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In the present work we want to show how to get in an operative way the rela tive standard quantities already obtained geometrically by means of the projection method. Methods of measurement of the foregoing standard quantities are therefore shown: the spatial metric tensor, the deformation tensor, the vortex spatial tensor, the gravitational standard field vector, the standard time, the relative standard velocity, the standard mass, the whole standard energy.

General relations were derived by expanding the strain-energy density function in terms of the invariants of the deformation tensor. Some constitutive equations obtained by keeping a third term in the expanson in addition to the two terms retained in the Mooney-Rivlin equation were tested in the light of currently available experimental data. It is shown that by the retention of the third term the upswing in the Mooney stress at low values of γ−1 is successfully predicted, and the stress—strain behavior can be described with excellent accuracy up to break, even in carbon black-filled rubber which is difficult to describe by the Mooney-Rivlin equation.

The stress relaxation properties of SBR vulcanizates, those unfilled and those filled with carbon, under orthogonal-biaxial deformation, were subjected to observation in a region from 3.4 to 10 of I1 values where I1 was invariant of deformation tensor, and the contour maps of ∂W/∂I1 and ∂W/∂I2 were obtained as functions of time t. The following results have been obtained.The time dependence of both ∂W/∂I1 and ∂W/∂I2 of the unfiilled SBR vulcanizate are very similar within experimental error, but in the case of the filled SBR, those are quite different. In the filled SBR, the relaxation rate of ∂W/∂I1 decreases with increase in I1, and on the contrary, the rate of ∂W/∂I2 increases with increase in I1. The shape of ∂W/∂I1 surface of the filled SBR is similar to that of the unfilled SBR, but the former is on a considerably higher level than the latter, while the shape of ∂W/∂I2 surface of the filled SBR is different from that of the unfilled SBR.The values of ∂W/∂I1 and ∂W/∂I2 in the uniaxial deformation are obtained by the extrapolating method using the biaxial data. It is shown that the values of ∂W/∂I1 and ∂W/∂I2 in the uniaxial deformation decrease with increase in deformation. In this fact is suggested that strain energy function W cannot be estimated by Mooney-Rivlin plot, and that the constants C1 and C2 obtained from the plot are almost of no significance.

The Taylor theory of polycrystal plasticity was applied to three-axial deformation accomplished by glide on (111) [110] or (110) [111] glide systems. The orientation dependence of the Taylor factor was used to calculate the angular dependence of the relative strength of a textured material. The angular dependence of the strain ratio R — calculated from the minimum value of the Taylor factor — was compared with the measured strain ratio. The orientation changes of the crystallites of a polycrystalline aggregate after 1% plastic deformation were calculated and compared with experimental values. The orientations which are deformed without orientation changes were calculated and compared with those of maximum orientation density found by a three-dimensional texture analysis of cold rolled copper and iron. The dependence of the rotationless orientations on the axis ratio of the deformation tensor was calculated. This allows suggestions to be made on the dependence of the rolling textures on the lateral broadening of the sheet during rolling. Die Taylor-Theorie der Vielkristallplastizitat wurde auf dreiachsige Verformungszustande und Gleitung auf (111) [110]- oder (110) [111]-Gleitsystemen angewandt. Aus der Orientierungsabhangigkeit des Taylor-Faktors wurde die Winkelabhangigkeit der relativen Festigkeit eines Texturmaterials berechnet. Die Winkelabhangigkeit des R-Wertes der plastischen Verformung — berechnet aus dem Minimalwert des Taylor-Faktors — wurde mit den gemessenen R-Werten verglichen. Die Orientierungsanderungen der Kristallite eines vielkristallinen Aggregates nach 1% plastischer Verformung wurden berechnet und mit experimentellen Werten verglichen. Diejenigen Orientierungen, die sich ohne Orientierungsanderung verformen, wurden berechnet und mit den Orientierungen maximaler Orientierungsdichte verglichen, die sich aus der dreidimensionalen Texturanalyse kaltgewalzter Kupfer- und Eisenbleche ergaben. Die Abhangigkeit der drehungsfreien Orientierungen vom Achsenverhaltnis des Deformationstensors wurde berechnet. Daraus ergeben sich Schlusfolgerungen uber die Abhangigkeit der Walztexturen von der relativen Breitung des Bleches wahrend der Walzung.

Drag coefficients for spheres moving through five aqueous solutions of carboxymethylcellulose (CMC) were measured. With the assumption that the behaviour of these fluids could be approximated by the Ellis model, two dimensionless correlations for the drag coefficient in terms of a modified Reynolds number and two other dimensionless groups were prepared. The Ellis model is easy to use and provides a much better description of non-Newtonian behaviour than the more commonly used Ostwald-de Waele model. The correlation is of value for the prediction of the settling time of a sphere in a non-Newtonian fluid. The results seem to indicate that the behaviours of the CMC solutions studied are independent of the third invariant of the rate of deformation tensor.

Here aij(x, t), eij(x, t), ui(x, t)are, respectively, the components of the stress tensor (~(x, t), the deformation tensor e(x, t), and the displacement vector u(x, t); Fi(x, t), S/(x, t), ni(x) are the components of the body force vector F(x, t), the vector S(x, t) of the forces applied at the boundary, and the unitvector n(x); the integral operators Lmi contain the elastic constants aij of the material and one to three-fold Volterra type convolutions Hmi (see [1]); L = a T  (1 + H~ni); a T is the linear thermal expansion coefficient; T is the change in temperature at a point of the body, moreover the thermorheologiea l constants are not functions of the coordinates; x - (x 1, x2); a comma preceding the subscript j indicates differentiation with respect to the coordinates xj; i, j, k, l, m, n = 1.2; 6ij is the Kronecker symbol. By substituting the equation of state into the compatibility equation for the deformation components we obtain a nonlinear equation on the basis of which we have

Summary In order to develop the elastic energy up to terms of the third order inclusive, the expressions defining the cubic dilatation and the deformation are given, either directly in the derivatives of the displacements ∂ui/∂xk (§ 5) or in the isotropic invariants of the deformation tensor (§ 6). Starting from the energy of isotropic solids as depending on these two quantities alone, the mean tensions, acting on the material and moving elements of the medium, are arrived at for longitudional and transverse waves travelling in the direction of the X-axis. The results are generalised for crystals, and the components of the radiation tensions are related with the density of energy E and the dependence of the wave velocities in the medium on deformations in bulk: B i K = 1 2 E ρ V 2 . ∂ ( ρ V 2 ) ∂ ( ∂ u K / ∂ x i ) .

Summary. Each particle of a continuum is assigned a second order tensor which is taken as a measure of the deformation of some neighborhood of the particle, and which is determined by a functional depending on the configurations of that neighborhood. Two invariance restrictions are imposed on the functional whose values are spatial strain tensors, that is, associated with the deformed configuration. The first requirement is that a time shift and rigid transformation of the deformed configuration leave the spatial deformation tensor unaltered relative to it. The second requires that if particles of distinct continua undergo the same deformation, the corresponding deformation tensors should be the same. For the special case in which the functional depends on the deformation in the smallest neighborhood of a particle, the restrictions imply that the deformation tensors associated with the deformed and reference configurations arc isotropic functions of the ]eft and right CAITClVZGREEN tensors, respectively. Zusammenfassung. Jedem Teilchen eines Kontinuums wird ein Tensor zwciter Stufe als Mal3 fiir die Deformation einer gewissen Nachbarschaft dieses Teilchen zugeordnet, der durch ein Funktional bestimmt wird, das yon der I~onfiguration dieser Nachbarschaft abh~Lng~. Zwei Invarianzbcdingu ngen werden diesem Funktional, dessen Werte raumliche Verzerrungstenso ren darstellen, auferlegt, und zwar im Hinblick auf die deformierte Konfiguration. Die erste Forderung besagt, dal3 eine Zeitverschiebung und eine starre Transformation der deformierten ]~onfiguration den rgumlichen Verzerrungstenso r im Hinblick auf diese unge~ndert lassen. Die zweite Einschr~nkung besag% dal3 entsprechende Deformationstens oren von Partikcln verschiedener Kontinua, die dieselbe Verformung erlitten haben, gleich scin sollen. Im Spezialfall, dal3 die Funktionale nur yon der Deformation in der n~ichsten Umgebung des Partikels abh~Lngen, beinhalten die Einschrgnkungcn die Aussage, daf3 die mit dem deformierten ned dem undcformierten Zustand verkniipften Deformationstensoren nur isotrope Funktionen des linken und des rechten CAIse~u GREEN Tensors sein k6nnen.

The effect is considered of non-uniform deformations satisfying the condition λ|∇u αβ |≪|u αβ |, where λ is the free path length of an electron in the undeformed crystal and, u αβ is the deformation tensor, on the electrical conductivity of a metal crystal with alloying elements. Account is taken of the local change of the dispersion law for conductivity electrons and also of the dependence on the coordinates of the alloying element of the electron scattering cross-section on the alloying element in the non-uniformly deformed crystal. An explicit expression is obtained for the tensor of the effective conductivity, first introduced by Herring and connecting the average volume density of the electrical current with the average electrical field in the crystal. The local electron scattering cross-section on the charged alloying elements is evaluated on the Thomas-Fermi model taking into account the local change of the constant shielding. Account is also taken of the possibility of redistribution of the alloying elements in the field of the deformations. The dependence is found and discussed of the conductivity on the sign of the constant, introduced in the continuous approximation, of the interaction of the alloying element with the deformation field. By way of illustration of the results obtained the simple case of unidimensional deformation is considered.

In this introduction we shall be concerned with a simple approach to tensor algebra, and the tensor formulation of stress, deformation and energy, leading to Hooke’s law, the properties of anisotropy, and considerations of elastic symmetry.

Based on physical considerations, the mathematical model of a micropolar continuum is introduced. Coordinates, base vectors, and shifters are presented in Art. 1.2. The concept of directors, fundamental to micropolar bodies, are discussed in Art. 1.3. The motion, deformation, strains, their measures and geometrical meaning constitute the material of Arts. 1.4. and 1.5. The rate tensors appropriate to micropolar bodies are presented in Art. 1.6. The question of compatibility necessary for the singlevaluedness of the deformation and rotation fields closes the chapter. These ideas are essential also to dislocations and disclinations within the context of dislocation theory.

Consider a tetrahedron, centroid P, of continuous material, isolated in thought from the material which surrounds it. Let the lengths of the edges be infinitesimal of order l.

The deformation of a plane elastic flat body is characterized by a tensor (deformation tensor): $${\varepsilon _{Ij}} = \frac{1}{2}\left( {\frac{{\partial {u_i}}}{{\partial {x_j}}} = \frac{{\partial {u_j}}}{{\partial {x_i}}}} \right),i,j = 1,2. $$ (5.1.1)

A body is deformed when forces are applied to it; the distances between points in the body subject to the forces are somewhat different from those between the same points in the absence of the forces . The change in distance is due to displacement of one relative to another during the deformation. Let r0 be the radius vector of some point in the body (relative to some fixed point in space) before the deformation, and let r(xi) be the same after deformation. The difference between these vectors is the displacement vector of the point and is (1.1) or in coordinate form (1.2)

A component of the deformation tensor has been measured for part of the Fermi surface of tin and the corresponding attenuation coefficient for longitudinal ultrasonic waves has been deduced.