Pseudoelasticity of Shape Memory Alloys - a 1D Material Model and Finite Element Implementation

This contribution is concerned with the formulation of a 1D-constitutive model accounting for the pseudoelastic behavior of shape memory alloys. The stress-strain-relationship is idealized by a hysteresis both in the compression as in the tension loading range. It is characterized by an upper loading path, which is to be ascribed to the transformation of the lattice to a martensitic structure. Unloading the material, a lower path is described, because of the reverse transformation into austenitic lattice. The constitutive model is based on a switching criterion which serves as a potential function for the evolution of the internal state variables. The model distinguishes between local and global variables to describe the hysteresis effects for the compression and tension range. A strain driven algorithm which captures the complete nonlinear material behavior is presented. The boundary value problem is solved for a truss element applying the finite element method. A consistent linearization of the nonlinear equations is derived. Simple examples will demonstrate the applicability of the proposed model. For future developments the usage of shape memory alloys within civil engineering structures is aimed. The advantage of the material is the very good damping behavior and the potential to overcome great strains. Both properties are distinguished to be of engineering interest. (© 2010 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Theoretical and experimental investigation of anisotropic damage in textile-reinforced composite structures

In the present work, a phenomenological plane-stress damage-mechanics-based model for textile-reinforced composites is presented and its predictive capability is evaluated by carrying out a series of experimental tests. Damage variables are introduced to describe the evolution of the damage state and, as a subsequence, the degradation of material stiffness. For calculating the nonlinear stress and strain distribution of complexly loaded composites with a textile reinforcement, a special emphasis has to be placed on the interaction between the fiber failure due to the stress in the fiber direction and the matrix failure due to the transverse and shear stresses. This demands the formulation of realistic failure criteria taking into account the microstructural material behavior and different fracture modes. The new failure criteria, like the fracture mode concepts, consider these fracture modes, as well as further fracture types, in the reinforcement plane. The failure criteria are based on equations for failure surfaces in the stress space and damage thresholds in determining the stiffness degradation of the composite. The model proposed was used to characterize the strength and the failure behavior of carbon-fiber-reinforced composites. For this purpose, several unidirectional and bidirectional tests were performed to determine the specific properties of the material. The specimens were investigated by using acoustic emission techniques and strain-controlled tension and torsion tests.Russian translated published in Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 791–810, November–December, 2004.     

A homogenization method for heterogeneous materials using a multi-scale technique

For modern microelectronics solders lifetime and stability predictions are important. To perform such an analysis material properties are required. As electronic devices and the corresponding amount of matter used become smaller, the influence of a changing microstructure on mechanical properties must be considered. First some analytical methods were conducted for upper and lower bounds ignoring the exact geometric distribution of the solder phases. Second, analytical equations derived for geometries such as laminate structures were applied to examine the influence of the geometry on homogenized properties. Third, a multi-scale approach for periodic media was presented allowing for a more general analysis of structures. We assume that the solder material is composed of periodic cells, which represent the properties of the whole structure. Composite materials with periodic structures can be investigated by using at least two scales. A global scale is related to the whole piece of material whereas a local scale is related to the periodic cell only. The constitutive equations are stated and a homogenization technique for the elastic properties of arbitrary structures is derived. The resulting equations are solved numerically and results are presented. Again, for layered materials closed-form formulas are derived and compared to the numerical results. The method is also used to obtain effective mechanical properties for materials with linear hardening. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damageability of a material reinforced with unidirectional orthotropic fibers for an exponential function of long-term microstrength

The theory of long-term damageability of a homogeneous material is generalized to the case of an orthotropic fibrous composite material with a stochastic structure. Equations of mechanics of microinhomogeneous media of this structure form the base of the theory. The process of damage of components of a composite is modeled by the formation of stochastically located micropores. The criterion of fracture of a unit microvolume is characterized by its long-term strength determined by the dependence of the time of brittle fracture on the degree of closeness of the equivalent stress to its limit value, which characterizes the short-term strength on the basis of the Huber–von Mises criterion accepted as an arbitrary function of coordinates. Efficient deformation properties and the stress-strain state of an orthotropic fibrous composite with microdamages in components are determined on the base of stochastic equations of elasticity of orthotropic media. For given macrostresses and macrostrains and an arbitrary moment of time, balance equations of damage (porosity) of components are formulated. On the basis of the iteration method, we construct algorithms for calculating dependences of microdamage of components of an orthotropic fibrous material on time and dependences of macrostresses or macrostrains on time and obtain the corresponding curves for the case of a bounded function of the long-term microstrength, which is approximated by an exponential law.     

Investigation of the local behavior of different cohesive zone models

Cohesive interface elements are well suited for three-dimensional crack propagation analyses as long as the crack path is known. This is the case e.g. in delamination analyses of laminated composite structures or failure analyses of adhesively bonded joints. Actually, they are widely used in such applications for both brittle and ductile systems. As long as the strength and fracture toughness of the material are accurately captured it is generally accepted that the shape of the cohesive law has little to no influence on the mechanical behavior of the investigated structures. However, when having a look on the local behavior of different cohesive zone models, such as stress distribution in the fracture process zone, the results exhibit certain differences. These will be studied in the present contribution. Especially the local stress distribution will be investigated and the effect on the computational efficiency will be pointed out. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Alpha well-posedness,alpha stability,and the matrix theorems of H.-O. Kreiss

Summary For systems of partial differential equations with constant coefficients and for the corresponding difference equations the concepts of well-posedness and stability are introduced. These concepts are more general than strong well-posedness and stability on the one hand, and more restrictive than weak well-posedness (Petrovskii condition) and weak stability (von Neumann condition) on the other. Characterizations of these properties are established which partly extend the matrix theorems of H.-O. Kreiss. Also a Lax type theorem is valid in this setting.This author was supported by the Deutsche Forschungsgemeinschaft (Grant Go 261/4)     

Simulation of a Hybrid-Forming Process using Temperature- and Rate-Dependent Viscoplastic Material Models

Hybrid-forming processes for graded structures are quite innovative methods for the production of components with tailored properties, particularly tailored material properties and geometrical shape. In this contribution a hybrid-forming process based on the utilization of locally varying thermo-mechanical effects is investigated [1]. For process optimization and improvement of the resulting work piece the simulation of the entire forming process is necessary in modern engineering. The main topics of this contribution are the simulation of the cyclic thermal loaded forming tool and the simulation of the work piece treated at large deformations with phase transformations. For both materials temperature- and rate-dependent viscoplastic material models are applied and parameter identification using cyclic tensile-compression tests for the forming tool material and phase transformation tests for a low-alloy steel similar to the work piece material is presented. For validation of finite-element-calculations for the forming tool thermal shock experiments are performed with optical deformation measurements. For validation of finite-element-calculations for the work piece numerical results of geometry and structure after heating, forming and cooling are compared to experimental micro sections. Results concerning the forming tool will be used for future lifetime prediction and results concerning the work piece will be used for future specific setting of graded material properties. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Separation of linear components in nonlinear boundary heat control

By means of an additional substitution a parabolic control problem with some nonlinear boundary condition will be decoupled into some control problem with linear parabolic state equations and an appropriate nonlinear mapping. This separation allows the use of efficient techniques e.g. Fourier methods, to determine the solution of linear parabolic state equations. Essential properties of the mapping used in the transformation are studied. Further, the application of piecewise constant discretizations of the controls in connection with the proposed splitting is discussed.     

Asymptotic aspect of Drygas,quadratic and Jensen functional equations in metric abelian groups

The asymptotic stability behavior of Drygas, quadratic and Jensen functional equations is investigated. Indeed, we show that if these equations hold approximately for large arguments with an upper bound \({\varepsilon}\), then they are also valid approximately everywhere with a new upper bound which is a constant multiple of \({\varepsilon}\). These results will be applied to the study of asymptotic properties of Drygas, quadratic and Jensen functional equations. We also obtain some results of hyperstability character for these functional equations.     

A variant of the phenomenological theory of fracture

It is proposed to characterize the damage suffered by a material subjected to a long-time constant or variable load by a certain function on a sphere. This function may give the magnitude and direction of the damage at the point in question. The spherical function is a functional of the stresses calculated in the corresponding local coordinate system. It is assumed that failure occurs when the invariant local fracture characteristics reach a certain critical value. The proposed approach is compared with the tensor variant. The possibility of taking the effect of complex loading into account is discussed.Institute of Polymer Mechanics, Academy of Sciences of the Latvian SSR, Riga. Translated from Mekhanika Polimerov, Vol. 4, No. 4, pp. 638–647, July–August, 1968.     

Regularity properties of a class of hybrid methods

IntroductionIt is important that the discrete dynamical system given by a numerical method appliedto a continuous dynamical system can have the same dynamical properties as the underlyingcontinuous system. Recently, many authors[1--71 have investigated the conditions under whichspurious solutions are not introduced by time discretization, and many interesting results aboutRunge-Kutta methods, linear multistep methods and general linear methods applied to dynamical systems of ordinary different…     

Global and mid-range function approximation for engineering optimization

Global and mid-range approximation concepts are used in engineering optimisation in those cases were the commonly used local approximations are not available or applicable. In this paper the response surface method is discussed as a method to build both global and mid-range approximations of the objective and constraint functions. In this method analysis results in multiple design points are fitted on a chosen approximation model function by means of regression techniques. Especially global approximations rely heavily on appropriate choices of the model functions. This builds a serious bottleneck in applying the method. In mid-range approximations the model selection is much less critical. The response surface method is illustrated at two relatively simple design problems. For building global approximations a new method was developed by Sacks and co-workers, especially regarding the nature of computer experiments. Here, the analysis results in the design sites are exactly predicted, and model selection is more flexible compared to the response surface method. The method will be applied to an analytical test function and a simple design problem. Finally the methods are discussed and compared.     

Axisymmetric torsional fretting contact between a spherical punch and an FGPM coating

The axisymmetric torsional fretting contact between a rigid conducting spherical punch and a functionally graded piezoelectric material (FGPM) coating is studied in this paper. The exponential model is used to simulate the inhomogeneous electro-mechanical properties of the FGPMs coating. The conducting spherical punch with a constant surface electric potential is considered in the contact. A normal force and a cyclic torque are applied to the two contact bodies. The applied torque produces an outer annular slip area and an inner stick area. The torsion angle is produced within the inner stick area as a rigid body. With the aid of the Hankel integral transform technique, we can reduce the contact problem to the singular integral equations of the Cauchy type. Then the unknown electro-mechanical fields and stick/slip area can be obtained numerically. The effect of the gradient index on the surface electro-mechanical fields is discussed at loading and unloading phases. The Mises stress and principal stress at the contact surface are also discussed to predict the possible location of fretting damage and failure.     

Projectively flat Finsler 2-spheres of constant curvature

After recalling the structure equations of Finsler structures on surfaces, I define a notion of "generalized Finsler structure" as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of "generalized path geometry" analogous to that of "generalized Finsler structure". I use these ideas to study the geometry of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that, modulo diffeomorphism, there is a 2-parameter family of projectively flat Finsler structures on the sphere whose Finsler-Gauss curvature K is identically 1.     

Fracture Mechanical Investigation of Elastomeric Material

The increasing use of elastomeric components in advanced engineering applications requires a thorough understanding of the complex material properties and a reliable assessment of the quality and durability of the products. This contribution concentrates on the computational determination of fracture mechanical parameters for rubber material using the material force method. For dissipative, inelastic material, a distinction between two fracture mechanical parameters is presented. The time-dependent behaviour of these fracture mechanical parameters is illustrated by an application to the dwell-effect. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical solutions of general-relativistic field equations for rapidly rotating neutron stars

Stationary axial symmetric equilibrium configurations rapidly rotating with uniform angular velocity in the framework of general relativity are considered. Sequences of models are numerically computed by means of a computer code that solves the full Einstein equations exactly. This code employs Neugebauer’s minimal surface formalism, where the field equations are equivalent to two-dimensional minimal surface equations for 4 metric potentials. The calculations are based upon 10 different equations of state. Results of various structures of neutron stars and the rotational effects on stellar structures and properties are reported. Finally some limits to equations of state of neutron stars and the stability for rapidly rotating relativistic neutron stars are discussed.     

Bifurcation from two equilibria of steady state solutions for non-reversible amplitude equations

In this paper, bifurcation and stability of two kinds of constant stationary solutions for non-reversible amplitude equations on a bounded domain with Neumann boundary conditions are investigated by using the perturbation theory and weak nonlinear analysis. The asymptotic behaviors and local properties of two explicit steady state solutions, and pitch-fork bifurcations are also obtained if the bounded domain is regarded as a parameter. In addition, the stability of a new increasing or decaying local steady state solution with oscillations are analyzed.     

Macroscopic damage modeling for silicon nitride

Modeling the damage of brittle materials is of great importance considering a variety of structural components. Prominent examples are high strength engineering ceramics. The present work is concerned with silicon nitride, a material with increasing relevance in industrial applications. In the sense of a hierarchical model structure, effective properties of micromechanical simulations were applied to macroscopic, phenomenological damage models for monotonous and cyclic loading. In the following, both models are introduced and the application of the cyclic damage model to a four point bending test is discussed. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

FE‐formulation for plate and beam elements to simulate onset and propagation of discrete cracks in solid structures

The aim of this paper is to present some numerical aspects related to the modeling both the formation and the propagation of discrete cracks in solid structures. The presented formulation corresponds to the concept of embedded discontinuities [1], and will be applied to a plate and to a beam element. The failure of solid structures is often triggered by a highly localized pattern of inelastic deformation in the form of narrow bands. Characteristic examples are shear bands in metals and soils, or localized bands of cracking in brittle materials, like concrete or rocks. A well known difficulty associated with classical (local, rate‐independent) continuum theories with strain softening attributes is that numerical solutions are found to lack invariance with respect to the choice of spatial discretization. For quasi‐static boundary problems, this mathematical inconsistency causes the loss of ellipticity for the governing equations (material instability). To regularize this inconsistency, several strategies have been applied. In the presented formulation, additional degrees of freedom are considered. Within the concept of embedded discontinuities, the regular displacements are enriched by discontinuities. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)