Variational principles in dissipative electro‐magneto‐mechanics: A framework for the macro‐modeling of functional materials

This paper presents a general framework for the macroscopic, continuum‐based formulation and numerical implementation of dissipative functional materials with electro‐magneto‐mechanical couplings based on incremental variational principles. We focus on quasi‐static problems, where mechanical inertia effects and time‐dependent electro‐magnetic couplings are a priori neglected and a time‐dependence enters the formulation only through a possible rate‐dependent dissipative material response. The underlying variational structure of non‐reversible coupled processes is related to a canonical constitutive modeling approach, often addressed to so‐called standard dissipative materials. It is shown to have enormous consequences with respect to all aspects of the continuum‐based modeling in macroscopic electro‐magneto‐mechanics. At first, the local constitutive modeling of the coupled dissipative response, i.e. stress, electric and magnetic fields versus strain, electric displacement and magnetic induction, is shown to be variational based, governed by incremental minimization and saddle‐point principles. Next, the implications on the formulation of boundary‐value problems are addressed, which appear in energy‐based formulations as minimization principles and in enthalpy‐based formulations in the form of saddle‐point principles. Furthermore, the material stability of dissipative electro‐magneto‐mechanics on the macroscopic level is defined based on the convexity/concavity of incremental potentials. We provide a comprehensive outline of alternative variational structures and discuss details of their computational implementation, such as formulation of constitutive update algorithms and finite element solvers. From the viewpoint of constitutive modeling, including the understanding of the stability in coupled electro‐magneto‐mechanics, an energy‐based formulation is shown to be the canonical setting. From the viewpoint of the computational convenience, an enthalpy‐based formulation is the most convenient setting. A numerical investigation of a multiferroic composite demonstrates perspectives of the proposed framework with regard to the future design of new functional materials. Copyright © 2011 John Wiley & Sons, Ltd.     

A micromechanical model of distributed damage due to void growth in general materials and under general deformation histories

We develop a multiscale model of ductile damage by void growth in general materials undergoing arbitrary deformations. The model is formulated in the spirit of multiscale finite element methods (FE ²), that is, the macroscopic behavior of the material is obtained by a simultaneous numerical evaluation of the response of a representative volume element. The representative microscopic model considered in this work consists of a space‐filling assemblage of hollow spheres. Accordingly, we refer to the present model as the packed hollow sphere (PHS) model. A Ritz–Galerkin method based on spherical harmonics, specialized quadrature rules, and exact boundary conditions is employed to discretize individual voids at the microscale. This discretization results in material frame indifference, and it exactly preserves all material symmetries. The effective macroscopic behavior is then obtained by recourse to Hill's averaging theorems. The deformation and stress fields of the hollow spheres are globally kinematically and statically admissible regardless of material constitution and deformation history, which leads to exact solutions over the entire representative volume under static conditions. Excellent convergence and scalability properties of the PHS model are demonstrated through convergence analyses and examples of application. We also illustrate the broad range of material behaviors that are captured by the PHS model, including elastic and plastic cavitation and the formation of a vertex in the yield stress of porous metals at low triaxiality. This vertex allows ductile damage to occur under shear‐dominated conditions, thus overcoming a well‐known deficiency of Gurson's model. Copyright © 2012 John Wiley & Sons, Ltd.     

Yield and deformation of an assembly of disks subjected to a deviatoric stress loading

The deformation of an assembly of particles is examined using a discrete element method (DEM) numerical model. The particles are modeled as random-sized, rough, inelastic, circular two-dimensional disks. The simulations keep track of the displacements, velocities and contact forces of each particle in order to examine the local rearrangements of the particles during the deformation and to determine the bulk stress states. The behavior of cohesive materials can be examined by introducing tensile forces between particles. The tests are done by applying constant confining pressures on two parallel flexible boundaries and a constant displacement rate on the two other flat frictionless walls. During the loading, the axial stress on the moving walls reaches a peak value and afterwards remains essentially constant for large strains. Stress–strain curves are obtained for a large range of confining pressures. They show that the yield envelopes follow the linear Mohr–Coulomb criterion. The global angle of friction is determined for a large range of particle–particle friction angles and particle size distributions. It was found that the global friction angle φ_cv increased with interparticle friction angle φ_i for φ_i<6°; for larger values of φ_i, the global friction angle is essentially constant. As the spread in the particle size distribution increased, the magnitude of the internal angle of friction was found to increase for a given interparticle friction angle.     

A non‐linear programming approach to kinematic shakedown analysis of composite materials

Using a Representative volume element (RVE) to represent the microstructure of periodic composite materials, this paper develops a non‐linear numerical technique to calculate the macroscopic shakedown domains of composites subjected to cyclic loads. The shakedown analysis is performed using homogenization theory and the displacement‐based finite element method. With the aid of homogenization theory, the classical kinematic shakedown theorem is generalized to incorporate the microstructure of composites. Using an associated flow rule, the plastic dissipation power for an ellipsoid yield criterion is expressed in terms of the kinematically admissible velocity. By means of non‐linear mathematical programming techniques, a finite element formulation of kinematic shakedown analysis is then developed leading to a non‐linear mathematical programming problem subject to only a small number of equality constraints. The objective function corresponds to the plastic dissipation power which is to be minimized and an upper bound to the shakedown load of a composite is then obtained. An effective, direct iterative algorithm is proposed to solve the non‐linear programming problem. The effectiveness and efficiency of the proposed numerical method have been validated by several numerical examples. This can serve as a useful numerical tool for developing engineering design methods involving composite materials. Copyright © 2005 John Wiley & Sons, Ltd.     

Free and Forced Vibrations of a Segmented Bar by a Meshless Local Petrov–Galerkin (MLPG) Formulation

We use the meshless local Bubnov–Galerkin (MLPG6) formulation to analyze free and forced vibrations of a segmented bar. Three different techniques are employed to satisfy the continuity of the axial stress at the interface between two materials: Lagrange multipliers, jump functions, and modified moving least square basis functions with discontinuous derivatives. The essential boundary conditions are satisfied in all cases by the method of Lagrange multipliers. The related mixed semidiscrete formulations are shown to be stable, and optimal in the sense that the ellipticity and the inf-sup (Babuška-Brezzi) conditions are satisfied. Numerical results obtained for a bimaterial bar are compared with those from the analytical, and the finite element methods. The monotonic convergence of first two natural frequencies, first three mode shapes, and a static solution in the L ², and H ¹ norms is shown. The relative error in the numerical solution for a transient problem is also very small.     

Different approaches for mixed LSFEMs in hyperelasticity: Application of logarithmic deformation measures

We present geometrically nonlinear formulations based on a mixed least‐squares finite element method. The L²‐norm minimization of the residuals of the given first‐order system of differential equations leads to a functional, which is a two‐field formulation dependent on displacements and stresses. Based thereon, we discuss and investigate two mixed formulations. Both approaches make use of the fact that the stress symmetry condition is not fulfilled a priori due to the row‐wise stress approximation with vector‐valued functions belonging to a Raviart‐Thomas space, which guarantees a conforming discretization of H(div). In general, the advantages of using the least‐squares finite element method lie, for example, in an a posteriori error estimator without additional costs or in the fact that the choice of the polynomial interpolation order is not restricted by the Ladyzhenskaya‐Babu?ka‐Brezzi condition (inf‐sup condition). We apply a hyperelastic material model with logarithmic deformation measures and investigate various benchmark problems, adaptive mesh refinement, computational costs, and accuracy.     

Determination of Bulk Flow Property of tef Flour and Seed and Design of a Silo

The flowability of the bulk materials was determined after development of yield locus, wall yield locus, and effect of time consolidation was measured. Data from the obtained curves were read and important parameters such as unconfined stress, major consolidation stress, angle of internal friction, angle of wall friction, and effective angle of internal friction were for further design of silo calculation made ready. Diagrams and relationships developed by Jenike were used to find important silo discharge opening parameters such as the angle of the conical or wedge shaped opening and critical silo opening diameter and width. The relative difference between tef flour and tef seed was made in which the flour has a characteristics indicator of particle size distribution with lower dimension and the flour is cohesive and less flowable and accordingly needs more generous discharge parameters. The comparison between the flour and the seeds was shown in the measurement of flowability indexes and also reflected clearly on the designed silo discharge opening parameters.     

Taylor impact test for ductile porous materials—Part 1: theory

Taylor tests have been commonly employed to determine dynamic yield stress of solids at a high strain rate. In this paper, the original Taylor model is extended in order to provide a theoretical basis for testing ductile porous materials. The key difference between solids and porous materials in this respect is that porous materials are compressible and their density changes with the compressive strain. Calculations have been made for porous materials with a relative density that is a linear function of compressive strain e, i.e., ρ/ρ₀=1+ae. The final length of the projectile after impact, L₁/L, is plotted against parameter ρ₀U²/Y (see Fig. 9) and this plot is used in a Taylor test to determine the dynamic yield stress. The mean strain rate of the test can be estimated from Eqs. (21) or (22). In a companion paper (Int J Impact Eng), experiments for dynamic yield stress of porous materials will be reported based on the present theoretical analysis.     

Optimal and noise-robust extraction of Fracture Mechanics parameters from kinematic measurements

The paper is devoted to an optimal (i.e. noise-robust) determination of stress intensity factors and crack tip locations based on a displacement field measured over an arbitrarily shaped domain. As the minimization of the noise sensitivity is included within the proposed extraction technique, this is especially dedicated to corrupted displacement fields, e.g. as measured by an optical technique. The main idea is to construct for mode I and II fields an extracting function so that its L₂ scalar product with the actual displacement field yields the sought parameter. The extracting function is also constrained to be orthogonal to a set of admissible elastic fields. Two applications are considered to illustrate the technique. The first example deals with a fatigue crack in steel for which small scale yielding occurs. A second example with a low signal/noise ratio illustrates the capability of the approach to analyze a crack in silicon carbide with sub-pixel openings.     

MY准则解析受均布载荷简支斜板极限载荷

为了得到斜板极限载荷的解析解,用平均屈服(MY)准则,对受均布载荷的简支金属斜板进行了塑性极限分析.首次获得MY准则下斜板极限载荷的解析解,该解是斜板几何参数长l1,宽l2以及长宽夹角θ的函数.研究表明:随着θ的增大,极限载荷先增大而后减小;斜板面积增加,极限载荷减小.得到了菱形、矩形和方形板的解析解,并将方形板的解析解与Tresca、Mises以及TSS提供的极限载荷进行比较,对比表明:方板的极限载荷与边长成反比关系,Tresca屈服准则提供极限载荷的下限,TSS屈服准则提供上限,MY准则预测结果恰居二者中间,且最靠近Mises解.     

The Nonwetting Behavior of Dilute 3He-4He Mixtures on Cs

We present evidence for ³ He interfacial bound states at the Cs-liquid He interface and show their influence on the interfacial free energies and the nonwetting behaviour of dilute ³ He- ⁴ He mixtures on Cs. The bound state energy and effective mass for the ³ He atoms at this interface are determined and found to be in good agreement with theoretical predictions. From the temperature dependence of the contact angle, direct evidence for a first order wetting transition at the lower wetting temperature is found. A detailed model is presented for the Cs-He mixture interface free energy and the contact angle and this model is shown to be in excellent agreement with the measurements. There is a short discussion on hysteresis and the thin film state. Finally it is emphasised that the large measured contact angles are only consistent with ³ He bound states together with ripplons at the Cs-He mixture interface     

The Effect of Stress Triaxiality on the Local Cleavage Fracture Stress in a Granular Bainitic Weld Metal

The local cleavage fracture stress σ_F measured at the cleavage initiation sites of a granular bainitic weld metal is only an apparent fracture parameter dependent on stress triaxialities at the local sites in the propagation controlled cleavage regime. This dependence can be explained by the stress triaxiality modified Smith equation in which the intrinsic cleavage fracture stress σ _F ⁰ is introduced, considered to be an invariant characterizing a material's fracture property. In the nucleation controlled cleavage regime σ_F is temperature dependent and coincident with the local yield stresses defined by Von Mises criterion at the local sites. It is suggested that the modified Smith model would provide a unified physical basis for the intrinsic correlation among various macroscopic fracture properties as well as their temperature transition behaviors from the temperature dependence of material's yield strength. This revised version was published online in August 2006 with corrections to the Cover Date.     

On the L2 and the H1 couplings for an overlapping domain decomposition method using Lagrange multipliers

In this paper, a comparison of the L² and the H¹ couplings is made for an overlapping domain decomposition method using Lagrange multipliers. The analysis of the local equations arising from the formulation of the coupling of two mechanical models shows that continuous weight functions are required for the L² coupling term whereas both discontinuous and continuous weight functions can be used for the H¹ coupling. The choice of the Lagrange multiplier space is discussed and numerically studied. The paper ends with some numerical examples of an end‐loaded cantilever beam and a cracked plate under tension and shear. It is shown that the continuity enforced with the H¹ coupling leads to a link with a flexibility that can be beneficial for coupling a very coarse mesh with a very fine one. To limit the effect of the volume coupling on the global response, a narrow coupling zone is recommended. In this case, volume coupling tends to a surface coupling, especially with a L² coupling. Copyright © 2006 John Wiley & Sons, Ltd.     

On a local predictor of shear instability and the initiation of local turbulence

The stability of a local laminar shear flow and its transition into turbulent flow is considered as a local phenomenon. This transition may remain local, in which case the flow field is partially laminar and partially turbulent, or it may spread and make the whole field turbulent. One of the applications of this analysis is the prediction of local heat-convection rates, which are enhanced by local turbulence. Another application is in heart-lung blood pumps, where excessive shear rates are detrimental to red blood cells.The analysis is Lagrangian, which concentrates on the stability of a fluid particle in maintaining its position in a laminar shear flow. This stability is shown to depend on the magnitude of a non-dimensional parameter, namely the local Reynolds numberRe _L =ha ²/v whereh is the local shear rate,a is the particle radius andv is the fluid's kinematic viscosity. It is shown that when, locally,Re _L > 530, the flow is, locally, unstable. The application of this criterion is simple and direct, and in certain cases it can be shown that the resulting unstable flow is indeed turbulent.Because the analysis relies on an experimental coefficient which has been obtained for a rigid sphere, rather than for a fluid particle, the criterion is introduced at this stage as a conjecture. Several examples are presented which demonstrate the criterion's ability to yield correct predictions for instability.     

均布载荷作用下简支圆板塑性极限载荷的解析解

该文建立了受均布载荷作用简支圆板运动许可应变场,并首次以EA(等面积)屈服准则进行了塑性极限分析,获得了极限载荷的解析解。该解为圆板半径a、圆板厚度h以及屈服强度的函数。与Tresca、TSS以及Mises解比较表明,Tresca屈服准则预测极限载荷的下限,TSS屈服准则预测极限载荷的上限,EA和Mises屈服准则预测的极限载荷恰居二者中间,且EA解几乎与Mises解重合。此外,该文还讨论了挠度与相对位置r/a之间的变化关系。     

A local thermomechanical model of friction for computing structures under extreme loading conditions

A multiscale thermomechanical model of friction is proposed for metallic interfaces submitted to extreme loading conditions with large sliding velocities ( [v] > 100 ms^?1) and high contact pressures (P > 1 GPa). This model decomposes the friction problem into a global multidimensional structural problem and a local interface subproblem. At global scale, the structure behavior is governed by an elastoplastic model in large strains, and thermal conduction is neglected. The local model describes the micrometric scale and the underlying thermomechanical mechanisms: mechanical hardening, frictional heating, and plastic work. Using dimensional and asymptotic analysis, the corresponding set of equations at local scale is simplified and reduces to a one‐dimensional quasistatic thermoelastoplastic friction model. It is solved locally at each global point by a finite differences subgrid model using a nonlinear time‐implicit solver. This solver is coupled to a time‐explicit Lagrangian hydrocode used at the global structural scale. The coupling strategy between the solvers is force‐based: the frictional stress is evaluated at the local scale by our multiscale friction model and transferred to the global model that can then predict local velocity corrections. This coupling strategy has been successfully tested on real‐life cases. Copyright © 2011 John Wiley & Sons, Ltd.     

On some mixed finite element methods for incompressible and nearly incompressible finite elasticity

We compare some mixed methods based on different variational formulations, namely a displacement-pressure formulation employed by de Borst and coworkers, the three-field formulation investigated by Simo and Taylor and a two-field formulation which is directly based on an energy functional. It emerges that all these yield the same discrete results if the stored energy function contains a volumetric contribution 1/2k(J−1)² whereJ is the volume dilatation, i.e., the Jacobian determinant of the deformation, andk is the bulk modulus. The equivalence holds for arbitrary 3D and plane strain elements. In the numerical examples the mixed formulations are discretized by the quadrilateral Q1/P0 and Q2/P1 elements and the triangular Crouzeix-Raviart P2+/P1 element. We also compare with standard displacement elements and the enhanced strain Q1/E4 element. This work was supported by the German Research Foundation (DFG) under Grant No. Ste 238/35-1.     

Two features of the uniaxial compression of a glassy epoxy resin: the yield stress rate-dependence and the volumetric instability

We report the results of uniaxial compressive tests on a DGEBA epoxy resin at room temperature, well below its glass transition. We first focus on the strength, defined as the stress value corresponding to either a maximum or a flattening of the stress-strain curve, which, for this polymer, may be taken to be coincident with the yield stress, as often assumed for many thermosets. Within the strain rate range (1.E?6 s^?1, 2.E?3 s^?1) we confirm the linear trend relating the logarithm of the strain rate to the yield stress, as already been observed by other investigators even for the same epoxy resin; instead, at strain rates below \(\dot{\varepsilon} _{0} \approx 1.\mathrm{E}{-}6~\mathrm{s}^{-1}\), we found a negligible rate-dependence, as our data indicate a lowest limit of the yield stress, of about 87 MPa. On the basis of these results, we propose how to extend to the viscoplastic regime of deformation a nonlinear viscoelastic model previously put forward.Secondarily, within the viscoelastic range, at a stress level significantly lower than the yield stress, our measurements show a mild volumetric instability, allowed by the free lateral expansion, not ascribable to any macroscopic structural effect; such a behaviour has never been reported in the literature, to the best of our knowledge.