scholarly journals Phase-Field Fracture Modelling of Thin Monolithic and Laminated Glass Plates under Quasi-Static Bending

Materials ◽  
2020 ◽  
Vol 13 (22) ◽  
pp. 5153
Author(s):  
Jaroslav Schmidt ◽  
Alena Zemanová ◽  
Jan Zeman ◽  
Michal Šejnoha
Keyword(s):  
Fracture Energy ◽  
Phase Field ◽  
Dimensional Reduction ◽  
Scale Parameter ◽  
Length Scale Parameter ◽  
Multiple Cracks ◽  
Laminated Glass ◽  
Length Scale ◽  
Thin Glass ◽  
Glass Plates

A phase-field description of brittle fracture is employed in the reported four-point bending analyses of monolithic and laminated glass plates. Our aims are: (i) to compare different phase-field fracture formulations applied to thin glass plates, (ii) to assess the consequences of the dimensional reduction of the problem and mesh density and refinement, and (iii) to validate for quasi-static loading the time-/temperature-dependent material properties we derived recently for two commonly used polymer foils made of polyvinyl butyral or ethylene-vinyl acetate. As the nonlinear response prior to fracture, typical of the widely used Bourdin–Francfort–Marigo model, can lead to a significant overestimation of the response of thin plates under bending, the numerical study investigates two additional phase-field fracture models providing the linear elastic phase of the stress-strain diagram. The typical values of the critical fracture energy and tensile strength of glass lead to a phase-field length-scale parameter that is challenging to resolve in the numerical simulations. Therefore, we show how to determine the fracture energy concerning the applied dimensional reduction and the value of the length-scale parameter relative to the thickness of the plate. The comparison shows that the phase-field models provide very good agreement with the measured stresses and resistance of laminated glass, despite the fact that only one/two cracks are localised using the quasi-static analysis, whereas multiple cracks evolve during the experiment. It was also observed that the stiffness and resistance of the partially fractured laminated glass can be well approximated using a 2D plane-stress model with initially predefined cracks, which provides a better estimation than the one-glass-layer limit.

Dynamic stiffness formulation for a micro beam using Timoshenko–Ehrenfest and modified couple stress theories with applications

2021 ◽  
pp. 107754632110482
Author(s):  
J Ranjan Banerjee ◽  
Stanislav O Papkov ◽  
Thuc P Vo ◽  
Isaac Elishakoff
Keyword(s):  
Free Vibration ◽  
Couple Stress ◽  
Scale Parameter ◽  
Couple Stress Theory ◽  
Dynamic Stiffness ◽  
Modified Couple Stress Theory ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Stress Theory ◽  
Modified Couple Stress

Several models within the framework of continuum mechanics have been proposed over the years to solve the free vibration problem of micro beams. Foremost amongst these are those based on non-local elasticity, classical couple stress, gradient elasticity and modified couple stress theories. Many of these models retain the basic features of the Bernoulli–Euler or Timoshenko–Ehrenfest theories, but they introduce one or more material scale length parameters to tackle the problem. The work described in this paper deals with the free vibration problems of micro beams based on the dynamic stiffness method, through the implementation of the modified couple stress theory in conjunction with the Timoshenko–Ehrenfest theory. The main advantage of the modified couple stress theory is that unlike other models, it uses only one material length scale parameter to account for the smallness of the structure. The current research is accomplished first by solving the governing differential equations of motion of a Timoshenko–Ehrenfest micro beam in free vibration in closed analytical form. The dynamic stiffness matrix of the beam is then formulated by relating the amplitudes of the forces to those of the corresponding displacements at the ends of the beam. The theory is applied using the Wittrick–Williams algorithm as solution technique to investigate the free vibration characteristics of Timoshenko–Ehrenfest micro beams. Natural frequencies and mode shapes of several examples are presented, and the effects of the length scale parameter on the free vibration characteristics of Timoshenko–Ehrenfest micro beams are demonstrated.

scholarly journals A Modified Couple Stress Elasticity for Non-Uniform Composite Laminated Beams Based on the Ritz Formulation

Molecules ◽  
2020 ◽  
Vol 25 (6) ◽  
pp. 1404 ◽  
Author(s):  
Farajollah Zare Jouneghani ◽  
Hamidraza Babamoradi ◽  
Rossana Dimitri ◽  
Francesco Tornabene
Keyword(s):  
Boundary Conditions ◽  
Couple Stress ◽  
Scale Parameter ◽  
Laminated Composite ◽  
Length Scale Parameter ◽  
Smart Devices ◽  
Small Scale ◽  
Length Scale ◽  
Couple Stress Elasticity ◽  
Modified Couple Stress

Due to the large application of tapered beams in smart devices, such as scanning tunneling microscopes (STM), nano/micro electromechanical systems (NEMS/MEMS), atomic force microscopes (AFM), as well as in military aircraft applications, this study deals with the vibration behavior of laminated composite non-uniform nanobeams subjected to different boundary conditions. The micro-structural size-dependent free vibration response of composite laminated Euler–Bernoulli beams is here analyzed based on a modified couple stress elasticity, which accounts for the presence of a length scale parameter. The governing equations and boundary conditions of the problem are developed using the Hamilton’s principle, and solved by means of the Rayleigh–Ritz method. The accuracy and stability of the proposed formulation is checked through a convergence and comparative study with respect to the available literature. A large parametric study is conducted to investigate the effect of the length-scale parameter, non-uniformity parameter, size dimension and boundary conditions on the natural frequencies of laminated composite tapered beams, as useful for design and optimization purposes of small-scale devices, due to their structural tailoring capabilities, damage tolerance, and their potential for creating reduction in weight.

A Modified Gradient Plasticity Theory for Micro-Bending and Micro-Torsion Size Effects

2004 ◽  
Author(s):  
George Z. Voyiadjis ◽  
Rashid K. Abu Al-Rub
Keyword(s):  
Size Effects ◽  
Scale Parameter ◽  
Plasticity Theory ◽  
Material Behavior ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Gradient Plasticity ◽  
Non Local ◽  
Material Length Scale ◽  
Material Length

The definition and magnitude of the intrinsic length scale are keys to the development of the theory of plasticity that incorporates size effects. Gradient plasticity theory with a material length scale parameter is successfully in capturing the size dependence of material behavior at the micron scale. However, a fixed value of the material length-scale is not always realistic and that different problems could require different values. Moreover, a linear coupling between the local and non-local terms in the gradient plasticity theory is not always realistic and that different problems could require different couplings. A generalized gradient plasticity model with a non-fixed length scale parameter is proposed. This model assesses the sensitivity of predictions in the way in which the local and non-local parts are coupled. The proposed model gives good predictions of the size effect in micro-bending tests of thin films and micro-torsion tests of thin wires.

Static Deflection and Pull-In Instability of the Electrostatically Actuated Bilayer Microcantilever Beams

2015 ◽  
Vol 07 (06) ◽  
pp. 1550090 ◽  
Author(s):  
M. Mojahedi ◽  
M. Rahaeifard
Keyword(s):  
Scale Parameter ◽  
Couple Stress Theory ◽  
Beam Theory ◽  
Modified Couple Stress Theory ◽  
Length Scale Parameter ◽  
Length Scale ◽  
Static Deflection ◽  
Static Behavior ◽  
Stress Theory ◽  
Electrostatically Actuated

This paper deals with the static behavior of an electrostatically actuated bilayered microswitch on the basis of the modified couple stress theory. The beam is modeled using Euler–Bernoulli beam theory and equivalent elastic modulus and length scale parameter are presented for the bilayer beam. Static deflection and pull-in voltage of the beam is calculated using numerical and analytical methods. The numerical method is based on an iterative approach while the homotopy perturbation method (HPM) is utilized for the analytical simulation. Results show that there is a very good agreement between these methods even in the vicinity of the pull-in instability. Moreover, the effects of different parameters such as thicknesses of layers and length scale parameter on the static deflection and instability of the microcantilever are studied. Results show that for the cases with the equivalent length scale parameter comparable to the thickness of beam, the size-dependency plays significant roles in the static behavior of the bilayer microcantilevers.

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