## Abstract

The chapter contains information that forms the basis of a new direction in the nonlinear theory of elasticity. The theory, having adopted the mathematical apparatus obtained in the middle of the last century, after its analysis, is used with significant changes. This concept allows us to more accurately reveal the mechanism of deformation of materials, the elastic nature of which significantly depends on the type of stress state, due to the growth of additional volumetric deformation associated with the accumulation of defects, called dilatation. The work is original — after abandoning the elasticity characteristics in the form of modules - constants, the main role is assigned to material functions, which represent statistical characteristics. Their relation can be considered a coefficient of variation and a parameter of tensor nonlinearity, which makes it possible to represent the deformation in the form of two parts, different in origin.

### Keywords

- dilatancy
- volume deformation
- shape change
- phase similarity of deviators
- volume deformation
- coefficient of variation
- tensor nonlinearity
- anisotropy
- variable elasticity parameter

## 1. Introduction

Experimental studies of well-known mechanics with various materials already in the eighteenth century revealed numerous nonlinear effects described in the book [1]. From the standpoint of the linear theory of elasticity, many of them could not be explained, so they were called second-order effects, as not significant. However, in the middle of the twentieth century, they pushed M. Rayner [2], and a little later, V. V. Novozhilov [3], to the need to develop a theory based on a new concept of tensor-nonlinear equations [4, 5] that more accurately reflect the nonlinearity of materials. The widespread introduction of composite media and the study of their mechanical properties began at the end of the last century. In the same years, a lot of experimental works appeared to study the mechanical properties of various composites, illuminating the properties of not only reinforced materials, but also grain composites, which differ in different reactions to tension and compression. This property is possessed by media whose longitudinal modulus of elasticity and other characteristics depend on the type of stress state, determined at values of deformations close to zero. It should be called the work of Tolokonnikov L. A., Makarov E. S. [6] and many others who have devoted research to the properties of these media, in which the presence of damage to internal connections and loosening, that is, the development of dilatancy, is stated. The theories put forward by them are based on tensor-linear equations. As a rule, in them all the characteristics of different-modulus media are determined from the condition of the existence of a specific deformation potential.

In this paper, in continuation of the study [7], to take into account the noted effects, such a transformation equations was found, which made it possible to develop methods for determining the elasticity characteristics. These equations presented for the main deformations made it possible not only to describe the deformation of the shape change, the coefficients of transverse deformations along different axes, to determine the volume deformation depending on the average stress, but also the dilatancy associated with the shape change.

## 2. About of different-module materials

The development of methods was carried out based on the results of studies of grain composite [8], and in earlier works of gray cast iron, using the research of [9]. The first is a hardened mechanical mixture of a mineral filler with a polymer matrix, the test results and information about its properties are published in [8, 10, 11, 12]. These materials have not only the presence of divergence of the initial longitudinal modules under tension and compression, but also show the dependence of elastic properties on time; therefore, in this work, the test results obtained at a single strain rate are used. The nonlinearity of the diagrams of a grain composite is clearly represented by the results of testing cross-shaped samples under repeated static stretching. It has a high malleability at normal temperature. The main purpose of testing such samples was to more fully reveal the mechanism of deformation of different-modulus materials. Figure 1a shows the curve 1—the ascending branch at the first cycle of active deformation along the axis 1–1 represents the initial properties of the material. Where P is the force in H,

The difference between the ascending branches of the first and second stretching cycles along the 1–1 and 2–2 axes is a real one, called [3] by V. V. Novozhilov “real” anisotropy. The second cycle shows that the material has noticeably softened, the slope of the curve has decreased, but the tangential longitudinal elastic modules manifest themselves on the second part of the branch as increasing, differing from the first cycle. This emphasizes the fact that the links are divided into “short” and “long”—stronger, although in [13] a more detailed gradation of links is given, which will be superfluous for this work.

Both in [8, 12], it is noted that stretching is accompanied by a noticeable increase in volume. The same is observed with compression, although to a lesser extent. The loss of bonds and softening are the cause of the loss of elastic energy, which is taken into account by the mathematical model with a proportional increase in stresses only by the growth of additional volume deformation, as in the deformation theory, plastic shifts. For practical calculations, test diagrams of standard samples were used according to the method described in [8]. The tensile diagram for testing along the 1–1 axis, curve 1, Figure 1a, is a sequence of limit values of groups of bonds that are close in strength. The same is true for other types of loading, but to a lesser extent.

The purpose of this work is to fully reveal the possibilities tensor-nonlinear equations: transformed to a form convenient for the formulation of material functions, analysis, and processing of test results. On their basis, to develop methods for calculating all characteristics, including the coefficients of transverse deformations, elastic modulus, and compliance, as well as parameters that characterize the loosening of the structure and the change in elastic properties both with increasing load and with a change in the type of stress state.

## 3. On tensor-nonlinear equations

To describe the deformation of different-modulus materials, considering them isotropic, we used tensor-nonlinear equations of the connection of the strain deviator

In the left part:

Strain intensity. In the right part:

Abandoning the constancy of the phase similarity diverters ω, which was proposed in [4], the generalized modulus G and the phase can be expressed through the coefficients of the tensor arguments:

For this we can use Eq. (1) presented for the main component of the deviator of the strain

The coefficients X and Y can be given an unambiguous physical meaning and formulas for determining them can be derived. Using three shear pliabilities

Thus, the analysis of the Eq. (1) allows, without any assumptions, to be free from uncertainty and to find an approach to the characterization of the deformation

The sum of the squares of the differences of the main values of the deformation deviator

leads to the need to calculate the relations: ∑

It leads to generalized malleability:

as a function of the angle ξ, and the inverse of the malleability to the generalized modulus of elasticity under shear:

It follows from this relation that the modulus clearly depends on the type of stress state, and it can be a constant value only in the special case, as it was envisaged in [4]. After replacing the second invariants on the stress intensity and strain intensity and replacing the sequence of main stresses:

After replacing the third invariants, the formulas for the angles take the form: the first

The exact definition of which is given below. Performing trigonometric transformations taking into account the new sequence of principal stresses, the material functions in Eq. (8) can be represented:

where they acquire values that have a physical meaning of average and standard compliance, manifesting themselves by statistical characteristics. The deviatory part of M. Rayner’s equations [2] leads to the same results of the functions

## 4. Initial data

Due to the lack of proven methods, the first calculations in [8] used only the results of tensile and compression tests. Generalized compliance is determined by the relation (6), which for these states is taken by simple expressions:

Assuming the independence of these functions from the type of stress state, we find a simple way to approximate the calculation of the shear modulus and the phase similarity of deviators according to the formula (6). The form change for any stress state, although approximate, can be described. To refine it, you can use the same ratio, but for a pure shift. At the same time, difficulties arose due to the fact that the tests were usually carried out on other equipment and other means of measuring deformations, so the lack of initial data was compensated by algorithms that were derived from the same equations converted to equations for anisotropic media [8, 10].

Experimental data obtained by tensile testing and compression of grain composite [8, 11], which has the maximum deformation under compression

where

Since the material functions exhibit a statistical character, and its values correspond to the condition: * p* < 1. The study of its extremum shows that the derivative with respect to the angle

The graphs for the phase differ slightly from the half-wave of the sine wave when the angle

For phase values other than zero, the ratios of the deviator components belonging to the same stress state are not equal:

Solid lines represent two diagrams, after the refinement performed according to formulas (11). The dependence of

Graphs for the coefficient of variation p (dashed line), the maximum values of the similarity phase of the deviators

## 5. On the equations for form-changing

The rejection of the constancy of the phase gives the ratio of (6), which after the transition to the second sequence of the principal stresses is the law of deformation:

where the main characteristic becomes generalized compliance (7):

as the inverse of the generalized shift modulus of G, they are represented in a discrete (digital) form by a mathematical model, as well as material functions. After replacing the sequence of main stresses, sin3ξ in the expression (6) is transformed in the ratio (15) into

where the compliance for the second part is the value

From the ratio (15) for stretching and compression, it also follows:

where

It protects the characteristics of the shape change from errors in their calculations:

The results of calculations for two variants according to the formulas (12) and (17) showed that they differ only by the fifth significant digit after the decimal point for any loading stage. It is for checking the postulates that duplication is necessary. If there is a coefficient of variation, the calculation of material functions for any other states is significantly simplified: first,

## 6. On the equation for volumetric deformation

The derivation of equality (21), as an additional part of the deformation of the form change, is proposed as an unknown formula for dilatancy, as a part of the volume deformation, consistent with the previously expressed idea that the parameter p allows the deformation, divided into two parts. This thought, the results of experimental studies and already published works allow us to propose an equation for the volumetric strain in the following form:

The first part

The process of transformation of the tensor-nonlinear equations mentioned above is covered in sufficient detail in [7 , p. 56] and probably first implemented in [10]. The equations for coupling the strain tensor to the stress tensor (8), together with the equation for average strain with average stress (18), lead to the equations for coupling the strain tensor to the stress tensor

The equations reduced to the principal deformations are used for the matrix transformation:

with the known specifications for the diagonal components:

and non-diagonal matrix components:

where

where

Pairs of coefficients

## 7. Supplement to the methodology

The high values of the theoretical modulus of volumetric elasticity, but low for compliance with tension, and low for compression, can be explained by a simple transformation of the ratio (18), if we isolate from it

It follows from the first that the second term reduces the flexibility for stretching, and the value of the theoretical module, on the contrary, increases as an inverse value. In the second formula, the second term increases the malleability for compression, although dilatancy is present. The second terms in these relations allow us to quantify its influence on the values of theoretical compliance. From the second formula, for compression, greater malleability is required, although dilatancy is present. The second terms in these relations allow us to quantify its influence on the values of theoretical compliance. Since the pliability of

It follows from the relations (24) and (25) that in the process of converting tensor-nonlinear equations to matrix equations, the pliabilities

## 8. On deformation anisotropy

V. V. Novozhilov in his work [3] expressed his opinion about this phenomenon, for the description of which the mathematical apparatus of tensor-nonlinear equations can be used, as an “important phenomenon,” without emphasizing on what characteristics it manifests itself. The studies show that the effect of dilatancy on the longitudinal elastic moduli

The lower the values of the last points of the curve for

Figure 3b shows graphs of the dependence of transverse deformations during compression. The line shown by the dots refers to the main direction coinciding with the voltage

The deformation anisotropy is more clearly shown on the graphs for the pliability of the bulk elasticity in the direction of the main stresses. The total volume deformation is determined by the formula (22), where

which determines the directions of the axes. Give

defining them as the degree of deviation from the theoretical volumetric compliance, which is the average,

The behavior of the curves for the parameters

Briefly still on the shape change, it should be noted that the initial values of shear moduli

The solution to this problem is formulated using tensor-nonlinear Equations [15]. Using the material functions of the proposed equations, finding the difference of Lode parameters,

where the former repeats the same fraction with the principal stresses by which it is determined. The problem of the researchers was to determine

## 9. Conclusion

A variant of the tensor-nonlinear equations, which can become the main direction in the nonlinear theory of elasticity, is proposed for wide use. This concept leads to taking into account dilatancy and strain anisotropy, about which Novozhilov V.V. prudently expressed in his work. They were used to study the properties of different-module materials and show that this mathematical apparatus is suitable not only for describing second-order smallness effects but also for describing effects associated with changes in the material structure. The influence of dilatancy on all the characteristics of form change and bulk elasticity is revealed, since its development with proportional stress growth is the main cause of deformation anisotropy, both of transverse strain coefficients and of bulk elasticity yields (or modules), which are directly related to the changing elasticity parameter, which is a quantitative estimate of these changes. In tensile and near-tensile states, its values significantly exceed unity. This can be explained by the fact that, in the first direction, dilatancy, being transverse for the other directions, causes transverse strain coefficients with values exceeding the number 0.5. The assumption of dilatancy to elastic deformations is an unavoidable step to trace the behavior of all deformations along the three directions. The exact coincidence of the total bulk strain as the sum of its components in the direction of the principal stresses, or, as the sum of linear-elastic and dilatancy, indicates recognition of the fact that the apparatus of the proposed equations may be a major trend in nonlinear elasticity theory. Whatever concepts other elasticity theories may adhere to, taking into account the real values of transverse strain coefficients in tension and compression will implicitly lead to the consideration of dilatancy and, consequently, to the difference in the values of the bulk elasticity characteristics. The next stage in the development of the nonlinear theory of elasticity is the involvement of the apparatus of thermodynamics.

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