Hadamard’s variational formula in terms of stress and strain tensors

Starting from a Lagrangian action functional for two scalar fields we construct, by variational methods, the Laplacian Green function for a bounded domain and an appropriate stress tensor. By a further variation, imposed by a given vector field, we arrive at an interior version of the Hadamard variational formula, previously considered by P. Garabedian. It gives the variation of the Green function in terms of a pairing between the stress tensor and a strain tensor in the interior of the domain, this contrasting the classical Hadamard formula which is expressed as a pure boundary variation.

Evaluation of heterogeneous mechanical tests for model calibration of sheet metals

The accuracy of strategies combining heterogeneous mechanical tests and full-field strain measurement techniques is dependent on many factors. Recently, many heterogeneous mechanical tests with different specimen shapes have been proposed using optimization techniques or empirical knowledge. However, a comparison of heterogeneous mechanical tests is a difficult task because studies use different materials and different representations of the strain and stress states. This work discusses metrics, calculated from the stress and strain tensors, to evaluate heterogeneous mechanical tests and proposes a metric to evaluate the tests’ sensitivity to anisotropy. To illustrate the approach, four heterogeneous mechanical tests are evaluated through the use of the suggested metrics. Results show that the use of various metrics provides a good basis to evaluate heterogeneous mechanical tests. Moreover, this work identifies a heterogeneous mechanical test achieving a high range of mechanical states and high values of equivalent plastic strain.

Mapping and classifying large deformation from digital imagery: application to analogue models of lithosphere deformation

Summary Particle Image Velocimetry (PIV), a method based on image cross-correlation, is widely used for obtaining velocity fields from time series of images of deforming objects. Rather than instantaneous velocities, we are interested in reconstructing cumulative deformation, and use PIV-derived incremental displacements for this purpose. Our focus is on analogue models of tectonic processes, which can accumulate large deformation. Importantly, PIV provides incremental displacements during analogue model evolution in a spatial reference (Eulerian) frame, without the need for explicit markers in a model. We integrate the displacements in a material reference (Lagrangian) frame, such that displacements can be integrated to track the spatial accumulative deformation field as a function of time. To describe cumulative, finite deformation, various strain tensors have been developed, and we discuss what strain measure best describes large shape changes, as standard infinitesimal strain tensors no longer apply for large deformation. PIV or comparable techniques have become a common method to determine strain in analogue models. However, the qualitative interpretation of observed strain has remained problematic for complex settings. Hence, PIV-derived displacements have not been fully exploited before, as methods to qualitatively characterize cumulative, large strain have been lacking. Notably, in tectonic settings, different types of deformation - extension, shortening, strike-slip - can be superimposed. We demonstrate that when shape changes are described in terms of Hencky strains, a logarithmic strain measure, finite deformation can be qualitatively described based on the relative magnitude of the two principal Hencky strains. Thereby, our method introduces a physically meaningful classification of large 2D strains. We show that our strain type classification method allows for accurate mapping of tectonic structures in analogue models of lithospheric deformation, and complements visual inspection of fault geometries. Our method can easily discern complex strike-slip shear zones, thrust faults and extensional structures and its evolution in time. Our newly developed software to compute deformation is freely available and can be used to post-process incremental displacements from PIV or similar autocorrelation methods.

ON DIFFERENT DEFINITIONS OF STRAIN TENSORS IN GENERAL SHELL THEORIES OF VEKUA-AMOSOV TYPE

Several possible definiions of strains in a general shell theory of I.N. Vekua – A.A. Amosov type are considered. The higher-order shell model is definedon a two-dimensional manifold within a set of fieldvariables of the firstkind determined by the expansion factors of the spatial vector fieldof the translation. Two base vector systems are introduced, the firs one so-called concomitant corresponds to the cotangent fibrtion of the modelling surface while the other is defind on a surface equidistant to the modelling one. The distortion appears as a two-point tensor referred to both base systems after covariant differentiationof the translation vector feld. Thus, two main definition of the strain tensor become possible, the firstone referred to the main basis whereas the second to the concomitant one. Some possible simplificationsof these tensors are considered, and the interrelation between the general theory of A.A. Amosov type and the classical ones is shown.

MATHEMATICAL MODEL OF MATERIALS OF STRUCTURES AND ELEMENTS OF RESTORED SOIL DAMS FOR NUMERICAL CALCULATIONS

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.

Hooke’s law and elastic constants

Hooke’s law and elastic constants are introduced. The symmetry of the elastic constant tensor follows from the symmetry of stress and strain tensors and the elastic energy density. The maximum number of independent elastic constants is 21 before crystal symmetry is considered, and this leads to the introduction of matrix notation. Neumann’s principle reduces the number of independent elastic constants in different crystal systems. It is proved that in isotropic elasticity there are only two independent elastic constants. The directional dependences of the three independent elastic constants in cubic crystalsare derived. The distinction between isothermal and adiabatic elastic constants is defined thermodynamically and shown to arise from anharmonicity of atomic interactions. Problems set 3involves the derivation of elastic constants atomistically, the numbers of independent elastic constants in non-cubic crystal symmetries, Cauchy relations, Cauchy pressure, invariants of the elastic constant tensorand compatibility stresses.

A Mathematical Model of the Field of the Strain Tensor in Terms of Deformation of the Flat Shell Structures

The article considers the topic of mechanics of a deformable solid. Mathematical modeling of the process of deformation of shell structures by deforming a flat sample (part of the shell) under conditions of a plane stress state with a discontinuous field of displacement velocities is considered. Analytical solutions were obtained for the fields of strain tensors observed in shells of various materials during their deformation. A comparative analysis is carried out under various deformation states. Such decisions are due to the need to obtain deformation fields at various points of the shells used at objects of various directions (military, industrial, etc.) and there are significant difficulties in determining the strain fields by numerical methods (for example, the finite element method).