Wolfgang S. Kreher

Residual stresses and stored elastic energy of composites and polycrystals

Abstract R esidual Stresses in heterogeneous materials may arise because of differential or anisotropic thermal expansion of constituents. The paper is concerned with thermoelastic solids whose material properties fluctuate on the microscopic scale. Rigorous general relations between stored elastic energy and statistical averages (mean values and fluctuations) of residual stresses are derived. These results are applied to two-phase composites and to materials where the fluctuations of elastic constants can be neglected. One obtains exactly the stored energy, certain conditional mean values and the covariance matrix of the residual stresses. Under the assumptions of statistical homogeneity and isotropy, the results hold for any type of heterogeneous microstructure.

Modelling piezoelectric modules with interdigitated electrode structures

The Interdigitaded Electrode Design, which allows to utilise the primary piezoelectric effect in longitudinal direction for piezoelectric sheet actuators, is investigated with the help of a detailed finite element analysis. Two different set-ups have been studied, based on (a) a piezoelectric bulk film and (b) a composite with piezoelectric fibres as the piezoelectric active component. The models allow to investigate the influence of various geometrical and material parameters on the deformation performance and on failure hazards due to field concentrations. For the bulk film actuator case, the development of an inhomogeneous poling state from the poling process of the piezoceramic material may be included into the model in an approximate way. For the composite actuator case, special emphasis has been laid on the problem of indirect contact between electrodes and embedded piezoelectric fibres. Specific proposals for the parameter design could be derived, which have been used as a starting point for the practical optimisation process.

Residual stresses in polycrystals as influenced by grain shape and texture

Abstract R esidual stresses in polycrystals arise because of anisotropic thermal expansion of constituent crystals. Due to the random crystal arrangement, the internal stresses also become random quantities, which have to be described by probability distributions or statistical moments. Conditional first order moments (averages) and second order moments (fluctuations) can be derived if the elastic energy stored in the residual stress field is known. By applying the Green function method, an analytical expression for this energy is obtained under some mild assumptions concerning the neglected elastic anisotropy and crystallographic correlations between the crystals. From the stored energy we calculate the average crystal stresses as depending on shape, orientation and polycrystalline texture. Furthermore we obtain a scalar second order moment of the residual stress, which also depends on grain shape and texture. The results arc evaluated for alumina ceramics and zircalloy.

Influence of domain switching zones on the fracture toughness of ferroelectrics

The paper presents a fracture model for ferroelectric materials taking into account the hysteretic domain switching processes near to the tip of a macroscopic crack. The model is based on the balance of energy supplied by the driving forces, on the one hand, and the total of energies either dissipated by domain switching, stored in the crack wake region or consumed by the formation of new fracture surface, on the other hand. An internal variable theory describes the nonlinear coupled electromechanical material response within the framework of a three-dimensional continuum model. For simplicity, the complex orientation distribution function of domains in a polycrystalline ceramic is approximated by only six representative space orientations. The theory predicts certain dimensionless material parameter combinations which govern the change of fracture toughness under the application of different mechanical and electrical loadings. A comparison with data available in the literature for barium titanate ceramics yields a reasonable coincidence.

Internal residual stresses in heterogeneous solids—A statistical theory for particulate composites

Abstract We consider a linearly elastic composite medium, which consists of a homogeneous matrix containing a homogeneous and statistically uniform random set of ellipsoidal inclusions. Because of the differential thermal expansion, a microstructural residual stress state arises. By means of the “multiparticle effective field” method we first derive the functional relation between the stored elastic energy and the thermoelastic constants of the components. Using this result an exact estimation of all components of the statistical second moment tensor of the stress fields is given. Furthermore, an expression for the second moment of the stress in the matrix in the vicinity of the ellipsoidal inclusion and a correlation function of internal stresses is obtained. The application of the theory is demonstrated by some numerical results for a WC-Co composite.

Modelling linear and nonlinear behavior of polycrystalline ferroelectric ceramics

A micro-electromechanical model is presented, which describes the macroscopic behavior of ferroelectric ceramics under weak signal and unipolar large signal loading conditions. The approach is based on a laminar domain structure and a hierarchical homogenization procedure, which considers grains as spherical inclusions within a homogenized effective medium. The model predicts the extrinsic effects of mobile domain walls, hysteresis at large fields, and ferroelectric fatigue by unipolar cycling. Additionally, internal fields can be estimated. As an example, numerical results and available experimental data for barium titanate ceramics are discussed.

Effective properties of composites utilising fibres with a piezoelectric coating

Composites with piezoelectric fibres are promising new materials, combining the beneficial properties of very different constituents. Recently hybrid fibres with an inactive core and a piezoelectric coating have been developed. For conventional two-phase systems the correlation between component properties and effective composite behaviour is well approximated using effective field or self-consistent models. Since the latter approaches are commonly based on solutions for homogeneous inclusions, they cannot be directly employed for heterogeneous particles as coated fibres. Different methods are employed to estimate the relevant effective electromechanical parameters of composites continuously reinforced with coated piezoelectric fibres: (a) a unit cell finite element model, (b) an effective field approach built on a rigorous solution for a coated fibre in an infinite matrix and (c) a simple homogeneous field approximation. The results of the approaches are compared and discussed.

On microstructural residual stresses in particle reinforced ceramics

Abstract Residual microstructural stresses are shown to influence the crack propagation in particle reinforced ceramics. The dispersion of second phase particles with different coefficients of thermal expansion generates a spatially fluctuating stress field. In this paper deterministic and statistical solutions are considered for the determination of residual stresses. Numerical results are presented for SiC-platelet reinforced Si 3 N 4 and correlated to mechanical properties. It is concluded that the toughening mechanisms of particle reinforced composites are influenced decisively by the spatially fluctuating stress field.

Self-consistent modelling of non-linear effective properties of polycrystalline ferroelectric ceramics

A constitutive model for the non-linear effective behaviour of ferroelectric ceramics is presented. The model is developed on the basis of the effective medium approximation (EMA), which describes the interaction of the crystallites in a statistical way. Additionally, a particular simplified domain configuration within the crystallites and the possibility of domain wall motion are taken into account. In connection with a thermodynamic criterion for the domain wall displacement the volume fractions of the domains can be calculated dependent on the crystallite orientation and the applied load in a self-consistent manner. This mechanism leads to an extrinsic contribution to the effective behaviour. If the domain wall displacement is associated with energy dissipation the macroscopic behaviour is non-linear and hysteretic.

Smart structures by integrated piezoelectric thin fibres (II): Properties of composites and their physical description

Abstract The present paper describes the performance of glass fibre reinforced polymers (GFRP) with integrated piezoelectric transducers formed by aligned piezoceramic fibres (50 μm, type A) or microrods (120 μm, type B), both terminated by an interdigital electrode. The following results have been obtained experimentally: demonstration of electromechanical coupling of a 5 cm × 12 cm vibrating plate via transducers A and B; demonstration of impact sensing capability of transducer A integrated in a 19 cm × 20 cm plate and demonstration of passive piezoelectric damping using a shunted transducer B with relative dissipation of power (e.g. related to the power stored within the transducer) of 12 %.