Pure cross-anisotropy for geotechnical elastic potentials

The pure cross-anisotropy is understood as a special scaling of strain (or stress). The scaled tensor is used as an argument in the elastic stiffness (or compliance). Such anisotropy can be overlaid on the top of any elastic stiffness, in particular on one obtained from an elastic potential with its own stress-induced anisotropy. This superposition does not violate the Second Law. The method can be also applied to other functions like plastic potentials or yield surfaces, wherever some cross-anisotropy is desired. The pure cross-anisotropy is described by the sedimentation vector and at most two constants. Scaling with more than two purely anisotropic constants is shown impossible. The formulation was compared with experiments and alternative approaches. Static and dynamic calibration of the pure anisotropy is also discussed. Graphic representation of stiffness with the popular response envelopes requires some enhancement for anisotropy. Several examples are presented. All derivations and examples were accomplished using the algebra program Mathematica.

Две модели резинокордного слоя

Breker layers in a pneumatic tire are an important part in the tire construction. These layers have a metal cord resulting in substantial bending stiffness. When homogenizing such layers, a “shave” method is applied to the breaker layer. This results in a thinner layer having adequate stiffness in both tension and bending. In this work, a phenomenological approach is used to obtain the effective properties of a homogeneous anisotropic hyper elastic material of an equivalent layer. Two models utilize transverse isotropic or orthotropic potential used to describe the homogenized properties. Comparison is made between these models for the “shaved” rubber-cord layer based on numerical experiments. In both cases, the potentials are built on the basis of the Treloar or Mooney potentials. Note that in the case of an inhomogeneous thin layer, the traditional definition of homogenization needs to be modified. In previous works of the authors, it was proposed to determine 3D averaged elastic properties of a layer by surrounding it with a homogeneous material. This makes it possible to correctly take into account the fact that the boundary effect from the upper to lower surfaces that penetrates through the whole periodicity cell. A set of local problems formulated for the periodicity cell is proposed. This set is sufficient for determining elastic potential material parameters. Nonlinear local problems on a periodic cell are solved and the material constants of the elastic potential are determined. The applicability of the orthotropic potential (second model) is determined for the “shaved” layer. It was found that orthotropic properties are manifested relative to longitudinal shears. The results show the suitability of the proposed potential and the scheme for determining the material parameters.

Modelling of Flexible Adhesives in Simple Mechanical States with the Use of the Darijani–Naghdabadi Strain Tensors and Kirchhoff–de Saint-Venant Elastic Potential

Practical aspects of modelling of flexible adhesives with the energy conjugate measures of stress and strain of the Darijani–Naghdabadi (D-N) family are discussed. A possibility of description of materials exhibiting non-linear physical characteristics with the use of non-linear geometric relationships and linear elastic constitutive law is considered. Nominal stress vs. stretch relations are specified in cases of simple tension and simple shear with the use of the Kirchhoff–de Saint-Venant elastic potential and D-N energy conjugate stress and strain measures. Obtained theoretical estimates were compared with experimental results of simple tension and simple shear tests performed on Sika PM polyurethane (Cracow, Sika Poland). The deformation rate was fixed in order to minimize the influence of viscosity. Values of parameters in the definition of the D-N strain tensor were optimized in order to provide good agreement between model predictions and experimental results. Observed discrepancies indicate that the proposed approach is not appropriate for constitutive modelling of the PM polymer. The presented approach is proposed to be used as a simple design model providing practical formulas describing the behavior of materials of non-linear characteristics in chosen mechanical states. Admissible values of exponents are discussed regarding its bijectivity in a limited range of variation of principal stretches.

A gravity balancing assistant arm design in 3-D for rehabilitation of stroke patients

A gravity balancing assistant arm design in 3-D is a mechanical mechanism consisted of springs, rigid rods, joints and sliders, which can be modified to the geometry and inertia of the arm of stroke patients. This mechanism is designed without any controllers and motors, based solely on mechanical principles, to achieve a relative balance of gravitational potential energy and elastic potential energy, thereby reducing the burden on the arm of a stroke patient to facilitate rehabilitation. To achieve this function, first, the center of gravity of the patient’s arm will be positioned, and then the mounting position of the spring on the assistant arm will be determined. In this paper, the following objectives will be achieved: (i) the calculation of the gravitational potential energy and the elastic potential energy in the mechanism (ii) the simplification of the potential energy equation and the elimination of the coefficient of the items related to the angle. (iii) The comparison between 2-D and 3-D cases of the mechanism. (iv) The motion process of simulating the mechanism using MATLAB (v) Using MATLAB to create the energy plots (vi) Using SolidWorks to construct the prototype of the mechanism (vii) Describe the practical application and future extensions of this mechanism.

On the theory of linear micropolar hemitropic media

В статье рассматриваются вопросы применения относительных тензоров при моделирвоании гемитропных микрополярных сред. Вводится определяющая форма микрополярного упругого потенциала. С помощью принципа виртуальной работы получаются определяющие уравнения для силовых и моментных характеристик микрополярного континуума в терминах относительных тензоров. Приводятся уравнения движения микрополярного континуума в терминах относительных тензоров. Выводится финальная форма динамических уравнений для перемещений и микровращений в случае полуизотропной (гемитропной) симметрии. The paper deals with the application of relative tensors to modeling hemitropic micropolar media. The latter is of crucial importance for biomechanics, mechanics of growing solids and mechanics of metamaterials. The constitutive form of the micropolar elastic potential is discussed. The basic equations of micropolar continuum are derived due to the principle of virtual displacements. Differential equations of the micropolar continuum are given in terms of relative tensors. The final form of dynamic equations for displacements and microrotations in the case of semi-isotropic (hemitropic) micropolar continuum is derived and discussed.

Triple-Cell Origami Structure for Multistable Transition Sequences

With the infinite design space and the excellent folding-induced deformability, origami has been recognized as an effective tool for developing reconfigurable structures. Particularly, the multistable origami structure, which possesses more than one stable configuration that is distinct in shape and mechanical properties, has received wide research attention. Generally, the origami structure reaches a kinematic singularity point when switching among different stable configurations. At this critical state, multiple switching sequences are possible, and the actual transition is generally hard to predict. In this paper, evolving from the conventional bistable Miura-ori unit, a triple-cell origami structure with eight potential stable configurations is proposed, which serves as a platform for investigating the transition sequences among different stable configurations. To quantify the overall elastic potential of the structure, besides the conventional elastic energy originating from the rigid folding creases, extra elastic potential induced by the mismatch among the cells are introduced, so that folding of the triple-cell structure is no longer a strict single degree-of-freedom mechanism. Instead, the three cells can deform asynchronously to avoid reaching the kinematic singularity point. Hence, under displacement loading, the transition sequence of the multistable structure is predicted by performing optimization on the elastic potential energy. It shows that sequences with multifarious characteristics are possible, including reversible and irreversible transitions, and transitions with symmetric and asymmetric energy barriers. Considering that the fundamental transition mechanisms are of great significance in understanding the quasi-static and dynamic behaviors of multistable structures, the results could be potentially employed for developing morphing structures, adaptive metamaterials, and mechanical logic gates.