In this research work, biocomposites based on a ternary system containing softwood Kraft lignin (Indulin AT), poly-L-lactic acid (PLLA) and polyethylene glycol (PEG) have been developed. Two binary systems based on PLLA/PEG and PLLA/lignin have also been studied to understand the role of plasticizer (i.e., PEG) and filler (i.e., lignin) on the overall physicomechanical behavior of PLLA. All samples have been prepared by melt-blending. A novel approach has also been introduced to improve the compatibility between PLLA and PEG by using a transesterification catalyst under reactive-mixing conditions. In PEG plasticized PLLA flexibility increases with increasing content of PEG and no significant effect of the molecular weight of PEG on the flexibility of PLLA has been observed. Differential scanning calorimetry and size-exclusion chromatography along with FTIR analysis show the formation of PLLA-b-PEG copolymer for high temperature processed PLLA/PEG systems. On the other hand, binary systems containing lignin show higher stiffness than PLLA/PEG system and good adhesion between the particles and the matrix has been observed by scanning electron microscopy. However, a concomitant good balance in stiffness introduced by the lignin particles and flexibility introduced by PEG has been observed in the ternary systems. This study also showed that high temperature reactive melt-blending of PLLA/PEG leads to the formation of a segmented PLLA-b-PEG block copolymer

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated vertical constant load at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler-Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper. Precisely, the proposed problem is a generalization of similar cases in literature, consequently numerical comparisons are performed with these previous works, i.e. assuming linear elastic materials or assuming symmetric Ludwick type material with same behavior in tension and compression like aluminum alloy and annealed copper. After verifying a proper agreeing with the literature, in order to investigate the effect of the different material behavior on the horizontal and vertical displacements of the free end cross-section, numerical results are obtained for different values of elastic moduli and strain exponential coefficients. The arising conclusions are coherent with the assumed hypotheses and with similar works in literature.

The investigated cantilever beam is characterized by a constant rectangular cross-section and is subjected to a concentrated constant vertical load, to a concentrated constant horizontal load and to a concentrated constant bending torque at the free end. The same beam is made by an elastic non-linear asymmetric Ludwick type material with different behavior in tension and compression. Namely the constitutive law of the proposed material is characterized by two different elastic moduli and two different strain exponential coefficients. The aim of this study is to describe the deformation of the beam neutral surface and particularly the horizontal and vertical displacements of the free end cross-section. The analysis of large deflection is based on the Euler-Bernoulli bending beam theory, for which cross-sections, after the deformation, remain plain and perpendicular to the neutral surface; furthermore their shape and area do not change. On the stress viewpoint, the shear stress effect and the axial force effect are considered negligible in comparison with the bending effect. The mechanical model deduced from the identified hypotheses includes two kind of non-linearity: the first due to the material and the latter due to large deformations. The mathematical problem associated with the mechanical model, i.e. to compute the bending deformations, consists in solving a non-linear algebraic system and a non-liner second order ordinary differential equation. Thus a numerical algorithm is developed and some examples of specific results are shown in this paper.

Piezoelectric bimorph benders are a particular class of piezoelectric devices, which are characterized by the ability of producing flexural deformation greatly larger than the length or thickness deformation of a single piezoelectric layer. Piezoelectric bimorph benders were first developed by Sawyer in 1931 at the Brush Development Company. The performance of these actuators was rudimentary studied and improved much later, with the results of research on smart structures in 1980s. Piezoelectric benders have been used in different applications: in robotics, spoilers on missile fins, actuation for a quick-focusing lens, to control the vibration of a helicopter rotor blade and for many other purposes. Due to extensive dimensional reduction of devices and to high precision requested, the effect of erroneous parameter estimation and the fluctuation of parameters due to external reasons, sometimes, cannot be omitted. So, we consider mechanical, electrical and piezoelectric parameters as uniformly distributed around a nominal value and we calculate the distribution of natural frequencies of the device. We consider an efficient and accurate analytical model for piezoelectric bimorph. The model combines an equivalent single-layer theory for the mechanical displacements with layerwise-type approximation for the electric potential. First-order Timoshenko shear deformation theory kinematics and quadratic electric potentials are assumed in developing the analytical solution. Mechanical displacement and electric potential Fourier-series amplitudes are treated as fundamental variables, and full electromechanical coupling is maintained. Numerical analysis of simply supported bimorphs under free vibration conditions are presented for different length-to-thickness ratios (i.e., aspect ratio), and the results are verified by those obtained from the exact 2D solution. According to Timoshenko theory, a shear correction factor is introduced with a value proposed by Timoshenko (1922) and by Cowper (1966). Free vibration problem of simply supported piezoelectric bimorphs with series or parallel arrangement is investigated for the closed circuit condition, and the results for different length-to-thickness ratios are compared with those obtained from the exact 2D solution. Numerical examples are presented on bimorphs constituted by two orthotropic piezoceramic layers (PZT-5A material). The calculation of natural frequencies is based on a Weibull distribution, because it is capable to properly model a large class of stochastic behaviours. The effect of errors on the Weibull distribution of the natural frequencies is shown in terms of change of the Weibull parameters. The results show how the parameters errors are reflected on the natural frequencies and how an increment of the error is able to change the shape of the frequencies distribution.

Abstract:The aim of this paper is to calculate the horizontal and vertical displacements of a cantilever beam in large deflections. The proposed structure is composed with Ludwick material exhibiting a different behavior to tensile and compressive actions. The geometry of the cross-section is constant and rectangular, while the external action is a vertical constant load applied at the free end. The problem is nonlinear due to the constitutive model and to the large deflections. The associated computational problem is related to the solution of a set of equation in conjunction with an ODE. An approximated approach is proposed here based on the application Newton-Raphson approach on a custom mesh and in cascade with an Eulerian method for the differential equation.