Analysis For Post-Buckling In Micro-Composite Films

Credit: Pixabay

Multiscale structures are all around the world, ranging from the Statue of Liberty at macroscale to drug delivery or biomedical robotics at micro/nanoscale. In order to tune and optimize the mechanical performance of those structures, the mission of solid mechanics has been experiencing a paradigm shift from traditionally preventing structural failure to efficiently deploying advanced structures to achieve multifunctional performance. The future direction of solid mechanics is thus aimed to predict and tailor the properties of novel geometric forms for improved behavior and multifunctional features.

Structural instability, e.g., postbuckling, is one of the promising phenomena leads to those novel features. In particular, microelements, typically designed as slender microfilms subjected to axial compression, have been extensively implemented in micro/nanoscale actuators, sensors and various Micro/Nano-Electro-Mechanical Systems (MEMS and NEMS), e.g., switches, damage detection, remote sensing, energy harvesting.

Motivated by harnessing the postbuckling characteristics of advanced materials and structures at the multiscale, we conducted studies on, 1) fundamental understanding of axially loaded members subjected to bilateral constraints [1,2]; and 2) multifunctional applications such as energy harvesting [3] and damage sensing [4].

Fig. 1(a) demonstrates the mechanism of postbuckling-induced energy harvesting. Deploying postbuckling behavior of bilaterally confined macroscale beams subjected to axial compression, piezoelectric cantilevers attached to the beams are activated to generate electrical power for energy harvesting. Fig. 1(b) indicates the applications in structural health monitoring for charging wireless sensors [5].

It is of desire to sufficiently harness the mechanical response of the slender elastica in the postbuckling-induced energy harvester such that the generated electrical power can be maximized. Studies have been carried out to explore the controllability of the bilaterally confined beams at the multiscale.  Jiao et al. [5] reported theoretical and numerical models to predict and tune the postbuckling behavior, i.e., force-displacement relation and deflection configuration, of micro-composite films (MCF) under different boundary conditions.

Fig. 1. (a) Postbuckling-induced energy harvesting mechanism, and (b) applications in structural health monitoring (SHM) [3].
Fig. 2(a). Postbuckling-inspired energy harvesting technique using piezoelectric transducer subjected to axially quasi-static force, and (b) tailoring bilateral constraints of micro-composite beams to achieve tunable postbuckling response [5].
Fig. 2(a) illustrates the axially-loaded MCF subjected to discontinuously flexible constraints. A theoretical model is developed to investigate the postbuckling characteristics of the irregularly confined MCF subjected to dynamic loading. Hamilton’s principle is used to obtain the governing equations using the Euler-Bernoulli beam theory and modified couple stress theory. The theoretical model is solved using an energy method by minimizing the total energy of the deflected MCF. A novel discretization method is proposed to convert the irregular confinements into gap vectors, and the Nelder-Mead algorithm is expanded to numerically solve the constrained minimization.

Fig. 2(b) presents the discretization algorithm that converts the irregular constraints into gap vectors. A total of n discontinues confinements are discretized into q segments each. The gap vectors are obtained with respect to different constraint conditions, i.e.,

  • vertical displacement of the MCF wr (x) is smaller than the initial gap h;
  • wr (x) is larger than h, and the segment is constrained by the irregular walls; and
  • wr (x) is larger than h, but the segment is located in the space between the irregular walls.

Fig. 3 presents the highest achievable buckling mode of the irregularly-confined MCF solved using the proposed model. The theoretical predictions are obtained with respect to the material factor ϒ (i.e., the ratio of walls gap-to-material length factor ϒ = n/ς ) and the geometric factor α (i.e., the ratio of MCF length-to-thickness a = L/t). The highest reachable mode is critically increased when the gap is enlarged. However, when ϒ is adequately large (namely ϒ > 10) the achievable buckling mode is dramatically reduced. On the other hand, a positively affects the highest reachable mode of the flexibly constrained MCF.

The presented model accurately predicts and tunes the postbuckling response of the axially-loaded MCF subjected to irregularly bilateral constraints, which could be deployed for applications that require structural instability with multistable deflection configuration and full recovery after large deformation such as MEMS or wings of ultralight micro air vehicles (MAVs).


Fig. 3. Highest achievable buckling mode of the irregularly constrained MCF with respect to the material factor ϒ and geometric factor a.

These findings are described in the article entitled Micro-composite films constrained by irregularly bilateral walls: A size-dependent post-buckling analysis, recently published in the journal Composite StructuresThis work was conducted by Pengcheng Jiao from the University of PennsylvaniaAmir H. Alavi from the University of Missouri, and Wassim Borchani and Nizar Lajnef from Michigan State University


  1. Jiao, P., Borchani, W. and Lajnef, N. (2017). Large deformation solutions to static and dynamic instabilities of post-buckled beam systems. International Journal of Solids and Structures. 128(1): 85-98.
  2. Jiao, P., Alavi, A.H., Borchani, W. and Lajnef, N. (2017). Small and large deformation models of post-buckled beams under lateral constraints. Mathematics and Mechanics of Solids.
  3. Jiao, P., Borchani, W., Alavi, A.H. Hasni, H. and Lajnef, N. (2017). An energy harvesting and damage sensing solution based on post-buckling response of non-uniform cross-section beams. Structural Control and Health Monitoring. e2052: 1-19.
  4. Jiao, P., Borchani, W., Hasni, H. and Lajnef, N. (2017). A new solution of measuring thermal response of prestressed concrete bridge girders for structural health monitoring. Measurement Science and Technology. 28(8): 085005.
  5. Jiao, P., Alavi, A.H., Borchani, W. and Lajnef, N. (2018). Micro-composite films (MCF) constrained by irregularly bilateral walls: A size-dependent post-buckling analysis. Composite Structures. 195: 219-231.
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