Vibrations of Layered Beams under Conditions of Damaging and Slipping Interfaces

A nonlinear laminate layerwise beam theory is developed to simulate the effect of inelastic interlayer slip on the stiffness degradation of layered beam structures. Layerwise continuous and linear in-plane displacement fields are implemented. It is shown that by definition of an effective cross-sectional rotation the complex problem reduces to the simpler case of a homogenized shear-deformable beam with effective stiffness and a corresponding set of boundary conditions. Inelastic defects of the interlayer material are equivalent to eigenstrains in an identical but unlimited elastic background structure of the homogenized beam with proper effective virgin stiffness. Cross-sectional resultants of these eigenstrains are defined. Since the incremental response of the background to the given load and to the properly imposed eigenstrain increments is considered to be linear within a given time step, effective solution methods of the linear theory of flexural vibrations become applicable. This revised version was published online in July 2006 with corrections to the Cover Date.     

Eigenstrain induced vibrations of composite plates

Summary The paper is concerned with eigenstrain induced small amplitude vibrations of moderately thick composite plates. In case of thermal shock loading such eigenstrains are imposed as thermal strains. Within a multiple field approach, thermal strains, piezoelectrically induced strains, as well as additional fields of strains that are due to inelastic deformations are interpreted as eigenstrains acting in the associated linear elastic background structure. Polygonal simply supported composite plates with three asymmetrically arranged layers in perfect bond are considered. The homogeneous layers consist of dissimilar linear elastic isotropic materials. The Mindlin-Reissner kinematic assumptions are implemented separately to each layer. The continuity of the transverse shear stress across the interfaces is prescribed according to Hooke's law. For plates composed of layers with a uniform ratio of mass density to shear modulus and a common Poisson's ratio the lateral deflection can be calculated independently from the in-plane displacements. It is shown that the lateral deflection is governed by a boundary value problem of the fourth order. Alternatively, by neglecting the longitudinal as well as rotatory inertia the complex problem reduces to the simpler case of a homogenized shear-deformable plate with effective stiffness, a corresponding set of boundary conditions and effective eigenstrain resultants. The theory is applied to rectangular simply supported composite plates of various dimensions.     

A mixed finite element method for beam and frame problems

 In this work we consider solutions for the Euler-Bernoulli and Timoshenko theories of beams in which material behavior may be elastic or inelastic. The formulation relies on the integration of the local constitutive equation over the beam cross section to develop the relations for beam resultants. For this case we include axial, bending and shear effects. This permits consideration in a direct manner of elastic and inelastic behavior with or without shear deformation. A finite element solution method is presented from a three-field variational form based on an extension of the Hu–Washizu principle to permit inelastic material behavior. The approximation for beams uses equilibrium satisfying axial force and bending moments in each element combined with discontinuous strain approximations. Shear forces are computed as derivative of bending moment and, thus, also satisfy equilibrium. For quasi-static applications no interpolation is needed for the displacement fields, these are merely expressed in terms of nodal values. The development results in a straight forward, variationally consistent formulation which shares all the properties of so-called flexibility methods. Moreover, the approach leads to a shear deformable formulation which is free of locking effects – identical to the behavior of flexibility based elements. The advantages of the approach are illustrated with a few numerical examples. Dedicated to the memory of Prof. Mike Crisfield, for his cheerfulness and cooperation as a colleague and friend over many years.     

Multiple 3D inhomogeneous inclusions in a half space under contact loading

A semi-analytic solution is given for multiple three-dimensional inhomogeneous inclusions of arbitrary shape in an isotropic half space under contact loading. The solution takes into account interactions between all the inhomogeneous inclusions as well as the interaction between the inhomogeneous inclusions and the loading indenter. In formulating the governing equations for the inhomogeneous inclusion problem, the inhomogeneous inclusions are treated as homogenous inclusions with initial eigenstrains plus unknown equivalent eigenstrains, according to Eshelby’s equivalent inclusion method. Such a treatment converts the original contact problem concerning an inhomogeneous half space into a homogeneous half-space contact problem, for which governing equations with unknown contact load distribution can be conveniently formulated. All the governing equations are solved iteratively using the Conjugate Gradient Method. The iterative process is performed until the convergence of the half-space surface displacements, which are the sum of the displacements due to the contact load and the inhomogeneous inclusions, is achieved. Finally, the obtained solution is applied to two example cases: a single inhomogeneity in a half space subjected to indentation and a stringer of inhomogeneities in an indented half-space. The validation of the solution is done by modeling a layer of film as an inhomogeneity and comparing the present solution with the analytic solution for elastic indentation of thin films. This general solution is expected to have wide applications in addressing engineering problems concerning inelastic deformation and material dissimilarity as well as contact loading.     

Non-uniform eigenstrain induced anti-plane stress field in an elliptic inhomogeneity embedded in anisotropic media with a single plane of symmetry

A closed-form solution is derived for an anti-plane stress field emanating from non-uniform eigenstrains in an elliptic anisotropic inhomogeneity embedded in anisotropic media with one elastic plane of symmetry. The prescribed eigenstrains are characterized by linear functions of the inhomogeneity in Cartesian coordinates. By means of the polynomial conservation theorem, use of complex function method and conformal transformation, explicit expressions for stresses at the interior boundary of the matrix and the strain energy for the elastic inhomogeneity/matrix system are obtained in terms of coefficients in the linear functions. The coefficients are evaluated analytically using the principle of minimum potential energy of the elastic system, leading to the anti-plane stress field. The resulting solution is verified by means of the continuity condition for the shear stress at the interface between the elliptic inhomogeneity and matrix. The present solution is shown to reduce to known results for uniform eigenstrains with illustration by numerical examples.     

Effects of interfacial excess energy on the elastic field of a nano-inhomogeneity

A semi-analytical approach based on a variational framework is developed to obtain the three-dimensional solution for a nano-scale inhomogeneity with arbitrary eigenstrains embedded in a matrix of infinite extent. Both the inhomogeneity and the matrix can be elastically anisotropic. The Gurtin–Murdoch surface/interface model is used to describe the elastic behavior of the inhomogeneity/matrix interface. The displacement fields in the inhomogeneity and the matrix are represented, respectively, by two sets of polynomials. Coefficients of these polynomials are determined by solving a system of linear algebraic equations that are derived from minimizing the total potential energy of the system. In the case of an isotropic spherical inhomogeneity with dilatational eigenstrain in an isotropic matrix, our solution shows an excellent agreement with the corresponding analytical solution available in the literature. To demonstrate the capabilities of the method developed here and to investigate the effect of interfacial excess energy, numerical examples are also presented when the inhomogeneity and matrix are both elastically anisotropic. Both dilatational and pure shear eigenstrains are considered in these examples.     

Numerical investigation of dynamic shear bands in inelastic solids as a problem of mesomechanics

The main objective of the present paper is to discuss very efficient procedure of the numerical investigation of the propagation of shear band in inelastic solids generated by impact-loaded adiabatic processes. This procedure of investigation is based on utilization the finite element method and ABAQUS system for regularized thermo-elasto-viscoplastic constitutive model of damaged material. A general constitutive model of thermo-elasto-viscoplastic polycrystalline solids with a finite set of internal state variables is used. The set of internal state variables is restricted to only one scalar, namely equivalent inelastic deformation. The equivalent inelastic deformation can describe the dissipation effects generated by viscoplastic flow phenomena. As a numerical example we consider dynamic shear band propagation in an asymmetrically impact-loaded prenotched thin plate. The impact loading is simulated by a velocity boundary condition, which are the results of dynamic contact problem. The separation of the projectile from the specimen, resulting from wave reflections within the projectile and the specimen, occurs in the phenomenon. A thin shear band region of finite width which undergoes significant deformation and temperature rise has been determined. Shear band advance, shear band velocity and the development of the temperature field as a function of time have been determined. Qualitative comparison of numerical results with experimental observation data has been presented. The numerical results obtained have proven the usefulness of the thermo-elasto-viscoplastic theory in the investigation of dynamic shear band propagations.     

Adiabatic shear failure in brittle solids

Recent studies have revealed microscopic amorphous lamella resulting from inelastic deformation in the ballistic impact of boron carbide ceramic. The possibility that these deformation features are a consequence of adiabatic shear deformation in the impact event is explored. An early theory of adiabatic shear that was limited to the response of rigid-plastic deformation is expanded to include elastic strain energy. The study reveals that elastic strain energy is commonly a small, but not negligible, contribution to impact-induced adiabatic shear in metals. Elastic strain energy is paramount in brittle solids. Relations are developed from the theory to predict the nominal width and spacing of adiabatic shear-bands in brittle solids. Comparisons of the theoretical predictions are consistent with observations of impact-induced deformation features in boron carbide.     

Levy-type piezothermoelastic solution for hybrid plate by using first-order shear deformation theory

A Levy-type solution is presented for hybrid rectangular plates, with two opposite edges simply supported, made of a cross-ply composite laminate with attached piezoelectric layers, and subjected to thermoelectromechanical load. First-order shear deformation and classical lamination theories are used. A mixed formulation is employed for the solution. The effect of the width-to-depth ratio and aspect ratio on deflection and force resultants has been illustrated for a uniform load on plates with various boundary conditions. The effect of shear deformation on deflection and force resultants for moderately thick plates is generally more pronounced for the mechanical load case than for the self-straining cases of thermal and electric loads.     

Dynamic buckling of metal matrix composite plates and shells under cylindrical bending

A micro-to-macro analysis is offered to investigate the dynamic response and buckling of metal matrix composite cylindrical shells and plates under cylindrical bending. The micromechanical analysis relies on the elastic fibers and inelastic matrix material properties, and provides the bulk behavior of the metal matrix composite at room and elevated temperatures. The macromechanical analysis employs the classical and higher order plate theories in conjunction with a spatial finite difference and temporal Runge-Kutta integrations to provide the dynamic response of the structure. The effects of the metallic matrix inelasticity, material rate sensitivity, shear deformation, fiber orientation, and initial geometrical imperfection on the behavior of the metal matrix composite structures are studied.     

Shear band systems in plane strain extension: analytical solution and comparison with experimental results

Shear banding represents a local failure mechanism of a soil structure as a response to shear loading. In soil structures of different spatial scales systems of regularly spaced shear bands can be observed as a consequence of extensional loading. The phenomenon of single shear bands, defined as thin zones of localized deformation with a discontinuity of the strain field at its boundaries, is well understood. Inside the shear band the material undergoes inelastic strain softening accompanied by shearing and dilation, whereas the material outside the shear band unloads accompanied by elastic contraction in extension tests. Despite numerous experimental and numerical investigations, the physical mechanisms and parameters determining the spacing of parallel shear bands remained unknown. The paper in hand presents an analytical solution for the spacing of the shear bands and a comparison with a large base of experimental data gained from 1g and ng (geotechnical centrifuge) model experiments. The analytical solution is based on the assumption that the elastic energy rate in the unloaded zone between the shear bands tends to a minimum value. The spacing was calculated as the energetically preferred solution for a broad range of cohesive-frictional granular materials. The dependency of the calculated spacing on initial and boundary conditions as well as on material parameters was found to be in good agreement with the experimental results.     

Implementation of the transformation field analysis for inelastic composite materials

Conclusions The transformation field analysis is a general method for solving inelastic deformation and other incremental problems in heterogeneous media with many interacting inhomogeneities. The various unit cell models, or the corrected inelastic self-consistent or Mori-Tanaka fomulations, the so-called Eshelby method, and the Eshelby tensor itself are all seen as special cases of this more general approach. The method easily accommodates any uniform overall loading path, inelastic constitutive equation and micromechanical model. The model geometries are incorporated through the mechanical transformation influence functions or concentration factor tensors which are derived from elastic solutions for the chosen model and phase elastic moduli. Thus, there is no need to solve inelastic boundary value or inclusion problems, indeed such solutions are typically associated with erroneous procedures that violate (6₂); this was discussed by Dvorak (1992). In comparison with the finite element method in unit cell model solutions, the present method is more efficient for cruder mesches. Moreover, there is no need to implement inelastic constitutive equations into a finite element program. An addition to the examples shown herein, the method can be applied to many other problems, such as those arising in active materials with eigenstrains induced by components made of shape memory alloys or other actuators. Progress has also been made in applications to electroelastic composites, and to problems involving damage development in multiphase solids. Finally, there is no conceptural obstacle to extending the approach beyond the analysis of representative volumes of composite materials, to arbitrarily loaded structures.This work was supported by the Air Force Office of Scienctific Research, and by the Office of Naval Research     

Spall fracture in sapphire

The dynamic tensile strength of sapphire has been studied under conditions of impact loading in the region of dynamic (Hugoniot) elastic limit (HEL). In the absence of inelastic deformation, the spall strength of sapphire strongly depends on the loading time and exhibits a tendency to decrease with increasing compressive stress in the incident impulse. The development of inelastic deformation leads to the complete loss of material resistance to the tensile straining.     

DYNAMIC INELASTIC RESPONSE AND BUCKLING OF METAL MATRIX COMPOSITE INFINITELY WIDE PLATES DUE TO THERMAL SHOCKS

SUMMARY

A micromechanical approach is combined with a structural analysis in order to investigate the thermally induced response and dynamic thermal buckling of infinitely wide plates composed of elastic-viscoplastic matrix reinforced by elastic fibres. The micromechanical analysis relies on the thermoelastic and inelastic properties of the constituents of the composite, and provides the instantaneous effective relation for the metal matrix composite at any point of the structure. The structural analysis employs the classical and higher-order plate theories in conjunction with a spatial finite difference and temporal Runge-Kutta integrations to generate the dynamic response of the plate. Results are given that illustrate the effects of the metallic matrix inelasticity, shear deformation, fibre orientation and initial geometrical imperfection on the dynamic response of the metal matrix composite plate subjected to various types of thermal shocks.     

Dynamic mechanical response of elastic spherical inclusions to impulsive acoustic radiation force excitation

Acoustic radiation force impulse imaging has been used clinically to study the dynamic response of lesions relative to their background material to focused, impulsive acoustic radiation force excitations through the generation of dynamic displacement field images. Dynamic displacement data are typically displayed as a set of parametric images, including displacement immediately after excitation, maximum displacement, time to peak displacement, and recovery time from peak displacement. To date, however, no definitive trends have been established between these parametric images and the tissues' mechanical properties. This work demonstrates that displacement magnitude, time to peak displacement, and recovery time are all inversely related to the Young's modulus in homogeneous elastic media. Experimentally, pulse repetition frequency during displacement tracking limits stiffness resolution using the time to peak displacement parameter. The excitation pulse duration also impacts the time to peak parameter, with longer pulses reducing the inertial effects present during impulsive excitations. Material density affects tissue dynamics, but is not expected to play a significant role in biological tissues. The presence of an elastic spherical inclusion in the imaged medium significantly alters the tissue dynamics in response to impulsive, focused acoustic radiation force excitations. Times to peak displacement for excitations within and outside an elastic inclusion are still indicative of local material stiffness; however, recovery times are altered due to the reflection and transmission of shear waves at the inclusion boundaries. These shear wave interactions cause stiffer inclusions to appear to be displaced longer than the more compliant background material. The magnitude of shear waves reflected at elastic lesion boundaries is dependent on the stiffness contrast between the inclusion and the background material, and the stiffness and size of the inclusion dictate when shear wave reflections within the lesion will interfere with one another. Jitter and bias associated with the ultrasonic displacement tracking also impact the estimation of a tissue's dynamic response to acoustic radiation force excitation.     

Piezoelectric vibrations of composite beams with interlayer slip

Summary Actuating piezoelectric effects in two-layer beams with interlayer slip are described in detail, and special attention is given to the identification of the piezoelectric actuation as eigenstrains. It is demonstrated that piezoelectrically induced strains conveniently can be interpreted as eigenstrains acting in a background composite beam without piezoelectric actuators. The analogy between the piezoelectric effect and that of thermal strains is utilized in the present paper, where a layer-wise first-order flexural theory is applied to two-layer beams with various boundary conditions. The layers are assumed to be made of piezoelectric materials. Bernoulli-Euler hypothesis is assumed to hold for each layer separately, and a linear constitutive relation between the horizontal slip and the interlaminar shear force is considered. The governing sixth-order initial-boundary value problem is solved by separating the dynamic response in a quasistatic and in a complementary dynamic portion. The quasistatic solution that may also contain singularities or discontinuities due to sudden load changes is determinded in a closed form. The remaining complementary dynamic part is nonsingular and can be approximated by a truncated modal series of accelerated convergence. The proposed procedure is illustrated for piezoelectrically induced flexural deformations, where the forcing function is the piezoelectric curvature.     

Numerical modeling of elastohydrodynamic lubrication in point or line contact for heterogeneous elasto-plastic materials

A semi-analytic solution is developed for heterogeneous elasto-plastic materials with inhomogeneous inclusions under elastohydrodynamic lubrication in point contact or line contact. The inhomogeneous inclusions within a material are homogenized as homogeneous inclusions with properly determined eigenstrains based on the equivalent inclusion method, and the surface displacements induced by these eigenstrains are then introduced into the gap between the contact bodies to update surface geometry. The accumulative plastic deformation is iteratively obtained by a procedure involving a plasticity loop and an incremental loading process. The model takes into account the interactions among the contact bodies, the embedded inclusions, and the plastic zones, thus leading to a solution of the surface pressure distributions, film thickness profiles, plastic zones, and subsurface stress fields. This solution is of great importance for the analysis of elasto-plasto damage of heterogeneous materials subject to lubricated contact.     

An efficient implementation of stress resultant plasticity in analysis of Reissner-Mindlin plates

An efficient solution procedure for elasto-plastic analysis of Reissner-Mindlin plates has been proposed in this work. The main ingredients which render efficiency are: what appears to be an optimally tuned four-node plate element with assumed shear strains and incompatible bending modes; an elasto-plastic constitutive model given directly in terms of stress resultants; and efficient computation of plastic flow which simplifies to a solution of a single scalar equation and remaining state update computation. The performance of the element has been proved very satisfying in both elastic singularity-dominated and elasto-plastic problems.     

Solution for Eshelby's elastic inclusions in a finite plate using boundary integral equation method

This paper provides a solution for Eshelby's elastic inclusions in a finite plate based on the complex variable boundary integral equation (CVBIE) method. In the formulation, an inclusion with Eshelby's eigenstrains is embedded in an elliptic plate, and the exterior boundary is applied by some static loading. Two BIEs are suggested in the present study. One of BIEs is used for the finite inclusion region, and the other is used for region bounded by interface and the exterior boundary. After the discretization of BIEs, a numerical solution is suggested. In the solution, an inverse matrix technique is suggested which can eliminate one unknown vector in advance. Three numerical examples under different generalized loadings are provided. Interaction between the prescribed eigenstrains and the static loading along the exterior boundary is studied in detail.