Realization of active metamaterials with odd micropolar elasticity

Materials made from active, living, or robotic components can display emergent properties arising from local sensing and computation. Here, we realize a freestanding active metabeam with piezoelectric elements and electronic feed-forward control that gives rise to an odd micropolar elasticity absent in energy-conserving media. The non-reciprocal odd modulus enables bending and shearing cycles that convert electrical energy into mechanical work, and vice versa. The sign of this elastic modulus is linked to a non-Hermitian topological index that determines the localization of vibrational modes to sample boundaries. At finite frequency, we can also tune the phase angle of the active modulus to produce a direction-dependent bending modulus and control non-Hermitian vibrational properties. Our continuum approach, built on symmetries and conservation laws, could be exploited to design others systems such as synthetic biofilaments and membranes with feed-forward control loops.

On the flexible needle insertion into the human liver

In the present research, the navigation of a flexible needle into the human liver in the context of the robotic-assisted intraoperative treatment of the liver tumors, is reported. Cosserat (micropolar) elasticity is applied to describe the interaction between the needle and the human liver. The theory incorporates the local rotation of points and the couple stress (a torque per unit area) as well as the force stress (force per unit area) representing the chiral features of the human liver. To predict the deformation of the needle and the liver, the elastic properties of the human liver have been evaluated. Outcomes reveal that considering smaller deformations of the needle and the liver results in better needle navigation mechanism. The needle geometry can enhance the penetration.

Modeling of the Flexible Needle Insertion into the Human Liver

The insertion of the needle is difficult because the deformation and displacement of the organs are the key elements in the surgical act. Liver and tumor modeling are essential in the development of the needle insertion model. The role of the needle is to deliver into the tumor an active chemotherapeutic agent. We describe in this chapter the deformation of the needle during its insertion into the human liver in the context of surgery simulation of the high- robotic-assisted intraoperative treatment of liver tumors based on the integrated imaging-molecular diagnosis. The needle is a bee barbed type modeled as a flexible thread within the framework of the Cosserat (micropolar) elasticity theory.

Wave-like solutions in Cosserat micropolar elasticity

<p>The Cosserat model generalises an elastic material taking into account the possible microstructure of the elements of the material continuum. In particular, within the Cosserat model the structured material point is rigid and can only experience microrotations, which is also known as micropolar elasticity. The propagation of elastic waves in such a medium is studied and we find two classes of waves, transversal rotational waves and longitudinal rotational waves, both of which are solutions of the nonlinear partial differential equations. For certain parameter choices, the transversal wave velocity can be greater than the longitudinal wave velocity.  We couple the rotational waves to linear elastic waves to study the behaviour of the coupled system and find wave-like solutions with differing wave speeds. In addition we also consider the so-called Cosserat coupling term. In this setting we seek soliton type solutions assuming small elastic displacements, however, we allow the material points to experience full rotations which are not assumed to be small.</p>

Harmonic waves of a given azimuthal number in a micropolar cylindrical waveguide

Рассматривается система двух связанных векторных дифференциальных уравнений линейной теории микрополярной упругости, сформулированная в терминах перемещений и микровращений в случае гармонической зависимости перемещений и микровращений от времени. Вводятся потенциалы перемещений и микровращений. Выполнено расщепление связанных векторных дифференциальных уравнений микрополярной теории упругости для потенциалов на несвязанные винтовые уравнения, опираясь на пропорциональность (с разными масштабными факторами) вихревых составляющих перемещений и микровращений только одному вихревому винтовому полю. Найдено представление векторов перемещений и микровращений с помощью четырех винтовых векторов. Оно обеспечивает выполнимость связанных векторных дифференциальных уравнений линейной теории микрополярной упругости. Проблема нахождения вихревых составляющих перемещений и микровращений приведена к решению четырех несвязанных между собой векторных винтовых дифференциальных уравнений. Получено представление перемещений и микровращений с помощью двух несвязанных метагармонических векторов. Выполнено разделение пространственных переменных в уравнениях Гельмгольца в цилиндрической системе координат. Определены решения скалярного и векторного уравнений Гельмгольца в бесконечной цилиндрической области, содержащие ряд произвольных постоянных. В явном виде найдены представления векторов перемещений и микровращений в длинном линейном микрополярном цилиндре, содержащие восемь произвольных постоянных. Такого рода решения определяют формы гармонических волн перемещений и микровращений, распространяющихся вдоль оси длинного кругового цилиндра. Полученные представления для гармонических волн перемещений и микровращений имеют смысл только для волн, характеризующихся заданным азимутальным числом. The coupled system of vector differential equations of the linear theory of micropolar elasticity presented in terms of displacements and micro-rotations in the case of a harmonic dependence of physical fields on time is considered in the three different variants of which the two are due to W. Nowacki and H. Neuber. A new scheme of splitting the coupled vector differential equation of the linear theory of micropolar elasticity into uncoupled ones is proposed. The scheme is based on proportionality of the vortex parts of the displacements and micro-rotations to the single vector, which satisfies the screw equation. The problem of determination of the vortex parts of the displacements and micro-rotations fields is reduced to solution of four uncoupled screw differential equations. A new representation of displacement and micro-rotation vectors is obtained by using two uncoupled metaharmonic vectors. The separation of spatial variables in the Helmholtz metaharmonic equations in a cylindrical coordinate net is described. Solutions of the scalar and vector Helmholtz equations in an infinite cylindrical domain containing a series of arbitrary constants are obtained. Representation of displacement and micro-rotation vectors in a long micropolar cylinder containing eight arbitrary constants are explicitly found. The corresponding solutions are proved to determine the modes of harmonic waves of displacements and micro-rotations propagating along the axis of a long circular cylinder. The obtained modes of the harmonic displacements and micro-rotations waves are valid only for those characterized by a given azimuthal number.