Showing papers on "Displacement field published in 2014"

New Quasi-3D Hyperbolic Shear Deformation Theory for the Static and Free Vibration Analysis of Functionally Graded Plates

Abstract: In this paper, a new quasi-three-dimensional (3D) hyperbolic shear deformation theory for the bending and free vibration analysis of functionally graded plates is developed. By dividing the transverse displacement into bending, shear, and thickness stretching parts, the number of unknowns and governing equations of the present theory is reduced, and hence makes it simple to use. The present plate theory approach accounts for both transverse shear and normal deformations and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factor. Unlike any other theory, the number of unknown functions involved in displacement field is only five, as against six or more in the case of other shear and normal deformation theories. A comparison with the corresponding results is made to check the accuracy and efficiency of the present theory.

An efficient and simple refined theory for buckling and free vibration of exponentially graded sandwich plates under various boundary conditions

Abstract: In this paper, an efficient and simple refined shear deformation theory is presented for the vibration and buckling of exponentially graded material sandwich plate resting on elastic foundations under various boundary conditions. The displacement field of the present theory is chosen based on nonlinear variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theory is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton’s principle. Numerical results for the natural frequencies and critical buckling loads of several types of symmetric exponentially graded material sandwich plates are presented. The accuracy of the present theory is verified by comparing the obtained results with solutions available in the literature. Numerical results show that the present theory can archive accuracy c...

Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories

Abstract: The theoretical framework of the present manuscript covers the dynamic analysis of doubly-curved shell structures using the generalized displacement field of the Carrera Unified Formulation (CUF), including the Zig-Zag (ZZ) effect given by the Murakami’s function. The partial differential system of equations is solved by using the Generalized Differential Quadrature (GDQ) method. This numerical approach has been proven to be accurate, reliable and stable in several engineering applications. The current paper focuses on Functionally Graded (FG) doubly-curved shells and panels using various higher-order equivalent single layer theories, introduced and applied for the first time by the authors to completely doubly-curved shell structures, and different through-the-thickness volume fraction distributions, such as four-parameter power law, Weibull and exponential distributions. Moreover, the classic theory of mixtures is compared to the Mori–Tanaka scheme for the calculation of the mechanical properties of the materials. In particular, the numerical applications presented in this work are related to particular FG configurations in which it is possible to model a soft-core structure using a continuous variation of the mechanical properties of the materials at hand. The natural frequencies and mode shapes of several structures are presented and compared to numerical solutions taken from the literature.

Phase field modeling of fracture in rubbery polymers. Part I: Finite elasticity coupled with brittle failure

Abstract: This work presents a new phase field model for rate-independent crack propagation in rubbery polymers at large strains and considers details of its numerical implementation. The approach accounts for micro-mechanically based features of both the elastic bulk response as well as the crack toughness of idealized polymer networks. The proposed diffusive crack modeling based on the introduction of a crack phase field overcomes difficulties associated with the computational realization of sharp crack discontinuities, in particular when it comes to complex crack topologies. The crack phase field governs a crack density function, which describes the macroscopic crack surface in the polymer per unit of the reference volume. It provides the basis for the constitutive modeling of a degrading free energy storage and a crack threshold function with a Griffith-type critical energy release rate, that governs the crack propagation in the polymer. Both the energy storage as well as the critical energy release due to fracture can be related to classical statistical network theories of polymers. The proposed framework of diffusive fracture in polymers is formulated in terms of a rate-type variational principle that determines the evolution of the coupled primary variable fields, i.e. the deformation field and the crack phase field. On the computational side, we outline a staggered solution procedure based on a one-pass operator split of the coupled equations, that successively updates in a typical time step the crack phase field and the displacement field. Such a solution algorithm is extremely robust, easy to implement and ideally suited for engineering problems. We finally demonstrate the performance of the phase field formulation of fracture at large strains by means of representative numerical examples.

Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory

Abstract: It has been known since the pioneering works by Piola, Cosserat, Mindlin, Toupin, Eringen, Green, Rivlin and Germain that many micro-structural effects in mechanical systems can be still modeled by means of continuum theories. When needed, the displacement field must be complemented by additional kinematical descriptors, called sometimes microstructural fields. In this paper, a technologically important class of fibrous composite reinforcements is considered and their mechanical behavior is described at finite strains by means of a second-gradient, hyperelastic, orthotropic continuum theory which is obtained as the limit case of a micromorphic theory. Following Mindlin and Eringen, we consider a micromorphic continuum theory based on an enriched kinematics constituted by the displacement field u and a second-order tensor field ψ describing microscopic deformations. The governing equations in weak form are used to perform numerical simulations in which a bias extension test is reproduced. We show that second-gradient energy terms allow for an effective prediction of the onset of internal shear boundary layers which are transition zones between two different shear deformation modes. The existence of these boundary layers cannot be described by a simple first-gradient model, and its features are related to second-gradient material coefficients. The obtained numerical results, together with the available experimental evidences, allow us to estimate the order of magnitude of the introduced second-gradient coefficients by inverse approach. This justifies the need of a novel measurement campaign aimed to estimate the value of the introduced second-gradient parameters for a wide class of fibrous materials.

Isogeometric analysis of functionally graded plates using a refined plate theory

Abstract: We present in this paper a simple and effective approach that incorporates isogeometric finite element analysis (IGA) with a refined plate theory (RPT) for static, free vibration and buckling analysis of functionally graded material (FGM) plates. A new inverse tangent distributed function through the plate thickness is proposed. The RPT enables us to describe the non-linear distribution of shear stresses through the plate thickness without any requirement of shear correction factors (SCF). IGA utilizes basis functions namely B-splines or non-uniform rational B-splines (NURBS) which reach easily the smoothness of any arbitrary order. It hence satisfies the C1 requirement of the RPT model. The present method approximates the displacement field with four degrees of freedom per each control point allowing an efficient solution process.

An inverse finite element method for beam shape sensing: theoretical framework and experimental validation

Abstract: Shape sensing, i.e., reconstruction of the displacement field of a structure from surface-measured strains, has relevant implications for the monitoring, control and actuation of smart structures. The inverse finite element method (iFEM) is a shape-sensing methodology shown to be fast, accurate and robust. This paper aims to demonstrate that the recently presented iFEM for beam and frame structures is reliable when experimentally measured strains are used as input data.The theoretical framework of the methodology is first reviewed. Timoshenko beam theory is adopted, including stretching, bending, transverse shear and torsion deformation modes. The variational statement and its discretization with C0-continuous inverse elements are briefly recalled. The three-dimensional displacement field of the beam structure is reconstructed under the condition that least-squares compatibility is guaranteed between the measured strains and those interpolated within the inverse elements.The experimental setup is then described. A thin-walled cantilevered beam is subjected to different static and dynamic loads. Measured surface strains are used as input data for shape sensing at first with a single inverse element. For the same test cases, convergence is also investigated using an increasing number of inverse elements. The iFEM-recovered deflections and twist rotations are then compared with those measured experimentally. The accuracy, convergence and robustness of the iFEM with respect to unavoidable measurement errors, due to strain sensor locations, measurement systems and geometry imperfections, are demonstrated for both static and dynamic loadings.

Ground Movements due to Shallow Tunnels in Soft Ground. I: Analytical Solutions

Abstract: This paper presents simplified closed-form analytical solutions that can be used to interpret and predict ground movements caused by shallow tunneling in soft ground conditions These solutions offer a more comprehensive framework for understanding the distribution of ground movements than widely used empirical functions Analytical solutions for the displacement field within the ground mass are obtained for two basic modes of deformation corresponding to uniform convergence and ovalization at the wall of a circular tunnel cavity, based on the assumption of linear, elastic soil behavior Deformation fields based on the superposition of fundamental, singularity solutions are shown to differ only slightly from analyses that consider the physical dimensions of the tunnel cavity, except in the case of very shallow tunnels This work demonstrates a simplified method to account for soil plasticity in the analyses and illustrate closed-form solutions for a three-dimensional (3D) tunnel heading A companion paper describes applications of these analyses to interpret field measurements of ground response to tunneling

On a Theory of Thermoelastic Materials with a Double Porosity Structure

Abstract: In this article, we use the Nunziato–Cowin theory of materials with voids to derive a theory of thermoelastic solids, which have a double porosity structure. The new theory is not based on Darcy's law. In the case of equilibrium, in contrast with the classical theory of elastic materials with double porosity, the porosity structure of the body is influenced by the displacement field. We prove the uniqueness of solutions by means of the logarithmic convexity arguments as well as the instability of solutions whenever the internal energy is not positive definite. Later, we use semigroup arguments to prove the existence of solutions in the case that the internal energy is positive. The deformation of an elastic space with a spherical cavity is investigated.

Singularity-free dislocation dynamics with strain gradient elasticity

Abstract: The singular nature of the elastic fields produced by dislocations presents conceptual challenges and computational difficulties in the implementation of discrete dislocation-based models of plasticity. In the context of classical elasticity, attempts to regularize the elastic fields of discrete dislocations encounter intrinsic difficulties. On the other hand, in gradient elasticity, the issue of singularity can be removed at the outset and smooth elastic fields of dislocations are available. In this work we consider theoretical and numerical aspects of the non-singular theory of discrete dislocation loops in gradient elasticity of Helmholtz type, with interest in its applications to three dimensional dislocation dynamics (DD) simulations. The gradient solution is developed and compared to its singular and non-singular counterparts in classical elasticity using the unified framework of eigenstrain theory. The fundamental equations of curved dislocation theory are given as non-singular line integrals suitable for numerical implementation using fast one-dimensional quadrature. These include expressions for the interaction energy between two dislocation loops and the line integral form of the generalized solid angle associated with dislocations having a spread core. The single characteristic length scale of Helmholtz elasticity is determined from independent molecular statics (MS) calculations. The gradient solution is implemented numerically within our variational formulation of DD, with several examples illustrating the viability of the non-singular solution. The displacement field around a dislocation loop is shown to be smooth, and the loop self-energy non-divergent, as expected from atomic configurations of crystalline materials. The loop nucleation energy barrier and its dependence on the applied shear stress are computed and shown to be in good agreement with atomistic calculations. DD simulations of Lomer–Cottrell junctions in Al show that the strength of the junction and its configuration are easily obtained, without ad-hoc regularization of the singular fields. Numerical convergence studies related to the implementation of the non-singular theory in DD are presented.

A mesh-independent finite element formulation for modeling crack growth in saturated porous media based on an enriched-FEM technique

Abstract: In this paper, the crack growth simulation is presented in saturated porous media using the extended finite element method. The mass balance equation of fluid phase and the momentum balance of bulk and fluid phases are employed to obtain the fully coupled set of equations in the framework of $$u{-}p$$ formulation. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. The spatial discritization is performed using the extended finite element method, the time domain discritization is performed based on the generalized Newmark scheme, and the non-linear system of equations is solved using the Newton–Raphson iterative procedure. In the context of the X-FEM, the discontinuity in the displacement field is modeled by enhancing the standard piecewise polynomial basis with the Heaviside and crack-tip asymptotic functions, and the discontinuity in the fluid flow normal to the fracture is modeled by enhancing the pressure approximation field with the modified level-set function, which is commonly used for weak discontinuities. Two alternative computational algorithms are employed to compute the interfacial forces due to fluid pressure exerted on the fracture faces based on a ‘partitioned solution algorithm’ and a ‘time-dependent constant pressure algorithm’ that are mostly applicable to impermeable media, and the results are compared with the coupling X-FEM model. Finally, several benchmark problems are solved numerically to illustrate the performance of the X-FEM method for hydraulic fracture propagation in saturated porous media.

A microscale second gradient approximation of the damage parameter of quasi-brittle heterogeneous lattices

Abstract: Lattice models are powerful tools to investigate damage processes in quasi-brittle material by a microscale perspective. Starting from prior work on a novel rational damage theory for a 2D heterogenous lattice, this paper explores the connection between the series of critical strains at which the microcracks form (i.e. lattice links fail) and the second gradient of the microscale displacement field. Taking a simple tensile test as a representative case study for this endeavour, the analysis of accurate numerical results provides evidence that the second gradient of the microscale displacement field (notably the quantity | ∇ (∂ ux/∂ x)| for the specific example elaborated here) conveys indeed crucial information about the microcracks formation process and can be conveniently used to introduce simplifications of the rational theory that are of relevance by practical purposes as full field strain measurements become routinely possible with digital imaging correlation techniques. Note worthy, the results support the new view that the damage evolution is a three regimes process (I dilute damage, II homogeneous interaction, III localization.) The featured connection with the second gradient of the microscale displacement field is applicable in regions II–III, where microcracks interactions grow stronger and the lattice transitions to the softening regime. The potential impact of these findings towards the formulation of new and physically based CDM models, which are consistent with the reference discrete microscale theory, cannot be overlooked and is pointed out.

A refined and simple shear deformation theory for thermal buckling of solar functionally graded plates on elastic foundation

Abstract: A refined and simple shear deformation theory for thermal buckling of solar functionally graded plate (SFGP) resting on two-parameter Pasternak's foundations is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces. Therefore, there is no need to use shear correction factor. The number of independent unknowns of present theory is four, as against five in other shear deformation theories. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the present plate theory based on exact neutral surface position is employed to derive the governing stability equations. The nonlinear strain-displacement relations are also taken into consideration. The boundary conditions for the plate are assumed to be simply supported in all edges. Closed-form solutions are presented to calculate the critical buckling temperature, which are useful for engineers in design. The effects of the foundation parameters, plate dimensions, and power law index are presented comprehensively for the thermal buckling of solar functionally graded plates.

Crack-tip thermal and mechanical hysteresis in Shape Memory Alloys under fatigue loading

Abstract: Crack tip stress-induced phase transformation mechanisms in nickel–titanium alloys (NiTi), subjected to fatigue mechanical loads, have been analyzed by full field measurement techniques. In particular, Infrared thermography (IR) and Digital Image Correlation (DIC), have been applied to analyze the cyclic temperature and displacement evolutions in the crack tip region of a commercial pseudoelastic alloy, together with the associated thermal and mechanical hysteresis, by using Single Edge Crack (SEC) specimens. IR investigations revealed a global temperature variation of the specimen due to crack formation and propagation mechanisms, which is similar to common engineering metals, i.e. surface temperature rises quickly in an initial phase, then it reaches an almost constant value, and finally it increases rapidly as a consequence of the fatigue crack growth. In addition, cyclic thermal variation in the crack tip region and related hysteresis (temperature vs load) has been measured, which can been directly related to the thermal effects of the reversible stress-induced phase transformations. Furthermore, a proper experimental setup has been made, based on a reflection microscope, for direct measurements of the crack tip displacement field by the DIC technique. Furthermore, a fitting procedure has been developed to calculate the mode I Stress Intensity Factor (SIF), starting from the displacement field, and the related mechanical hysteresis (SIF vs load).

Displacement field analysis for cracking processes in bonded-particle model

Abstract: Bonded-particle model (BPM) is widely used to model the cracking processes in rock/rock-like materials. However, the discrete particles in the BPM cannot produce a continuous displacement field, which is comparable with those in physical experiments and numerical method such as the finite element method. Displacement trend lines are introduced to analyze the displacement field type of the BPM in the present study. Three displacement field types (DF_I, DF_II and DF_III) are defined and used in the analysis of the cracking processes. The study shows that the type of displacement field may evolve during the cracking processes. Researchers are advised to examine the displacement field before and right after a particular crack has formed. An examination of the associated type of displacement field can help reveal the nature of the crack.

Three-Dimensional Solid Brick Element Using Slopes in the Absolute Nodal Coordinate Formulation

Abstract: The present paper contributes to the field of flexible multibody systems dynamics. Two new solid finite elements employing the absolute nodal coordinate formulation are presented. In this formulation, the equations of motion contain a constant mass matrix and a vector of generalized gravity forces, but the vector of elastic forces is highly nonlinear. The proposed solid eight node brick element with 96 degrees of freedom uses translations of nodes and finite slopes as sets of nodal coordinates. The displacement field is interpolated using incomplete cubic polynomials providing the absence of shear locking effect. The use of finite slopes describes the deformed shape of the finite element more exactly and, therefore, minimizes the number of finite elements required for accurate simulations. Accuracy and convergence of the finite element is demonstrated in nonlinear test problems of statics and dynamics. [DOI: 10.1115/1.4024910]

Time-dependent elastic response to a local shear transformation in amorphous solids

Abstract: The elastic response of a two-dimensional amorphous solid to induced local shear transformations, which mimic the elementary plastic events occurring in deformed glasses, is investigated via molecular-dynamics simulations. We show that for different spatial realizations of the transformation, despite relative fluctuations of order one, the long-time equilibrium response averages out to the prediction of the Eshelby inclusion problem for a continuum elastic medium. We characterize the effects of the underlying dynamics on the propagation of the elastic signal. A crossover from a propagative transmission in the case of weakly damped dynamics to a diffusive transmission for strong damping is evidenced. In the latter case, the full time-dependent elastic response is in agreement with the theoretical prediction, obtained by solving the diffusion equation for the displacement field in an elastic medium.

A family of conforming mixed finite elements for linear elasticity on triangular grids

Abstract: This paper presents a family of mixed finite elements on triangular grids for solving the classical Hellinger-Reissner mixed problem of the elasticity equations. In these elements, the matrix-valued stress field is approximated by the full $C^0$-$P_k$ space enriched by $(k-1)$ $H(\d)$ edge bubble functions on each internal edge, while the displacement field by the full discontinuous $P_{k-1}$ vector-valued space, for the polynomial degree $k\ge 3$. The main challenge is to find the correct stress finite element space matching the full $C^{-1}$-$P_{k-1}$ displacement space. The discrete stability analysis for the inf-sup condition does not rely on the usual Fortin operator, which is difficult to construct. It is done by characterizing the divergence of local stress space which covers the $P_{k-1}$ space of displacement orthogonal to the local rigid-motion. The well-posedness condition and the optimal a priori error estimate are proved for this family of finite elements. Numerical tests are presented to confirm the theoretical results.

Origin and effect of nonlocality in a composite

Abstract: A simple demonstration of nonlocality in a heterogeneous material is presented. By analysis of the microscale deformation of a two-component layered medium, it is shown that nonlocal interactions necessarily appear in a homogenized model of the system. Explicit expressions for the nonlocal forces are determined. The way these nonlocal forces appear in various nonlocal elasticity theories is derived. The length scales that emerge involve the constituent material properties as well as their geometrical dimensions. A peridynamic material model for the smoothed displacement field is derived. It is demonstrated by comparison with experimental data that the incorporation of nonlocality in modeling dramatically improves the prediction of the stress concentration in an open hole tension test on a composite plate.

Numerical modeling tensile failure behavior of concrete at mesoscale using extended finite element method

Abstract: Uniaxial tensile behavior of concrete plays an important role in the behavior of concrete specimens as well as structural elements. A 2D mesoscale analysis is performed to investigate the fracture process of concrete subjected to uniaxial tension by using the extended finite element method. The concrete considered includes the hydrated cement paste, aggregate particles, and the interfacial transition zones. In the cracked domain, to represent the discontinuities of a displacement field, the Heaviside jump function is added to the common finite element approximation for the local enrichment based on the framework of partition of unity. A user-defined subroutine for the extended finite element method is implemented with FORTRAN codes and embedded into the commercial software ABAQUS. The failure process of the L-specimen is studied, and good agreement is obtained between the experimental observation and the present simulation results. Based on the extended finite element method, the fracture process of the c...