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2. Stressed State of Elastic bodles

2. S t r e s s e d S t a t e of N o n l i n e a r l y Elastic
B o d i e s w i t h D i s l o c a t i o n s and D i s c l i n a t i o n s
Finding the Stressed State of Elastic Body
with Given Characteristics of Isolated Defect;
Set u p o f the Problem
Given the Burgers' and Frank's vectors, bk and qk, let us consider the problem
of finding displacements and stresses in multiply connected nonlinear elastic
body with Volterra dislocations. There are two lines of attack on the problem.
On the first way, the displacement vector components are unknown variables satisfying the nonlinear equilibrium equations and force conditions on
the boundary surface of multiply connected volume v . We are setting up the
boundary value problem in a simply connected domain that is obtained by making cuts in v . On the cut surfaces, the displacements must satisfy the jump
conditions (1.3.6). It is clear that we can find the vector/~ instead of the displacement vector u . For unique solvability of the problem in question, we need
to determine values of the displacement vector and the rotation tensor at some
point of the body.
From continuity at the cuts of Cauchy strain measure, A , it follows continuity of Kirchhoff stress tensor, P . Thus by equations (1.1.11), (1.3.2), it follows
a relation for the jump on a cut in Piola stress vector, n . D ,
where n is the normal vector to the cut surface Tk •
On the second way, the boundary value problem is formulated in the multiply connected domain; now the components of Cauchy strain measure, G sk
, which are determined by the compatibility conditions (1.2.4) and the equilibrium equations (1.1.12)1, are unknowns. If there are no body forces these
equations are easily reduced to a form containing only unknown G sk • Indeed,
multiply equation (1.1.12)1 by C , with regard for D = P . C , we obtain
[div P . C)]. C -~ -- 0.
By equation (1.2.2), we have
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
div P + r k. P . Hk = 0.
Since the tensors Hk are expressed in terms of Gmn and the Kirchhoff stress
tensor for any material ( and thus for anisotropic elastic material) is expressed
in terms of Cauchy strain measure, we can see that the only unknown variables
in equations (2.1.3) are Gm~.
Assume that the surface load is of tracking type, i.e. the vector f in (1.1.12)2
is of the form
f(r) = fo(r, A(r)) • C(r).
In view of equation (2.1.4), the boundary conditions (1.1.12) are
n . P = f0,
so they are also expressed completely in terms of Gmn •
Combining the integral relations (1.3.13), (1.3.17), which express the defect parameters, bk, qk , in terms of the tensor field A , with equations (1.2.4),
(2.1.3) and the boundary conditions (2.1.5), we get the final set up of the problem of equilibrium of a body with isolated defects in Cauchy strain measure
Let us consider the case when the Frank's tensor of each defect is zero; all
the defects are translational dislocations. Now the strain gradient C is a singlevalued function in a multiply connected domain, and to specify the strained
state, one can formulate the boundary value problem with the components of
C taken as unknowns. The compatibility equations for the deformation gradient
consist of 9 linear equations,
rot C -- 0
Combining these with the equilibrium equations and boundary conditions
(1.1.12) and, besides, with the integral relations
~ dr . C = bk,
we get the set up of the problem on translational dislocations in terms of the
deformation gradient. In equations (2.1.7) % is a contour enclosing the line of
kth dislocation with Burgers' vector bk . Dependence of intensities of loads k
and f on the deformation gradient is arbitrary. In particular, the vectors k and
f may be some given functions of Lagrangian coordinates (the dead load).
Under dead load, the equilibrium equations (1.1.12)1 is the identity if
D = rot ,I, + D*,
where ,I~ is a twice-differentiable tensor of second order, D* is a particular
solution to the equations (1.1.12)1.
As well as for the given reference configuration, we can consider a case
of given deformed configuration with prescribed characteristics of defect and
2.1 Finding the Stressed State
compose equations with respect to components of Almansi strain tensor. Such
a set up, when Eulerian coordinates are independent variables, is possible only
for isotropic material because the Cauchy stress tensor is fully determined by
Almansi strain tensor only in isotropic elastic solids, that is
tMN= FMN(gps) ,
tMN= nM. T. R N.
The functions gps(Q,M) must satisfy the compatibility equations, the equilibrium equations,
DivT - 0,
the boundary conditions N . T = F and some relations which are similar to
the equations (1.3.13), (1.3.17). If equations (2.1.9) can be uniquely inverted
then the problem of Volterra dislocations can be stated in stresses. Now the
equilibrium equations (2.1.10) may be identically satisfied with the tensor of
stress functions, O = O T,
Rot O = R M × 0 0 / i ) Q M.
T = Rot ( Rot o)T,
The tensor O must satisfy the compatibility equations, boundary conditions, and integral relations determining Burgers' and Frank's vectors of given
Consider now formulation of the system of equilibrium equations for a nonlinearly elastic body with continuously distributed field of translational dislocations when the density c~ is given in advance. Note that in the constitutive
relation (1.1.20) for the Cauchy stress tensor, by (Vakulenko 1991), the deformation gradient must be replaced by the elastic distortion tensor, C (e) •
C (e)) T
In view of equations
takes the form
A(e) - C(e) • (c(e)) T
(1.1.19), (1.5.7), and (2.1.11),
the Piola stress tensor
On substituting equation (2.1.12) into equation (1.1.12)1, we obtain a vector
equation that, being combined with equations (1.5.8), forms the full system for
the unknowns C (p) and C (e).
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
2.2 Variational F o r m u l a t i o n of t h e P r o b l e m of Volterra
D i s l o c a t i o n s in N o n l i n e a r Elasticity
Let external forces be conservative; we apply Lagrange variational principle
to an elastic body with Volterra dislocations, that is the elementary work of
external forces in equation (1.1.9) is the variation of a functional ~, called the
potential of external loads, with respect to the displacement vector,
~ pk . ~Rdv + ~ f . SRda = 5E.
Transform the multiply connected volume v into a simply connected one
making a sufficient number of partitions and consider the potential energy functional for the elastic body,
A = .~,,YYdv - t~.
The functional is defined on the set of displacement fields, which are twice
differentiable in the simply connected domain and satisfy the jump conditions
(1.3.6) with given parameters bk, qk on the cuts 7k. Since 5u = 5R we can
consider the vector field R , specifying the location of body particles in the
deformed configuration, as an argument of the functional ,4 instead of the displacement field, u. The jump conditions for the vector R are equations (1.3.4).
With regard for (1.1.20), (2.2.1), we obtain the variation of functional A
(f,4= jfvtr (D T. grad~R) d v - ~ p k . ~ R d v - ~ f . S R d a .
Using the divergence theorem and the equation (1.3.4) in the stationary
condition, 5A = 0 we obtain
- ~(div D + pk). 3Rdv + f ( n . D - f ) . ~ R d v
D _ . ~R_ - nk. D+. QT. ~R_)da = O.
Here, nk is the unit normal to the partition Tk which directs from the partition
side marked by "-" to the "+"side. From the variational equation (2.2.3) it
follows the differential equilibrium equations (1.1.12)I, the boundary conditions
n. D = f on the body surface, and the dynamic conjugation conditions (2.1.1)
on the cuts which transform the domain to be simply connected.
Thus, the Euler equations for the potential energy functional are the equilibrium equations in displacements and the natural boundary conditions are the
force conditions and dynamic conjugation conditions on the cuts.
Now let us turn to the variational formulation of the problem of Volterra
dislocations which is based on the concept of complementary energy. The specific
2.2 Variational Formulation
complementary energy of elastic body is introduced (Zubov 1970) as a function
of Piola stress tensor. This function relates with the function of specific potential
energy, )/V(C), by Legendre transformation
I;(D) = tr (D. CT(D)) - W(D),
C = 01)/0D.
To construct the function ~;(D), we need to invert the relationship for Piola
stress tensor D(C) with respect to the deformation gradient, i.e., to express
the deformation gradient as a function of Piola stress tensor. For an isotropic
elastic body, the problem of representation of the deformation gradient in terms
of Piola stress tensor has been fully studied in (Zubov 1976). This paper states
in particular that there is a unique inverse of the relation D(C) if the rotation
angles of material fibers are not too large. As to anisotropic material, the above
problem remains to be unsolved. In this case, one can use a weaker formulation
(de Veubeke 1972; Christoffersen 1973) of the complementary energy principle,
in which the specific complementary energy is a function of two tensors: the
Piola stress tensor and the rotation tensor A. In this case, the constitutive
relation for elastic material takes the form
S=~1 ( p .
U + U . P) = ~1 (D. A T + A . D T) = 0)4;/OV.
So it follows
tr (D. C T) - 14; = tr (S. U ) - 14;.
This enables us to represent the specific complementary energy V as a function of the tensor S called Jaumann stress tensor:
l)(S) = tr (S. U(S)) - )4; [U(S)].
To construct the function (2.2.6), it is necessary to invert the dependence
S(U) linking two symmetric tensors. This problem is much easier than that of
defining the dependence C(D). In the weaker complementary energy principle,
Jaumann tensor is assumed to be expressed, in accordance with equation (2.2.5),
in terms of independent variations, Piola stress tensor, D, and the rotation
tensor, A. By varying equation (2.2.6), we obtain
(iV = tr (U. 5S) = t r ( U . A . 5D T) + tr [(A T. U . D ) . (A T. 5A)]
= t r ( C . 5D T) + tr [(C T. D ) . (A T. 5A)].
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
Piola stress tensor fields satisfying the equilibrium equations and the force
boundary conditions will be called statically admissible. These fields, accordingly
to equation (2.1.8), are expressed in terms of the tensor of stress functions, ,~,
which should meet the boundary conditions on the surface of multiply connected
n . (rot • + D*) = f.
We have assumed that the load is dead, i.e. the vectors f and k are some
given functions of Lagrangian coordinates. The expression (2.2.8) can be transformed into the following
V. ( e . ¢ ) = f - n . D * ,
e =
× E,
V ~ = (E - n n ) . grad ~,
V . q =_ r ~ . O~/Oy ~
where y~ are the Gaussian coordinates on the surface a; r ~ is the reciprocal
vector basis; V is the gradient operator on the surface a; e is the surface discriminant tensor (Zubov 1982). The tensor stress function satisfies equation
(2.2.9) and the dynamic conjugation conditions (2.1.1) on the cuts converting
the body into simply connected, that is
~7. (ek" O+) = V. (ek-O_). Qk,
ek = --nk × E.
The complementary energy functional for elastic body, A1, is defined on the
set of differentiable orthogonal tensors and twice differentiable tensors of stress
functions undergoing the restrictions (2.2.9) and (2.2.10). This functional is
AI[~I,, A ] = f V[D(~I,), A]dv+ E
bk "~k V " (ek " cI'+)da"
We will show that stationary points of .AI satisfies the compatibility equations (2.1.6), written in terms of the rotation tensor and stress function tensor,
the symmetry of the tensor C. D T and the jump conditions (1.3.4) for the
position vector R.
Using equations (2.1.8) and (2.2.7), for the variation of ~41 we obtain:
5A1 = ~ [ t r ( C T. rot~O) + t r ( C T. D . A T. ~A)] dv
+ E bk" ~k V . e k . 5"~+da
With regard for the identity
tr rot (x. y) = tr (y. rot x) - tr (x T. rot yT)
2.2 Variational Formulation
integration by parts in (2.2.12) implies
5A1 = ~ [tr(bO T. rotC) + tr(C T. D- A T. hA)] dv
- .~. tr (e. 5O. CT)da
[tr(ek. ~ + . C T) - tr(ek. ~ _ - c T ) ]
V . e k . 5O+da.
Let ~A1 = 0. First take 5~ = 5~+ = 5cI,_ = 0 on a and Tk (this is
compatible with the restrictions (2.2.9), (2.2.10)). In view of the fundamental
lemma of calculus of variations we derive the continuity equations (2.1.6). These
equations mean that in the domain occupied by the body there is a vector
field R, whose gradient is C(O,A). In a simply connected volume, I / i s a
single-valued function of coordinates, which is defined up to an additive vector
constant. Thus the stationary condition for A1 takes the form
tr (C T. D. A T. 6 A ) d v - £ , V. (e. ~ .
R)da + j f R . ( V . e . 6O)da
J~~k[R_. (V. (ek. 5O_)) - R+. (V. (ek. 5¢+))] da
~k V . e k . ~ + d a = 0.
Here, a' is a closed surface constituted by a and both sides of all the cuts
By the divergence theorem on a surface (Zubov 1982), the integral over a'
vanishes. The integral over a becomes zero because of equation (2.2.9). From
orthogonality of the tensor A it follows that A T. (~A is skew-symmetric. Hence
the stationary state of the functional A1 implies the tensor C T. D to be symmetric that is equivalent, in view of equation (1.1.19), to symmetry of Cauchy
stress tensor and provides the balance of moments of all the forces acting on any
part of the body (Lurie 1970). With regard for equation (2.2.10), the variational
equation becomes
- R _ . Qk - bk). [V. (ek. ~¢+))]da = 0.
Since ~ + on ~'k is arbitrary, we establish the jump condition (1.3.4).
On the basis of the two fundamental variational principles for bodies with
Volterra dislocations, discussed above, we can construct functionals for other
stationary principles similar to those in (Zubov 1971).
A limitation of above variational theorems is that the varied functions are
defined in a simply connected volume obtained by making cuts but not in the
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
initial multiply connected domain. The additional boundary surfaces complicate
the set up and solving the equilibrium problem and bring some elements of
artificiality. In a specific case of plane or generalized plane deformation without
disclinations, there is a variational principle without making the cuts, i.e., the
principle in which the functions to be varied are defined and single-valued in
multiply connected plane region.
Under deformation of the form (1.4.1), for isotropic, orthotropic and some
other materials, Piola stress tensor will have the representation
3 3
D = D{f~fl + D2flf2 + D~f2fl + D2f2f2 + Daf
D~¢ = D~(¢, ¢).
Let body forces be absent; in terms of the complex stress components the
equilibrium equations are
OD~/O-¢ + 0D~/0¢ = O.
On introduction of the complex function of stresses, ~(¢, ~),
D{ = 00/0¢,
D~ = -0(I)/0¢,
equation (2.2.14) becomes identical.
The force conditions on the boundary contour of the plane region, ¢(s), s
being the length parameter, can be written as follows
de D~ = iF1 ,
d---~ - ~s
F 1 = f " af1,
where F 1 is the complex component of the vector of external dead load. To
derive equation (2.2.16), we used a formula for the normal-to-contour vector,
o- -i :l + i :2
From equations (2.2.15), (2.2.16)we have
d--7 = iF1
on the domain boundary.
The boundary of multiply-connected plane domain w consists of the outer
contour, 70, and the inner contours, % (k - I, 2, ...m). In what follows we shall
suppose, for conciseness, that the boundaries of holes are load-free, i.e., F 1 - 0
along %. Therefore,
(I)IT k - - Olk~
2.2 Variational Formulation
where ak are complex constants. By equation (2.2.15), the stress function ¢ is
defined by the stress field only up to a constant, so we may arbitrarily choose the
outer-contour constant arising while we integrate equation (2.2.17). We choose
it such that ¢[7o = 0 when F 1 = 0; hence
~1-~o = i
If in the body there are only translational dislocations then the rotation angle X (see Sec.l.4) and Piola stress tensor are single-valued functions in multiply
connected domain. Now the integral in (1.4.20) is zero for any closed contour.
The complex parameters characterizing the Burgers' dislocation vectors are
~k = ~£ eix(U~d( + Uld~).
Instead of the contour % in equation (2.2.19), we can take any closed contour
enclosing k-th hole only.
If the relations (2.2.15) are considered to be equations to define the function
• when the Piola tensor is given, then they reduce to the problem of finding the
stress function with total differential given in advance. Using equation (2.2.16)
we can easily show that • is single-valued in the multiply connected domain if
and only if the principal vector of external load applied to each hole contour is
zero. This condition holds in the case under consideration (of load-free holes).
As in (2.2.6), we consider the specific complementary energy of an elastic
material under plane deformation,
V = S~U~ - IV,
S~ = f a . S. f~
(a,/3 = 1,2),
to be a function of Jaumann stress tensor, whose complex components are
expressed in terms of the rotation angle and the stress function as follows
S~ S~
S~ = S~ =
On the basis of (2.2.20), we have
= U.SS~ " =
U~5(e -~xo 1o + eiXO-~/O-~)-U~5(e-ixO'blO~)-U~5(eiXO-~lO~).
Let us consider the following functional in which the rotation fields and
stress functions, both single-valued in the multiply connected domain w, as well
as the contour constants, a k , are varied independently:
( / 3 k ~ k _ ~kak).
A2[~, ~:,al,...,a~] = Of Vdw + ~i E
The varied stress function must satisfy equation (2.2.18) and the conditions
= ak on the contours 7k.
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
It is easy to verify that at the stationary state of the functional (2.2.21)
it follows the compatibility equations (1.4.14) expressed in terms of the stress
function and the rotation field, the symmetry property of Cauchy stress tensor
Im ( V l e - i X O ~ / o q ¢
U2 e-iX o(~ / (~-~)=
and the integral relations (2.2.19).
2.3 The Semi-inverse M e t h o d for Solving Quasi-static
Problems of Nonlinear Mechanics of Solids
By semi-inverse methods in continuum mechanics one usually means the techniques for constructing specific solutions such that the system of resolving equations reduce to equations in a less number of independent variables. The semiinverse methods are of great importance when exact statements of boundary
value problems are so complex that cannot be solved by other methods.
It is well known a great role of the semi-inverse method, proposed by
St. Venant to solve the problems on torsion and bending of prismatic beams
in development of linear elasticity as applied science. By St. Venant, threedimensional problems of elasticity were reduced to two-dimensional ones for
the Laplace equation, that enabled us to obtain efficient solutions for a number
of problems of practical importance.
In theory of finite strains of elasticity, the use of semi-inverse methods is
primarily due to Rivlin, Ericksen, Green, Adkins, Lurie (Rivlin 1948; Ericksen
1977; Green, Adkins 1960; Lurie 1980). These authors found types of deformations for which the system of equilibrium equations reduces to one or more
ordinary differential equations. For incompressible materials, the method results in a series of exact solutions to large deformation problems of practical
It should be noted that using semi-inverse methods it is hard to expect to
satisfy boundary conditions exactly. On a part of the boundary we are forced
to "weaken" them, i.e. to satisfy these conditions approximately. For example,
in the classical St. Venant problem, the boundary conditions on the beam ends
are satisfied only in integral sense within the accuracy in forces up to a force
system being statically equivalent to zero.
Although for our needs it is enough to consider only elastic bodies, we shall
formulate the semi-inverse method for simple media with memory with regard
for links between mechanical and thermal effects (Zubov 1981).
Proposed here is the procedure for constructing such types of quasi-static
deformations of bodies with arbitrary memory as those for which equilibrium
equations have only one independent spatial variable. Unlike usual semi-inverse
approach, which consists in specifying the deformation type based on some
semi-intuitive and hardly formulated consideration and subsequent testing of
adopted assumptions, we suggest a procedure to construct a class of needed
deformations. This procedure is based on the analysis of equilibrium equations
2.3 The Semi-inverse Method
for continuum with rheology, which are written in a particular form using expansion of tensors in elements of dual Lagrange-Euler basis. In so doing, the
invariance of the equations with respect to certain transformations, conditioned
by flame-independence and material symmetry properties (isotropy, orthotropy,
etc.) of continuum is essentially used.
Consider the equation system describing a quasi-static process of deformation of thermomechanical continuum with memory. It consists of the equilibrium
equations, the energy conservation law, and the constitutive equations:
DivT + pk = 0,
Divh+ tr(T.6)+pq=p~,
2~: "-- C -1./~k. C -T.
P(t) = ¢[Zt(s)],
J(t)T(t) = cT(t) • P(t). C(t),
e(t) = ¢[Zt(s)],
C = grad R,
g = Grad0,
if(s) - f ( t - s),
J = det C,
s > O.
Here, T is the symmetric Cauchy stress tensor, P is the symmetric Kirchhoff stress tensor, p is the material density in the actual configuration, k is the
body force vector, h is the heat flux vector, q is the mass density of volume
heat sources, e, r/are the mass densities of the internal energy and entropy,
respectively, Grad is the spatial gradient operator, grad is the gradient operator in the reference configuration metric, R is the radius-vector of a particle
in the actual configuration, O is the absolute temperature, g is the temperature
gradient, A is the Cauchy deformation measure, e is the rate-of-strain tensor, t
is the time, if(s) is the history of f up to time t (this history also includes the
actual value of a function). The over-dot denotes the material derivative with
respect to time.
The relation (2.3.2) is the general form of constitutive equations for simple materials with memory which is consistent with the requirement of frameindifference (Truesdell 1977).
The operators ¢, ¢, ~, w, defined on the histories of the deformation gradient, the temperature, and the temperature gradient, must satisfy the restrictions
following from Clausius-Duhem inequality that is the most commonly used formulation of the second law of thermodynamics in continuum mechanics (Truesdell 1977),
p//>_ Div (O-lh) + O-ipq.
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
On introducing the Piola stress tensor, D = J C - T . T, and vector d J C - T . h, the equations (1.2.1) can be written as
div D + pok = 0,
div d + tr (D. c T ) + Poq = P0~,
P0 being the density in the reference configuration.
Let Yk (k = 1, 2, 3) be orthogonal curvilinear coordinates in the reference
configuration of material body (Lagrange's coordinates), Y~ (s = 1,2,3) arbitrary orthogonal curvilinear coordinates in the space (Euler's coordinates).
The Lame coefficients of these coordinates are denoted by am(yk) and An(Y~),
respectively, and the orthonormal vector bases associated with the above coordinates are im(yk) and ln(Ys), respectively. We assume that the vector bases
im and In are of the like orientation (the left or the right simultaneously).
We shall use the following decompositions of the above vectors and tensors
k = knln,
l = grad 0 = C . g = lmim,
h = hnln,
D = dmni,~ln,
C = Cmni~ln,
d = dmim,
A = Amnimin,
P = P,~nimin,
T = Tmnlmln.
From equation (2.3.5) it follows that
Tn~ = J-1CmnDm~ = J-1CmnP~kCks,
J = ICm l,
Amn = CmkCnk,
Dm~ = P,~nCn~,
hn = J-1Cmndm.
Although C ¢ (Cmn)" imIn, the identity t r ( D T. 12) = Dm,(Cmn)" holds.
The proof is based on the symmetry of tensor C T. D and orthonormality of
bases ik and In.
Let functions Y~ = Y~(yk,t) describe motion of continuum. There is the
Cks = a-klAs(OY~/Oyk)
(no summing over
s and k!).
= ala2a3.
The energy balance equation takes the form
1 0 (x/"gds ~ + Dkn(Ckn).Zc POq = PO~,
x/~ Oys as ,/
A special form of the equilibrium equations, found in (Zubov 1981), appeared to be convenient to construct semi-inverse solutions. These equations
2.3 The Semi-inverse Method
are formulated with respect to Piola tensor components in the dual LagrangeEuler basis and have the form
1 0
x/~ Oys
Dsm- ~Dsk
+pokm = 0.
In (2.3.8), (2.3.9), we use summation over indices s, k, n from 1 to 3 and no
summation over m. The easiest way to derive equation (2.3.9) is to use, with
regard for (2.3.7), the virtual work principle
f tr (D T. 3C)dv - / v pok. 3Rdv = 0.
In equation (2.3.10), dR = 0 on the body boundary.
We shall consider gyrotropic material media. The fact that material is
gyrotropic imposes the following restrictions on the material response operator
in equation (2.3.2)
¢[z:(~)] = o . ¢[z'(~)]. o ~,
¢[z:(~)] = ¢[z'(~)],
w[Z~(s)] = w[Zt(s)],
~[z,'(~)] = o . ~[z'(~)],
z:(~) = { o . A'(~). 0 ~, 0~(~), O. V(~)},
where O is an arbitrary properly orthogonal tensor, which is constant (i.e., independent of t and s). Write now the component representation of the constitutive
relation in orthonormal basis
¢~.( A t~,
The property of gyrotropness (2.3.11) means that this component representation does not depend on the choice of orthonormal basis of the like orientation.
In other words, the form of operators Cmn, i.e., dependence of components of
tensor P on pre-history of the Cauchy strain measure and the temperature gradient, is the same in all orthonormal bases of the like orientation. It follows, in
particular, that, in spite of dependence of vector basis im on the coordinates
Yk, the relation of components in this bases of P with components of the tensor
A t and the vector I t is the same in all points of homogeneous body.
A similar statement is true for operators ¢, w, ~.
In what follows, the coaxiality theorem will be of major importance.
Theorem. Assume that in some motion of gyrotropic medium, for a fixed
particle, there exists a unit vector i having properties: a) i and I are coUinear
vectors for all t, s; b) i is an eigenvector of tensor At(s) for all t, s. Then at
this particle the vector d(t) is coUinear to i, and i is an eigenvector of P(t).
To prove this, in (2.3.11) we take a proper orthogonal tensor O of the form
0 = 0 T = 2 i i - E.
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
Under the conditions of theorem, we have
O . At(s) • O T = At(s).
From equation (2.3.11), it now follows the relations
O . P(t) = P ( t ) . O,
O - d ( t ) = d(t).
Substituting equation (2.3.13) into these and multiplying the first equality
in (2.3.14) by the vector i, we obtain ( i . P . i ) i = i.P, that is i is an eigenvector of
tensor P. The second equality in equations (2.3.14) gives us i(i. d) = d. Q.E.D.
In the system of equations (2.3.8), (2.3.9), the Lagrangian coordinates,
Ys , and the time, t, are independent variables whereas Eulerian coordinates,
Yn(Ys, t), and the temperature, O(y~, t), are unknown functions. We will seek
the continuum motions for which the number of independent spatial variables
reduces to one, i.e., all the quantities in equations (2.3.8), (2.3.9) depend on the
only spatial coordinate yl and the time. We may satisfy this condition if the
coordinate frames y~, Yn are such that the corresponding Lame coefficients of
the systems depend on a single coordinate, am = am (Yl), An = An (]I1), respectively. Using the Lame relations, we may show that the only coordinates which
satisfy these requirements are Cartesian and circular cylindrical coordinates
and those which differ from them in transformation of the form: y~ = ~(yl),
y~ = c2y2, y~ = caYa, c2 , Ca being constants. In so far, it is necessary that the
given values of km, q do not depend on y~, Ya. For the Lame coefficients An to
be independent of y2, Ya in the Lagrangian representation, it should be taken
Y1 = c~(yl, t). By (2.3.7), it follows that C,1 = 0 (# = 2, 3).
Let the magnitudes of Cmn and 9 be functions of the coordinate yl and time
t only. From gyrotropy of medium, for homogeneous body, it follows that the
quantities Pm~, din, e, 77 (and thus all the values defined by equations (2.3.6))
will also depend on the coordinate yl and the time only. From equation (2.3.7)
it follows that Cta = 0 and Cmn are independent of y2, ya if and only if the
body motion is of the form
Y1 = o~(yl, t),
Y2 ~- ~(yl, t) + p2(t)y2 + T2(t)y3,
Y3 - "Y(yl, t)+ p3(t)y2 + T3(t)y 3.
The functions in (2.3.15) must satisfy the condition J > 0.
Under above conditions, both sides of inequality (2.3.3) do not depend on
y2, Ya. Thus, the above assumptions do not contradict the Clausius-Duhem
For motions of the form (2.3.15), the system of resolving equations (2.3.8),
(2.3.9) reduces to a system of four equations for four functions of a single spatial
variable and the time: a, /3, 7, 9. As boundary conditions on two surfaces yl
= constant, we can specify, say, the temperature and stress components Tim
(m = 1, 2, 3) being functions of the time only. On the rest part of body surface,
the boundary conditions may be satisfied in the integral (averaged) sense, by
matching the functions pi(t), ri(t).
2.3 The Semi-inverse Method
Note that the Lagrangian and Eulerian coordinates may be of different
type. For example, we can choose the Cartesian coordinates to be Lagrangian
whereas the cylindrical coordinates to be Eulerian, or vice versa. Therefore, the
representations (2.3.15), in spite of their simplicity in writing, contain a rather
wide set of types of deformation.
The semi-inverse method also enables us to find out the continuum strains
which satisfy identically some part of the equilibrium equations. For the motions
(2.3.15), let il be an eigenvector of A'(s). In view of C,1 = 0 (# = 2, 3), this
implies the equalities C12C22 ÷ C13C23 = 0, C12C32 ÷ Ci3C33 = 0, which hold if
C12 = C13 = 0. The general representation of deformation satisfying the above
conditions has the form of equation (2.3.15), with ~ and ~ depending on the
time only. By coaxiality theorem and relations (2.3.6), we have P1, = T1, =
D1, = d, = h , = 0 (m = 2, 3). If k2 = k3 = 0 and m = 2, 3, the equilibrium
equations (2.3.9) are identical. The system of resolving equations consists of the
only equilibrium equation and the heat influx equation (2.3.8). On the surfaces
yl = constant, it may be specified the uniform normal pressures, being the time
functions, as well as the temperature or the heat flux.
Let us list some important cases of deformation described by equations
(2.3.15) with ~ = ~ = 0:
inflation, torsion and extension of a hollow cylinder;
bending of a rectangular block into a sector of a circular cylinder or into a
complete circular-cylindrical tube;
straightening of a sector of circular cylinder into a rectangular block;
cylindrical bending of a hollow-cylindrical sector;
shearing of a sector of a circular tube along the axis;
eversion of a cylinder;
Using expressions (2.3.15) we can describe combinations of all these deformations.
In problems for circular cylinder, indefiniteness in deformation of the form
(2.3.15) is such that we can satisfy end force conditions in St. Venant sense.
In pure mechanical theory of viscoelasticity with no regard for thermal e f
fects, the constitutive equations for stresses do not include the temperature, and
no consideration is given to the heat influx equation. In this case for isotropic
homogeneous bodies, under the condition k2 = 0, it can be proved that the
second equilibrium equation (m = 2), is identically fulfilled in the motions of
the type (2.3.15) with ~ = P2 = r3 = 0 or ~ = P3 = r2 = 0. If ~ = P2 = r3 = 0
or "~ = P3 = r2 = 0, the third equilibrium equation is satisfied when k3 = 0.
These semi-inverse solutions were found by systematic trials, based on
search of tensors C such that one of ik is an eigenvector for At(s), using essentially the coaxiality theorem. In a similar way it can be proved that the
above semi-inverse solutions with a single independent spatial variable remain
valid for orthotropic bodies with predominant directions il, i2, i3 and arbitrary
non-homogeneity with respect to coordinate yl.
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
2.4 Combination of Screw Dislocation and Wedge
Disclination in Nonlinear Elastic Cylinder
Isolated defects such as screw dislocations, wedge disclinations or their combination are rather simple for investigation in the framework of nonlinear elasticity.
Let r, ~, z be the cylindrical coordinates in the reference configuration of
elastic body; R, ~, Z be the cylindrical coordinates in space. The unit vectors
tangential to the coordinate lines in reference configuration and in space will
be by denoted er, e,, ez and eR, e¢, ez , respectively. Consider the following
deformation of continuum
R = R(r),
¢ = ~ + Cz,
Z = (27r)-~b~ + az,
where ~, ¢, b, a are constants and a > 0. This deformation relates to the class of
continuum motions considered in Sec.2.3. For this class, the system of resolving
equations reduces to an ordinary differential equation.
When ~ > 1, relations (2.4.1) describe the deformation appearing if one has
removed the sector 27r~-1 _< ~ _< 27r from a circular hollow cylinder, rotated
the section ~ = 27r~-1 about the cylinder axis up to coincidence with the
plane ~ - 0, then made translational displacement of this section by b~ -1
along the cylinder axis, and at last fastened tightly the planes ~ = 0 and
= 27r~-1. Moreover, the cylinder is subjected to torsion with angle of twist ¢,
axial extension (or compression), and inhomogeneous radial deformation defined
by a function R(r).
If ~ > 1, we have: 0 _< ~ _ 27r~-1, 0 _< (I) _ 27r. This means that in the
reference configuration the body occupies a simply connected domain but in
the deformed configuration the domain is doubly connected.
The case 0 < ~ < 1 corresponds to addition of a material wedge into the
cylinder which was cut by the half-plane ~ - 0. In this case, 0 _ ~ < 27r,
0 _ • _< 2r~, i.e., the body without additional material occupies the doubly
connected domain in the reference configuration but in the deformed state the
domain is simply connected.
Using the formulae of Sec.2.3, let us find the deformation gradient associated
with transformation (2.4.1)
C = R'ereR + ~ e ~ e ¢
e~ez + CReze¢ + C~ezez,
det C =
RR'( be)
R' _- dR
The deformation measures, defined by equation (2.4.2), are as follows
2.4 Combination of Screw Dislocation and Wedge Disclination
A = R
( ab
erer +
b2 /
- ~ + 47r2r2
= R ,2 e R e R +
+ R2¢~
47r~r2 + a 2 ezez,
A = (R')-2eReR + (27ra~ -- b¢)2R ~
aCR+27rr 2 ( e c e z + e z e ~ ) +
c~2 + 47r~r9
ezez •
The invariants of the Cauchy strain measure, A, are
I1 -- tr A = R '2 + n 2--~ + ¢2R2 + 47r2r2 + a 2,
/2 - ~1 tr ( tr 2A - tr h 2)
2 R 2 R '2 a 2
b2 + ¢ 2 R 2 + n 2
~-~ +
+ 41r2r2
/3-= d e t A =
be) 2
If n > 1 the vectors eR, e,~ are single-valued and continuous in doubly
connected domain occupied by the deformed body. Hence from equation (2.4.4)
it follows that A -1 keeps continuity when it passes through the surface (I) a-lCZ which is the final position of the plane ~ = 0 after deformation whereas
the deformation gradient, C, takes a jump,
(~ --, Ca-~z + 0),
C+ -- lim C (~ --+ 27r + ~2o/-1Z - 0),
Q1 = Q~-T = (E - ezez)cos 27r(1 - n -1)
+ezez - ez x Esin21r(1
- n-l).
The relation (2.4.6) is tested by using equations (2.4.1), (2.4.2), and the
apparent representations
e~ = eR cos(~ -- ~) + e¢ sin(qo -- ~),
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
e~ = --eR sin(~ -- ~) + e~ cos(~o -- ~).
The formulae (2.4.4), (2.4.6) show that the Volterra dislocation occurs in
the cylinder. In this case, by equation (1.3.28), the Frank and Burgers vectors
take the form
bl "- bn-lez,
ql = - 2 t g ~ ( 1 - ~-l)ez.
Here, the origin lies on the cylinder axis.
If 0 < k < 1, the vectors er, e~ are single-valued and continuous in the
doubly connected domain occupied by the body in reference configuration. From
equation (2.4.4) it follows that the Cauchy strain measure, A, is continuous and
single-valued in this doubly connected domain. Thus, if 0 < n < 1, it is the
case of Volterra dislocation characterized by the jump equation (1.3.6) with the
b = b~-lez,
q = - 2 t g ~ ( 1 - n)ez.
In linear theory of isolated defects (de Wit 1977), the Volterra dislocation,
whose Frank vector is zero and the Burgers' vector is parallel to the dislocation
axis, is called the screw dislocation, and the defect with Burgers vector of zero
magnitude and the Frank vector being parallel to the defect axis is called the
wedge disclination. By analogy, the isolated defect under consideration can be
called the combination of screw dislocation and wedge disclination.
Negative values of the parameter n correspond to generation of screw dislocation and wedge disclination in the cylinder which is initially inverted.
From the constitutive relations of isotropic or orthotropic material, for deformation of the form (2.4.1) it follows that the Cauchy and Piola stress tensors
T = aR(r)eReR + a¢(r)e~e¢ + T¢z(r)(ecez + eze¢) + az(r)ezez,
D = DrR(r)e~en + D~(r)e~e~ + D~z(r)e~ez
+Dz~(r)eze¢ + Dzz(r)ezez.
The first two of the equilibrium equations in the form (2.3.9) are satisfied identically, whereas the third equation is a nonlinear ordinary differential
equation with respect to the function R(r), namely,
DrR- ~D~
In terms of physical components of Cauchy stress tensor, the equation (2.4.8)
aR -- a¢
2.4 Combination of Screw Dislocation and Wedge Disclination
Boundary conditions for equation (2.4.8) are the load-free conditions on the
lateral surfaces of the cylinder, DrR = 0. For a whole cylinder (without a hole),
the condition on the inner surface is replaced by the requirement R(0) = 0.
It can be shown that stresses in any cross-section of the cylinder (Z = const)
are statically equivalent to a longitudinal force applied at the center of crosssection and to a twisting torque. By matching the parameters a and ¢ we can
make these force factors to be zero.
Thus, the problem of determining stresses caused by an isolated defect,
being a combination of screw dislocation and wedge disclination, was reduced
to a boundary value problem for an ordinary differential equation.
For incompressible material, the Cauchy stress tensor is expressed in terms
of strains up to an arbitrary spherical tensor. Now the function R(r) is found
implicitly from the incompressibility condition, det C =. 1, and the spherical
tensor is defined by equilibrium equations (2.4.8) or (2.4.9) with quadratures.
The solution (2.4.1), describing formation of a screw dislocation and wedge
disclination in an elastic circular cylinder, relates with medium deformation
such that the system of equilibrium equations contains a single independent
spatial variable.
Let us now study a dislocation nonlinear problem wherein application of
the semi-inverse approach reduces to a three-dimensional problem to the twodimensional form, i.e., to a system of differential equations and boundary conditions in two independent variables.
Consider an elastic body that is a prismatic rod, i.e. noncircular cylinder,
in the reference configuration (undeformed state). Cartesian coordinates xl, x2
refer to the plane of prism cross-section whose center of inertia is the origin,
x3 refers to the rod axis. The coordinate basis vectors shall be denoted by ek
(k = 1, 2, 3). Xn (n = 1, 2, 3) stand for Cartesian coordinates of points in the
deformed body, they refer to the same directions ek and origin.
Using the complex coordinates and complex bases introduced in Section 1.4,
let us define deformation of the prismatic rod as follows
z -- z0(~, ~)e i¢x3,
X3 = c~x3 + ~ arg (~+ w(~, ~),
where ¢, a, and b are constants, z0 is a complex-valued function and w is a real
single-valued function. Geometrically, the representations (2.4.10) mean that in
the prismatic body there is a screw dislocation whose axis directs along the x3axis and the Burgers vector is equal be~; the bar cross-section, at a distance x~
from the origin, being deformed (plane deformation, given by the function z0) ,
rotates about the rod axis through the angle ¢x3, translationally moves along
the axis for distance (a-1)x3, and warps (the form of warping is described by the
function w). The deformation gradient C corresponding to the transformation
(2.4.10) is defined by
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
c = c~f-/~,
0Z0 _i¢x3
= C 2 = --~-e
-- -C3
~ --
1 :
OZo •
e ~¢~3,
--a Ow
= C2 = -~
The constitutive relations for isotropic material with respect to the Piola
stress tensor may be written as follows
D = a~ (Iq)C + a~(Iq)A. C
+ o/3(/q)C -T,
11 = tr A,
1 tr 2A - tr A2),
I~ = ~(
Ia = det A,
where aj (j = 1, 2, 3) are scalar functions of the invariants Iq (q = 1, 2, 3).
For complex components of Cauchy strain measure, by equations (2.4.11),
we have
All = A22 = OZo O-2-d Jr-~2
o¢ o¢
1 (OzoO~o
A13 =
A23 =
A31 =
0( 0~ { 0( 0-~
A32 -
Zo--~- - -20 0¢ ,] + a'7,
Aaa = ¢~Zo~o + a 9,
Ank = f , . A . f k,
"/= O~
The invariants Iq are as follows
11 -- 4A12 + Aaa,
(Ozo O~o
12 = 4[(A~2) 2 + A12A3a- AliA22- A~aA2a],
oozo) + 2¢ Im (7-2o-~
ozo- "yZoozo
0( ] "
0( 0-~
The relations (2.4.13), (2.4.14)indicate that under deformation (2.4.10), the
tensor A and its invariants do not depend on the coordinate x3. The tensors
A . C and C involved in the constitutive relations (2.4.12) take the form
2.4 Combination of Screw Dislocation and Wedge Disclination
X/~3C-T= Ls fm f s,
A . C = K~nfm f ~,
--~. [OZo (OZo___
1 2 )
O~o ,,0-2o0Zo
K1 = K2 = L o¢ \ o¢ ~ +-~-£(-~ + 2~-~+ -~¢ zo-~o
+ ~ 2 ~ - ~ + i¢~zo
- ~¢Zo-~
, =_K1rozo(.ozo
o oozo
= Loc: ~ - ~
+ - -o¢
-- £
e *~,
+ 2~+
( zo )
0¢ zo( o i,zoOZO
~ + iCzo--~ + lCZo-~-
+ ~ 27--0-~- + iCazo - ~¢z o--~j e'¢x3,
K~ - K3
+ 2~V-b-~- + iCzo(¢~zo~o + £ ) ei¢~,
( OzoOzo
= K~ = ~ 2 o¢ o~ + 4~2 + ~ i¢,:,,, zo--5-( - -~o-y(
(azo 0~o 0~oa~o
/(33 = 4a'y~ + iCzo (0-2o7
\--~ + -0~
-i*~o -~- + -O-~7) + a(¢2ZoZo + a2),
= L2 = c~-~ - iCzo"/ e*~
L~= L-~I = ( iCzo~-C~--~je
Ozo'~ '¢xa,
( Ozo
= L3 = 7"~- - V"~') e'¢~3,
L~ -3
= r~ =
1. ( 0zo
_~°~ + zo-b-(],
0zo 0-20 0-200Zo
The representations
D 1
- p.(~,
~)e i~,
D.2 -- p.(~',
~)e -i~x3
D~ = p~(¢,-~), (~ =
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
follow from the equations (2.4.11), (2.4.12), (2.4.15), where quantities p~ are
independent of the axial coordinate x3. With regard for (2.4.16), the equilibrium
equations are
Op~ Op1 i 1
O~ F - - ~ + ~¢P3 = O,
Op~ ~- Op~
--~ = O.
With given constants b, a, and ¢, the relations (2.4.17) make up a system
of equations with respect to the complex and real functions, z0 and w respectively, in two variables. To these equations it is necessary to associate boundary
conditions on the lateral surface of the prism which, for the free load surface,
O~ ~ = O,
~88/.,2 -- ~88b,1
0¢ ~3 = O,
~8/j2 -- ~88b,1
where ~(s) describes the contour of cross-section of the bar, s is the arc-length
of the contour.
If z0 and w is a solution to the boundary value problem (2.4.17), (2.4.18)
it is easy to see that the functions z0eiw and w + a with w and a being real
constants also satisfy the conditions (2.4.17), (2.4.18). This ambiguity can be
eliminated if we require
(~Z0 - l ° z °
1) da = O,
f~ wda = O,
a being a cross-section of the undeformed bar. Under the conditions (2.4.19),
we can expect uniqueness of solution to the problem (2.4.17), (2.4.18), at least
for not too large strains.
Assume that the bar cross-section is centrally symmetric, i.e. remains the
same under rotation about the rod axis through 180 degrees (an example is
a cross-section of the form of the letter "Z"). Cross-sections with two axes of
symmetry are obviously in this class too.
Make the following changes of independent variables and unknown functions
in the system (2.4.17) and boundary conditions (2.4.18):
~' - -~,
~ " = -~,
-~' -
-~" = ~,
Zo' = -z0,
z0" = -zo,
w' = w,
~" = - w .
There is the following
Theorem. For homogeneous isotropic elastic body, under transformation
(2.~.20), the boundary value problem (2.~.17)-(2.~.19) in a centrally symmetric
2.4 Combination of Screw Dislocation and Wedge Disclination
domain is invariant. If the domain a possesses two axes of symmetry coincident with the axes x l and x2 , the boundary value problem is invariant under
transformations (2.~.20), (2.~.21).
The theorem is proved by substituting equations (2.4.20), (2.4.21) into the
boundary value problem.
Let z0 = h(¢,~), w = m(~,~) be a solution to the boundary value
problem (2.4.17)-(2.4.19). By the theorem, the functions Zo = - h ( - ~ , - ~ ) ,
w = m ( - ~ , - ~ ) satisfy the same boundary value problem. Uniqueness of the
solution implies h(~, ~) = - h ( - ~ , - ~ ) , m(~, ~) = m ( - ~ , - ~ ) .
So, if the cross-section is centrally symmetric, the solution possesses the
~ o ( - ~ , - ~ ) = -zo(¢, ~), ~ ( - 4 , - ~ )
= w(¢, ~).
For bisymmetric cross-section, in addition to (2.4.22), there is
zo(¢, ¢) = ~o(¢, ¢), ~(¢, ;) = -w(¢, ¢),
From the equations (2.4.11)-(2.4.15), (2.4.22)it follows
p~(-¢,-¢) =p~(¢,¢),
p ~ ( - ~ , - ¢ ) = -p~(¢,¢),
p~ (_if, _~) = _pa (~.,~)
(o~,/3= 1, 2),
p~(-¢, -~) = p](¢, ~).
For the bisymmetric cross-section, besides, there holds
p~(~, ~) = -p~ (~, ~),
P3(~, ~) = Pa(~, ~).
The principal vector F and principal moment M of forces acting in a rod
section, xa = const, are
R = Xmem = zkfk,
f fa'Dx(R-Ro)da,
RO = X3(0, 0, xa)f3,
R being the radius-vector of a point of deformed body; Ro is the radius-vector
of the reference point which is the center of inertia of the deformed bar section.
Based on equations (2.4.10), (2.4.16), and (2.4.26), we have
2. Stressed State of Nonlinearly Elastic Bodies with Dislocations
F = f~/~1,~i¢x3
re3 ~ f l + ~'3
f2 + Pafa)
l ( z o p ~ - -z0p1)
- zoP3) e -'~3 f2] da,
+ (w • 1 - z0p ) e
= w(~, ~) - w(O, 0).
Using equations (2.4.22), (2.4.24) and (2.4.27) we obtain
The relations (2.4.28) mean that under deformation of the form (2.4.10),
the stresses in a cross-section of the cylinder can be reduced to a longitudinal
force F3 and a torque M3. These quantities remain constant in all prism crosssections; on solving the boundary value problem (2.4.17)-(2.4.19), they become
known functions of the parameters ¢, a in accordance with formulae
F3 = F . f3 = f~ p~da,
M3 = M . f3 = - f~ Im (zop~)da.
Thus, the assumptions (2.4.10) for a cylinder with centrally symmetric crosssection enable us, by solving two dimensional boundary value problem, to satisfy
the equilibrium equations and boundary conditions on the lateral surface exactly, whereas the boundary conditions at the bar ends only in the integral
sense, matching the constants ~, a.