Strain Gradient Notation for Finite Elements Polynomials
JOÃO ELIAS ABDALLA Fo
Programa de Pós-Graduação em Engenharia Mecânica
Pontifícia Universidade Católica do Paraná
Rua Imaculada Conceição, 1155 Prado Velho, Curitiba, Paraná
BRASIL
Abstract: - This paper describes a physically interpretable notation named strain gradient notation for writing finite
elements polynomials that model the behavior of solid and structural mechanics problems. The displacement and
strain fields are written in terms of coefficients whose physical meanings are apparent revealing modeling
capabilities and deficiencies of the element under construction. Spurious terms responsible for the phenomenon of
artificial stiffening can be identified precisely. The finite element can be improved by simple removal of these
spurious terms, not being necessary the application of any method of reduced integration.
Key-Words: - Finite element, Strain Gradient notation, artificial stiffening, parasitic shear
1 Introduction
The finite element method has been widely used to
solve complex problems in physics and engineering.
Briefly, a continuous medium will be discretized by a
mesh comprised of many simple shaped elements
which will attempt to correctly model the problem at
hand. Each finite element must be described by
polynomials that represent the physics of the problem
at best while maintaining certain mathematical
requirements which assure convergence such as
completeness and compatibility [1]. The number of
terms in the polynomials depends on the number of
degrees-of-freedom of the chosen element. For
instance, a displacement-based six-node triangle to
model plane elasticity problems has six degrees-offreedom in both x- and y-directions. Therefore, the
corresponding displacements u and v are represented
by six-terms polynomials as shown by the equations
below:
u ( x, y )  a1  a 2 x  a3 y  a 4 x 2  a5 xy  a 6 y 2
(1)
v( x, y )  b1  b2 x  b3 y  b4 x 2  b5 xy  b6 y 2
(2)
where ai and bi are unknown coefficients often
referred to as generalized coordinates. For a given
problem, the magnitude of these coefficients are
determined during the numerical analysis. Their
physical meanings, however, remain hidden. This
paper describes a notation to write finite elements
polynomials, strain gradient notation, which is
physically interpretable. That is, through the notation,
the physical meanings of the coefficients are
displayed, revealing the modeling capabilities as well
as the modeling deficiencies embedded into the
polynomial representations. In the following sections,
the procedure for determining the contents of the
unknown coefficients will be described. A finite
element will be developed using the notation, and its
polynomials will be inspected. Further, spurious
terms will be identified and removed. Finally, results
of numerical analysis run with elements containing
the spurious terms and then with elements free of
those terms will be compared to validate the
procedure.
2 Strain Gradient Formulation
Technique
The purpose of this section is to show how the
unknown coefficients of the polynomial expansions
can be evaluated in terms of kinematic quantities
which represent the behavior of the continuum. That
is, the coefficients will be written in terms of strain
states the continuum is able to undergo. These strain
states, which are here called strain gradients, serve as
a linearly independent basis set for the strain content
of a particular finite element. These strain gradients
are the kinematic quantities that can be represented
by the finite element. The displacement expansions,
such as the complete second order polynomials in
equations 1 and 2, are Taylor series expansions
evaluated at a particular origin with the unknown
coefficients being displacement derivatives or strain
gradient quantities. The determination of the
coefficients requires the evaluation of displacements
and their derivatives at the element´s local origin.
Then,
u0,0  u0  a1
v0,0  v0  b1
(3)
(4)
where [u]0 and [v]0 are the rigid body displacement
modes evaluated at the origin.
The normal strains are given by:
u
 a 2  2a 4 x  a5 y
x
v
 y x, y    b3  b5 x  2b6 y
y
 x x, y  
(5)
(6)
 0  b3
 y 0,0   y
1  v u 
 

2  x y 
1
 b2  a3   2b4  a5 x  b5  2a 6  y 
2
 u v 
 xy x, y     
 x y 
 a3  b2   a5  2b4 x  2a 6  b5  y 
(7)
(8)
(9)
(10)
which are evaluated at the origin to yield
r 0,0  r 0 
1
b2  a3 
2
 xy 0,0   xy  a3  b2
 0
(11)
(12)
Solution of equations 11 and 12 yields



0
a3   xy 2  r
0
b2   xy 2  r

(15)
 0
a 6   xy , y   y , x  2
0
b4   xy , x   x, y  2
0
b5   y , x 
0
b6   y , y 2
0
(16)
(17)
(18)
(19)
(20)
Therefore, all the coefficients have been evaluated in
terms of strain gradients, rendering displacement
polynomial expansions whose physical meanings are
transparent to the analyst. Substitution of the results
above into equations 1 and 2 yields



u ( x, y )  u 0   x 0 x   xy 2  r y   x, x / 2 0 x 2
0
The small-displacement rotation and shear strain are
given by:
r  x, y  

a 4   x, x / 2 0
a5   x , y

which are evaluated at the origin to yield
 x 0,0   x 0  a 2
Evaluation of the higher-order derivatives of the
strains at the origin yields after manipulation
(13)
(14)
 0 xy   xy , y   y, x  20 y 2
  x, y

(21)
0  0 y 
(22)
 xy, x   x, y  20 x 2   y, x 0 xy   y, y 20 y 2
v( x, y )  v 0   xy 2  r x   y
These equations show that the displacements depend
mostly on rigid body translations [u]0, [v]0 and rigid
body rotation [r]0, and on constant normal and shear
strains [x]0, [y]0 and [xy]0. Also, they show that the
displacements depend on first gradients of these
normal and shear strains, i.e., [x,x]0, [x,y]0, [y,x]0 ,
[y,y]0 , [xy,x]0 and [xy,y]0.
This procedure has been carried out for the complete,
three-dimensional,
fourth-order
displacement
polynomials to determine the contents of the
coefficients in terms of strain gradients [2]. The
results have been conveniently organized in table
form allowing for the construction of virtually any
finite element displacement approximation function.
In the process of formulating a given finite element
using strain gradient notation, nodal displacements
are written in terms of strain gradient quantities
through the following expression:
d   sg
(23)
where d is the vector of nodal displacements, sg is
the vector of strain gradient quantities, and  is the
matrix relating nodal displacements to strain
gradients. The columns in  are all linearly
independent once the strain gradients form a linearly
independent basis set for the finite element. These
column vectors of the transformation matrix 
contain the nodal displacements and rotations
required to produce a given strain state.
The continuum elastic strains are related to the
individual strain gradients through
Substituting sg from equation 23 into equation 26
gives
  Tsg  sg
the element´s stiffness matrix is then calculated by
1
 T C dV

V
2
(25)
where C is the constitutive matrix, which relates
stresses to strains. Substitution of equation 24 into 25
yields
U scalar 
1
T T
 sg
Tsg C Tsg  sg dV

V
2
(26)
Considering that the strain gradients do not vary with
the volume of the element, the strain energy matrix is
defined as
U
1
TsgT C Tsg dV

V
2
1 T T
d  U  1d
2
(28)
Recalling that
U scalar 
1 T
d Kd
2
(29)
(24)
where  is the vector of continuum elastic strains, and
Tsg is the transformation matrix between these strains
and the strain gradients. The contents of this matrix
results from the substitution of the displacement
expansions into the strain definitions.
Further, the general strain energy equation for a
deformable body is
U scalar 
U scalar 
(27)
The strain energy matrix is a symmetric matrix of
strain energy quantities. Its diagonal terms are
interpreted as the amount of strain energy contained
in the element when it is subjected to a deformation
associated to one of the strain gradients. This
deformation is the displacement vector in matrix 
corresponding that strain gradient. The off-diagonal
terms are the strain energy associated to the coupling
of two deformation modes which in turn are
associated to two strain gradients.
K   T U  1
(30)
3 A-Priori Evaluation of Modeling
Characteristics
The transparency of the strain gradient notation
allows for the determination of the modeling
characteristics of a finite element prior to numerical
analysis. Evaluation of the displacement polynomials
of the six-node triangle (equations 21 and 22) shows
for instance that rigid body and constant strain modes
contribute to the field. Therefore, the element
possesses the theoretical requirements for
convergence [1].
According to equations 5, 6 and 10, the strains are
given by
 0 y
(31)
 0   y,x 0 x   y, y 0 y
(32)
 x ( x, y )   x 0   x, x 0 x   x, y
 y ( x, y )   y
 0   xy,x 0 x   xy, y 0 y
 xy ( x, y )   xy
(33)
which show that every term in the polynomials is a
legitimate one because all of them belong to the
Taylor series expansions associated to the
corresponding strain quantity on the left-hand side.
This explains why the six-node triangle is such an
efficient element presenting a high rate of
convergence. The reason this element does not
contain spurious terms in its polynomial expansions
is that it is derived from complete polynomials. In
fact, any element whose polynomials are complete do
not contain erroneous terms, thus not developing
parasitic shear.
Let´s now consider the four-node quadrilateral
element. Its displacement polynomials are given by


u ( x, y )  u 0   x 0 x   xy 2  r y
0
 
  x, y xy
0
v( x, y )  v 0   xy 2  r x   y y
0
0
  y , x xy
 0


 
(34)
(35)
and the corresponding strain polynomials are given
by
 0 y
 y ( x, y )   y    y , x  x
0
0
 x ( x, y )   x 0   x, y
 0   x, y 0 x   y, x 0 y
 xy ( x, y )   xy
(36)
(37)
(38)
Inspection of equations 36 and 37 shows that the
normal strains are pure Taylor series expansions.
However, the shear strain expression contains
derivatives of normal strains, i.e. flexural terms,
which are not terms of the Taylor series expansion of
xy. This gives coupling between normal and shear
strains when the element is subjected to bending.
That is, during bending these erroneous terms are
activated increasing the value of xy unduly which in
turn increases the stiffness of the element. This
phenomenon is called artificial stiffening. Because
these spurious terms contaminate the shear strain
expression they are referred to as parasitic shear
terms, and their effect of artificial stiffening is named
parasitic shear.
When regular notation is used such as in
isoparametric finite elements [1] the modeling
characteristics of the polynomial expansions are not
revealed a-priori such as it was done above. In that
case, artificial stiffening has to be dealt with during
numerical analysis through some form of the reduced
integration technique [1]. On the other hand, when
strain gradient notation is employed as was done
above, the modeling characteristics of the element
become transparent to the analyst, and erroneous
terms which are inherent to the formulation can be
precisely identified. Because the strain gradients
form a set of independent quantities, the erroneous
terms can be simply removed from the shear strain
expansion without reducing the modeling capabilities
of the element. Hence, the corrected shear strain
expression of the four-node quadrilateral element is
 xy ( x, y )   xy 0
(39)
which results from eliminating the two spurious
flexural terms from equation 38. At this point, it is to
be noted that those terms were not removed from the
basis set of the element as they still appear in the
normal strains expressions and in the displacements
expressions. This ensures that the element will
perform at its best.
As far as eliminating errors is concerned, the same
result is obtained when reduced integration is used
when integrating the stiffness matrix terms of a fournode isoparametric quadrilateral. More specifically,
integration with one-point reduced order Gauss
quadrature rule eliminates the effects of the parasitic
shear terms. Therefore, it is an alternative and
effective approach. However, reduced integration is
not as effective when employed in analyses with the
eight-node isoparametric quadrilateral element.
Either a portion of the added strain energy due to
parasitic shear is left in the strain energy expression
or the strain energy due to required terms is removed.
In the former case, the element is still overly stiff,
whereas in the latter the strain energy is reduced and
a new error known as spurious zero energy modes is
introduced. Further details about this matter can be
found in the work of Dow and Abdalla [3].
This case of the eight-node quadrilateral indicates the
importance of a physically interpretable notation
such as strain gradient notation. The correcting
procedure allowed by it is always effective ensuring
good quality of results in the sense that all erroneous
terms can be fully removed without introducing any
other deleterious effects.
The strain gradient notation was first used in the
analysis of continuum lattice structures [4, 5]. Then,
it has been used in finite elements analysis to model
plane elasticity problems [2, 6], plate problems [2, 7],
and laminated composite plate [8] and laminated
composite beam problems [9].
4 Numerical Validation
The effectiveness of the procedure for removing
parasitic shear terms from the shear strain expression
of the four-node quadrilateral to render a better finite
element is demonstrated in this section. A cantilever
beam subjected to an end shear load of unit value, as
shown in figure 1, is analyzed using four-node
quadrilateral meshes both with and without the
parasitic shear terms identified earlier. This problem
is heavily influenced by bending, which creates an
environment where the artificial stiffening produced
by parasitic shear terms is very significant.
Table 1 Results of Cantilever Beam Analysis
Modeled with 4-Node Plane Quadrilateral Elements.
MAXIMUM DEFLECTION
MESH Element
Error
Element
Error
with
(%)
without
(%)
parasitic
parasitic
shear
shear
1x2
0.376
90.67
3.439
14.69
1x10
2.716
32.63
3.657
9.28
2x20
3.581
11.17
3.915
2.88
4x40
3.901
3.23
3.994
0.92
Correct Deflection = 4.0312
elasticity modulus of the material, G is the shear
modulus of the material, and K is the shear correction
factor. In this example, the values used are P = 1.0, L
= 10.0, E = 1,000.00, G = 384.615, A = 1.0, I = 1/12,
and K = 5/6.
The values displayed in table 1 clearly show the
difference between results obtained with elements
containing parasitic shear terms and those obtained
with elements corrected for this deficiency. The
coarser the mesh the more evident is that difference.
It is also shown that refinement tends to attenuate the
artificial stiffening caused by parasitic shear and that
eventually convergence will be attained. However,
convergence occurs much sooner when corrected
elements are employed. This is easily seen by
studying the percent error columns. The error in the
results using the coarser mesh is 90.67% when
parasitic shear is present, and only 14.69% after
removal of parasitic shear. These error values are
very effective to demonstrate the strength of parasitic
shear in finite element analyses and also to validate
the error elimination procedure allowed by strain
gradient notation. Further, the third mesh, which
contains only 40 elements, produces results with
11.17% error when parasitic shear is present, and
results with only 2.88% error after removal of
parasitic shear. Finally, the finer mesh, which
contains 160 elements, presents results with only
0.92% error after removal of parasitic shear while the
error is 3.23% with parasitic shear.
It can be concluded from these analyses that if the
element is corrected for parasitic shear, coarser
meshes already produce acceptable results. On the
other hand, if the effects of parasitic shear are to be
attenuated by refinement, much finer meshes may be
necessary for accurate results. Furthermore, the error
elimination procedure of simply removing the
spurious terms from the shear strain polynomial
works well as the numerical analysis results converge
to the correct analytical solution.
The correct deflection is given by the following
analytical expression
5 Summary and Conclusions
1,0
1,0
10,0
Fig.1 Cantilever beam modeled by strain gradient
quadrilateral plane elements.
Four meshes are constructed, namely; 1x2, 1x10,
2x20, and 4x40. The first number is the number of
subdivisions of the mesh along the vertical direction
while the second number is the number of
subdivisions along the horizontal direction. The
result used for comparison is the maximum
deflection of the beam. The results of the analyses as
well as the percent error contained in those results are
shown in Table 1 below.
PL3
PL


3EI KGA
(40)
where P is the applied load, L is the length of the
beam, A is its cross-section area, I is the moment of
inertia of the cross-section, E is the longitudinal
This paper presents a physically interpretable
notation, strain gradient notation, for expressing the
polynomial expansions which represents the
displacement and strain fields of solid mechanics
problems to be modeled by finite elements. It is
shown that the unknown coefficients of these
polynomial expansions are expressed in terms of the
kinematic quantities of the continnum, that is, the
quantities which contribute to the deformation of the
solid body. These quantities are rigid body
displacements, constant strains, and gradients of
strains, which, in the present context, have been
called strain gradients.
It has been demonstrated here that strain gradient
notation allows the analyst to determine a-priori
which are the modeling capabilities of a given finite
element. By the same token, he or she can identify
spurious terms which are responsible for artificial
stiffening. In the present work, the procedure for
identifying and eliminating parasitic shear terms
effectively using a widely employed finite element
has been shown. That procedure is valid for any
finite element containing spurious terms.
The conclusion is that strain gradient notation is
effective for formulating finite elements. This is so
mainly for allowing the identification of inherent
erroneous terms which are the sources of artificial
stiffening. This is important because only those terms
are eliminated in the process of correcting the
element´s polynomial expansions. That is, genuine
terms are not removed blindly such as is the case of
the eight-node quadrilateral when reduced integration
is employed.
Strain gradient notation through this paper has been
shown once again. Even though it has been used for
over fifteen years in finite element analysis, the
notation is not known by many members of the
computational mechanics community, and its
advantages not necessarily appreciated. It is hoped
that the efforts being made in developing and
applying strain gradient notation further will render it
a widely well recognized procedure.
References:
[1] O. C. Zienkiewicz and R. L. Taylor, The Finite
Element Method, Fourth Edition, Vol. 1, McGraw-Hill
Book Company, London, 1989.
[2] D.E. Byrd, Identification and Elimination of Errors
in Finite Element Analysis, Ph.D. Dissertation,
University of Colorado, Boulder, CO, 1988.
[3] J.O. Dow and J.E. Abdalla Fo, Qualitative Errors in
Laminated Composite Plate Models, International
Journal for Numerical Methods in Engineering, Vol.
37, 1215-1230, 1994.
[4] J.O. Dow, T.H. Ho, and H.D. Cabiness, A
Generalized Finite Element Evaluation Procedure,
ASCE Journal of Structural Engineering, Vol.111,
No.2, pp 435-452, 1985.
[5] J.O. Dow and S.A. Huyer, Continuum Models of
Space Station Structures, ASCE Journal of Aerospace
Engineering, Vol.2, No.4, pp 212-230, 1989.
[6] J.O. Dow, H.D. Cabiness, and T.H. Ho, A Linear
Strain Element with Curved Edges, ASCE Journal of
Structural Engineering, Vol.112, No.4, pp 692-708,
1986.
[7] J.O. Dow and D.E. Byrd, Error Estimation
Procedure for Plate Bending Elements, AIAA Journal,
Vol.28, No.4, pp 685-693, 1990.
[8] J.E. Abdalla Fo, Qualitative and Discretization
Error Analysis of Laminated Composite Plate Models,
Ph.D. Dissertation, University of Colorado, Boulder,
CO, 1992.