1 Model of Block Media Taking into Account Internal Friction

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Model of Block Media Taking into Account

Internal Friction

N. I. Aleksandrova

Chinakal Institute of Mining of the Siberian Branch of the RAS, Novosibirsk, 630091 Russia

e-mail: nialex@misd.ru

Abstract ⎯ The block medium is modeled by a discrete-periodic spatial lattice of masses connected by

elastic springs and viscous dampers. To describe the viscoelastic behavior of the interblock layers, a

rheological model of internal friction with two Maxwell elements and one Voigt element with the quality

factor of the material as the determining parameter is proposed. Numerical experiments show that, within

the framework of this interlayer model, it is possible to select the viscosity and stiffness of the Maxwell

and Voigt elements so that the quality factor of the material differs from the given constant value by no

more than 5%. In the one-dimensional case, within the framework of the proposed model, the influence of

the quality factor on the dispersion properties of a block medium is studied and it is shown that the greatest

effect of the quality factor on the dispersion is observed in the low-frequency part of the spectrum. In the

three-dimensional case, within the framework of the proposed model, some geomechanical problems are

numerically studied for a block half-space under the action of a surface concentrated vertical load. Namely,

the attenuation of the velocity amplitudes of surface blocks was studied depending on the Q-factor under

step action and under the action of a Gaussian pulse. In addition, we study a layer on the surface of a half-

space under the action of a concentrated vertical impulse load in the case when both the layer and the half-

space are block media but have different properties.

Keywords: internal friction, block medium, Lamb problem, half-space, layer on half-space, wave motion,

numerical simulation

DOI: 10.3103/S0025654422030025

1. INTRODUCTION

According to modern ideas developed in the works of Sadovsky [1] and his followers, rocks

are a hierarchical system of blocks of different scale levels. Blocks of the same level are separated by

interlayers of rocks with weakened mechanical properties. It was noted in [2, 3] that the sizes of

blocks change on a scale from fractions of a rock mass to geoblocks of the earth’s crust. In the

experimental work [4], it was shown on a two-dimensional model of a block medium (a brick wall),

that for a real geomedium it is possible to determine the sizes of the characteristic blocks of the rock

mass according to seismic logging data, using the relation discovered in [5] that relates the value of

the propagation velocity of a low-frequency wave, the frequency limiting its spectrum, and the

longitudinal size of the blocks. As shown in [2, 3, 6], the motion of a block medium can be represented

as the motion of rigid blocks due to the deformation of the interlayers. As a result, the dynamics of a

block medium can be studied in the pendulum approximation, when it is assumed that the blocks are

incompressible, and all deformations and displacements occur due to the compressibility of the

interlayers (see, for example, [8, 9]). In [7–9], a block medium is modeled as a three-dimensional

lattice of masses connected by Voigt elements in axial and diagonal directions. In [9], the qualitative

correspondence of the finite-difference solution of the Lamb problem for a block medium according

to this model with the results of field experiments carried out in a limestone quarry is shown. An

alternative approach is based on a mathematical model of a block medium with elastic blocks

interacting through compliant interlayers [5, 10, 11]. To describe interlayers, various versions of the

model were proposed in [10, 11], in which interlayers between elastic blocks can be elastic,

viscoelastic, plastic, and porous.

Fig. 1. Interlayer model between blocks.

In this article, the 3D model proposed in [8] is modified. In the new model, internal friction in

the layers between the blocks is modeled by the Maxwell and Voigt elements. Zener [12] and Biot

[13] were among the first to include the Maxwell and Voigt elements in models to describe the

inelastic behavior of materials. The model with two Maxwell elements and one Voigt element was

first proposed by Biot [13] and also presented by Fang [14]. Many researchers then used various

combinations of Maxwell and Voigt elements to account for inelastic losses. In this case, an important

task for several decades in each model was to limit the number of relaxation mechanisms to obtain a

satisfactory solution, i.e., to obtain an almost constant quality factor of the material Q. In [15], a

model with internal friction was studied to describe the propagation of waves in homogeneous

inelastic media. This model includes two Maxwell elements and one Voigt element. As shown in

[15], this model makes it possible to choose the parameters of viscosity and stiffness of these

elements, so that the quality factor of the material Q differs from the prescribed constant value by no

more than 5% in the frequency range from 3% to 100% of the maximum frequency of interest. An

important property of this model is a small number of additional variables with a maximum coverage

of the frequency range of interest. The use of a minimum number of Maxwell elements limits the

memory and computational resources required in computer programs for the numerical simulation of

three-dimensional problems.

Below, to describe the viscoelastic behavior of the interblock layers, we use the internal friction

model with two Maxwell elements and one Voigt element with a quality factor Q as the determining

parameter. This model is used to solve geomechanical problems of wave propagation.

2. ONE-DIMENSIONAL MODEL OF A BLOCK MEDIUM TAKING INTO

ACCOUNT INTERNAL FRICTION

Let us first demonstrate this model using the example of a discrete-periodic one-dimensional

chain of masses connected by viscoelastic interlayers. The rheological model of the interlayers

consists of two Maxwell elements and one Voigt element (Fig. 1). All three elements are connected

in parallel. Each Maxwell element consists of a spring and a damper, which are connected in series.

The Voigt element also consists of a spring and a damper, but they are connected in parallel. Here k,

k1, k2 are the spring stiffnesses in the Voigt element and in two Maxwell elements, respectively,



are the damper viscosities in the Voigt element and in two Maxwell elements.

λ1

λ2

The equations of one-dimensional motion of blocks with this rheological model of interlayers

have the following form:

1 1 1 1

[( 2 ) ( 2 )

j j j j j j j

Mu K u u u u u u

   

     

(1.1)

1 1 1 1 2 2 1 1

( 2 ) ( 2 )], 1,2,...,

j j j j j j j

   

               

0 1 0 1 0 1 1 1 0 2 2 1 0

[( ) ( ) ( ) ( )] ( )Mu K u u u u P t            

e e ( )

  

   



e e ( )

  

   



1 2 1 2

1 2 1 2 1 2

, , , , ,

k k k k K k k k

K K K



            



Here, uj are the displacements of rigid blocks,

()Pt

is the actual load applied to the block j = 0; K is

the total stiffness of the springs. In (1.1), two additional variables



and



are introduced, which

depend on the displacement function uj.

Let us apply the Fourier transform in time to the equations of motion (1.1). In the frequency

domain, the force acting on the block from the side of the interlayer can be expressed by the formula:

( ) ( ) ( )P F u   

where

1 1 2 2 1 1 2 2

2 2 2 2 2 2 2 2

1 2 1 2

( ) 1F K i



   

       

        



   

           

   



is a normalized impedance function [16].

Internal attenuation in a medium is usually determined by the quality factor Q(ω) or its

reciprocal Q−1(ω), which is described by the formula:

2 2 2 2

1 1 1 2 2 2

2 2 2 2 2 2

1 1 1 2 2 2

( ) ( )

Im ( )

() Re ( ) 1 ( ) ( )



            



  

            

. (1.2)

For soils and rock materials, it is customary to assume that the target quality factor of the

material remains constant over a wide range of frequencies of interest, i.e.,

()QQ

, see [17].

To relate the parameters





and



of the rheological model to Q−1(ω) using simple

frequency independent approximations, we introduce new dimensionless parameters







and the frequency



using the formulas:

1 1 0 2 2 0 1 1 max 2 2 max max 0 max

ˆ ˆ ˆ ˆ ˆ

, , / , / , , /Q Q Q                    

, (1.3)

where

max



is the maximum angular frequency that is of interest for modeling.

In expression (1.2) for the coefficient

1()Q

, we pass to the normalized parameters (1.3). Then

formula (1.2) takes the form:

2 2 2 2

11 0 1 1 1 2 2 2

1 2 2 2 2 2 2

0 1 1 0 1 2 2 0 2

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

( ) ( )

( , )

ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

1 [ ( )] [ ( )]

Q Q Q







             



           

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1ModelofBlockMediaTakingintoAccountInternalFrictionN.I.AleksandrovaChinakalInstituteofMiningoftheSiberianBranchoftheRAS,Novosibirsk,630091Russiae-mail:nialex@misd.ruAbstract⎯Theblockmediumismodeledbyadiscrete-periodicspatiallatticeofmassesconnectedbyelasticspringsandviscousdampers.Todescribethevisco...

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