Components Of Strain Tensor Research Articles

Application of Flow Theory Relations for Solving Problems of Steady-State Crack Growth

To represent local displacement fields in the problem of the steady-state growth of a crack, which contains a plate of incompressible material, the strain intensity formula is used in the form of a polynomial of the second degree. The case of plane deformation for an elastoplastic material is considered. The solution is obtained by the method of asymptotic expansions. Numerical analysis is carried out for the first term of the expansion. The aim of the work is the process of obtaining analytical solutions to applied problems of the theory of plasticity: finding the components of stress and strain tensors. The paper considers a variant of the method of asymptotic expansions and its application for the problem of the distribution of the stress-strain state in an elastoplastic specimen with a crack. The method of asymptotic expansions has some advantages over the numerical approach in studying the stress-strain state in the vicinity of a crack. It allows to establish exact quantitative relationships between the radial component, the angle, and the components of the stress and strain tensor. Another advantage of this method is the possibility of compiling the mechanical characteristics of an object at the design stage. A system of differential equations has been developed that contains V0 and its derivatives up to the third order. An example of stress distribution in the vicinity of a crack tip in a steel sample, obtained in a computer system by a numerical method, is given. The deformation diagram has been constructed for the material steel 40. The research results can be used to construct stress and strain fields in the vicinity of a crack, as well as to predict the further direction of crack development.

Whispering gallery modes for 3D strain measurement

This paper deals with the use of Whispering Gallery Modes to measure the six components of an homogeneous strain tensor within a sphere undergoing a uniform strain field. The device is a silica sphere in which six WGMs resonate, describing six circles positioned on the sphere’s surface in planes perpendicular to the normals of a dodecahedron. The direct model relating the strain to the resonant wavelength variation is established for one circle and then linearised in accordance with the intended application, i.e. the realisation of a sensor. The expressions are then generalised to six circles and inverse system computed to obtain an expression for the six components of the strain field as a function of six wavelength shifts of WGMs. Finally, the theoretical performances of the sensor are assessed. The study shows that using a conventional 1 pm resolution detector, the uncertainty on the strain components is, after correction of systematic errors, equal to 2με.

Three‐Dimensional Modeling of Hysteresis in Rocks

AbstractHysteresis in rock is considered to be as one of the basic possible material behaviors. McCall and Guyer treat rocks as materials consisting of hysteretic mesoscopic elastic units (HMEU). Hysteresis with discrete memory can be well reflected by the density of HMEU in Preisach‐Mayergoyz space (PM space). However, this model is limited to one‐dimensional case and unable to obtain all the six components of strain tensor. We revisit and extend their model to three‐dimensional case by introducing a type of initially closed crack, and by redefining the hysteretic behavior of the initially open/closed mesoscopic cracks. Together with the directional probability density function E(n), our analysis leads to an explicit expression of the strain tensor of a rock under triaxial loading/unloading. The ratio of closed cracks, the direction of cracks and a parameter related to the equilibrium length of cracks are included in this three‐dimensional model.

Finite element modeling of the processes of elastic-plastic deformation of reclamation objects of the agro-industrial complex

Annotation. To study the processes of nonlinear deformation of reclamation objects and engineering systems of the agro-industrial complex, taking into account the plastic stage of the used structural material, a finite element model was created based on a volumetric prismatic discretization element with quadrangular bases. The plastic stage of deformation of the applied structural material of the object is taken into account on the basis of the provisions of the deformation theory of plasticity. The plasticity matrix at the (j + 1)-th stage of sequential loading was compiled as a result of applying the operation of differentiating the stress tensor components accumulated over the j previous stages of sequential loading with respect to the strain tensor components. The stiffness matrix and the column of nodal forces of a prismatic discretization element with quadrangular bases at the (j + 1)-th stage of loading were obtained by minimizing the Lagrange functional. Numerical experiments on the calculation of a tubular outlet proved the adequacy of the developed finite element model and the accuracy of calculating the required strength parameters of the studied reclamation objects and other engineering systems of the agro-industrial complex, sufficient for engineering practice.

Experimental evaluation of the coefficients of thermal expansion in a composite laminate using digital image correlation

A procedure for measuring simultaneously the coefficients of thermal expansion (CET) in the in-plane and the thickness directions in L-shaped composite laminated samples is presented. Measurements are carried out using digital image correlation, from which the displacements in one of the faces of the samples are measured when de sample undergoes a temperature increment inside an oven. From the in-plane displacement measurements, the in-plane strain tensor components and the corresponding CTE are determined. Results show that the presented technique is capable of determining these CTE with high accuracy. Displacement measurements in the direction normal to the face, enables determining the rotation of the face and consequently the difference between the CTE in the thickness direction and the CTE in the in-plane direction affected by the curvature. Results obtained for the CTE in the thickness direction show, first, values significantly higher than the CTE of the material in the in-plane direction normal to the fibres and, second, a high dispersion from one stacking sequence to another. These discrepancies were expected and are caused by the misalignment of the fibres in the corner zone of the samples.

Constitutive relations for materials with strain state dependent properties

Many materials demonstrate a dependence of mechanical properties on the type of stressed or deformed states. This is most noticeable in the dependence of the processes of shear and bulk deformation. Such materials include rocks, structural graphite, concrete, some grades of steel, cast iron, and aluminum. The main properties of these materials are an absence of a "single curve" relationship between the intensity of stresses and the intensity of deformations. Under shear conditions, bulk deformations can occur. Such materials can be described by constitutive equations that depend on the parameter of the type of a stress state, which is the ratio of the first invariant of the stress tensor to the stress intensity. Thus, these defining relations give the dependence of the strain tensor components on the stress tensor components. Such defining relations can be quite cumbersome, and therefore do not allow an analytical treatment to obtain defining relations that give the dependence of the components of the stress tensor on the components of the strain tensor. The paper proposes the constitutive relations obtained from the analysis of test results of various materials, which properties depend on the type of deformed state. Conditions are derived for material constants that ensure the uniqueness of the solution of boundary value problems. Based on experimental data obtained under the conditions of the proportional loading of various rocks: limestone and talcochlorite, as well as the results of mechanical tests of several grades of concrete, the constants of the mathematical model are determined. The results of the experimental studies are compared with theoretical dependencies predicted by the model. The limited applicability of the proposed constitutive relations is established.

(Invited) Magnetoelectrics at Nanometer and Gigahertz Scales for Advanced Spintronic Computing Applications

Since the work of Oersted and Ampère in the early 19th century, magnetism and magnetic fields have been connected to electric currents. By contrast, the control of magnetism by electric fields has elicited interest in the scientific and engineering communities only much more recently. Although originally postulated by Pierre Curie in 1894 and later discussed in more detail by Debye in the 1920s, the first experimental demonstration of a magnetoelectric effect occurred only in 1960. While single phase multiferroic materials were initially at the center of interest, the focus shifted rapidly in the 1970s towards magnetoelectric compound materials that offer larger magnetoelectric coupling strengths. Recently, magnetoelectricity has seen a renaissance due to both technological and theoretical progress [1,2]. The main interest in magnetoelectric systems lies in the potentially much higher energy efficiency with respect to current- and even current-density-based magnetization control. Therefore, magnetoelectricity may become an enabling step for novel low-power spintronic devices [3,4].Magnetoelectric compound materials are layered composites that consist of piezoelectric and magnetostrictive films. The magnetoelectric coupling occurs via strain: applying an electric field across the piezoelectric generates strain, which is then transferred to the adjacent magnetostrictive. This leads to an effective magnetic field that can influence the magnetization of the system. An analogous inverse effect exists also that links changes in the magnetization direction to electric fields. Since the coupling between voltage and strain in piezoelectric films can be efficient, magnetoelectric composites are promising as transducers between electric and magnetic domains with high energy efficiency and large output signal.For microelectronic applications, magnetoelectric composites must be integrated into devices and miniaturized to the nanoscale. In this presentation, we review our recent work on submicron magnetoelectric devices and discuss the effect of scale on the voltage coupling. We demonstrate that the miniaturization can lead to an enormous increase in the voltage response of magnetoelectric structures. Moreover, many applications require fast responses of the devices with targeted operational frequencies in the GHz range. Such high frequencies can strongly modify the mode of operation of magnetoelectric devices since the coupling may now occur between acoustic waves or resonator modes in nanoscale devices and dynamic magnetic excitation, such as ferromagnetic resonance or spin waves [5,6]. We introduce the basic concepts of magnetoelectric spin wave transducers and demonstrate their operation. Finally, we discuss the potential of magnetoelectric transducers for spin wave computing applications [7].This work has been funded by the European Union’s Horizon 2020 research and innovation program within the FET-OPEN Project CHIRON under Grant Agreement No. 801055as well as by imec’s industrial affiliate program on beyond-CMOS logic.[1] M. Fiebig, J. Physics D: Appl. Phys. 38, R123 (2005).[2] G. Srinivasan, S. Priya, N. Sun, Composite Magnetoelectrics: Materials, Structures, and Applications, (Woodhead Publishing, Boston, MA, 2015).[3] S. Manipatruni, D.E. Nikonov, I.A. Young, Nature Phys. 14, 338 (2018).[4] S. Manipatruni, D.E. Nikonov, C.-C. Lin, T.A. Gosavi, H. Liu, B. Prasad, Y.-L. Huang, E. Bonturim, R. Ramesh, I.A. Young, Nature 565, 35 (2019).[5] F. Vanderveken, H. Ahmad, M. Heyns, B. Sorée, C. Adelmann, F. Ciubotaru, J. Phys. D: Appl. Phys. 53, 495006 (2020).[6] F. Vanderveken, J. Mulkers, J. Leliaert, B. Van Waeyenberge, B. Sorée, O. Zografos, F. Ciubotaru, C. Adelmann, Phys. Rev. B. 103, 054439 (2021).[7] A. Mahmoud, F. Ciubotaru, F. Vanderveken, A.V. Chumak, S. Hamdioui, C. Adelmann, S. Cotofana, J. Appl. Phys. 128, 161101 (2020).Figure: Top: Schematic of a nanoscale magnetoelectric spin wave transducer on top of a magnetostrictive waveguide including simulated strain tensor components due to resonator modes at 10 GHz. Bottom: Micromagnetic simulation results of the magnetization dynamics generated by a magnetoelectric transducers at 10 GHz [5]. Figure 1

Confined vortex surface and irreversibility. 1. Properties of exact solution

We revise the steady vortex surface theory following the recent finding of asymmetric vortex sheets (Migdal, 2021). These surfaces avoid the Kelvin–Helmholtz instability by adjusting their discontinuity and shape. The vorticity collapses to the sheet only in an exceptional case considered long ago by Burgers and Townsend, where it decays as a Gaussian on both sides of the sheet. In generic asymmetric vortex sheets (Shariff, 2021), vorticity leaks to one side or another, making such sheets inadequate for vortex sheet statistics and anomalous dissipation. We conjecture that the vorticity in a turbulent flow collapses on a special kind of surface (confined vortex surface or CVS), satisfying some equations involving the tangent components of the local strain tensor. The most important qualitative observation is that the inequality needed for this solution’s stability breaks the Euler dynamics’ time reversibility. We interpret this as dynamic irreversibility. We have also represented the enstrophy as a surface integral, conserved in the Navier–Stokes equation in the turbulent limit, with vortex stretching and viscous diffusion terms exactly canceling each other on the CVS surfaces. We have studied the CVS equations for the cylindrical vortex surface for an arbitrary constant background strain with two different eigenvalues. This equation reduces to a particular version of the stationary Birkhoff–Rott equation for the 2D flow with an extra nonanalytic term. We study some general properties of this equation and reduce its solution to a fixed point of a map on a sphere, guaranteed to exist by the Brouwer theorem.

End-to-end deep learning method to predict complete strain and stress tensors for complex hierarchical composite microstructures

Due to the high demand for materials with superior mechanical properties and diverse functions, designing composite materials is an integral part in materials development. However, due to the heterogeneity and complexity of composites, measurements of properties and designs of optimal structures are often experimentally or computationally intractable. Here we report complete strain and stress tensors predictions given input composite geometries, based on a deep learning approach. The model not only predicts all strain and stress tensor components accurately, but also observes rigorous continuum mechanical principles. Furthermore, through tuning data statistics, the model performance is enhanced even with a small amount of data in terms of handling material microstructures with varying constituent ratios and hierarchical geometries. The method vastly improves the efficiency of predicting comprehensive mechanical behaviors of composite materials and provides a powerful tool for design problems such as multifunctional composites design and hierarchical structures optimization.

Decomposition of elastic potential energy and a rational metric for aftershock generation

SUMMARY The occurrence of earthquakes can be regarded as shear fracture releasing the elastic potential energy stored in the Earth. The potential energy density of linear elastic forces is generally represented in the quadratic form of strain tensor components with the fourth-order coefficient tensor of elastic stiffness. When the material is isotropic, since the stiffness tensor is expressible as a linear combination of two independent symmetric tensors, we can decompose the elastic potential energy density into two independent parts, namely the volumetric part and the shearing part. By definition, the partial derivatives of the elastic potential energy density with respect to volumetric and shearing deformations give the corresponding generalized forces in the sense of Lagrangian mechanics: specifically, one-third of the first invariant of stress tensor (equivalent to the mean stress) to volumetric deformation and the square root of the second invariant of deviatoric stress tensor (equivalent to $\sqrt {3/2} $ times the octahedral shear stress). With these generalized forces instead of the normal and tangential stresses on a specific fault plane, we correctly represented the original concept of Coulomb's failure criterion (shear failure occurs when shearing stress is equal to shearing strength) and defined energetics-based failure stress (EFS). The change in EFS associated with the occurrence of a main fracture (ΔEFS) gives a rational metric for aftershock generation, which can be reduced to previously proposed various metrics in special cases. For example, when the level of background deviatoric stress is much higher than the magnitude of coseismic stress changes, the expression of ΔEFS is reduced to a similar form to the well-known Coulomb failure stress change (ΔCFS). Even in the energetics-based metric, the effects of pore-fluid pressure changes are essential. We theoretically examined the mechanical effects of induced and enforced pore-fluid pressure changes and elucidated that the difference between them is reflected in the focal mechanisms of aftershocks.

Imitating Simulation of Thermomechanical Processing of Metals

Thermomechanical processing of metals by cutting is a complicated technological problem that is difficult for mathematical simulation. The various phenomena observed in this process are so closely intertwined with each other and their interaction is so complex that eleven relatively independent theories not coming yet to a holistic unity are focused on the cutter edge. These are a theory of chip formation, metal cutting mechanics, a friction theory in metalworking, thermodynamics of cutting, a theory of wear and resistance of cutting tools.
 A mesh-free method of Smoothed Particle Hydrodynamics (SPH) has been used for simulation in this paper. The SPH-based simulation in LS-DYNA is performed to predict cutting forces and plastic deformations for machining processing of metals by cutting. The results characterizing the distribution patterns of the strain tensor components and the temperature field at different points in time and space have been presented. The performed studies have demonstrated that it is possible to use changes in the temperature fields as a criterion for estimating the elastic-plastic deformations.

Features of determining the deformed state of a particle material during hot stamping of porous moldings

The paper describes main methods for assessing the deformed state of porous body metal frames developed by different authors based on the analysis of yield conditions and governing equations, using the principle of equivalent strains and stresses, and studying the kinetics of metal strain during pressing. Formulas were derived to determine the components of the powder particle material strain tensor through dyads, as scalar products of the basis vectors of the convected coordinate system at each moment of porous molding strain. The expediency of using the analytical expressions developed to determine the deformed state of the particle material was experimentally substantiated subject to the known displacement vector parameters of representative elements (macrostrains) of porous billets. The applications of well-known analytical expressions were established, and the proposed formulas proved applicable for the deformed state assessment of particle metal during the pressure processing of powder products of different configurations and designing billets with a defined porosity and geometric parameters as a basis for compiling software algorithms for the computer simulation of porous molding hot stamping.

Three-dimensional strain descriptors at the Earth’s surface through 3D retrieved co-event displacement fields of differential interferometric synthetic aperture radar

In deformation analysis, invariant strain descriptors that are independent of the coordinate system are significant. An approach is proposed to determine the strain tensor and invariant strain quantities through the 3D retrieved co-event displacement of differential interferometric synthetic aperture radar (DInSAR). An optimal cell count is selected based on the proposed rule-of-thumb formula. This set of coherent cells with prescribed acceptable minimum value coherence is distributed on the domain through a quasi-random number generator. The recursive moving least squares method is applied to estimate the components of strain and differential rotation tensors. In practice, this method is not limited to the homogeneous strain field hypothesis. The objective here is to propose an approach to determine the local and regional strain patterns in SAR’s image resolution details. This approach is tested on both synthetic and real data. The real data consist of 3D displacements of 19 global navigation satellite system stations and SAR images of Kilauea Volcano eruption in July 2007. Results indicate that the areas with maximum shear strain and maximum dilatation are located exactly near the east rift zone of the volcano.

Measuring residual stress in Ti-6Al-4V with HR-EBSD, using reference patterns from annealed material

Elastic strain and residual stress values in an engineering material were determined by high-resolution EBSD (HR-EBSD) using reference patterns from annealed, recrystallized material. Specifically, rolled Ti-6Al-4V sheet was analyzed using HR EBSD in both an as-annealed and a strained state, the latter established via a static two-point bend fixture. The commercial software CrossCourt4 (CC4) was used to evaluate the relative intragranular strain tensor components of the alpha grains mapped by HR-EBSD. The intragranular reference patterns identified in each grain by the software were then replaced with similarly-oriented patterns from scans of the as-annealed, recrystallized material. The CC4 analysis was then repeated to provide calibrated strain and residual stress values. Elastic strains on the order of 10−3 were detected, with corresponding residual stress values on the order of 1 GPa. Subsequent analyses using high energy synchrotron x-ray diffraction showed good correlation with the elastic strains measured by HR-EBSD using the annealed reference patterns. Additionally, the maximum macroscopic stress analytically predicted for two-point bending was in agreement with the residual stress observed. Good repeatability of the methodology and insight into the state of individual grains were also observed. The findings provide encouraging evidence that accurate intragranular and long-range residual stress measurements of engineering significance can be made with HR-EBSD using a physical reference pattern library.

Analysis of strain changes caused by placing the substrate-bonded optical fiber sensor on the structure surface

The paper proposes a mathematical model for estimating changes in the strain fields caused by the installation of a substrate with an optical fiber sensor on the surface of the controlled structure. It also analyzes the information on changes in the strain tensor components in the zone of sensor location at different dimensions of the structure. It has been shown that these changes depend on the ratio of mechanical characteristics of the substrate and the material of the structure with the fiber optic strain sensor placed on its surface. A comparative analysis of strain changes is made under different loading conditions of the structure with the substrate-bonded optical fiber sensor placed on its surface.

МАТЕМАТИЧЕСКАЯ МОДЕЛЬ МАТЕРИАЛОВ КОНСТРУКЦИЙ И ЭЛЕМЕНТОВ ВОССТАНАВЛИВАЕМЫХ ГРУНТОВЫХ ПЛОТИН ДЛЯ ЧИСЛЕННЫХ РАСЧЕТОВ

Identified in the process of analyzing the operation of the structure, in the conditions of its operation, allow to assess the actual reserves of the bearing capacity of the structure and take effective measures to restore the operational parameters. The main criteria influencing the choice of mathematical models of materials for structures and elements of soil dams are more consistent with the model of the equation of state connecting the components of stress and strain tensors, as well as the rate of their change, which are obtained and tested for numerical calculations and have a full set of constants for materials used in the calculations of earth dams, the choice of their structures (concrete,reinforced concrete, soils, etc.). Reliable operation of soil dams is possible only if all proper conditions are met. The causes of dam accidents and their damage must be known not only to eliminate errors at the design and construction stages, but also during their operation. In order to exclude the negative impact of operational factors on the safety of soil HTS, it is necessary not only to strictly observe the rules of technical operation and take measures to exclude the possibility of an emergency situation during technological operations at facilities, but also to have methods for predictive justification of the restoration of strength and operational indicators of structures and elements of soil dams.

The method of reconstruction of residual stresses and plastic deformations in thin-walled pipelines in the delivery state and after bilateral vibro-shock surface hardening with a shot

We suggest the phenomenological method of reconstructing the fields of residual stresses and plastic deformations in thin-walled cylindrical tubes made of Х18N10Т steel in the delivery state and after a simultaneous bilateral surface plastic hardening by the vibration-shot blasting of the surface with beads on a special vibrating stand. A cylindrical container filled with three-millimeter beads was attached to it. The tubes were 50 % filled with one-millimeter beads, and they were placed inside the container. The axis of the tube and the container coincided. The space between the tube and the container was 80 % filled with beads. The vibrational frequency of the stand was 18.5 KHz, the hardening time was 20 minutes. The tube in the container was rotated to ensure uniform hardening. We determined the experimental values of residual stresses σθ and σ z in the surface layers using the method of rings and strips with the procedure of the layer-by-layer electrochemical picking of the hardened layers. For this purpose, the experimentally measured values of the beam-strip deflection and the angular opening of the cut ring (changing the diameter) were used. The hardening anisotropy parameter which relates the axial and circumferential components of plastic deformation was introduced into the mathematical model. In solving the stated problems the hypotheses of plastic incompressibility of the material, the absence of secondary plastic deformations of the material in the compression region of the surface layer, as well as the hypothesis of flat sections and straight radii were used. We described the method aimed at solving this type of boundary value problems of reconstructing stress-strain states, which makes it possible to determine the missing component σ r and all the components of the tensor of residual plastic deformations (off-diagonal components of the tensors of stresses and deformations were not considered). The method of reconstructing the stress-strain state is universal, because it has shown its operability both in determining the technological fields of residual stresses, as well as the irreversible strains in the samples in the delivered state after mechanical operations, and after bilateral surface plastic deformation. The adequacy of the calculated data was verified, which was obtained using the phenomenological method of reconstructing the stress and strain fields of the experimental data for the samples in the delivery state and after hardening. The correspondence of the calculated and experimental data was matched. The numerical values are given for the anisotropy parameter connecting the circumferential and axial irreversible strains, for samples, in the delivery state, its numerical value is 0.1, and, for the hardened samples, it is 4.2. This indicates a significant anisotropy of the distribution of the axial and circumferential components of the residual strain tensor. It has been established that the compressive residual stresses are observed in the delivery state in the region adjacent to the inner surface, and the tensile stresses are observed in the layer on the outer surface. Only compressive stresses are observed in both regions after hardening, which significantly exceed in module similar stresses for the samples in the delivery state. The main results are illustrated by the tabular data and the corresponding diagrams of the distribution of residual stresses along the depth of the hardened layer.

A variant of the thermoplasticity theory for monotonic and cyclic processes of nonisothermal loads

We revealed some features and differences in isotropic and anisotropic hardening under monotonic and cyclic loads by analyzing the experimental results of the samples made of 12X18H10T stainless steel under a rigid (controlled) deformation process, which includes a sequence of monotonic and cyclic loading modes under uniaxial tension-compression and different temperature levels. To describe these features with the theory of thermoplasticity, which belongs to the class of flow theories for combined hardening, a memory surface is introduced in the space of the plastic strain tensor components that separates the processes of monotonic and cyclic deformations. The main assumptions and equations of the thermoplasticity theory are formulated. To describe the transition from the monotonic to the cyclic and from the cyclic to the monotonic deformations, the evolutionary equations are formulated for the parameters of isotropic and anisotropic hardening. The basic experiment, which determined the material functions, consists of three stages, such as cyclic loading, monotonic loading and the subsequent cyclic up to destruction. The method of identifying the material functions based on the results of the basic experiment is considered. The material functions that close the thermoplasticity theory at different temperature levels are determined for 12X18H10T stainless steel due to the basic experiment and identification method. We considered the results of the computational and experimental studies of the rigid cyclic deformation under isothermal and non-isothermal loadings up to destruction of 12X18H10T stainless steel. The kinetics of the stress range and the average stress during isothermal and non-isothermal cyclic loadings are analyzed. A reliable compliance of the computational and experimental results is obtained.

A three-dimensional surface elastic model for vibration analysis of functionally graded arbitrary straight-sided quadrilateral nanoplates under thermal environment

AbstractIn this paper, a three-dimensional (3D) size-dependent formulation is developed for the free vibrations of functionally graded quadrilateral nanoplates subjected to thermal environment. The plate model is constructed within the frameworks of the Gurtin–Murdoch surface and the 3D elasticity theories. In this way, the effect of surface free energy and all the components of stress and strain tensors are considered without any initial assumption on them as there is no need to assume the variation of transverse normal stress inside the bulk material in advance. The variational differential quadrature approach and the mapping technique are applied to derive a discretized weak form of the governing equations. The present solution method bypasses the transformation and discretization of the higher order derivatives appearing in the equations of the strong form. The effects of surface stress, thermal environment, material gradient index and geometrical properties on the size-dependent vibrational behavior of quadrilateral nanoplates are investigated. It is observed that the thermal load intensifies the effect of surface free energy on the natural frequency of the nanoplates. The present model is exact in the extent of the continuum models and can be employed for structures with any thickness–span ratios.

3D Muscle Deformation Mapping at Submaximal Isometric Contractions: Applications to Aging Muscle.

3D strain or strain rate tensor mapping comprehensively captures regional muscle deformation. While compressive strain along the muscle fiber is a potential measure of the force generated, radial strains in the fiber cross-section may provide information on the material properties of the extracellular matrix. Additionally, shear strain may potentially inform on the shearing of the extracellular matrix; the latter has been hypothesized as the mechanism of lateral transmission of force. Here, we implement a novel fast MR method for velocity mapping to acquire multi-slice images at different % maximum voluntary contraction (MVC) for 3D strain mapping to explore deformation in the plantar-flexors under isometric contraction in a cohort of young and senior subjects. 3D strain rate and strain tensors were computed and eigenvalues and two invariants (maximum shear and volumetric strain) were extracted. Strain and strain rate indices (contractile and in-plane strain/strain rate, shear strain/strain rate) changed significantly with %MVC (30 and 60% MVC) and contractile and shear strain with age in the medial gastrocnemius. In the soleus, significant differences with age in contractile and shear strain were seen. Univariate regression revealed weak but significant correlation of in-plane and shear strain and shear strain rate indices to %MVC and correlation of contractile and shear strain indices to force. The ability to map strain tensor components provides unique insights into muscle physiology: with contractile strain providing an index of the force generated by the muscle fibers while the shear strain could potentially be a marker of lateral transmission of force.