2GOVERNING EQUATIONS FOR MOTIONAND DEFORMATION OF BLOCK SYSTEMSAND HEAT TRANSFER

The DEM techniques were developed primarily for mechanical deformation/motion processes of

particle or block assemblages. The governing equations are the equations of motion of systems of rigid or

deformable bodies or particles. Due to the demands for modeling coupled hydro-mechanical, thermo-

mechanical and THM processes of fractured rocks for civil engineering (e.g. slopes, tunnels, hydropower

dams and rock foundations), energy resources engineering (e.g. geothermal reservoirs and underground

gas or oil storage caverns) and environmental engineering projects (e.g. underground radioactive waste

repositories), the effects of fluid and heat flows through the fractured rock masses become more and

more important issues for the design, operation and performance assessments of structures in rock

masses. The general conservation equations of mass, momentum and energy in continuum mechanics

are the guiding principles.

A distinct feature of DEM for the coupled THM processes is that fluid flow is usually assumed to be

dominated by the connected fracture networks, and the heat conduction through the rock matrix

dominates the heat transfer process, due mainly to the fact that the volume of fluid is very small

compared to that of rock matrix and the fluid velocity is also very low, in hard crystalline rocks. This

assumption may be less appropriate for very porous rocks like sandstones, but then the DEM is mainly

developed for fractured hard rocks that have large differences in fluid conductivities between the

fractures and rock matrix. The matrix flow and heat convection due to fluid movement are therefore

usually ignored. Accumulated experience in both field experiments and numerical modeling work has

shown that this assumption is basically acceptable, especially for heat transfer processes in fractured

rocks.

Going one step further in the detail, we may summarize the main governing equations as the

Newton–Euler equations of motion for rigid bodies, the Cauchy equations of motion for deformable

bodies, the Nervier–Stokes equation for fluid flow through fracture networks, the heat transfer equation

based on Fourier’s law and various constitutive equations of the rock matrix and fractures. The hydro-

mechanical coupling is represented by the effect of rock deformation on the variation of fracture

apertures (therefore the transmissivity) and the additional boundary stresses on blocks due to fluid

pressure. The effects of pore pressure on rock (matrix) deformation or the effects of stresses on

porosity/permeability of the rock matrix are not considered in this book at this stage since the rock

matrix is assumed to be impermeable.

The treatment of thermal effects on hydraulic and mechanical processes in most of the currently

available DEM codes is through consideration of the thermally induced variation of fluid viscosity,

volume expansion and thermal stresses in the rock matrix, in addition to temperature distribution and

evolution with time during transient heat conduction. The effects of mechanical deformation and fluid

flow on heat transfer, such as conversion from the dissipated energy by mechanical work to heat and heat

convection due to fluid flow in rock are usually ignored in practice.

25

The purpose of this chapter is to present a brief coverage of the governing equations most widely

adopted in the current DEM approaches. We assume that the general reader is familiar with the basics of

continuum mechanics, the FEM as well as tensor analysis techniques. For details on the complete

definitions of the basic concepts and fundamental relations in solid and continuum mechanics, the

readers are encouraged to look in the classical works by Fung (1969), Wang (1975), McDonough

(1975) and Shabana (1998) for the equations of motions, Lai et al. (1993) for heat transfer and general

continuum mechanics principles.

2.1 Newton’s Equations of Motion for Particles

A particle is defined as a body with a constant mass, m, but its volume and shape have little effect on its

dynamical behavior, according to classical mechanics. Denoting the product of mass and velocity,

pi ¼ mvi ði ¼ 1; 2; 3Þ ð2:1Þ

as the linear momentum of a particle of a finite mass m (but negligible volume) and velocity vi, Newton’s

second law of motion states that the resultant force that acts on the particle and causes its motion equals

the rate of variation in its linear momentum, i.e.,

fi ¼ _pi ¼dp

dt¼ m _vi ¼

dðmviÞdt

ð2:2Þ

In Newtonian (non-relativistic) mechanics, the mass of a particle remains constant during motion,

therefore the law can be expressed as

fi ¼dp

dt¼ m _vi ¼ m

dðviÞdt¼ m

d2ui

dt2¼ mai ð2:3Þ

where ui and ai are the displacement and acceleration vectors of the particle, respectively. Newton’s

equations of motion (2.3) consider only the translational motion of a particle since the rotation of the

particle is eliminated by ignorance of its volume (therefore also the shape). This law, expressed by Eqn (2.3),

is a statement of the law of linear momentum conservation.

The above equations are valid when the particle’s motion is represented in a single fixed inertial frame.

If, on the contrary, the particle is fixed in an arbitrarily moving (both translating and rotating) frame o–xyz

relative to a fixed global inertial frame O–XYZ, with angular velocity components ðWx;Wy;WzÞ of the

moving frame relative to the fixed inertial frame and defined by using Euler’s angles (�, , �) (Fig. 2.1),

the expression for the inertial-space acceleration vector of the particle is then written as (Wells, 1967)

Y

a′ z

yx

O

m

f

a

Z

X

z

y

x

o

ψ

φ

θ Ω

N

Fig. 2.1 Motion of a particle in a dual coordinate system.

26

ax ¼ a0x þ €x � x½ ðWy Þ2 þ ðWz Þ2� þ y½WxWy � ð _W zÞ� þ z½WxWz þ ð _W yÞ� þ 2½_zWy � _yWz�

ay ¼ a0y þ €y þ x½WxWy þ ð _W zÞ� � y½ ðWx Þ2 þ ðWz Þ2� þ z½WyWz � ð _W zÞ� þ 2½ _xWz � _zWx�

az ¼ a0

z þ €z þ x½WxWz � ð _W yÞ� þ y½WyWz þ ð _W xÞ� � z½ ðWx Þ2 þ ðWy Þ2� þ 2½ _yWx � _xWy�

8>><>>: ð2:4Þ

where ða0x ; a0y ; a

0zÞ are the translational accelerations of the origin o of the moving frame (o–xyz) relative to

the fixed inertial frame (O–XYZ), and the angular velocities are given by

Wx ¼ _c sin � sin�þ _� cos�

Wy ¼ _c sin � cos�� _� sin�

Wz ¼ _� þ _c cos �

8><>: ð2:5Þ

2.2 Newton–Euler Equations of Motion for Rigid Bodies

A rigid body characterized by a domain W of constant volume V and mass M does not deform. The

distance between any two points in a rigid body remains unchanged. A rigid body is, of course, an

idealization since all bodies deform, more or less, under the action of external forces. However, this

idealization is acceptable in many rock engineering problems, especially large-scale block movements

under low stress conditions. Rigid body dynamics is governed by Newton’s law of motion and Euler’s

rotations of rigid bodies.

2.2.1 Moments and Products of Inertia

In rigid body dynamics, the rotation of the body must be taken into account. The volume and

geometric shape of the bodies therefore become important. The most important properties of a rigid

body are its moments and products of inertia. Assume dm as the mass of a differential element in

the rigid body (Fig. 2.2), dm ¼ � dV ¼ � dx dy dz, its position is represented by vector

r ¼ rif g ¼ðx; y; zÞ and its perpendicular distance to an arbitrary line through the origin OB is d.

The direction cosines of the line OB are then given by ðl;m; nÞ ¼ cos�; cos�; cos�ð Þ in the

adopted inertial frame O–XYZ. The moment of inertia of the element about axis OB is then

given by

dIOB ¼ ðdmÞd2 ¼ � ð jr j Þ2 � ðOPÞ2� �

dV ¼ �½ðx2 þ y2 þ z2Þ � ðlx þ my þ nzÞ2�dV ð2:6Þ

γ

β α

B

dm r

P

d

Z

Y

X

O

Fig. 2.2 Moments of inertia of a rigid body about an arbitrary axis OB.

27

Recall l2 þ m2 þ n2 ¼ 1, then the above equation can be rewritten

dIOB ¼ � ðx2 þ y2 þ z2Þðl2 þ m2 þ n2Þ � ðlx þ my þ nzÞ2½ �dV

¼ � l2ðy2 þ z2Þ þ m2ðx2 þ z2Þ þ n2ðx2 þ y2Þ � 2lmðxyÞ � 2lnðxzÞ � 2mnðyzÞ½ �dVð2:7Þ

The moments of inertia of the whole body are then

IOB ¼Z

V

�½l2ðy2 þ z2Þ þ m2ðx2 þ z2Þ þ n2ðx2 þ y2Þ � 2lmðxyÞ � 2lnðxzÞ � 2mnðyzÞ�dV

¼ �½l2Ixx þ m2Iyy þ n2Izz � 2lmIxy � 2lnIxz � 2mnIyz� ð2:8Þ

where

Ixx ¼Z

V

ðy2 þ z2ÞdV ; Iyy ¼Z

V

ðx2 þ z2ÞdV ; Izz ¼Z

V

ðx2 þ y2ÞdV ð2:9aÞ

are called the moments of inertia about the x-axis, y-axis and z-axis, respectively, and

Ixy ¼Z

V

xy dV ; Ixz ¼Z

V

xz dV ; Iyz ¼Z

V

yz dV ð2:9bÞ

are called the products of inertia of the body, respectively. The products of inertia are symmetric,

i.e., Ixy ¼ Iyx; Ixz ¼ Izx; Iyz ¼ Izy. The collection of moments and products of inertial of a rigid

body is also often expressed by an inertia tensor, Iij, given by

Iij ¼Ixx Ixy Ixz

Iyx Iyy Iyz

Izx Izy Izz

24

35 ¼ Ixx Ixy Ixz

Ixy Iyy Iyz

Ixz Iyz Izz

24

35 ð2:10Þ

which is a second-rank tensor. Its three principal moments of inertia, Ipx ; Ip

y ; Ipz , are given by three

non-trivial roots of the equation

Ip � Ixx Ixy Ixz

Ixy Ip � Iyy Iyz

Ixz Iyz Ip � Izz

24

35 l

m

n

8<:

9=; ¼ 0 ð2:11Þ

The necessary condition for the non-trivial solution of Eqn (2.11) is

Ip � Ixx Ixy Ixz

Ixy Ip � Iyy Iyz

Ixz Iyz Ip � Izz

������������ ¼ 0 ð2:12Þ

The principal directions of inertia, lk; mk; nk, (k = x, y, z), can be obtained by inserting the principal moments of

inertia, Ipk (k = x, y, z), into Eqn (2.11), resulting in the three orthogonal principal axes ðxp; yp; zpÞ.

2.2.2 Mass, Linear and Angular Moments of Rigid Bodies

For a rigid body of mass density � and volume V, its mass M, linear momentum pi and angular

momentum hi are defined by,

M ¼ZZZ

V

�dV ¼ �V ð2:13Þ

28

pi ¼ZZZ

V

�vi dV ¼ �Vvci ¼ Mvc

i ð2:14Þ

hi ¼ZZZ

V

�eijkxjvk dV or

ZZZV

� r� vÞdVð ð2:15Þ

where vi is the velocity of a point x in V with coordinates xi, vci the linear velocity vector at the mass

center of the body, � the mass density and eijk is the permutation tensor. The angular momentum hi is

measured with respect to the origin of an inertial (global) frame. Vector r ¼ ðx1; x2; x3Þ is the position

vector and v ¼ ðv1; v2; v3Þ is the velocity vector in Eqns (2.14) and (2.15).

2.3 Newton’s Equations of Motion for Rigid Body Translations

Let fi denote the resultant force vector of a set of external forces acting on a rigid body of mass M.

The principle of the linear momentum conservation is expressed as

dpi

dt¼ fi ð2:16Þ

or

Mdvc

i

dt¼ Mac

i ¼ fi ð2:17Þ

where aci is the acceleration vector of the mass center of the rigid body. Besides the resultant force fi as

shown above, the set of external forces may also produce resultant torque causing rotational motions as

well. However, Eqns (2.16) and (2.17) are valid regardless of the rotational motions that will be described

by the equations of rotational motions as given below.

2.4 Euler’s Equations of Rotational Motion – The Generaland Special Forms

For a representative differential element of volume dm in a rigid body with an embedded frame o–xyz

with its origin located at an arbitrary point o (Fig. 2.3), its general acceleration vector can be

directly deduced from Eqn (2.4), with the following simplifications: (i) x, y and z are constant and

fa

φψ

o

y x

zZ

Y

X

O

dm

x

yz

N

θ

Fig. 2.3 Rotation and Euler angles of a rigid body.

29

(ii) _x ¼ _y ¼ _z ¼ €x ¼ €y ¼ €z ¼ 0, since no relative movements of material points in the rigid body are

allowed. The angular velocity ðWx;Wy;WzÞ of the moving frame is now the angular velocity ð!x; !y; !zÞof the embedded body frame o–xyz relative to the global inertial frame (O–XYZ). The acceleration of

the element is written as (Wells, 1967)

ax ¼ a0x � xðð!y Þ2 þ ð!z Þ2Þ þ yð!x!y � _!zÞ þ zð!x!z þ _!yÞ

ay ¼ a0y þ xð!x!y þ _!zÞ � yðð!x Þ2 þ ð!z Þ2Þ þ zð!y!z � _!zÞ

az ¼ a0

z þ xð!x!z � _!yÞ þ yð!y!z þ _!xÞ � zðð!x Þ2 þ ð!y Þ2Þ

8><>: ð2:18Þ

where x, y and z are the coordinates of the embedded (local) body frame, whose origin has the coordinates

ðXo; Yo; ZoÞ in the global O–XYZ and acceleration ða0x ; a0y ; a

0zÞ.

For the differential element of volume dm in a rigid body as shown in Fig. 2.3, with acceleration

a ¼ ðax; ay; azÞ and resultant (internal) force f ¼ ðf x; f y; f zÞ, the ‘free particle’ equations of motion are

written

ðdmÞax ¼ fx; ðdmÞay ¼ fy; ðdmÞaz ¼ f z ð2:19Þ

and the moments of force f about the body-embedded coordinate axes are

ðdmÞðazy� ayzÞ ¼ fzy� fyz ¼ dTx

ðdmÞðaxz� azxÞ ¼ fxz� fzx ¼ dTy

ðdmÞðayx� axyÞ ¼ fyx� fxy ¼ dTz

8><>: ð2:20Þ

and the integration over the whole body leads toZV

�ðazy� ayzÞdV ¼ZV

�ðf zy� fyzÞdV ¼ Tx

ZV

�ðaxz� azxÞdV ¼ZV

�ðfxz� f zxÞdV ¼ Ty

ZV

�ðayx� axyÞdV ¼ZV

�ðfyx� fxyÞdV ¼ Tz

8>>>>>>>>><>>>>>>>>>:

ð2:21Þ

Equation (2.21) is the basic form of the equations of rotational motion. Substituting Eqn (2.18) into

Eqn (2.21) and eliminating all acceleration components results in the general form of Euler’s equations of

motion for rigid body rotation

Mða0zyc � a0y zcÞ þ Ixx _!x þ ðIzz � IyyÞ!y!z þ Ixyð!x!z � _!yÞ � Ixzð!x!y þ _!zÞ þ Iyzð!2

z � !2yÞ ¼ Tx

Mða0x zc � a0zxcÞ þ Iyy _!y þ ðIxx � IzzÞ!x!z þ Iyzð!y!x � _!zÞ � Ixyð!y!z þ _!xÞ þ Ixzð!2

x � !2z Þ ¼ Ty

Mða0y xc � a0

x ycÞ þ Izz _!z þ ðIyy � IxxÞ!x!y þ Ixzð!y!z � _!xÞ � Iyzð!x!z þ _!yÞ þ Ixyð!2y � !2

xÞ ¼ Tz

8>><>>:

ð2:22Þ

where ðxc; yc; zcÞ are the coordinates of the center of mass of the rigid body.

Equations (2.22) and (2.17) determine completely the motion of a rigid body, with origin of the body-

embedded frame o located at any point. The LHS of Eqn (2.22) is the summation of the moments of

inertial forces about coordinate axes and the RHS are the summation of moments of the applied forces

about the corresponding axes. The equation is therefore an expression of the ‘principle of moments

conservation’. The resultant moments of forces T ¼ ðTx; Ty; TzÞ can be calculated in the global inertial

frame from the applied external forces by

30

Tx ¼XðfZY � fY ZÞ; Ty ¼

XðfXZ � fZXÞ; Tz ¼

XðfY X � fXYÞ ð2:23Þ

The general form of Eqn (2.22) can be simplified into special forms under certain conditions.

(a) Assuming that the origin o is located arbitrarily in the body and the local embedded coordinate

axes x, y and z are oriented along the principal axes of inertia through origin o, the products of inertia

Ixy ¼ Ixz ¼ Iyz ¼ 0 and the rotational equations of motion become

Mða0z yc � a0

y zcÞ þ Ixx _!x þ ðIzz � IyyÞ!y!z ¼ Tx

Mða0x zc � a0z xcÞ þ Iyy _!y þ ðIxx � IzzÞ!x!z ¼ Ty

Mða0y xc � a0x ycÞ þ Izz _!z þ ðIyy � IxxÞ!x!y ¼ Tz

8>><>>: ð2:24Þ

(b) Assuming that the origin o of the embedded coordinate system (o–xyz) is located at the center of

mass of the body (even though they may not be body-fixed), then xc ¼ yc ¼ zc ¼ 0. On the contrary, if

the origin o is arbitrarily located but fixed in space (but can rotate around o), then a0x ¼ a

0y ¼ a

0z ¼ 0. In

either case, Eqn (2.22) is simplified to

Ix _!x þ ðIzz � IyyÞ!y!z þ Ixyð!x!z � _!yÞ � Ixzð!x!y þ _!zÞ þ Iyzð!2z � !2

yÞ ¼ Tx

Iy _!y þ ðIxx � IzzÞ!x!z þ Iyzð!y!x � _!zÞ � Ixyð!y!z þ _!xÞ þ Ixzð!2x � !2

z Þ ¼ Ty

Iz _!z þ ðIyy � IxxÞ!x!y þ Ixzð!y!z � _!xÞ � Iyzð!x!z þ _!yÞ þ Ixyð!2y � !2

xÞ ¼ Tz

8>><>>: ð2:25Þ

(c) If the origin o of the body is either fixed in space or located at its center of mass, and the body-embedded

frame is oriented along the principal inertial directions at the same time, Eqn (2.22) is then simplified to

Ipx _!x þ ðIp

z � Ipy Þ!y!z ¼ Tx

Ipy _!y þ ðIp

x � Ipz Þ!x!z ¼ Ty

Ipz _!z þ ðIp

y � Ipx Þ!x!y ¼ Tz

8><>: ð2:26Þ

where ðIpx ; I

py ; I

pz Þ are the principal moments of inertia along principal axes ðxp; yp; zpÞ.

The integralsR

xmyndV ,R

ymzndV ,R

zmxndV (m, n = 0, 1, 2) are termed the integral properties of a

rigid body and can be analytically evaluated for two-dimensional polygonal bodies using the simplex

integration approach (Shi, 1993), as given in Chapter 9, when the coordinates of their vertices are

known. During motion, these coordinates change continuously, so that their re-evaluation at each time

step is computationally costly. This is the reason why for some DEM codes the calculations are

simplified by using equivalent circular discs or spheres replacing the generally shaped polygons or

polyhedra (but of identical areas and volumes) so that Eqn (2.26) is used for rotational calculations with

constant inertial matrices. For rock engineering problems using tightly packed particle systems of little

rotation, this simplification may be acceptable for some practical circumstances. However, it is theore-

tically incorrect and could not be applied for fundamental studies of the general behavior of granular

media when rotation and rotational moments are critical issues that need to be considered, such as DEM

simulations for equivalent Cosserat media where couple stresses due to particle rotation are the most

important variable for the media’s mechanical behavior, see Chapter 11 for more details. Other

techniques using numerical integration based on triangularization of surfaces of polyhedra by triangle

elements and constructive solid geometry (CSG) approaches using assembled regularly shaped solid

parts for the representation of bodies of complex shapes can be seen in Messner and Taylor (1980) and

Lee and Requicha (1982a,b).

31

2.5 Euler’s Equations of Rotational Motion – Angular MomentumFormulation

The general form of Euler’s rotational equation of motion for a rigid body is established using the

concept of angular velocity, Euler angles and moments and products of inertia. A different form of the

rotational equations of motion can be formulated using the concept of angular momentum. Referring to

Fig. 2.4, regard the O–XYZ frame as the inertial and the o–xyz frame as body-embedded (but the origin o

may or may not be attached to the body) and the two frames are parallel, with

X ¼ X0 þ x; Y ¼ Y 0 þ y; Z ¼ Z 0 þ z ð2:27Þ

where ðX 0;Y 0;Z 0Þ are the coordinates of the origin o of the body frame in the inertial frame.

Regarding f ¼ ð fx; fy; fzÞ as the resultant force acting on the differential element dm, the equations of

motion of the element, taking it as a free particle, can be written as

ðdmÞ€X ¼ fx; ðdmÞ€Y ¼ fy; ðdmÞ€Z ¼ fz ð2:28Þ

using accelerations

€X ¼ @2x

@t2; €Y ¼ @

2y

@t2; €Z ¼ @

2z

@t2

Similarly the moments of the body in the inertial frame can be writtenZV

�ð€ZY � €Y ZÞdV ¼ d

dt

ZV

�ð _ZY � _Y ZÞdV ¼Z

V

�ðf zY � fyZÞdV ¼ TX

ZV

�ð€XZ � €ZXÞdV ¼ d

dt

ZV

�ð _XZ � _ZXÞdV ¼Z

V

�ðfxZ � fzXÞdV ¼ TY

ZV

�ð€Y X � €XYÞdV ¼ d

dt

ZV

�ð _Y X � _XYÞdV ¼Z

V

�ðfyX � fxYÞdV ¼ TZ

8>>>>>>>>><>>>>>>>>>:

ð2:29Þ

where ðTX ; TY ; TZÞ are the torques generated by the applied external forces about the X, Y and Z

axes, respectively. Recall the definition of angular momentum from Eqn (2.15), and writing

_X ¼ vX ; _Y ¼ vY ; _Z ¼ vZ as velocity components, Eqn (2.29) can be rewritten as

x

f

X ′

oY′

Z ′Z

Y

X

O

dm

y

z

Fig. 2.4 Coordinate systems for the angular momentum formulation of Euler’s rotational equations of

motion of a rigid body.

32

_hX ¼d

dt

ZV

�ðvZY � vY ZÞdV ¼Z

V

�ðfzY � fyZÞdV ¼ TX

_hY ¼d

dt

ZV

�ðvXZ � vZXÞdV ¼Z

V

�ðfxZ � fzXÞdV ¼ TY

_hZ ¼d

dt

ZV

�ðvY X � vXYÞdV ¼Z

V

�ðfyX � fxYÞdV ¼ TZ

8>>>>>>>>><>>>>>>>>>:

ð2:30Þ

This is the basic form of Euler’s rotational equations of motion of a rigid body defined in the inertial

frame, which uses no angular velocity, moments or products of inertia.

Substitution of relation (2.27) into Eqn (2.29) leads to

ZV

�ð€ZY 0�€Y Z 0ÞdV þZV

�ð€zy� €yzÞdV þZV

�ð€Z 0y� €Y 0zÞdV ¼ TX

ZV

�ð€XZ 0�€ZX 0ÞdV þZV

�ð€xz 0�€zxÞdV þZV

�ð€X 0z� €Z 0xÞdV ¼ TY

ZV

�ð€Y X 0�€XY 0ÞdV þZV

�ð€yx� €xyÞdV þZV

�ð€Y 0x� €X 0yÞdV ¼ TZ

8>>>>>>>>><>>>>>>>>>:

ð2:31Þ

Recall the relations

ZV

�€Z 0ydV ¼ M€Z 0yc;

ZV

�€Y 0zdV ¼ M€Y 0zc;

ZV

�€X 0zdV ¼ M€X 0zc

ZV

�€Z 0xdV ¼ M€Z 0xc;

ZV

�€Y 0xdV ¼ M€Y 0xc;

ZV

�€X 0ydV ¼ M€X 0yc

ð2:32aÞ

and

TX ¼ZV

�ð fzY � fyZÞdV ¼ZV

�½ fzðY 0þ yÞ � fyðZ 0þ zÞ�dV

TY ¼ZV

�ð fxZ � fzXÞdV ¼ZV

�½ fxðZ 0þ zÞ � fzðX 0þ xÞ�dV

TZ ¼ZV

�ð fyX � fxYÞdV ¼ZV

�½ fyðX 0þ xÞ � fxðY 0þ yÞ�dV

8>>>>>>>>><>>>>>>>>>:

ð2:32bÞ

where ðxc; yc; zcÞ are the coordinates of the center of mass of the body in the moving frame, substitution

of Eqns (2.28) and (2.32) into Eqn (2.31) leads to

Tx ¼ZV

�ð fzy� fyzÞdV ¼ Mð€Z 0yc � €Y 0zcÞ þd

dt

ZV

�ð_zy� _yzÞdV

Ty ¼ZV

�ð fxz� fzxÞdV ¼ Mð€X 0zc � €Z 0xcÞ þd

dt

ZV

�ð _xz� z _xÞdV

Tz ¼ZV

�ð fyx� fxyÞdV ¼ Mð€Y 0xc � €X 0ycÞ þd

dt

ZV

�ð _yx� _xyÞdV

8>>>>>>>>><>>>>>>>>>:

ð2:33Þ

33

Defining rate of angular moments in the moving (local) frame by

_hx ¼Z

V

�ð€zy� €yzÞdV ; _hy ¼Z

V

�ð€xz� z€xÞdV; _hz ¼Z

V

�ð€yx� €xyÞdV ð2:34Þ

then Eqn (2.33) can be rewritten as

Tx ¼ Mð€Z 0yc � €Y 0zcÞ þ _hx

Ty ¼ Mð€X 0zc � €Z 0xcÞ þ _hy

Tz ¼ Mð€Y 0xc � €X 0ycÞ þ _hz

8<: ð2:35Þ

and is the usual form of Euler’s rotational equations of motion of rigid blocks, based on the angular

momentum concept. No moments and products of inertia or angular velocities of the block are required

for numerical solutions. All quantities, except for the global coordinates (X0, Y0, Z0) and their deriva-

tives, are evaluated in the local body-embedded frame. If either the block is fixed at one point in space

or the origin o of the moving frame is located at the mass center of the rigid body, Eqn (2.35) is

simplified to

Tx ¼ _hx

Ty ¼ _hy

Tz ¼ _hz

8<: ð2:36Þ

2.6 Cauchy’s Equations of Motion for Deformable Bodies

Unlike a rigid body, a deformable body may translate, rotate and deform, i.e., the body changes from

one configuration to another and has an infinite number of degrees of freedom.

Let a continuous body W have volume V, boundary surface S and undergo both translation and

rotational motions under the resultant force fi and resultant moment li about the origin of an inertial

reference frame. As illustrated in Fig. 2.5, let ða1; a2; a3Þ denote the coordinates of a point x in the body in

the reference configuration at time t = 0. At a later time, the point is moved to another position

ðx1; x2; x3Þ referred to in the same coordinate system, then the mapping

xi ¼ xiða1; a2; a3; tÞ ð2:37Þ

(a1, a2, a3)

configuration at t = 0

configuration at t = t

x2, a

2x1, a

1

x1 = x1 (a1, a2, a3, t)x2 = x2 (a1, a2, a3, t)x3 = x3 (a1, a2, a3, t)

X3, a3

Fig. 2.5 Labeling of particles during motion.

34

links the instantaneous configurations of the body at different instants of time, t. The functions

xiða1; a2; a3; tÞis called the deformation function. When ða1; a2; a3Þ and time, t, are considered as

independent variables, the mapping by Eqn (2.37) gives the instantaneous configurations of the body

at different instants of time, t. Then the description of the mechanical evolution is called a material

description or Lagrangian description. If, on the contrary, the spatial location ðx1; x2; x3Þ and time, t, are

taken as independent variables to describe the process, the description is called spatial description or

Eulerian description. The latter is more convenient because it interprets the mechanical event that occurs

at certain places, rather than following the movements of the particles.

In the material description, the velocity vi and acceleration Ai at point ða1; a2; a3Þ are defined as

viða1; a2; a3; tÞ ¼@xi

@tð2:38Þ

Aiða1; a2; a3; tÞ ¼@vi

@tð2:39Þ

while holding ai constant. The deformation gradient of the body, fij can be expressed as

fij ¼@xi

@aj

ð2:40Þ

and

det f ij

� �> 0 ð2:41Þ

while holding time t constant.

In the spatial description, the inverse of the mapping (2.37) is used and the velocity and acceleration

are, by using the chain rule, written as

vi ¼@xi

@tþ @ai

@xj

@xj

@t¼ Dxi

Dtð2:42Þ

ai ¼@vi

@tþ @ai

@xj

vj ¼Dvi

Dtð2:43Þ

The symbol D(�)/Dt in Eqns (2.42) and (2.43) are the material derivatives.

The deformation gradient, the velocity and acceleration fields are the basic kinematics quantities of

the motion of a continuum body. The other kinematics quantities, such as momentum and energy, can be

defined from these basic quantities.

The equation of continuity is

DM

Dt¼ D

Dt

ZZZ� dW ¼

ZZZ@�

@tdWþ

ZZ�vini dS ¼ 0 ð2:44Þ

where S is the surface of a representative differential volume of the continuum and ni the unit normal

vector of S. The differential form of Eqn (2.44) is then written as

@�

@tþ @ð�viÞ

@xi

¼ 0 ð2:45Þ

and is the expression for the principle of mass conservation.

35

The law of balance of linear and angular momentum may be written as

Dpi

Dt¼ D

Dt

ZZZ�vi dW ¼ fi ð2:46Þ

Dhi

Dt¼ D

Dt

ZZZeijkxjvk dW ¼ li ð2:47Þ

The resultant force and moment of the body about the origin of the inertial frame can then be

expressed as

fi ¼ZZ

ti dSþZZZ

bi dW ð2:48Þ

li ¼ZZ

eijkxjtk dSþZZZ

eijkxjbk dW ð2:49Þ

where bi is the body force.

Applying Gauss’s theorem and Cauchy’s stress formula,

ti ¼ �ijnj ð2:50Þ

where �ij (i, j = 1, 2, 3) denote the components of the Cauchy stress tensor acting on an elementary area

of the deformed body. Relations (2.48) and (2.49) then become

fi ¼ZZ

ti dSþZZZ

bi dW ¼ZZZ

@�ij

@xj

þ bi

� �dW ð2:51Þ

li ¼ZZ

eijkxjtk dSþZZZ

eijkxjbk dW ¼ZZZ

eijkxj

@�ik

@xi

þ bk

� �� �dW ð2:52Þ

where S and W follow the deforming body. Substitution of Eqns (2.51) and (2.52) into Eqns (2.46) and

(2.47) and using continuity Eqn (2.45) lead to the equations of motion for continuum bodies:

D

Dt

ZZZ�vi dW ¼

ZZZ@�ij

@xj

þ bi

� �dW ð2:53aÞ

D

Dt

ZZZ�eijkxjvk dW ¼

ZZZeijkxj

@�ik

@xi

þ bk

� �� �dW ð2:53bÞ

The differential forms of Eqns (2.53) and (2.54) may be written as

�Dvi

Dt¼ @�ij

@xj

þ bi ð2:54Þ

eijk�jk ¼ 0 ð2:55Þ

Equation (2.55) indicates that if the stress tensor is symmetric, i.e., �ij ¼ �ji, the law of balance of

angular momentum is satisfied at a point inside a continuum body. Equations (2.54) and (2.55) are the

equations of motion for deformable bodies, usually called Cauchy’s equations of motion.

36

In many applications for dynamic or quasi-static analyses of block or structural systems, damping is

often used to describe the resistance effects to motions by viscous fluid, such as air. The most common

formulation is to assume that damping is proportional to the velocity of the motion and the equation of

motion for the damped body then becomes

�Dvi

Dtþ cvi ¼

@�ij

@xj

þ bi ð2:56Þ

where parameter c is called the damping coefficient that needs to be determined by experiments that

could be very difficult to conduct for complex structures containing multiple deformable bodies. The

damping can also serve as an artificially added force term to reach a static steady-state solution for a

dynamic equation of motion. The damping term in such a case becomes a factor for a more stable

numerical solution technique, rather than a physically meaningful mechanism; therefore a trial-and-error

procedure may be used to reach a numerically appropriate damping coefficient value. The damping

coefficient then simply plays a role as an artificial acceleration parameter for the convergence of quasi-

static problems of blocks systems in DEM, see the details in Chapter 8.

2.7 Coupling of Rigid Body Motion and Deformationfor Deformable Bodies

2.7.1 Complexities Caused by Rigid Body Motion

and Deformation Coupling

The equations of motion for the deformable bodies, Eqns (2.55) and (2.56), are acceptable descriptions

if the ‘small displacement’ assumption is accepted. This assumption means that, for a deformable body, the

general size and shape of the body before and after the deformation process have negligible differences and

that the strains caused by external and/or internal loads are small. This is an acceptable assumption for

many practical problems where the total displacements are very small compared to the problem size. In

other cases, rigid body movements can also be acceptable approximations for rock engineering problems if

the main contributions of the deformation come from fracture displacements and the rock block deforma-

tions are rather small, such as in wedge sliding of blocks on rock slopes.

However, the small deformation or rigid body assumptions are just two extreme cases of uncoupled

deformation-motion conditions, and are not necessarily universally valid. Under certain circumstances,

deformable bodies may undergo large-scale displacements but have small strains that need to be taken

into account (such as rock block motions during blasting, large-scale landslides or slope failures with

internal rock deformation and fracturing processes, and movement and deformation/fracturing/splitting

of falling rocks in highway engineering design, etc.). Under such conditions, the large-scale rigid body

motion mode and the deformation mode of the bodies are coupled.

The motion and deformation of bodies undergoing large displacements have the following characteristics:

(1) The inertia of the body is no longer constant but is a function of time and displacement/

deformation paths.

(2) The equations of motion become highly non-linear because of the finite rotation of the body

relative to the inertial frame.

(3) The deformation of the body depends not only on the constitutive behavior of the material and the

loads but also on the gross rigid body motion of the body relative to the inertial frame, i.e., the

coupling between the rigid body motion mode and deformation mode.

37

The treatment of this coupling between rigid body motion and deformation has been an important

aspect of engineering mechanics, especially in multibody system dynamics. The simplest method is to

assume that the bodies are linear elastic materials following, therefore, the generalized Hooke’s law. The

resulting subject is called the linear theory of elastodynamics (Eringen, 1974, 1975; Shabana, 1998)

which plays an important role in mechanical engineering and the aerodynamics of flight vehicles. The

basic technique in solving the equations of motion for such problems is a three-step algorithm:

(1) assuming that the system consists of an assemblage of rigid bodies and solving the equations of

motion to produce the inertial and interaction forces for each of the bodies and the gross rigid

body translational and rotational displacements of the body as a whole;

(2) introducing these inertial and interaction forces to each of the bodies, but regarding them as

elastically deformable bodies, to determine their deformation (displacement and strain) and stress

fields according to analytical (if the geometry of the body permits) or numerical methods (FEM,

for example);

(3) superposition of the small elastic deformation fields over the gross rigid body motion displace-

ments.

The rigid body motion and elastic deformation are therefore de-coupled in linear elastodynamics. The

rigid body motion and the static linear elasticity then can be seen as two extreme cases in general

elastodynamics. The former governs the cases where the deformability and stress of the bodies are not of

concern. The gross motion is the objective. Problems governed by the latter need only be concerned with

the linear deformation of the body and the induced stress without gross motion.

Stepwise linearization processes can easily simulate the non-linearity of the materials without extra

difficulties. Such a treatment, however, may not be suitable for problems with high motion velocities and

non-linear deformations since deformation and gross rigid body mode of motion becomes significantly

coupled. However, in most rock engineering problems, the cases of combinations of high velocities of

motions and large, non-linear deformations are rare. Therefore using the linear elastodynamics principles

can be taken as an acceptable approximation.

In the sections below, we first review the complete equations of motion of deformable bodies based

on Cauchy’s classical Eqns (2.54) and (2.55), and then we present the numerical treatment of the

equations of motion in linear elastodynamics.

2.7.2 Extension of Equations of Motion of Deformable Bodies

with Large Rotations

The complete formulation of the equations of motion, which combines the gross rigid body motion

mode and deformation mode, without specifying any specific material behavior is presented in

McDonough (1975). We accept that Cauchy’s equations of motion, Eqns (2.55) and (2.56), are valid,

but we further assume that the motion of the body (and all kinematic parameters) has also been defined

relative to a non-inertial reference frame which is firmly associated with the body, but may or may not be

fixed at the body, and has general translational and rotational motions relative to the inertial frame.

Similar to the case of Euler’s general form of rotational equations of motion for rigid bodies, the

translations and rotations of the non-inertial reference frame thus represent the gross rigid body motion

of the deformable body. The deformational motion of the particles inside the body, at a much smaller

scale, is defined relative to the body-associated non-inertial reference frame. In the following develop-

ment, we denote (X, Y, Z) as the inertial frame coordinates and �x;�y;�zÞð are the coordinates measured in

38

the non-inertial reference frame. The position vector in the inertial and non-inertial frames are denoted as

R ¼ ðX; Y ; ZÞ or Ri ¼ Xi, or r ¼ ð�x;�y;�zÞ or ri ¼ �xi, (i = 1, 2, 3), respectively. The relation between the

position vectors between the two frames for the same material point is given by (Fig. 2.6)

Ri ¼ CiðtÞ þ ri or Xi ¼ CiðtÞ þ �xi ð2:57Þ

where C(t) is the position vector of the origin of the non-inertial reference frame relative to the origin of

the inertial frame and is a function of time, t. The material derivatives of an arbitrary vector, v, relative to

the two frames are given by

_vi ¼DRvi

Dt¼ eijk!jvk þ

Dr _vi

Dtð2:58Þ

where DRð�Þ=Dt and Drð�Þ=Dt are the material time derivatives relative to the inertial frame and non-

inertial reference frame, respectively, and !! ¼ !X ; !Y ; !Zf g is the angular velocity vector of the

non-inertial reference frame in the inertial frame, which is also a function of time. Insertion of relation

(2.58) into Eqn (2.57) then results in

DRRi

Dt¼ DRCiðtÞ

Dtþ eijk!jrk þ

@rri

@tð2:59Þ

D2RRi

Dt2¼ D2

RCiðtÞDt2

þ eijk

DR!j

Dtrk þ eijk!jð!k þ rkÞ þ 2eijk!j

@rri

@tþ @

2r ri

@t2ð2:60Þ

Note that, because of the invariance of vectors and tensors transformed between the two frames, we have

�ij ¼ ��ij; fi ¼ �fi ; "ij ¼ �"ij ð2:61Þ

for the stress tensor, force vector and strain tensor defined in the inertial (without the short bar on the top)

and non-inertial (with the short bar on the top) frames.

Recall Eqns (2.46, 2.47, 2.51 and 2.52) for the laws of linear and angular momentum conservation

and apply them for a deformable body relative to a non-inertial reference frame moving relative to an

inertial frame, then one has similarly

Dr

�pi

Dt¼ Dr

Dt

ZZZ��vi dW ¼ �fi ð2:62Þ

Dr

�hi

Dt¼ Dr

Dt

ZZZeijk�xj�vk dW ¼ �li ð2:63Þ

C(t)

R o

y x

zZ

Y

X

O

dm

r

Fig. 2.6 Deformable body coordinates.

39

�fi ¼ZZ

�ti dSþZZZ

�bi dW ¼ZZZ

@��ij

@xj

þ �bi

�dW

�ð2:64Þ

�li ¼ZZ

eijk�xj�tk dSþ

ZZZeijk�xj

�bk dW ¼ZZZ

eijk�xj

@ _�ik

@xi

þ �bk

��dW

��ð2:65Þ

Substitution of relations (2.57, 2.59 and 2.60) into Eqns (2.46, 2.47, 2.51 and 2.52) and recalling the

invariant relation (2.61), one obtains the following relations after rearrangement of the terms

fi ¼ �fi ð2:66Þ

li ¼ eijkCj�fk þ�li ð2:67Þ

pi ¼ MDRCi

Dtþ eijk!jGk þ �pi ð2:68Þ

hi ¼ eijkCjpk � eijkCjGk þ !i�Iij þ �hi ð2:69Þ

DRpi

Dt¼ M

D2RCi

Dt2þ eijk!jGk þ eijk!jðelmn!mGn þ 2�pi Þ þ

Dr�pi

Dtð2:70Þ

DRhi

Dt¼ eijkCj

DRpk

Dt� eijk

D2RCj

Dt2Gk þ

DR!i

Dtð�Iij!jÞþ eijk!jð!l

�IlkÞ þ !i�Iij þ eijk!j

�hk þDr

�hi

Dtð2:71Þ

where M is the mass of the body, �Iij the inertial tensor relative to the non-inertial reference frame and

Gi ¼ZZZ

W

�ri dW; ¼Mrci ð2:72Þ

where rci ¼ �xic are the coordinates of the mass center in the non-inertial reference frame. The equations

of motion of the deformable body defined in the inertial frame can then be written as the linear and

angular momentum conservation laws:

fi ¼@�ij

@xj

þ bi ¼ MD2

RCi

Dt2þ eijk!jGk þ eijk!jðelmn!mGn þ 2�piÞ þ

Dr�pi

Dtð2:73Þ

li ¼ eijkxj

@�ik

@xi

þ bk

� �¼ eijkCj

DRpk

Dt� eijk

D2RCj

Dt2Gk þ

DR!i

Dtð�Iij!jÞ þ eijk!jð!l

�IlkÞ

þ!i�Iij þ eijk!j

�hk þDr

�hi

Dtð2:74Þ

Using relations (2.61), (2.66), (2.67) and (2.70), the equations of motion of a deformable body

defined in the non-inertial reference frame can be derived as

�fi ¼ fi ¼@�ij

@xj

þ bi ¼ MD2

RCi

Dt2þ eijk!jGk þ eijk!jðelmn!mGn þ 2�pi Þ þ

Dr�pi

Dtð2:75Þ

40

�li ¼ �eijkCj fk þ eijkxj

@�ik

@xi

þ bk

� �¼ �eijkCj fk þ eijkCj

DRpk

Dt� eijk

D2RCj

Dt2Gk þ

DR!i

Dtð�Iij!jÞ

þ eijk!jð!l�IlkÞ þ !i

�Iij þ eijk!j�hk þ

Dr�hi

Dtð2:76Þ

The above equations can be simplified if the non-inertial reference frame is carefully selected, based

on selecting C(t).

When C(t) is fixed at the mass center of the body, the coordinates of C(t) in the inertial frame are

given by

Ci ¼ Xci ¼

1

M

ZZZW

�Ri dW ð2:77Þ

and in the non-inertial reference frame

Gi ¼ZZZ

W

�ri dW;¼ �pi ¼Dr�p

Dt¼ 0 ð2:78aÞ

and

�x c ¼ �y c ¼ �z c ¼ 0 ð2:78bÞ

Eqns (2.66) and (2.67) are then reduced to

@�ij

@xj

þ bi ¼ MD2

RCi

Dt2ð2:79Þ

eijk �xj

@�ij

@xiþ bk

� �¼ DR!i

Dtð�Iij!jÞ þ eijk!jð!l

�IlkÞ þ !i�Iij þ eijk!j

�hk þDr

�hi

Dtð2:80Þ

It is clear from the above derivation and Eqn (2.80) that the main complication in the coupling of

rigid body motion and deformation concerns the coupling between rotation and deformation, since the

inertial tensor is now a function of both time and deformation. This is the so-called issue of inertial

coupling or co-rotation in FEM methods for problems with large rotations of deformable bodies and is

especially important for problems with high velocities and slender bodies.

2.7.3 Treatment of Inertial Coupling of Motion

and Deformation using FEM

The most common numerical technique for solving problems of elastodynamics of multibody systems

is the FEM which is an effective technique for the treatment of inertial coupling of motion and

deformation of deformable bodies. Using the FEM technique, Eqns (2.79) and (2.80) can be formulated

into a matrix equation in a partitioned form (Shabana, 1998) for body i after FEM discretization

Mirru

ir þMi

rfuif ¼ Fi

r

Mifru

ir þMi

rfuif þKi

ffuif ¼ Fi

f

(ð2:81Þ

where Mirr is the mass matrix for rigid body mode, Mi

ff the mass matrix for deformation mode,

Mirf ¼ ðMi

fr ÞT are the inertial coupling matrices, Kiff is the stiffness matrix, ui

r and uif are the generalized

coordinate (unknown) vectors for the partitioned rigid body motion mode and deformation mode (with the

two dots indicating second-order partial differentiation with respect to time), and Fir and Fi

f are the

41

partitioned generalized force vectors, respectively. The subscripts r and f indicate the partition of rigid

body motion and deformation modes, respectively.

Assuming a linearized elastodynamic simplification, the above equation is reduced to

Mirru

ir ¼ Fi

r

Mirfu

if þKi

ffuif ¼ Fi

f �Mifru

ir

(ð2:82Þ

since the contribution of the elastic deformation to the change of generalized coordinates is negligible.

The first equation in Eqn (2.82) can then be solved by considering just rigid body motions alone, but

the impact of the inertial coupling must be included for elastic deformation calculations, as indicated in

the last term in the RHS of the second equation in Eqn (2.82). The key element is to ensure that the

condition of zero strain generation by the rigid body motion in the body prevails – the co-rotation strain

constraint.

It should be noted that the inertial coupling is most important for simulating coupled motion and

deformation of slender bodies using FEM, such as beam, plate and shell structures where the linear

dimensions of the bodies are much larger in one (beam) or two (shell) dimensions than that in other

dimensions, where conventional FEM discretization cannot cope with large rotations since infinitesimal

rotations are used as general nodal unknowns. For general elastic bodies with full FEM discretization

using standard meshing and using displacements as the only nodal unknowns, the inertial coupling is

automatically ensured. An example is given below in Fig. 2.7 as a demonstration using the standard

4-noded plane FEM elements, as shown in Shabana (1998).

For the four-noded rectangular FEM element as shown in Fig. 2.7 with the nodal displacement

components forming the unknown vector U ¼ ðu1x ; u

1y ; u

2x ; u

2y ; u

3x ; u

3y ; u

4x ; u

4yÞT , the geometry matrix of the

element is given by

S ¼ N1 0 N2 0 N3 0 N4 0

0 N1 0 N2 0 N3 0 N4

� �ð2:83Þ

with the following shape functions

N1 ¼1

4bcðb� xÞðc� yÞ; N2 ¼

1

4bcðbþ xÞðc� yÞ ð2:84Þ

2b

2c

Y

X

yx

Global (inertial) system

Local system

u1y

u1x

u 3y

u 3x

u 2y

u 2x

u 4y

u 4x

Fig. 2.7 A rectangular FEM element undergoing large rotation (Shabana, 1998).

42

N3 ¼1

4bcðbþ xÞðcþ yÞ; N4 ¼

1

4bcðb� xÞðcþ yÞ ð2:85Þ

where b and c are the length and width of the element (Fig. 2.7), and the sum of the shape functions is

equal to unity. For a rigid body motion described by two translational components Rx and Ry, and a finite

rotation angle �, the nodal displacement vector U then becomes

U ¼

u1x

u1y

u2x

u2y

u3x

u3y

u4x

u4y

8>>>>>>>>>>><>>>>>>>>>>>:

9>>>>>>>>>>>=>>>>>>>>>>>;

¼

Rx � b cos �þ c sin �Ry � b sin �� c cos �Rx þ b cos �þ c sin �Ry þ b sin �� c cos �Rx þ b cos �� c sin �Ry þ b sin �þ c cos �Rx � b cos �� c sin �Ry � b sin �þ c cos �

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

ð2:86Þ

The product of the shape function matrix S and displacement vector U leads to

SUT ¼ Rx þ x cos �� y sin �Ry þ x sin �þ y cos �

� �ð2:87Þ

where x and y are the coordinates of the embedded local frame in the element and the relation

shows an exact rigid body motion as required. The zero strain condition is met since the rectangle

elements are conforming elements of completeness and compatibility, as defined in Bathe and

Wilson (1976).

2.8 Equations for Heat Transfer and CoupledThermo-Mechanical Processes

Heat is transferred in three modes: conduction, convection and radiation. For fractured rocks,

conduction and convection through fluid movement are the major modes of heat transfer. In this section,

only the basic equations for conductive heat transfer and some key thermal properties (thermal con-

ductivity and heat capacity (specific heat)) are presented. The convective heat transfer due to fluid flow

through fractures will be described in Chapter 8 for DEM presentations.

2.8.1 Fourier’s Law and the Heat Conduction Equation

The basic constitutive law for heat conduction in a continuum is Fourier’s law. It states that the heat

flux, qhi , across a cross-sectional surface of unit area in a continuum is proportional to the gradient of the

temperature field, T, with a proportional coefficient (W/m � K), called thermal conductivity,

qhi ¼ �

@T

@xj

ð2:88Þ

Ignoring the conversion of the mechanical work into heat (which is usually very small for general rock

engineering practice), the energy conservation equation is usually given by

�cp

@T

@t¼ �ðqh

i Þ; j þ sh ð2:89Þ

43

where sh is the source term (W/m3) and cp is called the specific heat of the medium. Substitution of

Eqn (2.88) into Eqn (2.89) then leads to

T ; ii ¼ r2T ¼ �cp

@T

@t� sh

¼ 1

�

@T

@t� sh

ð2:90Þ

and this is called the heat conduction (or diffusion) equation. The symbol � ¼ =�cp is the thermal

diffusivity of the medium. For steady-state problems with no source term conditions, the equation is

similarly reduced to Laplace’s equation

r2T ¼ 0 ð2:91Þ

2.8.2 Thermal Strain and the Constitutive Equation of Thermo-Elasticity

During a coupled thermo-mechanical process, the total linear strain of a material point is assumed

(but well proven in practice) to be the sum of two components: the mechanical strain "Mij caused by

external forces and thermal strain "Tij caused by the temperature gradient field

"ij ¼ "Mij þ "T

ij ð2:92Þ

Assuming elastic behavior of the rock, the mechanical strain follows Hooke’s law of elasticity with

respect to stress, given by

"Mij ¼

1

2G�ij �

3þ 2G�ij�kk

� �ð2:93Þ

where and G are Lame’s elasticity constants and �ij is the Kronecker delta.

The thermal strain is given by

"Tij ¼ �ðT � T0Þ�ij ð2:94Þ

where � is the thermal expansion coefficient, T is the current temperature and T0 is the initial (reference)

temperature. Substitution of Eqns (2.93) and (2.94) into Eqn (2.92) lead to

"ij ¼ "Mij þ "T

ij ¼1

2G�ij �

3þ 2G�ij�kk

� �þ �ðT � T0Þ�ij ð2:95Þ

This is the Duhamel–Neumann relation for thermoelasticity. Its reciprocal is the constitutive equation for

the thermoelasticity of a continuum

�ij ¼ �ij"kk þ 2G"ij � ð3þ 2GÞ � �ðT � T0Þ�ij ð2:96Þ

The equations of motion of elastically deformable blocks of density � with heat transfer is the same as

before, Eqn (2.79), but in a simpler and more commonly seen form

@�ij

@xj

þ �bi ¼ �€ui ð2:97aÞ

but the terms in the stress tensor given by Eqn (2.96) should be used. When the equation is written in

terms of displacement instead of stress, the equation of motion becomes

Gui ; jj þ ðþ GÞuj ; ji þ �bi þ ð3þ 2GÞ�ðT � T0 Þ; j�ji ¼ �€ui ð2:97bÞ

44

2.8.3 Heat Conduction and the Energy Conservation Equation

If the transition of internal energy into heat is to be considered, the heat transfer Eqn (2.83) needs to

be modified. Introducing the heat capacity of the medium as

cp ¼ �1

�

@T

@t

� ��1

� �@qh

i

@xi

ð2:98Þ

with unit (J/kg �C) and assuming that the internal energy of the continuum is a function of both strain and

temperature, then the energy balance equation representing the first law of thermodynamics can be

written as

T; kk þ sh ¼ �cp

@T

@tþ ð3þ 2GÞ�T0

@ _"kk

@tð2:99Þ

where sh is the heat source term.

The effect of mechanical deformation (represented by the volumetric strain rate as the last term in the

RHS of (2.99) on the temperature field) is therefore included. If this effect is negligible, the equation is

reduced to the heat conduction Eqn (2.89).

For an anisotropic media in two dimensions, the transient heat conduction equation is simply

extended as

@T

@t¼ 1

�Cp

@

@xkx

@T

@x

� �þ @

@yky

@T

@y

� �� �ð2:100Þ

where ki (i = x, y) are the heat conductivities of the rock material in direction i.

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