Finite Strain Tensor Research Articles

Finite strain and finite rotation of solids

The so-called 'infinitesimal' strain tensor is shown to be a valid finite strain tensor for all symmetric deformations (which involve no rotation). Comparison with a similar result for the triangular matrix of the Voigt strain parameters shows that either description is valid for arbitrary six-parameter deformations of invariant algebraic form.

A simple algebraic method for strain estimation from deformed ellipsoidal objects. 1. Basic theory

A new algebraic method is developed to determine the shape and orientation of the strain ellipsoid by using deformed ellipsoidal objects as strain markers. It is assumed that objects are of truly ellipsoidal shape with random orientation in the undeformed state, and that they deform homogeneously with their matrix. This part presents basic theories for: 1. (1) the determination of the strain ellipse on a plane section from deformed elliptical objects; 2. (2) determination of the strain ellipsoid from measurements of lengths and orientations of all principal axes of deformed ellipsoidal objects; and 3. (3) construction of the strain ellipsoid from the two-dimensional analysis on three mutually orthogonal planes. General, finite, homogeneous deformation is treated, and the analysis gives the deviatoric natural strains in the principal directions of the Eulerian finite-strain tensor. The simplicity and usefulness of our method is demonstrated by the two-dimensional, pure-shear deformation of model objects. Evaluation of error and optimum sample size, and geological applications of the method will be discussed in detail in subsequent papers.

A new method of determining finite strain in models of geological structures

A new method of constructing models of cylindrical geological structures has been developed which allows the finite strain to be determined for each of a large number of small, initially square, elements. Each material layer is made up of uniformly thick, alternately colored laminations which are oriented vertically in one half of the model and horizontally in the other half. After the model has been deformed it is cut into slices perpendicular to the axis of the cylindrical structure, and the lamination patterns from the two model halves are superimposed graphically to reveal the strained shapes of the initially square elements. The three-dimensional finite-strain tensor of each element can be easily determined algebraically from measurements of its deformed shape, provided that the deformation involved rotation about no more than one principal direction. As an example the method is applied to a silicone putty model of a diapiric ridge which was formed by spinning the model in a centrifuge.

Invariant functions and homogeneous bases of irreducible representations of the crystal point groups, with applications to thermodynamic properties of crystals under strain

A general method is described for deriving the integrity bases of finite transformation groups. A generalization of an integrity basis which determines the homogeneous bases of irreducible representations of such groups is also given. It is shown that such bases can be expressed as linear combinations of a finite number of these bases with coefficients which are homogeneous polynomials that are invariant to the Coxeter group containing the group in question. Tables are given for the integrity bases for polynomial functions of a symmetric second-order tensor which are invariant to the crystal point groups. It is shown how these results may be applied to the finite strain tensor and hence to the thermodynamics of anisotropic crystals.

A Correlation Study of Formulations of Incremental Deformation and Stability of Continuous Bodies

In the past a number of different linearized mathematical formulations of the infinitesimal incremental deformations of continuous bodies under initial stress have been proposed. The best-known formulations are reviewed, tabulated, and subjected to a comparative study. It is demonstrated that they can be derived as special cases of a unified general formulation, and are all correct and mutually equivalent. In each formulation, the incremental elasticity constants and the incremental material stress tensor have a different significance. Their mutual relationships are established. Thus the analysis of a problem which has already been solved according to one formulation need not be repeated for another formulation. Furthermore, the connections to the various definitions of the objective stress rate are shown. The arbitrariness of choice between the infinitely many possible forms of incremental equilibrium equations corresponds to the arbitrariness in the definitions of (a) the finite strain tensor, (b) the material stress tensor, (c) the objective stress rates, (d) the stability criterion, and (e) the elastic material in finite strain. For demonstration of the differences, the problems of surface buckling of an orthotropic half space and a column with shear are studied. It is shown that the predicted buckling stresses can differ almost by a ratio of 1:2 if the proper distinction between various formulations is not made.

The moiré-grid analyzer method for the analysis of finite strain

Elements of Lagrangian finite-strain tensor are derived in terms of moire-fringe intercepts with grid analyzer axes (Ξ,η); they are shown to be independent of the relative angle between the analyzer and the undeformed specimen grids.

Equations of state of continuous matter in general relativity

The invariant forms that equations of state of continuous matter may take in general relativity, when the rheological behaviour of matter at any event may depend on previous rheological states through which that matter has passed, are discussed. A complete set of variables, needed in the general case to specify the relevant information about the matter at event x i , is first obtained as a set of space-tensors and then as a set of four-tensors orthogonal to the four-velocity at x i . These variables represent proper measures of deformation history, mechanical-stress history, temperature history, proper-time lag and physical constants of the material. An unambiguous definition is given of a physical constant (tensor) of the material (equation (122)). Elasticity, viscosity, and all possible combinations of these properties are within the scope of the theory. A detailed discussion is included of the processes of differentiation and integration of tensor quantities with respect to proper-time, following a particle along its world-line, such as will occur in equations of state in the general case. A convected integral with respect to proper-time is expressed (in equation (111)) in terms of displacement functions X 'm , which relate events x i and x 'm on the world-line of the same particle (such that x 'm is earlier than x i by an interval of proper-time t—t' ) through equations x 'm = X 'm (x i , t — t' ) . A convected derivative with respect to proper-time is expressed (in equation (82)) in terms of a Lie derivative defined with respect to the velocity vector field. Successive convected differentiation of a finite-strain tensor, defined in relation to an arbitrary reference configuration of a material element, gives rise to a sequence of rate-of-strain tensors.

On the bifurcation of the equilibrium state of a three-dimensional isotropic elastic solid under large subcritical deformations: PMM vol. 34, n≗6, 1970, pp. 1113–1125

General questions and the derivation of three-dimensional linearized equations for arbitrary subcritical deformations (linearized equilibrium equations in stresses) have been examined in a number of papers [1–11]. Particular forms of the elastic potetial [3,9, 12–16] have been applied in the derivation of the linearized equations in displacements for arbitrary subcritical deformations and in the solution of problems for 3 three-dimensional elastic isotropic solid in the majority of papers. For some of these problems, generala solutions have been constructed in the case of homogeneous subcritical states [3,14]. The linearized equations in displacements for an arbitrary form of the elastic potential have been obtained in [8], and a number of problems have been considered in such an approach [8,17,18]. Following [4], three-dimensional linearized equation in stresses are examined below, where components of the Green's finite strain tensor are chosen as strain characteristics. Linearized equations in displacements are obtained for an arbitrary form of the elastic potential. In the particular case of homogeneous subcritical states, general solutions are constructed which have been expressed in terms of the solution of second order equations for a solid with arbitrary cross-sectional outline. Proceeding from the general solutions, characteristics equations are obtained for a number of problems for an arbitrary form of the elastic potential. Certain forms of the elastic potential utilized in the literature are presented. Numerical examples are considered.

Hot high-nitrogen gas in a metal mine

Strain is an important component of the total displacement field in the emplacement of a thrust sheet. The finite strain tensor in a penetratively deformed thrust sheet is a spatial variable. I describe a method for quantitative estimation of the finite strain variation in thrust sheets by applying spatial statistics analysis on strain data collected from a part of the Sheeprock thrust sheet, in the southern Sheeprock Mountains and the West Tintic Mountains, north-central Utah. Strain was measured in the quartzites of the Sheeprock thrust sheet and the spatial statistics method is illustrated using the XZ strain axial ratios.The Sheeprock thrust sheet was penetratively deformed during Sevier-age fault propagation folding. I quantified finite strain from quartzites using the modified normalized Fry method and calculated the three-dimensional strain ellipsoid from the quartzites using three orthogonal thin-sections from each oriented field sample. The variation of finite strain in the Sheeprock thrust sheet was best represented by an exponential semivariogram model, which I used to predict values of strain from unsampled locations by ordinary kriging. Cross-validation showed that, in general, the predicted and measured values show good agreement (within 1% of each other). The sampled space was contoured using the measured and predicted strain values to obtain a detailed finite strain variation pattern in a part of the Sheeprock thrust sheet.