A consistent higher-order shear deformation non-linear theory is developed for shells of generic shape, taking geometric imperfections into account. The geometrically non-linear strain–displacement ...relationships are derived retaining full non-linear terms in the in-plane displacements; they are presented in curvilinear coordinates in a formulation ready to be implemented. Then, large-amplitude forced vibrations of a simply supported, laminated circular cylindrical shell are studied (i) by using the developed theory, and (ii) keeping only non-linear terms of the von Kármán type. Results show that inaccurate results are obtained by keeping only non-linear terms of the von Kármán type for vibration amplitudes of about two times the shell thickness for the studied case.
Postbuckling analysis of functionally graded ceramic–metal plates under edge compression and temperature field conditions is presented using the element-free
kp-Ritz method. The first-order shear ...deformation plate theory is employed to account for the transverse shear strains, and the von Kármán-type nonlinear strain–displacement relationship is adopted. The effective material properties of the functionally graded plates are assumed to vary through their thickness direction according to the power-law distribution of the volume fractions of the constituents. The displacement fields are approximated in terms of a set of mesh-free kernel particle functions. Bending stiffness is estimated using a stabilised conforming nodal integration approach, and, to eliminate the membrane and shear locking effects for thin plates, the shear and membrane terms are evaluated using a direct nodal integration technique. The solutions are obtained using the arc–length iterative algorithm in combination with the modified Newton–Raphson method. The effects of the volume fraction exponent, boundary conditions and temperature distribution on postbuckling behaviour are examined.
A four-field mixed variational principle is proposed for large deformation analysis of Kirchhoff rods with the lowest order C0 mixed FE approximations. The core idea behind the approach is to ...introduce a one-parameter family of points (the centerline) and a separate one-parameter family of orthonormal frames (the Cartan moving frame) that are specified independently. The curvature and torsion of the curve are then related to the relative rotation of neighboring frames. The relationship between the frame and the centerline is then enforced at the solution step using a Lagrange multiplier (which plays the role of section force). In case of Kirchhoff rods, the cross sectional orientation can be described using frames like the Frenet–Serret, which are defined only using the centerline, thereby demanding higher-order smoothness for the centerline approximation. Decoupling the frame from the position vector of the base curve leads to a description of torsion and curvature that is independent of the position information, thus allowing for simpler interpolations. The four-field mixed variational principle we propose has the frame, section force, extension strain and position vector as input arguments. We discretize the position vector using linear Lagrange shape functions, while the frames are interpolated as piecewise geodesics on the rotation group. Finite element approximations for extensional strain and section force are constructed using constant shape functions. Using these discrete approximations, a discrete mixed variational principle is laid out which is then numerically extremized. The discrete approximation is then applied to a few benchmark problems. Vis-á-vis most available approaches, our numerical studies reveal an impressive performance of the proposed method without numerical instabilities or locking.
The paper analyzes the elastic–plastic buckling behavior of thick, rectangular nanoplates embedded in a Winkler–Pasternak foundation, adopting the Reddy third-order plate theory in nonlocal ...elasticity. Elasto-plasticity is accounted for by considering two alternative plasticity theories, namely the J2 flow incremental and the J2 deformation theory, with material properties defined by a Ramberg–Osgood relation. An iterative procedure is proposed to obtain the critical load, and the corresponding critical mode, of plates simply supported on two opposite edges under applied uniaxial and biaxial loading conditions. Extensive analysis investigates the effects of geometrical, constitutive, and nonlocal parameters on the critical behavior of plates with different boundary conditions. To the best of the authors’ knowledge, there are no findings about elastoplastic buckling of nanoplates in the existing literature. It is therefore hoped that the results obtained may provide a helpful basis for comparison for future investigations.
•Buckling analysis of elastoplastic thick nanoplates.•Closed form solution for Third-order Reddy plate theory.•Investigation on the effects of geometrical, constitutive, nonlocal parameters.•Elastoplasticity modeled in both flow incremental and deformation theories.
In the present work, a nonlinear thermo-electro-mechanical response of functionally graded piezoelectric material (FGPM) actuators is investigated. The theoretical formulation is based on the ...Timoshenko beam theory with the von Kármán nonlinearity (in the form of midplane stretching), and a microstructural length scale is incorporated by means of the modified couple stress theory. A power-law distribution of thermal, electrical, and mechanical properties through beam thickness (or height) is assumed. The governing equations are derived using the principle of virtual displacements. A displacement finite element model of the theory is developed, and the resulting system of nonlinear algebraic equations is solved with the help of Newton's iteration method. Numerical results are presented for transverse deflection as a function of load parameters and out-of-plane boundary conditions. The parametric effects of microstructural length scale parameter, power-law index of the material distribution across the thickness, boundary conditions, beam geometry, and applied actuator voltage on the beam response are investigated through various numerical examples. The results reveal the existence of bifurcation (or critical states) for certain types of in-plane loads. For other load types, including out-of-plane loads, the beam undergoes a unique and stable deflection path that does not contain any critical point.
•Accounts for shear deformation with geometric nonlinearity and size effects.•Buckling analysis with identification of critical states is carried out.•Includes the effects of microstructure dependency, and material distribution.•Includes the effects of boundary conditions, beam geometry, and applied voltage.
It has been well established that the internal length scale related to the cell size plays a critical role in the response of architected structures. It this paper, a Volterra derivative-based ...approach for deriving nonlocal continuum laws directly from an energy expression without involving spatial derivatives of the displacement is proposed. A major aspect of the work is the introduction of a nonlocal derivative-free directionality term, which recovers the classical deformation gradient in the infinitesimal limit. The proposed directionality term avoids issues with correspondences under nonsymmetric conditions (such a unequal distribution of points that cause trouble with conventional correspondence-based approaches in peridynamics). Using this approach, we derive a nonlocal version of a shear deformable beam model in the form of integro-differential equations. As an application, buckling analysis of architected beams with different core shapes is performed. In this context, we also provide a physical basis for the consideration of energy for nonaffine (local bending) deformation. This removes the need for additional energy in an ad hoc manner towards suppressing zero-energy modes. The numerical results demonstrate that the proposed framework can accurately estimate the critical buckling load for a beam in comparison to 3-D simulations at a small fraction of the cost and computational time. Efficacy of the framework is demonstrated by analysing the responses of a deformable beam under different loads and boundary conditions.
AbstractThis paper is concerned with the bending response of nonlocal elastic beams under transverse loads, where the nonlocal elastic model of Eringen, also called the stress gradient model, is ...used. This model is known to exhibit some paradoxical responses when applied to beams with certain types of boundary conditions. In particular, for clamped-free boundary condition, this nonlocal model is not able to predict scale effects in the presence of concentrated loads, or it leads to an apparent stiffening effect for distributed loads in contrast to other boundary conditions for which softening effect is observed. In the literature, these paradoxes have been resolved by changing the kernel of the nonlocal model or by modifying the standard boundary conditions. In this paper, the paradox is solved from the nonlocal differential model itself via some related discontinuous nonlocal kinematics. It is shown that the kinematics related to the nonlocal constitutive law lead to the use of moment or shear discontinuities. With such a nonlocal differential model coupled with the nonlocal discontinuity requirements, the beam effectively shows a softening response irrespective of the boundary conditions studied, including the clamped-free boundary conditions, and thereby resolves the paradox. The model is also compared to lattice-based solutions where an excellent agreement between the present nonlocal model and the lattice one is obtained. Finally, the stress gradient model is shown to be cast in a stress-based variational framework, which coincides with a Timoshenko-type model where the shear effect is shown to play the nonlocal role.
The uncontrolled proliferation of cancer cells causes the growth of the tumor mass. Consequently, the normal surrounding tissue exerts a compressive force on the tumor mass to oppose its expansion. ...These stresses directly promote tumor metastasis and invasion and affect drug delivery. In the past, the mechanical behavior of solid tumors has been extensively studied using linear elastic and nonlinear hyperelastic constitutive models. In this study, we develop a two-dimensional biomechanical model based on the biphasic assumption of the solid matrix and fluid phase of the tissues. Heterogeneous vasculature and nonuniform blood perfusion are also investigated by incorporating in the model a necrotic core and a well-vascularized zone. The findings of our study demonstrate a significant difference between the linear and nonlinear tissue responses to stress, while the interstitial fluid pressure (IFP) distribution is found to be independent of the constitutive model. The proposed biphasic model may be useful for elasticity imaging techniques aiming at predicting stress and IFP in tumors.
In this paper, a new kinematic beam model based on a five-parameter displacement field is proposed in a geometric nonlinear framework. The proposed displacement field enriches the classical ...Timoshenko beam displacement field with two additional parameters, accounting for the Poisson effect. The equilibrium equations have been derived through a variational approach, and the linearized equations are solved analytically. The adoption of the linear solution as approximation functions for the nonlinear case allows prediction of nonlinear response of problems involving complex geometries with a relatively small computational effort. Several numerical examples of benchmark problems are analyzed, highlighting the characteristic features of the proposed five-parameter model and comparing the results with those obtained using the classical Bernoulli beam model and 3D finite element model. Numerical results show that, although for thin beams the differences in generalized displacement are generally negligible, the proposed model predicts a comparable stress field only when the Poisson ratio is equal to zero. Conversely, the stress field is meaningfully enriched by the new parameters, with significant differences where the Poisson effect is more pronounced.
A peridynamic theory for linear elastic shells Chowdhury, Shubhankar Roy; Roy, Pranesh; Roy, Debasish ...
International journal of solids and structures,
05/2016, Letnik:
84
Journal Article
Recenzirano
Odprti dostop
A state-based peridynamic formulation for linear elastic shells is presented. The emphasis is on introducing, possibly for the first time, a general surface based peridynamic model to represent the ...deformation characteristics of structures that have one geometric dimension much smaller than the other two. A new notion of curved bonds is exploited to cater for force transfer between the peridynamic particles describing the shell. Starting with the three dimensional force and deformation states, appropriate surface based force, moment and several deformation states are arrived at. Upon application on the curved bonds, such states yield the necessary force and deformation vectors governing the motion of the shell. By incorporating a shear correction factor, the formulation also accommodates analysis of shells that have higher thickness. In order to attain this, a consistent second order approximation to the complementary energy density is considered and incorporated in peridynamics via constitutive correspondence. Unlike the uncoupled constitution for thin shells, a consequence of a first order approximation, constitutive relations for thick shells are fully coupled in that surface wryness influences the in-plane stress resultants and surface strain the moments. Our proposal on the peridynamic shell theory is numerically assessed against simulations on static deformation of spherical and cylindrical shells, that of flat plates and quasi-static fracture propagation in a cylindrical shell.
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