Applied Mathematical Sciences, Vol. 10, 2016, no. 56, 2789 - 2799
HIKARI Ltd, www.m-hikari.com
http://dx.doi.org/10.12988/ams.2016.67220
The Newton–Kantorovich Method in Analytical
Solution of Plane Elasticity Problems
under Superimposed Finite Strains
Vladimir Anatolievich Levin
Lomonosov Moscow State University
Department of Mechanics and Mathematics
119991 Moscow, Leninskie Gory, 1, MSU Main Building, Russian Federation
Konstantin Moiseevich Zingerman
Fidesys Limited, Office 402, 1 bld. 77, MSU Science Park, Leninskie Gory
Moscow 119234, Russian Federation
c 2016 Vladimir Levin and Konstantin Zingerman. This article is distributed
Copyright under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The specific features of the application of modified Newton’s method
to approximate analytical solution of a class of nonlinear elasticity problems under finite plane strains are described. This class of problems
includes problems of stress distribution around holes in infinite nonlinear elastic bodies. The problems are solved within the framework of
the theory of superimposed finite strains. It is assumed that holes are
originated in previously loaded bodies undergoing finite homogeneous
initial strains. The specific features mentioned above are related with
the necessity to obtain functions in right parts of linearized equations in
a form that admits analytical integration. For this purpose, the equations and boundary conditions are written in a special form using the
Cayley–Hamilton theorem and some changes of variables. The analysis
is performed for the materials of Mooney type. Some numerical results
are given.
Mathematics Subject Classification: 74B20, 74H10
2790
Vladimir Levin and Konstantin Zingerman
Keywords: stress concentration, nonlinear elasticity, superimposed finite
strains, the Newton–Kantorovich method
1
Introduction
The paper deals with the approximate analytical solution of a class of nonlinear elasticity problems. This class includes problems of stress distribution
around holes that are originated in preliminarily strained bodies undergoing
finite initial strains. The problems are formulated and solved within the framework of the theory of superimposed finite strains [2, 5]. The solution of these
problems may be used for modelling of fracture and for strength analysis of
solids in which defects are originated after loading.
The theory of superimposed finite strains describes multi-stage loading of
bodies. Some states (configurations) of a body are considered, and it is assumed that the body passes from each state to the next one after the application of external forces to this body, or after removal of a part (parts) of the
body, or after changing of material properties in a part (parts) of the body
(the origination of inclusions) [10], or after the junction of parts of the body
[6].
The problems of superimposed finite strains can be solved analytically
[2, 3, 4, 10] or numerically [5]. An effective method for approximate analytical solution of these problems is the Newton–Kantorovich method [1]. This
method reduces the solution of a nonlinear problem to the solution of a sequence of linearized problems. If the modified Newton–Kantorovich method is
used, these linearized problems can be solved analytically using the Kolosov–
Muskhelishvili technique [9]. However, there is a need for some transformations
of equations and boundary conditions. These transformations permit one to
obtain functions in right parts of linearized equations and boundary conditions in a form that admits analytical integration. These transformations are
considered in detail in this paper, and some numerical results are presented.
Consider now the mechanical statement of problems of stress distribution
around holes that are originated in preliminarily strained bodies [3].
2
Problem statement
Assume large plane static strains and stresses are brought about by external
forces in a non-linear elastic body that was in the initial (unstressed) state. The
body passes to the first intermediate state. Then a closed surface is imagined
in the body, and its contents, bounded by this surface, is removed, and the
effect of the removed part of the body on the remainder is replaced by forces,
distributed over this surface. This transformation doesn’t change the stress
The Newton–Kantorovich method in plane elasticity
2791
and strain states in the body. Then these forces, changed to the category of
external forces, are reduced to zero quasistatically. It raises large strains and
stresses that are superimposed on the large initial strains and stresses already
existing in the body. The body passes to the final state. The shape of the
introduced boundary surface is changed, and one can predict the shape of this
surface either in the intermediate state or in the final state, and these two
problems are different.
The following notation is used in the theory of superimposed finite strains.
· is the sign of tensor contraction,
: is the sign of double tensor contraction,
I is the second-rank identity tensor.
The states are numbered consecutively from 0 (the initial state) to N (the
final state). The states from 1 to N − 1 are considered as intermediate states.
n
∇ is a gradient operator in coordinates of the n-state;
n
R is the position vector of a particle in the n-th state;
n
Γ is a boundary of a body in the n-state;
un+1 (t) is a displacement vector defining the transition from the n-th state
to (n + 1)-th state;
Ψm,n is the deformation gradient in transition from the m-th state to the
n-th state;
1 + ∆m,n is the relative volume variation in transition from the m-th state
to the n-th state;
Gm,n = Ψm,n · ΨTm,n is a tensor defining the strains associated with the
transition of a body from the m-th state to the n-th state (the generalization
of the Cauchy–Green deformation tensor);
Fm,n = ΨTm,n · Ψm,n is a tensor defining the strains associated with the
transition of a body from the m-th state to the n-th state (the generalization
of the Finger deformation tensor);
σ 0,n is the total true stress tensor for the n-th state;
m
S0,n+1 is the generalized stress tensor in the base of the m-th state under
transition from the initial state to the (n + 1)-th state; at m = 0, this tensor
is the second Piola–Kirchhoff stress tensor;
p0,n is the Lagrange multiplier for the n-th state.
For simplicity we consider only the problems of creation (origination) of
one hole, when there is the single superposition of finite strains. The contour
shape can be given at the moment of creation or at the final state. Let us
assume that the mechanical material properties are described by Mooney [8]
potential. The initial strains are assumed to be homogeneous.
Let us give the problem statement in concerned cases.
1. The hole shape is given at the moment of creation.
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Vladimir Levin and Konstantin Zingerman
The equilibrium equation:
1
1
∇ · Σ0,2 · Ψ1,2 = 0 .
(1)
(the incompressibility constraint is taken into account).
The incompressibility constraint:
1 + ∆0,1 = 1 ,
1 + ∆1,2 = 1 .
Boundary conditions:
1
1
1
−1
∗ −1
N2 · Σ0,2 1 = −P 1 + ∆0,2 N2 · Ψ1,2 · Ψ1,2 ,
(2)
(3)
Γ2
σ 0,2 |∞ = σ ∞
0,2 .
(4)
The relation between the true stress tensor and the total stress tensor is
the following [subject to the incompressibility constraint (2)]:
1
−1
∗ −1
Σ0,2 = Ψ1,2 · σ 0,2 · Ψ1,2 .
(5)
Constitutive equations:
µ
(1 + β)F0,n + (1 − β) (F0,n : I) F0,n − F0,n 2 −
σ 0,n =
2
− p0,n I (n = 1, 2) ,
(6)
where
F0,n = Ψ∗0,n · Ψ0,n
(n = 1, 2) .
(7)
Geometric equations:
1 + ∆0,1 = det Ψ0,1 ,
1 + ∆1,2 = det Ψ1,2 ,
Ψ0,2 = Ψ0,1 · Ψ1,2 ,
1
Ψ1,2 = I + ∇u2 .
(8)
(9)
(10)
2. The hole shape is given at the final state.
The equilibrium equation:
2
∇ · σ 0,2 = 0.
(11)
The incompressibility constraint:
1 + ∆0,1 = 1 ,
1 + ∆1,2 = 1 .
(12)
The Newton–Kantorovich method in plane elasticity
2793
Boundary conditions:
2
N2 · σ 0,2 2 = −N2 P,
(13)
σ 0,2 |∞ = σ ∞
0,2 .
(14)
2
Γ2
Constitutive equations:
σ 0,n =
µ
(1 + β)F0,n + (1 − β) (F0,n : I) F0,n − F0,n 2 −
2
− p0,n I (n = 1, 2) ,
(15)
where
F0,n = Ψ∗0,n · Ψ0,n
(n = 1, 2) .
(16)
Geometric equations:
1 + ∆0,1 = det Ψ0,1 ,
1 + ∆1,2 = det Ψ1,2 ,
Ψ0,2 = Ψ0,1 · Ψ1,2 ,
−1
2
Ψ1,2 = I − ∇u2
.
3
(17)
(18)
(19)
Application of the Newton-Kantorovich
method to the problem solving
Let us consider the application of the Newton-Kantorovich method [1] to the
solution of the problem of stress concentration near the hole, which is originated in the previously loaded body made of nonlinearly elastic material in
the case of finite strains.
In this section the following additional notation is used.
The upper index in parentheses denotes the number of approximation.
(i)
un is the i-th approximation for displacement vector un , characterized the
conversion from the (n − 1)-th to the n-th state;
b (i)
u
n is the correction to the displacement vector un in the (i + 1)-th approximation;
(i)
pbn is the correction to the Lagrange multiplier p0,n in the (i + 1)-th approximation (for incompressible materials);
k
∇ is the gradient in the base of the k-th state;
∇ is the gradient in the base of the state, in which coordinates the problem
is solved;
Γn is the body border in the n-th state in coordinates of the state, in which
the problem is solved;
2794
Vladimir Levin and Konstantin Zingerman
Nn is the normal to Γn ;
Ψ(i)
q,p is the i-th approximation for the displacement dyadic Ψq,p ;
(i)
Fm,n is the i-th approximation for the tensor measure of deformation Fm,n ,
described the the strain changing in going from state m to n and corresponds
to the Finger measure;
(i)
∆m,n is the i-th approximation for the relative change of the volume ∆m,n
in going from the m-th to the n-th state;
(i)
σ 0,n is the i-th approximation for the true stress tensor σ 0,n , described the
cumulative stresses in the body in going from the initial to the n-th state;
m (i)
m
Σ 0,n is the i-th approximation of the generalized stress tensor Σ 0,n (total
for the n-th state), defined in coordinate base of some m-th state;
L[u, p] = µ(1 − β)(∇ · u)I + µ(∇u + u∇) − pI;
(i)
fn is the vector of the fictitious body forces for the i-th approximation;
(i)
Qn is the vector of the fictitious surface forces, applied to the hole borders,
for the i-th approximation.
Let us modify the relations in the statement of considered problems to avoid
the necessity to inverse tensors and to divide some expressions by scalar functions in the process of implementation of the Newton–Kantorovich method.
Functions of complex structure will appear in the right-hand parts of the linearized equations at each step of the Newton-Kantorovich method, if we do
not modify the statement of problems. These functions are not analytically
integrable. The Cayley–Hamilton theorem [7] can be used to realize these
modifications. According to this theorem, the following identity is true for any
nonsingular second-rank tensor T:
1
1
1
2
2
2
−1
T − (T : I) T + (T : I) I −
T :I I .
(20)
T =
det T
2
2
Consider now the modifications of the problem statement for two cases
considered below.
1. The hole shape is given at the intermediate state. Using (20) and taking
into account the incompressibility constraint det Ψ1,2 ≡ 1, one can write
1
1
2
−1
2
2
Ψ1,2 = Ψ1,2 − (Ψ1,2 : I) Ψ1,2 + (Ψ1,2 : I) I −
Ψ1,2 : I I .
(21)
2
2
The substitution of the last equation in the relation (5) and the boundary condition (3) results to the modified version of the problem statement.
The obtained equations do not contain negative powers of tensors and scalar
functions.
2. The hole shape is given at the final state. The following notation is used
in this case:
2
Ψ2,1 = I − ∇u2 ,
(22)
2795
The Newton–Kantorovich method in plane elasticity
then the relation (19) can be written in the form
Ψ1,2 = Ψ2,1 −1 .
Taking into account the Cayley–Hamilton theorem and the incompressibility constraint, one can write the last equation as follows:
1
1
2
2
2
Ψ2,1 : I I .
(23)
Ψ1,2 = Ψ2,1 − (Ψ2,1 : I) Ψ2,1 + (Ψ2,1 : I) I −
2
2
Replacement of the relation (19) by (22), (23) results to the equations of
the modified problem statement. These equations do not contain negative
powers of tensors and scalar functions.
The modified Newton-Kantorovich method is used to solve the considered
problems [1]. The modified Newton-Kantorovich method is chosen because the
unmodified version of this method requires to solve a linearized problem with
the coordinate-dependent modulus of elasticity on each step of this method,
with the except of the first step. It is difficult to solve this linearized problem
by analytical methods.
The initial approximation is chosen as follows:
(0)
u1 = 0 ,
(0)
p1 = 0 ,
(0)
u2 = 0 ,
(0)
p2 = 0
(24)
The problem of calculation the initial strains Ψ0,1 by the given initial
stresses is reduced to the solution of the system of nonlinear algebraic equations, because initial strains are homogeneous. The solution of this system by
the modified Newton-Kantorovich method is so simple that there is no necessity to consider it in detail. The initial approximation for this system is chosen
(0)
as follows: Ψ0,1 = I.
Consider now the problem of calculation of stress strain state in the vicinity
of the hole, i.e. the problem of finding of u2 . Choosing th initial approximation in the form (24), one can write the linearized problem for the (i + 1)-th
approximation of the modified Newton-Kantorovich method in the following
way:
h
i
k
(i)
b (i)
,
p
b
= f (i) ,
∇·L u
2
2
(i)
b 2 = h(i)
∇·u
h
i
k
(i) (i) b 2 , pb2 k = Q(i) ,
N2 · L u
(25)
(26)
(27)
Γ2
h
i
(i) b (i)
L u
,
p
b
2
2 ∞ (i)
∞
= σ 0,2 ,
In relations (25)–(28) L[u, p] is a linear operator:
(28)
2796
Vladimir Levin and Konstantin Zingerman
L[u, p] = µ(1 − β)(∇ · u)I + µ(∇u + u∇) − pI;
k is the number of state, in which coordinates the problem is solved (if the
hole shape is given at the intermediate state then k = 1, and if in the final
(i)
b (i)
state then k = 2); u
b2 are the corrections to the solution in the (i + 1)-th
2 , p
approximation. After their calculation this approximation is defined by the
formulae
(i+1)
(i)
(i+1)
(i)
(i)
b (i)
u2
= u2 + u
p2
= p2 + pb2 .
(29)
2 ,
Let us notice that the general form of the linearized boundary-value problem, solved for each approximation, is the same as in case of perturbation
method. Only functions in the right parts of the equations and the boundary
conditions differ. Consider the formulae to determine these functions for each
case concerned in this section.
1. The hole shape is given at the moment of its creation. In this case the
relations (1)–(10) with (21) result to
1 (i)
1
(i)
(i)
f = −∇ · Σ0,2 · Ψ1,2 ,
(i)
h(i) = 1 − det Ψ1,2 ,
1
1 (i)
1
(i) ∗
(i)
Q(i) = −N2 · Σ0,2 − P N2 · Ψ2,1 · Ψ2,1 ,
where
(i)
σ 0,2 =
h
io
µn
(i)
(i)
(i) 2
(i)
(1 + β)F0,2 + (1 − β) F0,2 : I F0,2 − F0,2
− p0,2 I ,
2
1 (i)
(i) ∗
(i)
(i)
Σ0,2 = Ψ2,1 · σ 0,2 · Ψ2,1 ,
(i) ∗
(i)
(i)
Ψ2,1 =
(i) 2
(i)
Ψ1,2 − Ψ1,2
(i)
(i)
F0,2 = Ψ0,2 · Ψ0,2 ,
1 (i) 2 1 (i) 2
(i)
: I Ψ1,2 +
Ψ1,2 : I I −
Ψ1,2 : I I ,
2
2
(i)
1
(i)
Ψ0,2 = Ψ0,1 · Ψ1,2 ,
(i)
Ψ1,2 = I + ∇u2 .
2. The hole shape is given at the final state. As follows from the relations
(11)–(18), (22), (23) in this case
2
(i)
f (i) = −∇ · σ 0,2 ,
(i)
h(i) = det Ψ2,1 − 1 ,
2
(i)
2
Q(i) = −N2 · σ 0,2 − N2 P ,
2797
The Newton–Kantorovich method in plane elasticity
where
(i)
σ 0,2
h
io
µn
(i)
(i)
(i)
(i) 2
(i)
(1 + β)F0,2 + (1 − β) F0,2 : I F0,n − F0,n
− p0,2 I ,
=
2
(i)
(i) ∗
(i)
(i)
(i)
F0,2 = Ψ0,2 · Ψ0,2 , Ψ0,2 = Ψ0,1 · Ψ1,2 ,
1 (i) 2
1 (i) 2 (i)
(i) 2
(i)
(i)
Ψ1,2 = Ψ2,1 − Ψ2,1 : I Ψ2,1 +
Ψ2,1 : I I −
Ψ2,1 : I I ,
2
2
(i)
2
(i)
Ψ2,1 = I − ∇u2 .
These relations (together with the relations that are used for solution of
the linearized problems) underlie the algorithm of the solution of the boundaryvalue problems (1)–(10), (11)–(19) by the modified Newton–Kantorovich method
for the case in which the hole contour can be mapped conformally on the unit
circumference using the functions of the form
ω(ξ) = β−1 ξ + β0 + β1 ξ −1 + · · · + βn ξ −n .
(30)
The detailed approach to the solution of linearized problems is considered
in [4].
4
Results
Let us give the results of the calculation of the stress strain state in the vicinity
of the hole, that is originated in the previously loaded body and has the circular
shape at the final state, as an example. Initial uniaxial load is applied in the
direction of y axis: (σ0,1 )11 = (σ0,1 )12 = 0, (σ0,1 )22 = p, p/µ = 0.7, β = 1.
Fiures 1, 2 show the distribution of stresses along the x axis. R is the hole
radius in the final state. The characters 1–4 are approximation numbers.
One can see from the figures that the difference between the third and the
fourth approximation is small. In addition, it is clear that significant nonlinear
effects take place, especially in the vicinity of the hole. The correction for
nonlinear effects for the stress σ22 is about 65 % at the point of maximal stress
concentration (x = R).
5
Conclusion
The approximate analytical method is developed for solution of a class of
nonlinear elasticity problems of stress distribution near holes originating in
preliminarily strained infinite bodies. This method is based on the modified
Newton–Kantorovich method and the Kolosov–Muskhelishvili technique. The
transformation of differential equations permits one to to obtain functions in
2798
Vladimir Levin and Konstantin Zingerman
Figure 1: Distribution of true stress σ11 along the x axis.
Figure 2: Distribution of true stress σ22 along the x axis.
right parts of linearized equations in a form that admits analytical integration.
Some numerical results show the reliability of the proposed approach.
Acknowledgements. The research for this article was performed in FIDESYS LLC and financially supported by the Russian Ministry of Education
and Science (project No. 14.579.21.0007, project ID RFMEFI57914X0007).
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The Newton–Kantorovich method in plane elasticity
2799
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Received: August 8, 2016; Published: September 17, 2016