence properties with respect to 1 the approximation of displacement and tractions and 2 the rve size for random realizations of the microstructure

on the direct determination of invariant parameters governing anisotropic plate bending problems abstract many papers deal with the determination of flexural rigidities of anisotropic plates from only one specimen subjected to one or several loading conditions in such a problem one of the main questions is the best identification of the unknown parameters according to the type of material the shape of the specimen and the loading conditions it seems that no general rule giving the best configuration leading to an accurate determination has been established one of the main reasons is the anisotropic nature of the tested plates which leads the mechanical configuration to be dependent on the material itself this problem would be avoided by the direct measurement of invariant parameters from which the rigidities are easily deduced these invariant parameters reveal the intrinsic elastic behavior of the tested plate contrary to the rigidities usually considered which are directional quantities this paper describes a procedure for the identification of invariant parameters governing the bending of anisotropic plates it is shown that the whole displacement field processing of the surface strain fields obtained in two particular loading cases directly provides those parameters a numerical simulation shows the stability of the present method

on the inverse problem of identifying lame coefficients in linear elasticity an output least squares type functional is employed to identify the lame parameters in linear elasticity to be able to identify even the discontinuous lame parameters the regularization is performed by the bv seminorm finite element discretization is used and convergence analysis is given numerical examples are given to show the feasibility of the approach

a stress resultant shell theory based on multiscale homogenization in this paper we propose a multiscale method based on computational homogenization for simulating the mechanical response of a thin walled porous structure due to the inhomogeneous nature of the porous material in the thickness direction the length scale of the deformation field variations in the thickness direction is in the same order of magnitude as in the microstructure to resolve this issue a higher order stress resultant shell formulation with linear variation of the thickness stretch based on multiscale homogenization is considered the microscale is accounted for by calculating the micro fluctuations from a boundary value problem over the domain of a 3d representative volume element rve the breadth and width of the rve are determined by the representative microstructure of the porous material whereas in the thickness direction of the shell the rve completely resolves the microscopic variation as a result the macroscopic stress resultants are obtained as volume averages through the entire thickness of the shell the nested solution scheme is quite computationally demanding and it should only be applied where it is needed the paper is concluded by a couple of numerical examples that illustrate the method and support the arguments put forward in the paper comparison is made to a 2d plane strain reference solution with complete resolution of the microstructure over the domain also a 3d case is considered showing the significance of resolving the microscopic fluctuations

computational homogenization for heterogeneous thin sheets in this paper a computational homogenization technique for thin structured sheets is proposed based on the computational homogenization concepts for first and second order continua the actual three dimensional 3d heterogeneous sheet is represented by a homogenized shell continuum for which the constitutive response is obtained from the nested analysis of a microstructural representative volume element rve incorporating the full thickness of the sheet and an in plane representative cell of the macroscopic structure at an in plane integration point of the macroscopic shell the generalized strains i e the membrane deformation and the curvature are used to formulate the boundary conditions for the microscale rve problem at the rve scale all microstructural constituents are modeled as an ordinary 3d continuum described by the standard equilibrium and the constitutive equations upon proper averaging of the rve response the macroscopic generalized stress and the moment resultants are obtained in this way an in plane homogenization is directly combined with a through thickness stress integration from a macroscopic point of view a numerical generalized stress strain constitutive response at every macroscopic in plane integration point is obtained additionally the simultaneously resolved microscale rve local deformation and stress fields provide valuable information for assessing the reliability of a particular microstructural design

an eikonal based formulation for traveltime perturbation with respect to the source location traveltime calculations amount to solving the nonlinear eikonal equation for a given source location the relationship between the eikonal solution and its perturbations is analyzed with respect to the source location and a partial differential equation is developed that relates the traveltime field for one source location to that for a nearby source this linear first order equation in one form depends on lateral changes in velocity and in another form is independent of the velocity field and relies on second order derivatives of the original traveltime field for stable finite difference calculations this requires the velocity field to be smooth in a sense similar to ray tracing requirements our formulation for traveltime perturbation has several potential applications such that as traveltime calculation by source location perturbation velocity independent interpolation including datuming and velocity estimation additionally higher order expansions provide parameters necessary for gaussian beam computations

on the elastic characterization of composite plates with vibration data this paper takes a global look at a method of obtaining the elastic constants of the material of a rectangular plate sample by explicitly inputting the geometrical shapes and frequencies of a sufficient number of experimental vibration modes into a computational procedure that essentially compares experimental frequencies with analytical predictions there are many aspects of this method that have important bearing on accuracy ease of use and computational economy the issues that arise include vibration representation functions selection of experimental mode shapes and frequencies for analysis goodness of experimental data frequency sensitivity of the elastic constants influence of diagonal modes where relevant plate thickness plate aspect ratio material orthotropy ratio and orientation of reinforcing fibers in laminate composites whereas some of these factors have been discussed separately in other works this paper attempts bring the method completely into focus in one place with an examination of all the factors believed to influence the efficiency of the method

multi scale constitutive modelling of heterogeneous materials with a gradient enhanced computational homogenization scheme a gradient enhanced computational homogenization procedure that allows for the modelling of microstructural size effects is proposed within a general non linear framework in this approach the macroscopic deformation gradient tensor and its gradient are imposed on a microstructural representative volume element rve this enables us to incorporate the microstructural size and to account for non uniform macroscopic deformation fields within the microstructural cell every microstructural constituent is modelled as a classical continuum and the rve problem is formulated in terms of standard equilibrium and boundary conditions from the solution of the microstructural boundary value problem the macroscopic stress tensor and the higher order stress tensor are derived based on an extension of the hill mandel condition this automatically delivers the microstructurally based constitutive response of the higher order macro continuum and deals with the microstructural size in a natural way several examples illustrate the approach particularly the microstructural size effects

optimal solid shells for non linear analyses of multilayer composites ii dynamics abstract we are presenting a simple low order solid shell element formulation having only displacement degrees of freedom dofs i e without rotational dofs that has an optimal number of parameters to pass the patch tests and thus allows for efficient and accurate analyses of large deformable multilayer shell structures using elements at extremely high aspect ratio with the dynamics referred to a fixed inertial frame the elements can be used to analyze multilayer shell structures undergoing large overall motion the formulation of this element is based on the mixed hu washizu variational principle leading to a novel enhancing strain tensor enhanced assumed strain eas method that renders the computation particularly efficient with improved in plane and out of plane bending behavior poisson thickness locking especially in refined analyses of composite structures involving a large number of high aspect ratio layers the energy momentum conserving algorithm in the context of current solid shell element is presented we discuss the eas formulation based on the displacement gradient and its complexity compared to formulation on the green lagrange strain shear locking and curvature thickness locking are treated using the assumed natural strain ans method the element has an optimal combination of the ans method and the minimal number of eas parameters required to pass the plate bending patch test numerical examples involving dynamic analyses with conservation of energy and momentum of multilayer shell structures having a large range of element aspect ratios are presented several implicit direct integration methods with without numerical dissipation are used and compared in terms of the accuracy stability and cost in multilayer shell structure finally we note that the topic in this paper is a fitting dedication to professor ekkehard ramm who has made important pioneering contributions in this research direction

regularized virtual fields method for mechanical properties identification of composite materials abstract this paper presents a new material parameters identification method based on the virtual fields method vfm where a regularization term coming from a micromechanical model has been added in this regularized virtual fields method rvfm the system of equations associated to the vfm is solved within a constrained optimization framework the method was applied to virtual scanning electron microscope in situ tests aiming at determining the local properties of uni directional composites loaded in the transverse direction and for which the strain fields were obtained through digital image correlation dic the method was tested for composites with various volume fractions and properties contrasts and under different boundary conditions the potentials and limitations of the algorithm in the presence of noisy images were investigated and compared to that of the vfm and the other finite element based methods the corresponding results indicate that the proposed rvfm is more efficient than the other identification techniques in terms of both accuracy and computational time an optimum size of region of interest was also determined by taking into account dic magnification requirements and based on the accuracy of the identified parameters this kind of study can be very useful for determining a priori the level of resolution in the imaging system required for a favorable accuracy of the identified parameters

extended digital image correlation with crack shape optimization the methodology of extended finite element method is applied to the measurement of displacements through digital image correlation an algorithm initially based on a finite element decomposition of displacement fields is extended to benefit from discontinuity and singular enrichments over a suited subset of elements this allows one to measure irregular displacements encountered say in cracked solids as demonstrated both in artificial examples and experimental case studies moreover an optimization strategy for the support of the discontinuity enables one to adjust the crack path configuration to reduce the residual mismatch and hence to be tailored automatically to a wavy or irregular crack path copyright 2007 john wiley sons ltd

two point paraxial traveltime formula for inhomogeneous isotropic and anisotropic media tests of accuracy on several simple models of isotropic and anisotropic media we have studied the accuracy of the two point paraxial traveltime formula designed for the approximate calculation of the traveltime between points s 0 and r 0 located in the vicinity of points s and r on a reference ray the reference ray may be situated in a 3d inhomogeneous isotropic or anisotropic medium with or without smooth curved interfaces the twopoint paraxial traveltime formula has the form of the taylor expansion of the two point traveltime with respect to spatial cartesian coordinates up to quadratic terms at points s and r on the reference ray the constant term and the coefficients of the linear and quadratic terms are determined from quantities obtained from ray tracing and linear dynamic ray tracing along the reference ray the use of linear dynamic ray tracing allows the evaluation of the quadratic terms in arbitrarily inhomogeneous media and as shown by examples it extends the region of accurate results around the reference ray between s and r and even outside this interval obtained with the linear terms only although the formula may be used for very general 3d models we concentrated on simple 2d models of smoothly inhomogeneous isotropic and anisotropic 8 and 20 anisotropy media only on tests in which we estimated twopoint traveltimes between a shifted source and a system of shifted receivers we found that the formula may yield more accurate results than the numerical solution of an eikonal based differential equation the tests also indicated that the accuracy of the formula depends primarily on the length and the curvature of the reference ray and only weakly depends on anisotropy the greater is the curvature of the reference ray the narrower its vicinity in which the formula yields accurate results

kirchhoff migration using eikonal based computation of traveltime source derivatives abstractthe computational efficiency of kirchhoff type migration can be enhanced by using accurate traveltime interpolation algorithms we addressed the problem of interpolating between a sparse source sampling by using the derivative of traveltime with respect to the source location we adopted a first order partial differential equation that originates from differentiating the eikonal equation to compute the traveltime source derivatives efficiently and conveniently unlike methods that rely on finite difference estimations the accuracy of the eikonal based derivative did not depend on input source sampling for smooth velocity models the first order traveltime source derivatives enabled a cubic hermite traveltime interpolation that took into consideration the curvatures of local wavefronts and can be straightforwardly incorporated into kirchhoff antialiasing schemes we provided an implementation of the proposed method to first arrival traveltimes by modifying the fast marching eikonal solver severa

direct identification of elastic constants of anisotropic plates by modal analysis experimental results abstract the determination of the six elastic stiffnesses of thin anisotropic plates from vibration tests is usually performed with numerical procedures based on the finite element or the rayleigh ritz method which both require assumptions concerning the studied mode shapes the present paper describes a method based on the measurement and the processing of natural frequencies as well as mode shapes of the vibrating tested plate as a result the unknown stiffnesses are determined directly without any iterative calculations the experimental aspects of the method are presently described and several results illustrate the relevance of the approach