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INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
Int. J. Numer. Meth. Engng 2006; 68:338–380
Published online 3 April 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/nme.1711
Elasto-plastic finite element analysis of shells with damage
due to microvoids
Pawel Woelke1, ‡, ¶ , George Z. Voyiadjis2, ∗, † and Piotr Perzyna3, §
1Weidlinger
Associates, Inc., Applied Science Department, 375 Hudson Street, 12 FL, New York,
NY 10014-3656, U.S.A.
2 Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge,
LA 70803-6405, U.S.A.
3 Institute of Fundamental Technological Research, Polish Academy of Sciences, Swietokrzyska 21,
00-049 Warsaw, Poland
SUMMARY
This paper presents a non-linear finite element analysis for the elasto-plastic behaviour of thick/thin
shells and plates with large rotations and damage effects. The refined shell theory given by Voyiadjis
and Woelke (Int. J. Solids Struct. 2004; 41:3747–3769) provides a set of shell constitutive equations.
Numerical implementation of the shell theory leading to the development of the C 0 quadrilateral shell
element (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical shells.
2006, submitted) is used here as an effective tool for a linear elastic analysis of shells. The large
rotation elasto-plastic model for shells presented by Voyiadjis and Woelke (General non-linear finite
element analysis of thick plates and shells. 2006, submitted) is enhanced here to account for the
damage effects due to microvoids, formulated within the framework of a micromechanical damage
model. The evolution equation of the scalar porosity parameter as given by Duszek-Perzyna and Perzyna
(Material Instabilities: Theory and Applications, ASME Congress, Chicago, AMD-Vol. 183/MD-50,
9–11 November 1994; 59–85) is reduced here to describe the most relevant damage effects for isotropic
plates and shells, i.e. the growth of voids as a function of the plastic flow. The anisotropic damage
effects, the influence of the microcracks and elastic damage are not considered in this paper. The
damage modelled through the evolution of porosity is incorporated directly into the yield function,
giving a generalized and convenient loading surface expressed in terms of stress resultants and stress
couples. A plastic node method (Comput. Methods Appl. Mech. Eng. 1982; 34:1089–1104) is used to
derive the large rotation, elasto-plastic-damage tangent stiffness matrix. Some of the important features
of this paper are that the elastic stiffness matrix is derived explicitly, with all the integrals calculated
analytically (Woelke and Voyiadjis, Shell element based on the refined theory for thick spherical
∗ Correspondence
to: George Z. Voyiadjis, Department of Civil and Environmental Engineering, Louisiana State
University, Baton Rouge, LA 70803-6405, U.S.A.
† E-mail: [email protected]
‡ E-mail: [email protected]
§ E-mail: [email protected]
¶ Previously at Louisiana State University, Baton Rouge, LA 70803-6405, U.S.A.
Contract/grant sponsor: Air Force Institute of Technology, WPAFB; contract/grant number: F33601-01-P-0343
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Received 18 September 2005
Revised 7 February 2006
Accepted 13 February 2006
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
339
shells. 2006, submitted). In addition, a non-layered model is adopted in which integration through
the thickness is not necessary. Consequently, the elasto-plastic-damage stiffness matrix is also given
explicitly and numerical integration is not performed. This makes this model consistent mathematically,
accurate for a variety of applications and very inexpensive from the point of view of computer power
and time. Copyright 䉷 2006 John Wiley & Sons, Ltd.
KEY WORDS:
thick plates and shells; elasto-plastic analysis; kinematic hardening; large displacements;
ductile damage analysis; microvoids; isotropic damage
1. INTRODUCTION
1.1. Motivation and scope
Shells are very important structures for both onshore and offshore engineering applications.
Analysis and design of these is therefore continuously of interest to the scientific and engineering
communities. Accurate and conservative assessments of the maximum load carried by the
structure, as well as the equilibrium path in both elastic and inelastic ranges are therefore of
paramount importance. Although still there is room for improvement, the elastic behaviour of
shells has been very thoroughly investigated, mostly by means of the finite element method.
Inelastic analysis on the other hand, especially accounting for damage effects, received much
less attention from researchers [1–3]. One of the major difficulties in computations for the
inelastic behaviour of structures is the fact that they are based on incremental and/or iterative
algorithms, which may require prohibitively large storage in the computer. Thus, computational
efficiency needs special attention in non-linear modelling of shells. Voyiadjis and Woelke [4]
presented a general, accurate and very efficient finite element model for the elasto-plastic
analysis of thick/thin, isotropic plates and shells, including geometric non-linearities, isotropic
and kinematic hardening rules and featuring an explicit form of the tangent stiffness matrix.
The objective of the present work is to extend the work of Voyiadjis and Woelke [4] to
account for the damage effects due to growth of microvoids. Damage is modelled here as an
isotropic process induced by the plastic flow. Static loading conditions are considered, with
both plasticity and damage treated as rate-independent processes.
1.2. Shell theory
The problem of shell constitutive equations could be avoided by following a layered approach,
also referred to as ‘through-the-thickness-integration’ [5–8]. This procedure is conceptually
close to the analysis of shells by means of three-dimensional, solid ‘brick’ elements, and
is best suited for composite plates and shells. In the case of isotropic materials, the layered approach, although accurate, is unnecessarily complicated and computationally expensive.
A non-layered finite element approach, on the other hand, requires a general and accurate
shell theory. Voyiadjis and Woelke [4] presented a refined theory for thick shells, which was
developed based on analytical closed-form solutions for thick containers. This theory proves to
be very efficient in the treatment of both thin and thick shells of general shape, and is therefore
adopted in this work. It accounts for the effect of transverse shear deformation and distribution
of radial stresses. These are very important features for the thick shells. In addition, the initial curvature effect is included, which not only contributes to the stress resultants and stress
couples, but also leads to non-linear distribution of the in-plane stresses across the thickness
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
340
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
of the shell. The resulting constitutive equations offer very good approximations for extremely
thick (R/t = 3) and very thin shells of general shape, as well as for plates and beams. A brief
outline of the theory is provided in Section 2.
1.3. Shell finite element
The numerical implementation of the aforementioned theory provides a shell finite element
[9, 10], which is a very effective tool for the elastic analysis of the structures under consideration. The quasi-conforming technique given in Reference [11] is an extension of the assumed
strain fields method [12, 13], and it has been successfully applied to overcome the shear and
membrane locking phenomena [14, 15]. The appropriate choice of the strain fields provides
an adequate representation of the rigid-body modes and allows one to avoid spurious energy
modes. The biggest advantage of this technique, when compared with the most widely used
selective integration technique [16–20] is the fact that the stiffness matrix of the element is
given explicitly. Thus, this method is very attractive for non-linear calculations because the
element matrices are evaluated many times during the analysis. Moreover, the selective integration technique requires an explicit segregation of transverse shear terms from bending and
membrane terms, which is not possible when a coupling between these exists, as is mostly the
case for non-linear analysis [21]. This problem was solved by a generalization of the selective
integration procedure [16]. The quasi-conforming technique is chosen here for its simplicity
and low computational cost. As a result of that, and application of the non-layered approach,
numerical integration is not performed in the present procedure at any stage of the analysis.
All the integrals are calculated analytically, and later introduced into a computer code. This
makes the current formulation consistent mathematically and extremely efficient from the point
of view of computer time and power.
1.4. Geometrical non-linearities
Geometrical non-linearities are crucial in the elasto-plastic and damage modelling of shell [2].
Displacements at the regions of the structure, which undergo inelastic deformations, can be very
large. Thus, to achieve the desired accuracy, geometric non-linearities must be considered. The
Updated Lagrangian description, which has proven to be a very effective method [4, 6, 22–25]
is adopted here. The element local co-ordinates and the local reference frame are continuously
updated during the deformation. We consider large rotations and rigid translations here, but
small strains with the total rotations decomposed into large rigid rotations and moderate relative
rotations. The relative rotations and the derivatives of the in-plane displacements from two
consecutive configurations may be considered small [24, 25]. Consequently, the quadratic terms
of the derivatives of the in-plane displacement are negligible. We therefore have a non-linear
analysis with large displacements and rotations, but small strains. The transformation matrix
given in Reference [26] is employed to handle large rigid rotations. The assumed strain finite
element with an explicit form of the stiffness matrix, as described above, provides the linear
part of the element tangent stiffness matrix.
1.5. Material non-linearities—plasticity
The experimental results [27–29] show that the degradation of material properties of ductile
metals in the elastic range due to the damage effects is negligible. Hence, the damage is
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
341
considered here as a phenomenon induced by the plastic strain. A reliable elasto-plastic procedure is needed in order to perform damage.
In the layered plastic model a plate or a shell is divided into layers where stresses are
calculated and the yield condition is checked at each layer separately. The forces and moments
are then calculated by integration through the thickness. Although this method can give very
accurate results, it can also be very demanding in terms of computational power. If, on the
other hand, a ‘non-layered’ approach is adopted, the yield function is expressed in terms of
the stress resultants and couples. Numerical integration of the stresses is not necessary in this
case, which makes the ‘non-layered’ formulation much cheaper computationally. Voyiadjis and
Woelke developed a very accurate, non-layered elasto-plastic model for shells with an isotropic
and a new kinematic hardening rule [4, 10]. An Iliushin’s yield function [30] is employed in
this work, modified to account for the progressive development of the plastic curvatures across
the thickness of the shell, as shown in Reference [31], and the transverse shear forces, which
may significantly affect the plastic behaviour of both thick and, for certain loading conditions,
thin shells. The evolution of damage is directly linked to the plastic strain, hence all the factors
that affect plastic behaviour, are also very important for the damage analysis. Both isotropic and
kinematic hardening rules are defined, with the latter correctly representing the rigid translation
of the yield surface during the non-elastic deformation in the stress resultant space, and thus
capturing the Bauschinger effect.
The stiffness matrix in Reference [4] is derived by means of the principle of virtual work
and the plastic node method [32], which considers the inelastic deformations to be concentrated
in the plastic hinges. This method originates from the analytical limit analysis of structures
performed under the assumption of elastic-perfectly plastic behaviour of the material [33–35].
Following the work of Shi and Voyiadjis [31, 3] the plastic node method is adopted here
to derive the elasto-plastic-damage stiffness matrix of the element. The explicit form of the
stiffness matrix is therefore preserved.
1.6. Damage analysis
A ductile metal or structure is capable of undergoing large inelastic deformations. The plastic
strains can induce changes in the microstructure of the material, leading to its softening. These
changes in the microstructure of the material are irreversible thermodynamic processes and result
in the progressive degradation of the material properties [3]. The experimental investigations
[36–38] show that the softening of the material triggered by inelastic strains is mainly due
to the nucleation, growth and coalescence of microvoids and microcracks (sometimes thermal
effects are also pronounced) [39, 40]. This process is called ductile plastic damage. Damage
in the elastic region is mostly negligible in ductile materials. Modelling of damage is aimed
at the assessments of the influence of microvoids, microcracks and other microdamages on the
degradation of the material properties.
The investigations of the damage accumulation and evolution can be carried out following
a micromechanical approach (micromechanical damage models) or continuum damage theory
(phenomenological damage model). The latter approach is based on the pioneering work of
Kachanov [41], who introduced the effective stress concept, as well as a scalar damage variable
representing the effective surface density of microdamages per unit volume [42, 43]. The effective stress concept involves comparison of the actual damaged configuration with the fictitious
undamaged configuration [41, 44]. Many authors used a phenomenological approach as a basis
for modelling of damage [44–56].
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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342
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
An isotropic scalar damage parameter, based on the concept of Kachanov [41], was used
by many authors [48, 57, 58]. In this method, the stiffness of the material is reduced according
to the same relation in all the directions. For a better description of the anisotropic effects, a
second-order damage tensor, capable of representing different levels of material degradation in
different directions is often employed [43–45, 50–57, 59–63]. An anisotropic damage variable
poses however the problem which is not often addressed. For the appropriate depiction of
directional dependency of the evolution of damage, it is necessary to determine the material
constants, which define the evolution laws in different directions. Extensive experimental data
are needed to calibrate these constants with sufficient accuracy and consistency. The isotropic
damage formulation requires determination of fewer constants (two in the case of the current
analysis), while at the same time it delivers very accurate results for a variety of structural
applications. Moreover, it would be unrealistic to include in the investigation of structures all
effects observed experimentally on the level of the material behaviour. Constitutive modelling is
understood as a reasonable choice of effects, which are the most important for explanation of the
phenomenon described [40]. For the current work, concerning the investigation of the behaviour
of isotropic plates and shells, the use of the isotropic scalar parameter in the representation
of damage is deemed satisfactory. The effects of anisotropy of damage are not accounted for
here. The validity of these assumptions is verified by the discriminating numerical examples.
Micromechanical damage models are based on the observations of the material at the microscale. The observations of ductile fracture in metals [64] lead to a conclusion that this
process can involve the generation of considerable porosity through nucleation and growth of
voids [65]. Gurson developed a mathematical model [65, 66] describing the damage effects
through the evolution of porosity, which was incorporated into the yield function. He investigated a yield criterion and flow rule for porous ductile materials. Various modifications of
Gurson’s formulation appeared later in the literature [67], as well as the articles discussing the
model [68, 69]. Further investigations of ductile fracture aimed at explanation of the formation
of white-etching bands, commonly referred to as shear bands. A general conclusion from the
experimental results by Giovanola [70] is that the thermomechanical strain localization and
microdamage mechanisms become the main co-operative phenomena responsible for adiabatic
shear band formation and localized fracture [40]. Based on the microscopic observations of the
shear bands [71], it was found that the fracture preceded by the shear band formation, occurred
through nucleation, growth and coalescence of voids. An extensive study of the shear bands
and fracture phenomena, followed by the development of microdamage model by means of the
porosity function, was performed by Duszek-Perzyna et al. [72–78], and Perzyna [40, 79–88].
Duszek-Perzyna and Perzyna presented a theoretical formulation for the description of the
intrinsic microdamage process through evolution of the isotropic scalar damage variable, i.e.
the porosity parameter [76]. Similar to Gurson’s model [65, 66], the porosity was incorporated
directly into the yield function, obtaining a consistent and convenient procedure for the elasticviscoplastic damage analysis of ductile solids, with a coupling between plasticity and damage.
The evolution of porosity reduced to a rate-independent case, consisted of three terms responsible for the cracking of the second-phase particles, debonding of the second-phase particles
from the matrix material, and the void growth assumed to be controlled only by plastic flow
phenomena. The first term (cracking of the second-phase particles) was only dependent on the
stress, which allowed for variation of damage, even without occurrence of the plastic flow.
This made the formulation universal and capable of describing correctly the material behaviour
under all loading conditions, including the hydrostatic stress. In the present paper, the elastic
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
343
damage is regarded negligible. In addition, we consider only damage resulting from microvoids,
neglecting the effects of microcracks.
The porosity parameter defined by Duszek-Perzyna and Perzyna [76] is used here to describe
damage effects in shells. We only consider a rate-independent case here, and the evolution of
porosity, which as previously mentioned, accounts for the cracking of the second-phase particles,
debonding of the second-phase particles from the matrix material, and void growth is reduced
to represent void growth only, since we investigate isotropic plates and shells. The effects of the
microcracks are not considered in this work. Since void growth is a phenomenon induced by the
plastic deformation, elastic damage is neglected here. The yield function given in Reference
[76], which could be directly related to Gurson’s model [65, 66], is expressed in terms of
the stress resultants and stress couples, similar to Iliushin’s yield function [30], following the
procedure outlined by Bieniek and Funaro [89]. The yield surface derived here is very similar
to the one presented by Voyiadjis and Woelke [4], with kinematic hardening parameters in
the form of residual normal and shear forces, and residual bending moments. It is however
enhanced here to account also for damage effects, leading to the reduction of the stiffness, by
means of the porosity parameter.
The current formulation is an attempt to deliver a very simple and convenient way of a
detailed analysis of shells. It is, at the same time, mathematically consistent and produces
accurate results. One of the biggest advantages of this work is its simplicity and computational
efficiency. The stiffness matrix is given here explicitly, and calculated without performing
numerical integration. This is due to the application of the quasi-conforming technique in
derivation of the elastic stiffness matrix, where all the integrations are computed analytically.
In this non-linear analysis, the non-layered and plastic node methods are employed, with the
yield surface defined in the stress resultant space, and the damage parameter incorporated into a
yield function. This approach is very advantageous from the point of view of structural analysis.
The validity of the assumptions and the derivation presented here is verified through a series
of discriminating examples. We solve a plate and a spherical shell problem focusing mainly
on the representation of damage as this is the most important feature of the current work.
This paper is divided into eight sections. After the Introduction, the shell constitutive equations are briefly introduced. In Section 3, we present shell kinematics. Section 4 is devoted
to the linear stiffness matrix of the shell element. Section 5 gives a description of material
non-linearities, with the definition of the porosity function as a scalar damage parameter, the
yield surface, the flow and hardening rules. The elasto-plastic-damage stiffness matrix of the
element is derived in Section 6. In Section 7 we present an outline of a numerical procedure
and discriminating examples, demonstrating that the current computational model gives good
results for a variety of problems in elasto-plastic and damage analysis of shells and plates.
Finally, in Section 8 we summarize the results and draw the conclusions.
2. SHELL CONSTITUTIVE EQUATIONS
Voyiadjis and Woelke [4] presented a detailed derivation of the shell constitutive equations
adopted for the finite element formulation. Only the final set of relations is given here for
self-completeness. The refined theory accounts for the effect of transverse shear deformation,
distribution of radial stresses and initial curvature of the shell, which results in a non-linear
distribution of the in-plane stresses across the thickness of the shell.
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
344
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
The main features of the shell equations are the following:
(1) assumed out-of-plane stress components that satisfy the given traction boundary conditions. These are due to a closed-form elasticity solution for thick walled spherical
containers under internal and/or external uniform pressure, obtained by Lame [90];
(2) three-dimensional elasticity equations with an integral of the equilibrium equations; and
(3) stress resultants and stress couples acting on the middle surface of the shell together
with average displacements along a normal of the middle surface of the shell and the
average rotations of the normal [91].
The membrane strains and curvatures in a rectangular co-ordinate system (x, y, z) are given
by Equations (1)–(6).
x =
*u w
+
R
*x
(1)
*v
w
+
R
*x
1 *u *v
+
xy =
2 *y
*x
* *w
*x
u
=
− xz −
x =
R
*x
*x *x
y =
y =
*y
=
*
*y
*w
v
− yz −
R
*y
*y
*y
1 *x
+
xy =
2 *y
*x
(2)
(3)
(4)
(5)
(6)
where x , y , xy are normal and shear strains and x , y , xy are curvatures at the midsurface
in planes parallel to the xz, yz and xy planes, respectively; u, v, w are the displacements along
the x, y, z axes, respectively (Figures 2, 3); xz , yz are transverse shear strains in xz and yz
planes (Figure 1); x , y are angles of rotations of the cross-sections that were normal to the
midsurface of the undeformed shell; R is a radius of a shell.
The stress resultants and couples Mx , My , Mxy , Nx , Ny , Nxy , Qx , Qy shown in Figure 1, can
be expressed in terms of the strains given above as follows:
Mx = D[x + y ]
(7)
My = D[y + x ]
(8)
Mxy = D(1 − )xy
(9)
Nx = S[x + y ]
(10)
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
345
Figure 1. Stress resultants on a shell element.
Figure 2. Local co-ordinate system and normal vector es3 .
Figure 3. Incremental degrees of freedom of the shell element in local co-ordinates.
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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346
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
Ny = S[y + x ]
(11)
Nxy = S(1 − )xy
(12)
Qx = T xz
(13)
Qy = T yz
(14)
where
D=
Eh3
,
12(1 − 2 )
S=
Eh
,
(1 − 2 )
T=
5 Eh
12 (1 + )
(15)
and E is the Young’s Modulus, h is the thickness of the shell, and is the Poisson’s ratio.
These constitutive equations reduce to those given by Flugge [92] when the shear deformation
and radial effects are neglected. We use the above equations to formulate the coupled strain
energy density and derive the corresponding stiffness matrix of the element.
3. SHELL KINEMATICS
As in the case of the plastic analysis of shells by Voyiadjis and Woelke [4], the Updated
Lagrangian method is employed in the present study of large displacements and rotations of
the shell element. The co-ordinates of the nodal points are continuously updated during the
deformation. The rotations are additively decomposed into large rigid rotations and moderate
relative rotations [25].
The structure under consideration is defined in the global, fixed co-ordinate system X. We
also have the local co-ordinate system x, surface co-ordinates at any nodal point xs , and base
co-ordinates, which serve as a reference frame for the global degrees of freedom (Figure 2).
• In order to obtain the unit vector in the direction normal to the plane of the element, in
→
→
the local co-ordinate system, we first define two vectors, 41 and 42 connecting the origin
of the co-ordinate system (point 4) to points 1 and 2, respectively. The cross-product of
these two vectors, divided by its length, gives e3 , as shown in Figure 3 and given by
Equation (16):
→ →
41 × 42
e3 = → →
| 41 × 42|
(16)
The unit vector e2 can be similarly obtained as a cross-product of e3 and e1 .
We can now determine the relation between the global co-ordinates X and element local
co-ordinates in configuration k:
k
e = k RE
(17)
where k e is the unit base vector of the local co-ordinates in configuration k, E is the unit
base vector of the global co-ordinates; R is a transformation matrix from local to global
co-ordinates.
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
347
• The surface co-ordinate system xS originates at each node of the element. As defined
by Shi and Voyiadjis in Reference [25], the position and direction of this system are
functions of rotations. Surface co-ordinates translate and rigidly rotate with the element.
Consequently, xS3 is always normal to the surface of the element.
The finite rigid-body rotation vector V is given by
⎡ ⎤
1
⎢ ⎥
⎥
(18)
V= ⎢
⎣ 2 ⎦
3
where 1 , 2 , 3 are rigid-body rotations around x, y, z axes, respectively. The transformation matrix of large rotations T , given by Argyris [26] is used here:
˜
T = exp()
(19)
with:
˜ = ˜ij = eij k k ,
k = 1, 2, 3
(20)
where ˜ is a skew symmetric matrix and eij k is the permutation tensor. In the above
equation, the indicial notation is used with Einstein’s summation convention. The transformation of the surface co-ordinates is therefore
V = T V
(21)
where V is a rigid-body rotation vector transformed into a new position. Similarly, we can
write a transformation of the surface co-ordinates for a given rotation vector j resulting
from configuration k − 1 to k at node j :
k
es = Tk−1
j es
(22)
where k es are the unit base vectors of the surface co-ordinates at configuration k. Defining
the transformation between E and k es as
k
es = k R s E
(23)
we can rewrite Equation (22) as follows:
k
k
k Tk
k k
es = Tk−1
j Rs E = Rs R e = Sj e
(24)
where k RT is the transpose of k R defined in Equation (17) and k Sj is a transformation
matrix from local to the surface co-ordinate system. It is worthwhile to note that 0 Rs is
a 3 × 3 identity matrix for a flat plate.
• The base co-ordinates as defined as by Horrigmoe and Bergan [23] are adopted here as
a common reference frame to which all element properties are transformed, prior to the
assembly of the stiffness matrices. The base co-ordinates are defined by the combination
of the fixed global and base co-ordinates.
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
348
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
The global degrees of freedom at node j are the incremental translations: Uj , Vj , Wj
in directions of global co-ordinates X, Y, Z and rotations xj , yj around xS , yS . The
local degrees of freedom at node j are the incremental translations uj , vj , wj in
directions of local co-ordinates x, y, z and rotations xj , yj around x, y, respectively.
The transformation of the increments of the displacements at node j from the local
co-ordinate system qej , to the corresponding base co-ordinates, qbj can be written as
⎧
⎧
⎫
⎫
uj ⎪
Uj ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
v
Vj ⎪ j ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
k
T
⎨
⎬
⎬
R
0 ⎨
w
W
(25)
qbj =
=
= k Tbj qej
j
j
k
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
0
s
j
⎪
⎪
⎪
⎪
⎪ ⎪
⎪ ⎪
⎪
⎪
⎪
⎪
xj ⎪
xj ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎩
⎭
⎭
yj
yj
in which k sj is the upper left 2 × 2 submatrix of k Sj defined in Equation (24). The
transformation matrix for the nodal displacement vector can be written as
qb = k Tb qe
(26)
where k Tb is composed of k Tbj with j = 1, 2, 3, 4.
The vector of the local increments of nodal displacements is shown in Figure 3 and is
given by Equation (27):
qej = {uj , vj , wj , xj , yj }T
j = 1, 2, 3, 4
(27)
4. LINEAR ELEMENT STIFFNESS MATRIX
An accurate and efficient shell finite element was presented by Woelke and Voyiadjis [9].
It is an assumed strain type of element, free from locking and spurious energy modes. The
quasi-conforming technique [11] was used which gives an explicit form of the stiffness matrix,
as integrations are carried out directly.
The strain fields in the element are interpolated as follows:
• Linear bending strain field:
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
*x
⎫
⎧
*x
x ⎪
⎪
⎪
⎪
⎬
⎨
*y
y
b =
=
⎪
⎪
⎪
*y
⎪
⎪
⎭ ⎪
⎪
⎩
⎪
⎪
2xy
⎪
⎪
*y
*
⎪
⎪
⎩ x +
*y
*x
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎡
1 x y xy
⎢
=⎢
⎣ 0 0 0 0
⎪
⎪
⎪
⎪
⎪
0 0 0 0
⎪
⎪
⎪
⎪
⎭
0 0 0 0
1 x y xy
0 0 0 0
⎧
⎫
1 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
2
⎪
⎪
⎤⎪
⎪
⎪
0 0 0 ⎪
⎪
⎪ ⎪
⎪
⎨
⎥
3 ⎬
0 0 0⎥
= P b b
⎦⎪
⎪
⎪
···⎪
⎪
⎪
⎪
⎪
⎪
1 x y ⎪
⎪
⎪ ⎪
⎪
⎪
⎪
10
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
11
(28)
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
• Stretch strain field:
⎧
⎫
*u w ⎪
⎪
⎪
⎪
⎫ ⎪
⎪ *x + R ⎪
⎪
⎧
⎡
⎪
⎪
⎪
x ⎪ ⎪
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎬ ⎨ *v
⎨
w⎬ ⎢
+
y
m =
=
=⎢
⎣0
⎪
⎪
⎪
R ⎪
*y
⎪
⎪
⎪
⎪
⎭ ⎪
⎪
⎩
⎪
⎪
⎪
⎪
0
2xy
⎪
⎪
⎪
*u *v ⎪
⎪
⎪
⎩
⎭
+
*y
*x
y
0
0
0
1
x
0
0
0
• Constant transverse shear strain:
⎧
⎫
u
*w
⎪
⎪
⎪ ⎪
⎪
⎨ *x − x − R ⎪
⎬
xz
1
s =
=
=
⎪
⎪
*w
v⎪
yz
⎪
0
⎪
⎩
⎭
− y − ⎪
R
*y
0
1
⎧
⎫
12 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎤⎪
⎪
⎪
13 ⎪
0 ⎪
⎪
⎪
⎪
⎪
⎬
⎥⎨
⎥
0 ⎦ 14 = Pm m
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
1 ⎪
15 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎭
16
17
18
349
(29)
= P s s
(30)
where 1 , 2 , . . . , 18, are the undetermined strain parameters.
Let P be the trial function for the assumed strain field, i.e.:
= P
(31)
and N, the corresponding test function. We multiply both sides by the test function and integrate
over the element domain:
T
N d = NT P d
(32)
The strain parameter is determined from the quasi-conforming technique as follows:
= A−1 Cq
(33)
where q is the element nodal displacement vector given by Equation (27), and
NT P d
A=
(34)
Cq =
NT d
(35)
We may now express the strain field in terms of the nodal displacements as follows:
= P = PA−1 Cq = Bq
(36)
It is convenient to take P = N in order to obtain a symmetric stiffness matrix. This is the
case adopted in this formulation. Both matrices A and C can be easily evaluated explicitly.
Illustration of this procedure is given in References [9–11]. We therefore obtain
b = Pb A−1
b Cb q = Bb q
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P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
m = Pm A−1
m Cm q = Bm q
(38)
1
Cb q = Bs q
(39)
s =
where Bb , Bm , Bs are the strain displacement matrices related to bending, stretch and transverse
shear deformation, respectively.
The element stress resultants and stress couples given by Equations (7)–(14) can be rewritten
in terms of the strain fields b , m , s :
⎧
⎫
⎡
Mx ⎪
1
⎪
⎪
⎪
⎨
⎬
⎢
M = My
=D⎢
⎣
⎪
⎪
⎪
⎪
⎩
⎭
0
Mxy
⎧
⎫
⎡
Nx ⎪
1
⎪
⎪
⎪
⎨
⎬
⎢
N = Ny
=S ⎢
⎣
⎪
⎪
⎪
⎪
⎩
⎭
0
Nxy
Q=
Qx
Qy
=T
0
1
0
0
1 − /2
0
1
0
0
1 − /2
1
0
0
1
⎤
⎥
⎥ b = Db
⎦
(40)
⎤
⎥
⎥ m = Sm
⎦
(41)
s = Ts
(42)
where D, S, T are flexural, membrane and shear rigidities, respectively.
In order to determine the stiffness matrix of the element we make use of the strain energy
density, expressed as follows:
U = 21 (Mx x + My y + 2Mxy xy + Nx x + Ny y + 2Nxy xy + Qx xz + Qy yz )
(43)
Substituting Equations (1)–(14) into the above expression we obtain the following:
U = Ub + Um + Us
(44)
where Ub , Um , Us are, respectively: the bending component of the strain energy density function
(quadratic function of curvatures), the membrane component (quadratic function of membrane
strains) and the transverse shear component of the strain energy.
Using Equations (1)–(15) and (28)–(30) we may write the strain energy quantities Ub , Um ,
Us in the matrix forms as follows:
⎡
1
⎢
1
Ub = Tb D ⎢
⎣
2
0
Copyright 䉷 2006 John Wiley & Sons, Ltd.
0
1
0
0
1 − /2
⎤
⎥
⎥ b = 1 T Db
⎦
2 b
(45)
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
⎡
1
⎢
1
Um = Tm S ⎢
⎣
2
0
1
Us = Ts T
2
0
1
0
0
1 − /2
1
0
0
1
⎤
⎥
⎥ m = 1 T Sm
⎦
2 m
1
s = Ts Ts
2
The total strain energy e in the element domain may be written as
1
(T Db + Tm Sm + Ts Ts ) d
e =
2 b
or using Equations (37)–(39):
1
e = qT
2
351
(46)
(47)
(48)
(BTb DBb + BTm SBm + BTs TBs ) dq
(49)
which leads to
e = 21 qT [Kb + Km + Ks ]q
(50)
where Kb , Km , Ks are the element stiffness matrices related to bending, stretch, and transverse
shear deformation, given by
Kb =
BTb DBb d
(51)
Km =
BTm SBm d
(52)
BTs TBs d
(53)
Ks =
The elastic element stiffness matrix is then given by
K = Kb + Km + Ks
(54)
5. YIELD CRITERION AND HARDENING RULE
As discussed in the Introduction, a yield criterion for porous metals, expressed in terms of the
stress resultants and couples is used here, similar to the Iliushin’s yield function modified to
account for the shear forces, as given in Reference [31], the progressive development of the
plastic curvatures, and damage caused by growth of voids. The Iliushin’s yield function F can
be written as follows:
F=
M2
N2
1 |MN |
Y (k)
+ 2+√
− 2 =0
2
M
N
M0
N0
0
3 0 0
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P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
or
F=
|M| N 2
Y (k)
+ 2 − 2 =0
M0
N0
0
(56)
where
2
N 2 = Nx2 + Ny2 − Nx Ny + 3Nxy
(57)
2
M 2 = Mx2 + My2 − Mx My + 3Mxy
(58)
2
MN = Mx Nx + My Ny − 21 Mx Ny − 21 My Nx + 3Mxy
M0 =
0 h2
,
4
N0 = 0 h
(59)
(60)
and 0 is the uniaxial yield stress, Y (k) is a material parameter, which depends on isotropic
hardening parameter k, h is the thickness of the shell, and |.| denotes the absolute value.
The form of the yield condition given by Equation (55), can be easily derived from the von
Mises function and the definition of normal stresses at the top and the bottom surfaces of the
shell, as is shown in Reference [89]. Instead we use in this work the yield criterion for porous
ductile metals as originally proposed by Gurson [65, 66], and later modified by Perzyna [80]
and Dornowski and Perzyna [93]. Although it is of a form similar to von Mises equation, it
accounts for the isotropic damage effects through the dependence of the first invariant of stress
and the evolution of porosity. The plastic potential function defined by Dornowski and Perzyna
[93] can be written as
3
(61)
Sij Sij + n 2ii , i, j = 1, 2, 3
f=
2
where Sij is deviatoric stress tensor given by
Sij = ij − 13 kk
(62)
ij
ij is a stress tensor given by
ij =
Nij
6Mij
± 2
h
h
(63)
where Nij are normal forces; Mij are bending moments, h is a thickness of the shell and
ij is a Kronecker delta. The parameter n in Equation (61) is a material constant, determined
(for ductile metals) by Perzyna [80]: n = 1.2587. The symbol in Equation (61) is a porosity
parameter given by Gurson [65, 66] and modified by Duszek-Perzyna and Perzyna [76]:
p
p
= k1 ii + k2 ij ij + k3 ii
(64)
where k1 , k2 , k3 denote the material constants, and p are the increments of stress
and plastic strain, respectively. The first two terms in the above equation are responsible for
nucleation due to the cracking of the second-phase particles, and debonding of the second-phase
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
353
particles from the matrix material, respectively. The third term depicts the growth of voids, and
is controlled only by the plastic flow. The main term in the current work is the growth term.
We may assume that from the metallurgical investigations of the isotropic materials comprising
a plate or a shell, we can determine the initial porosity (t = 0) = 0 , and we shall consider
only the growth term in the evolution of porosity, i.e.:
p
= k3 ii
(65)
It is very important to determine or, in the absence of sufficient experimental data, to assume the
initial level of porosity in the virgin material. If nucleation is accounted for in the description
of damage, then it is possible to assume 0 = 0, which corresponds to a situation in which
there are no pores in the virgin material. Even though it is not a very realistic assumption,
since certain level of porosity exists in the undeformed material, through the representation of
nucleation in the damage model we could recognize the opening of voids at a certain level
of stress. The expansion of voids leading to localization and fracture may be approximated by
means of the growth term. In the current work however, the damage representation is reduced
to void growth only, hence the initial finite value of porosity must be determined or assumed.
Equations (61)–(65) are written using the indicial notation and a summation convention.
Rewriting Equation (65) in engineering notation yields:
p
p
p
= k3 (x + y + z )
p
p
(66)
p
where x ; y ; z are increments of the normal plastic strains due to both membrane and
p
p
bending actions in the x, y, z directions, respectively. x and y may be written as follows:
p
p
p
p
p
p
p
p
p
p
x = mx + bx = mx + zx
(67)
y = my + by = my + zy
p
p
where mx and my are the increments of plastic strains due to the membrane action only,
p
p
in the x, y directions; bx and by are the increments of plastic strains due to the bending
action only, in the x, y directions; z is the distance from the mid-plane to the plane under
p
p
consideration; and x , y are the increments of plastic curvatures at the midsurface in
planes parallel to the xz, yz planes, respectively. The maximum normal plastic strain caused
by bending will occur at z = h/2 which leads to
p
p
p
p
h p
x
2
h p
+ y
2
x = mx +
y = my
(68)
p
Substituting Equations (68) into Equation (66) and neglecting z we obtain
h
p
p
p
p
= k3 mx + my + (x + y )
2
(69)
We now proceed to the determination of the plastic potential function expressed in terms
of the stress resultants and couples. For the purpose of conciseness, we neglect radial and
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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354
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
transverse shear stresses in the current derivation. However, transverse shear forces are later
introduced into the yield condition. Equation (61) can be written using the engineering notation:
1 (70)
f = √ [(x − y )2 + 2x + 2y + 6 2xy + n (x + y )2 ]
2
where x , y are the normal stresses in the x, y directions, respectively, and
stress on the xy plane.
We can define the yield condition as follows:
1 √ [(x − y )2 + 2x + 2y + 6 2xy + n (x + y )2 ] = 0
2
xy
is a shear
(71)
where 0 denotes the uniaxial yield stress.
Substituting Equations (63) into (71) and performing some mathematical manipulations result
in the following relation:
N2
M2
NM
+
±2
=1
2
2
N
N0
M0E
0 M0E
(72)
where
2
N 2 = 1 + 21 n (Nx2 + Ny2 ) − (1 − n )Nx Ny + 3Nxy
2
M 2 = 1 + 21 n (Mx2 + My2 ) − (1 − n )Mx My + 3Mxy
(73)
1
N M = 1 + 21 n (Nx Mx + Ny My ) − (1 − n )(My Nx + Mx Ny ) + 3Nxy Mxy
2
and
N0 = 0 h,
M0E =
0 h2
6
(74)
Both the top and the bottom surfaces of the shell should be considered to obtain the larger
value of the term ±2(N M/N0 M0E ). We can ensure representation of the most negative effect
by writing Equation (72) in the following form [89, 94]:
N2
M2
|NM|
+
+2
=1
2
2
N0 M0E
N0
M0E
(75)
The yield surface given above is very similar to Iliushin’s yield function [30] given by
Equation (55). In order to derive Equation (55) we follow the procedure outlined by
Bieniek and Funaro [89], which is essentially the surface fitting approach. We write Equation (75) as follows:
a
M2
|NM|
N2
+
b
+c
=1
2
2
N0 M0E
N0
M0E
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
355
We determine the parameters a, b, c by considering the special loading cases separately. If we
account for membrane forces only, we see that for a = 1 we obtain the exact limit condition. Similarly, if we take a pure bending case, Equation (76) will produce exact results for
2 /M 2 . To find c we investigate the loading case corresponding to the maximum value
b = M0E
0
of the ratio N M/N0 M0E which occurs if Nx = Ny , Mx = My and Nxy = Mxy = 0. The stress
distribution in the cross-section in this case is as shown in Figure 4.
Based on the stress distribution in Figure 4, we can calculate the normal force:
Nx =
h/2
−h/2
x dz =
√
−h/2 3
−h/2
−0 dz+
h/2
−h/2
√
0 h
0 dz = √
3
3
(77)
Using Equation (74) we may write
Nx2 1
=
N02 3
(78)
Similarly, we may obtain
4M02
M2
=
2
2
M0E
9M0E
and
√ M0
NM
=2 3
N0 M0E
9M0E
(79)
Substitution of Equations (78)–(79) and previously determined parameters a = 1 and
2 /M 2 into Equation (76), yields
b = M0E
0
2 4M 2
√
M0
1 M0E
0
+ 2 3c
=1
+
2
2
3
9M0E
M0 9M0E
(80)
M0E
c= √
3M0
(81)
which leads to
Substituting the parameters a, b, c into Equation (76) we arrive at the limit yield surface as
defined by Iliushin:
F=
N2
1 |MN |
M2
+ 2+√
=1
2
M0
N0
3 M0 N0
(82)
The stress intensities are given by Equation (73) and unlike the original Iliushin yield function,
they account for the damage effects.
Voyiadjis and Woelke [4] introduced several other modifications to the Iliushin yield surface
for a better description of the plastic behaviour of shells. The damage variable is a function of
the plastic flow here, which makes the accuracy of the representation of plastic behaviour very
important. The same modifications of the yield function are therefore adopted in this work.
We can include the transverse shear forces Qx , Qy by expanding one of the stress intensities
given in Equation (73), cf. References [4, 31]:
2
N 2 = 1 + 21 n (Nx2 + Ny2 ) − (1 − n )Nx Ny + 3(Nxy
+ Q2x + Q2y )
(83)
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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356
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
√
Figure 4. Stress distribution corresponding to maximum NM/N0 M0E ( = h/2 3).
It was shown previously [4, 31] that the influence of the shear forces on the plastic behaviour
of thick plates and shells may be very important.
For a bending dominant situation, according to Equation (55) or (56), the structure will
deform linearly until the whole cross-section is plastic, i.e. the plastic hinge has formed. In
reality however, the plastic curvature develops progressively from the outer fibres of the shell
or plate and the material behaves non-linearly as soon as the outer fibres start to yield. To
account for the development of plastic curvature across the thickness, Crisfield [95] introduced
a plastic curvature parameter (¯ p ), into Equation (82):
F=
M2
N2
1 |MN|
Y (k)
+
+√
− 2 =0
2
2
2
M
N
M0
N0
0
3 0 0
(84)
|M|
N2
Y (k)
+ 2 − 2 =0
M0
N0
0
(85)
or
F=
where was chosen such that M0 follows the uniaxial moment–plastic curvature relation
= 1 − 13 exp − 83 ¯ p
(86)
and
¯ p =
Eh p
p
p
p
p
¯ p = √
((x )2 + (y )2 + x y + (xy )2 /4)1/2
30
p
p
(87)
p
The symbol ¯ p is the equivalent plastic curvature, and x , y and xy are the increments
of the plastic curvatures. We note that for ¯ p = 0, = 2/3 and we obtain M0 = 0 t 2 /6 which
represents first fibre yielding. If, on the other hand, ¯ p = ∞, = 1 and we obtain a fully
plastic cross-section. Therefore, through the introduction of the plastic curvature parameter we account for the progressive development of the plastic curvatures and correctly predict the
first yield.
We note that a material parameter Y (k), was employed in Equations (84)–(85) which depends
on the isotropic hardening parameter k, similar to Equations (55)–(56).
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
357
To model the elasto-plastic behaviour of shells subjected to reversing loads, one needs a
reliable kinematic hardening rule. Bieniek and Funaro [89] introduced residual bending moments
(‘hardening parameters’), allowing for the description of the Bauschinger effect. These were
later successfully applied to dynamic [94] and viscoplastic dynamic analysis of shells [96, 97].
To determine correctly the rigid translation of the yield surface in the stress resultant space, we
need not only residual bending moments, but also residual normal and shear forces. Voyiadjis
and Woelke [4] presented a new kinematic hardening rule for shells, with residual bending
moments and residual normal and shear forces as kinematic hardening parameters, related
directly to the backstress given by Armstrong and Frederick [98], and representing the centre
of the yield surface in the stress resultant space. Adopting that same hardening rule in the
current paper, we express the yield surface as follows:
F∗ =
|M ∗ | (N ∗ )2
Y (k)
+
− 2 =0
M0
N02
0
(88)
where
(N ∗ )2 = 1 + 21 n [(Nx − Nx∗ )2 + (Ny − Ny∗ )2 ]
− (1 − n )(Nx − Nx∗ )(Ny − Ny∗ )
∗ 2
) + (Qx − Q∗x )2 + (Qy − Q∗y )2 ]
+ 3[(Nxy − Nxy
(89)
(M ∗ )2 = 1 + 21 n [(Mx − Mx∗ )2 + (My − My∗ )2 ]
∗ 2
− (1 − n )(Mx − Mx∗ )(My − My∗ ) + 3(Mxy − Mxy
)
(90)
∗ , N ∗ , N ∗ , N ∗ , Q∗ , Q∗ are the above-described residual bending moments,
and Mx∗ , My∗ , Mxy
x
y
xy
x
y
normal and shear forces, respectively. It is worthwhile to mention that by setting the porosity
parameter to zero, i.e. = 0, the yield surface given by Equations (88)–(90) reduces to the one
given by Voyiadjis and Woelke [4], where the damage effects are not considered.
Detailed derivation of the kinematic hardening parameters is presented in Reference [4]. We
only briefly discuss the concept in the present paper. For the purpose of conciseness, we use the
indicial notation in the derivation, and only the final result is given employing the engineering
notation. Armstrong and Frederick’s evolution of the backstress ij is given by
ij
p
= cij − a
p
ij eq
where a and c are constants and the equivalent plastic strain increment is given by
p
p
p
eq = 23 ij ij
(91)
(92)
The backstress tensor represents the centre of the translated yield surface in the stress space.
It has the same dimension as the stress tensor. To compute the stress resultants we need to
integrate the stresses over the thickness of the shell. We use the same definition here to derive
the hardening parameters, which represents the centre of the yield surface in the stress resultant
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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358
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
space. We therefore need to integrate the backstress over the thickness of the plate or shell, in
order to obtain the residual normal and shear forces, and the bending moments. The definitions
of the increments of the hardening parameters are as follows [4]:
Nij∗
=
Mij∗ =
h/2
−h/2
h/2
−h/2
ij
dz
(93)
ij z dz
(94)
Substituting Equation (91) into Equations (93)–(94) and after some mathematical manipulations we obtain the definition of the increments of the kinematic hardening parameters in the
engineering notation as follows:
If F ∗ = 1 and ∇F ∗ > 0 (plastic loading)
N0
1
p
p
Nx∗ = 1 (1 − F )
x − Nx∗ eq
0
h
N0
1
p
p
Ny∗ = 1 (1 − F )
y − Ny∗ eq
0
h
N0
1 ∗ p
p
∗
xy − Nxy eq
Nxy = 1 (1 − F )
0
h
N0
1 ∗ p
p
∗
xz − Qx eq
Qx = 1 (1 − F )
0
h
N0
1 ∗ p
p
∗
yz − Qy eq
Qy = 1 (1 − F )
0
h
M0
6 ∗ p
p
∗
Mx = 2 (1 − F )
x − 2 Mx eq
0
h
M0
6 ∗ p
p
∗
My = 2 (1 − F )
y − 2 My eq
0
h
M0
6 ∗
p
p
∗
xy − 2 Mxy eq
Mxy = 2 (1 − F )
0
h
(95)
(96)
If F ∗ < 1 and ∇F ∗ 0 (unloading or neutral loading)
(97)
∗
Nx∗ = Ny∗ = Nxy
= Q∗x
= Q∗y
∗
= Mx∗ = My∗ = Mxy
=0
The parameters 1 and 2 in the above formulation control the membrane-force–membranestrain and moment–curvature relations. A value 1 = 2 = 2.0 was found to be of sufficient
accuracy in the representation of shells.
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
359
Figure 5. Yield surface on Nx Mx plane—interpretation of kinematic hardening parameters O is the
centre of the translated yield surface.
We therefore arrive at a final form of the yield function for ductile porous metals, given
by Equations (88)–(90) and (95)–(97), expressed in terms of the stress resultants and couples,
with both isotropic and kinematic hardening rules. This is a very convenient form of the yield
surface for the analysis of shells accounting for the damage effects through the evolution of
porosity. A graphic representation of the yield surface on the Nx Mx plane with = 1 and
Y = 20 is shown in Figure 5. Point O denotes the transferred centre of the yield surface.
6. EXPLICIT TANGENT STIFFNESS MATRIX
The plastic node method is employed here in the derivation of the stiffness matrix, i.e. the
plastic deformations and damage are considered to be concentrated in the plastic hinges. The
yield function is only checked at each node of the finite elements. If the combination of stress
resultants satisfies the yield condition, that node is considered to be plastic, which triggers the
void growth, as the porosity is a function of the plastic flow. Thus, in this method the inelastic
deformations are only considered at the nodes, while the interior of the element remains always
elastic. When node i of the element becomes plastic, the yield function takes the form
Fi∗ (Ni , Qi , Mi , Ni∗ , Q∗i , Mi∗ , ki , i ) = 0
(98)
where
⎧
⎫
Nx ⎪
⎪
⎪
⎪
⎨
⎬
Ni = Ny ;
⎪
⎪
⎪
⎪
⎩
⎭
Nxy
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Qi =
Qx
Qy
;
⎫
⎧
Mx ⎪
⎪
⎪
⎪
⎬
⎨
Mi = My
⎪
⎪
⎪
⎪
⎭
⎩
Mxy
Int. J. Numer. Meth. Engng 2006; 68:338–380
360
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
⎧ ∗⎫
Nx ⎪
⎪
⎪
⎪
⎬
⎨
∗
∗
N
;
Ni =
y
⎪
⎪
⎪
⎭
⎩ ∗ ⎪
Nxy
Q∗i =
Q∗x
Q∗y
⎧ ∗⎫
Mx ⎪
⎪
⎪
⎪
⎬
⎨
∗
∗
M
Mi =
y
⎪
⎪
⎪
⎭
⎩ ∗ ⎪
Mxy
(99)
At the same time the stress resultants must remain on the yield surface, i.e. the consistency
condition must be satisfied:
*F ∗
*F ∗
*Fi∗
*Fi∗
*Fi∗
*Fi∗
∗
∗
dMi + i dNi + i dQi +
dN
+
dQ∗i
∗ dMi +
i
*Mi
*Ni
*Qi
*Ni∗
*Q∗i
*M i
+
*Fi∗
*F ∗
dki + i di = 0
*ki
*i
(100)
We assume an additive decomposition of strains into elastic and plastic parts:
= e + p
(101)
The associated flow rule is used here to determine the increments of plastic strains:
p
x =
NPN
i=1
i
*Fi∗
*Mxi
and
p
x =
NPN
i=1
i
*Fi∗
*Nxi
(102)
where NPN is the number of plastic nodes in the element and di is a plastic multiplier. The
remaining increments of the plastic strains are obtained in the same way. The plastic strain
fields are interpolated as in the linear elastic analysis (Equations (28)–(30)) given here in the
incremental form:
⎧
⎧
p ⎫
p ⎫
x ⎪
x ⎪
⎪
⎪
p ⎪
⎪
⎪
⎪
⎬
⎬
⎨
⎨
xz
p
p
p
p
p
b =
(103)
, m =
, s =
y
y
p
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
yz
⎩
⎩
p ⎭
p ⎭
2xy
2xy
The evolution of the porosity parameter representing damage is given by Equation (69) repeated
here for convenience:
h
p
p
p
p
(104)
= k3 x + y + (x + y )
2
The assumption of an additive decomposition of strains can be extended to displacements
provided that the strains are small [31, 32]. Although geometric non-linearities are taken into
account in the current work, we only consider large rigid rotations and translations, but small
strains. Thus, we may write
q = qe + qp
(105)
Following the work of Shi and Voyiadjis [31] we approximate the increments of plastic
p
displacements by the increments of plastic strains. The plastic rotation x is a function of
p
p
both x and xy , as can be deduced from Equation (28). Assuming that the increment of
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
361
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
p
plastic nodal rotation xi is proportional to the increment of elastic nodal rotation xi we
can express the former as
2
p
p
p
xi
xi = lim
x +
2xy dx dy
→0
2xi + 2yi
i
= i
*Fi∗
*Fi∗
22xi
+
2
2
*Mxi
xi + yi *Mxyi
(106)
where i represents the infinitesimal neighbourhood of node i. The vector of the incremental
nodal plastic displacements of the element at node i can be then expressed as follows:
p
qi = ai i
(107)
with ai given by
aiT
=
*Fi∗
*Fi∗
*Fi∗ *Fi∗
*Fi∗ *Fi∗
+ pu
;
+ pv
;
+
;
*Nxi
*Nxyi *Nyi
*Nxyi *Qxi
*Qyi
*Fi∗ *Fi∗
*Fi∗
*Fi∗
+ p x
;
+ py
*Mxi
*Mxyi *Myi
*Mxyi
pu =
2u2i
u2i + vi2
;
pv =
2vi2
u2i + vi2
;
px =
(108)
22xi
2xi + 2yi
;
py =
22yi
2xi + 2yi
Equations (107) and (108) indicate that the plastic displacements at the nodes are only functions
of the stress resultants at this node [31]. Therefore, we may write the vector of the increments
of the nodal plastic displacements, as follows:
⎫
⎤⎧
⎡
0
1 ⎪
a1 0
⎪
⎪
⎪
⎬
⎥⎨
⎢
⎥
0
a
0
qp = ⎢
= a
(109)
i
i
⎦⎪
⎣
⎪
⎪
⎪
⎩
⎭
0 0 aNPN
NPN
In order to determine the tangent stiffness matrix of the element we define b , m , s
as virtual elastic bending, membrane and transverse shear strains, respectively ( -virtual), and
M, N, Q as stress couples and stress resultants of the element. We also make use of the
linearized equilibrium equations of the system at configuration k + 1 in the Updated Lagrangian
formulation, expressed by the principle of the virtual work, which in finite element modelling
takes the form:
( Tb Db + Tm Sm + Ts Ts ) dx dy +
Tk F dx dy
=k+1 R −
( Tb k M + Tm k N + Ts k Q) dx dy
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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362
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
where k+1 R is the total external virtual work at step k + 1 and is the slope vector and k F
is a membrane stress resultant matrix at step k given as follows:
⎧
⎫
*w ⎪
⎪
⎪
⎪
k
⎪
⎨ *x ⎪
⎬
Nx k Nxy
k
=
(111)
F=
,
k
k
⎪
*w ⎪
⎪
⎪
N
N
xy
y
⎪
⎪
⎩
⎭
*y
The slope field is evaluated in a similar way to the strain fields, using the quasi-conforming
technique [11]. A bilinear interpolation is used as in Reference [25] to approximate the slope
field:
⎧ ⎫
⎪
⎪
⎪
⎪ 1⎪
⎪
⎪
⎪
⎪
⎪
⎪
2 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪ ⎪
⎬
1 x y xy 0 0 0 0 ⎨ 3 ⎪
=
= P
(112)
.. ⎪
⎪
⎪
0 0 0 0 1 x y xy ⎪
⎪
⎪
.
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
7⎪
⎪
⎪
⎪
⎪
⎪
⎩ ⎪
⎭
8
with P denoting the trial function matrix and is a vector of undetermined parameters,
calculated in the same way as the vectors of strain parameters used to approximate the strain
fields (Equations (28)–(30)):
= A−1 Cqe , A =
PT P dx dy, Cqe =
PT dx dy
(113)
The slope field is therefore expressed in terms of the slope–displacement matrix G:
= PA−1 Cqe = Gqe
(114)
The cubic interpolation of w along the boundary of the elements, given by Hu [99] is used
here to evaluate the C matrix:
w(s) = [1 − + ( − 32 + 23 )]wi + [ − 2 + ( − 32 + 22 )]
lij
si
2
+ [ − ( − 32 + 23 )]wj + [− + 2 + ( − 32 + 22 )]
=
s
;
lij
0slij ;
01;
1
= D
1 − 12
T L2
lij
sj
2
(115)
where lij is the distance between nodes i and j , si , sj are tangential rotations at nodes i
and j , respectively, and D, T are flexural and transverse shear rigidities. The influence of the
parameter is explained in References [9, 10, 99].
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
363
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
Using Equation (114), the virtual work principle given by (110) may now be rewritten as
follows:
T
( Tb Db + Tm Sm + Ts Ts ) dx dy + qe Kg qe
=
k+1
R−
( Tb k M + Tm k N + Ts k Q) dx dy
(116)
where Kg is the initial stress matrix defined as
Kg =
GTk FG dx dy
(117)
Substituting Equations (37)–(39) on to the right-hand side of Equation (116), we may write:
( Tb k M + Tm k N + Ts k Q) dx dy = qT f
(118)
where f is the internal force vector resulting from the unbalanced forces in configuration k and
is expressed as follows:
f=
(BTb k M + BTm k N + BTs k Q) dx dy
(119)
We may now rewrite Equation (116) using Equations (40)–(42), (101) and (119) as follows:
T
T
T
T
pT
pT
pT
[( eb + b )M + ( em + m )N + ( es + s )Q] dx dy + qe Kg qe
= k+1 R − qT f
(120)
Re-arranging terms and writing the above equation in incremental form we obtain
T
T
T
( eb M + em N + es Q) dx dy
+
pT
pT
pT
( b M + m N + s Q) dx dy + qe Kg qe = k+1 R − qT f
T
(121)
Substituting Equations (102) into Equation (121) we obtain
T
T
T
( eb M + em N + es Q) dx dy
+
NPN
i=1
i
*Fi∗
*Fi∗
*Fi∗
T
dMi +
dNi +
dQi + qe Kg qe =
*Mi
*Ni
*Qi
k1
R − qT f
(122)
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
364
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
Making use of Equations (40)–(42), (48)–(50), as well as the consistency condition given by
Equation (100), we may write
eT
q (K + Kg )q −
e
NPN
i=1
i
*Fi∗
*Fi∗
*Fi∗
*Fi∗
*Fi∗
∗
∗
∗
dM
+
dN
+
dQ
+
dk
+
di
i
i
i
i
*Mi∗
*Ni∗
*Q∗i
*ki
*i
= k+1 R − qT f
(123)
where K is the linear elastic stiffness matrix given by Equation (54).
Similar to Equation (109) we define
⎡
abT =
⎢
*Fi∗
⎢
=
⎢ 0
*Mi∗ ⎣
0
⎡
asT
T
ab1
T
as1
*F ∗ ⎢
⎢
= i∗ = ⎢ 0
⎣
*Qi
0
⎤
0
0
T
abi
0
0
T
abNPN
0
0
T
asi
0
0
T
asNPN
⎥
⎥
⎥,
⎦
⎡
T
am
=
⎤
⎥
⎥
⎥,
⎦
T
am1
*Fi∗ ⎢
⎢
=⎢ 0
*Ni∗ ⎣
0
⎡
a
1
*F ∗ ⎢
a = i =⎢
⎣ 0
*i
0
T
⎤
0
0
T
ami
0
0
T
amNPN
0
a
0
0
0
i
a
⎥
⎥
⎥
⎦
⎤
(124)
⎥
⎥
⎦
NPN
and
∗
∗
∗
*F
*F
*F
i
i
i
T
=
abi
∗ ; *M ∗ ; *M ∗
*Mxi
yi
xyi
*Fi∗ *Fi∗ *Fi∗
T
ami =
∗ ; *N ∗ ; *N ∗
*Nxi
yi
xyi
*Fi∗
*Fi∗
T
=
;
asi
*Q∗xi
*Q∗yi
a i=
(125)
*Fi∗
* i
Substituting Equations (102) into (95) and (96) we obtain
dMx∗ = Mx∗
⎤
⎡
! ∗ 2 ∗
∗ 2
∗ 2
!2
M0
*F
6
*F
*F
*F
⎦
= 2 (1 − F )
⎣
− M ∗"
+
+
0
*Mx h2 x 3
*Mx
*My
*Mxy
Copyright 䉷 2006 John Wiley & Sons, Ltd.
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365
ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
and similarly for the remaining hardening parameters. The vectors of the hardening parameters
therefore yield:
⎧
∗ ⎫
Nxi
⎪
⎪
⎪
⎪
⎨
⎬
∗
∗
dNi = Nyi
= Am ;
⎪
⎪
⎪
⎪
⎩
⎭
∗
Nxyi
dQ∗i =
Q∗xi
Q∗yi
= As ;
⎧
∗ ⎫
Mxi
⎪
⎪
⎪
⎪
⎬
⎨
∗
∗
= Ab dMi = Myi
⎪
⎪
⎪
⎪
⎭
⎩
∗
Mxyi
(127)
where Am , As , Ab are given by
⎡
0
0
Ami
0
0
AmNPN
⎤
0
⎥
0 ⎥
⎦
Am1
⎢
Am = ⎢
⎣ 0
0
⎡
0
As1
⎢
As = ⎢
⎣ 0
Asi
0
0
⎤
⎥
⎥,
⎦
⎡
Ab1
⎢
Ab = ⎢
⎣ 0
0
0
0
Abi
0
0
AbNPN
⎤
⎥
⎥
⎦
(128)
AsNPN
and
⎧
⎪
⎪
N0
⎪
⎪
1 (1 − F )
⎪
⎪
⎪
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
N
Ami = 1 (1 − F ) 0
⎪
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
N0
⎪
⎪
⎪
1 (1 − F )
⎪
⎩
0
Asi =
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎡
⎤
⎪
!
⎪
2
2
2
⎬
∗
∗
∗
∗
!
*Fi
*F
*F
*F
1
2
i
i
i
∗
"
⎣
⎦
− Nyi
+
+
⎪
h
3
*Nyi
*Nxi
*Nyi
*Nxyi
⎪
⎪
⎪
⎪
⎪
⎪
⎤
⎡
⎪
⎪
!
⎪
2
2
2
∗
∗
∗
∗
⎪
!2
*F
*Fi
*F
*F
1
i
i
i
∗
⎪
"
⎦⎪
⎣
⎪
− Nxyi
+
+
⎪
⎭
h
3
*Nxyi
*Nxi
*Nyi
*Nxyi
⎤
! ∗ 2
∗ 2
∗ 2
!2
*F
*F
*F
*Fi∗
1
i
i
i
∗"
⎦
⎣
− Nxi
+
+
h
3
*Nxi
*Nxi
*Nyi
*Nxyi
⎡
⎧
⎡
⎤⎫
! 2 2 ⎪
⎪
∗
∗
∗
⎪
⎪
!
⎪
⎪
*Fi
*Fi
N0 ⎣ *Fi
1 ∗ "2
⎪
⎦⎪
⎪
⎪
(1
−
F
)
−
+
Q
⎪
⎪
1
xi
⎪
⎪
0 *Qxi
h
3
⎪
⎪
*Qxi
*Qyi
⎨
⎬
⎡
⎪
⎤ ⎪
! ⎪
⎪
∗
∗ 2
∗ 2
⎪
⎪
!
⎪
⎪
*F
*F
*F
N0 ⎣ i
1 ∗ "2
⎪
⎪
i
i
⎪
⎪
⎦
(1
−
F
)
−
+
Q
⎪ 1
⎪
yi
⎪
⎪
⎩
⎭
0 *Qyi
h
3
*Qxi
*Qyi
Copyright 䉷 2006 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Engng 2006; 68:338–380
366
Abi
P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
⎧
⎪
⎪
M0
⎪
⎪
2 (1 − F )
⎪
⎪
⎪
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
M
= 2 (1 − F ) 0
⎪
0
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
M0
⎪
⎪
⎪
(1 − F )
⎪
⎩ 2
0
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎡
⎤
⎪
!
⎪
2
2
2
⎬
∗
∗
∗
∗
!
*Fi
*F
*F
*F
6
2
i
i
i
∗"
⎣
⎦
− 2 Myi
+
+
⎪
3
h
*Myi
*Mxi
*Myi
*Mxyi
⎪
⎪
⎪
⎪
⎪
⎪
⎡
⎤
⎪
⎪
!
⎪
2
2
2
∗
∗
∗
∗
⎪
!
*Fi
*F
*F
*F
6
2
⎪
i
i
i
∗ "
⎪
⎣
⎦
⎪
− 2 Mxyi
+
+
⎪
⎭
3
h
*Mxyi
*Mxi
*Myi
*Mxyi
⎤
! ∗ 2
∗ 2
∗ 2
!2
*F
*F
*F
*Fi∗
6
i
i
i
∗"
⎦
⎣
− 2 Mxi
+
+
3
h
*Mxi
*Mxi
*Myi
*Mxyi
⎡
(129)
The evolution equation for the porosity parameter may be written by substituting
Equations (102) into (104):
*Fi∗
*Fi∗
*Fi∗
h *Fi∗
d i = i = k3 i
(130)
+
+
+
= A i i
2 *Mxi
*Nxi
*Nyi
*Myi
As previously done, we apply the plastic node method to derive the matrix form of the above
equation:
di = i = A where:
⎡
A
⎢
A =⎢
⎣ 0
0
1
0
A
0
0
0
i
A
⎤
⎥
⎥
⎦
and
A i = k3
(131)
*Fi∗
*Fi∗
h
+
+
2
*Nxi
*Nyi
*Fi∗
*Fi∗
+
*Mxi
*Myi
(132)
NPN
Following the work of Shi and Voyiadjis [31] we also define the isotropic hardening
parameter as
⎧
⎫
*F1∗
⎪
⎪
⎪
⎪
⎪
dk1 ⎪
⎪
⎪
⎫
⎪
⎪
⎡
⎤⎧
*k1
⎪
⎪
⎪
⎪
H1 0
0
1
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎬
⎨
⎬
∗
⎢
⎥
*F
i
⎥
0
H
0
(133)
H = ⎢
=
−
dki
i
i
⎣
⎦⎪
⎪
⎪
⎪
*k
⎪
⎪
⎪
⎪
i
⎩
⎭
⎪
⎪
⎪
⎪
⎪
⎪
0
0 HNPN
NPN
⎪
⎪
∗
⎪ *FNPN
⎪
⎪
⎪
⎪
dkNPN ⎪
⎩
⎭
*kNPN
We may now substitute Equations (124), (127), (131) and (133) into (123) to obtain
T
qe (K + Kg )qe + T [H − abT Ab − am
Am − asT As − aT A ] = k+1 R − qT f
T
(134)
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
Using Equations (105) and (107) in Equation (134) we may write:
T
T
Am − asT As − aT A ]
( qT − qp )(K + Kg )qe + T [H − abT Ab − am
− k+1 R + qT f= qT [(K + Kg )qe − k+1 R ∗ + f] + T [−aT (K + Kg )qe
T
+ (H − abT Ab − am
Am − asT As − aT A )] = 0
(135)
with
k+1
R = k+1 R ∗ q
(136)
By the virtue of the variational method equation (135) we obtain
(K + Kg )qe − k+1 R ∗ + f = 0
T
−aT (K + Kg )qe + (H − abT Ab − am
Am − asT As − aT A ) = 0
(137)
Substituting (105) and (107) into the above equations we get
(K + Kg )qe − k+1 R ∗ + f = (K + Kg )(q − a) = k+1 R ∗ − f
(138)
T
−aT (K + Kg )(q − a) + (H − abT Ab − am
Am − asT As − aT A ) = 0
(139)
Equation (139) leads to the following expression:
T
Am − asT As − aT A )]−1 aT (K + Kg )q
= [aT (K + Kg )a + (H − abT Ab − am
(140)
Equation (138) becomes
Kepdg q = k+1 R ∗ − f
(141)
where Kepdg is the elasto-plastic damage large displacement stiffness matrix of the element,
given by
Kepdg = (K + Kg ){I − a[aT (K + Kg )a
T
Am − asT As − aT A )]−1 aT (K + Kg )}
+ (H − abT Ab − am
(142)
The tangent stiffness matrix given by Equation (142) is similar to the one presented by
Shi and Voyiadjis [31]. The present formulation accounts for large displacements. Consequently,
the stiffness matrix of the element contains the initial stress matrix Kg . The above-derived
stiffness matrix describes not only the isotropic hardening, by means of the parameter H, but
also the kinematic hardening, through parameters Ab , Am , As , which are not determined by
curve fitting, but derived explicitly from the evolution equation of the backstress given by
Armstrong and Frederick [98].
The most important characteristic of the current work is a consistent and convenient
incorporation of the damage effects into the yield condition and stiffness matrix, by means
of the A matrix. We therefore have a non-layered finite element formulation with shell
constitutive equations, a yield condition for porous ductile metals, the flow and hardening
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P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
rules expressed in terms of the membrane and shear forces and the bending moments. All the
variables used here, namely the porosity function, the stress resultants and couples, as well
as the residual stress resultants and couples, representing the centre of the yield surface, are
derived in a very rigorous manner.
A very important feature of the derived tangent stiffness is its explicit form. The linear elastic
stiffness matrix and initial stress matrix are determined by the quasi-conforming technique,
which allows all the integrations to be performed analytically. The hardening parameters are
also given explicitly. In addition, the through-the-thickness-integration is not employed here
either, since the current model is the non-layered model with the yield condition expressed in
terms of stress couples and resultants. The benefits of an explicit stiffness is low computational
cost. Numerical integration of the stiffness matrix using a standard Gaussian Quadrature method
requires a number of floating-point operations of the order O(p 6 ) where is the order of the
interpolation functions (p = 3 in the current model). The Vector Quadrature technique [100]
performs fewer floating-point operations than standard Gaussian Quadrature but has the same
level of complexity [101]. In the current model, these operations are not performed at all,
since the stiffness matrix is given explicitly. This leads to substantial savings in computer time.
Actual time savings are dependent on the implementation method, the problem solved and a
processing unit. In the non-linear simulations, where the stiffness matrix is evaluated many
times during the analysis, the improvement in computational efficiency is even more apparent.
7. NUMERICAL EXAMPLES
For the purpose of computational implementation of the proposed model, a finite element
computer program previously developed for the elasto-plastic analysis of shells using the
programming language Fortran 95 is enhanced to account for the damage effects. The modified
Newton–Raphson technique is employed to solve a system of non-linear, incremental equations. In order to overcome a singularity problem appearing at the limit point, the arc-length
method [102] is adopted to determine the local load increment for each iteration. The return
to the yield surface algorithm is also implemented [102]. The numerical results delivered by
the current model are computed using a personal computer. Some of the reference solutions
obtained with the layered approach (ABAQUS) are determined using a Silicon Graphics Onyx
3200 system.
The accuracy of the description of the elasto-plastic and damage behaviour of shells is
verified through a series of discriminating numerical examples. This paper is a continuation of
the previous work of the authors [4, 9], where linear elastic and elasto-plastic formulations are
given. The most important novel feature of the present algorithm is the description of isotropic
damage in plates and shells. Thus, the examples presented here are selected to challenge mainly
the representation of the evolution of damage in shells and the associated reduction in stiffness.
Table I lists the references used here, and their corresponding abbreviations used later in
the text.
7.1. Clamped square plate subjected to a central point load
In this example we consider a square plate with all the edges fixed, with an aspect ratio of
L/ h = 20, where L is the length of the plate and h is the thickness. The plate is subjected
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
369
Table I. Listing of the models used with abbreviations.
Name
Description
W&V-E
W&V-EP
W&V-EPD
The present formulation—elastic analysis
The present formulation—elasto-plastic analysis
The present formulation—elasto-plastic, damage analysis
2
s
n
2
Figure 6. Clamped square plate subjected to a central point load—geometry and material properties.
to a central point load. Only a quarter of the plate needs to be examined due to symmetry.
This problem was analysed by Shi and Voyiadjis [3], by means of the 4 × 4 mesh of finite
elements. The same mesh of 4 × 4 elements per quarter of the plate is employed here. The
geometry of the problem and the material properties are given in Figure 6.
The equilibrium path is a universal curve providing most of the information regarding the
functioning of the model independently of whether the deformation of the structure is governed
by the bending, membrane or shear strains. Thus, we study the equilibrium path for the
problem described above. The material parameters n and k3 appearing in Equations (61) and
(65) are: n = 1.2587, k3 = 0.09, as determined by Perzyna [80]. The central deflection of the
plate as a function of the applied load is given in Figure 7. Shi and Voyiadjis [3] solved this
problem using a phenomenological damage model with isotropic damage parameter defined
by Lemaitre [49]. The ultimate load for this problem, given in Reference [3], without the
influence of damage is Pc = 10M0 . Shi and Voyiadjis also showed the substantial reduction
in stiffness of the structure when damage was considered. The ultimate load of the damaged
plate was about Pc = 8M0 . The current model is based on micromechanical observations of
the material and features more physically sound interpretation of the damage variable, i.e. the
porosity. The result of the current analysis with the influence of damage considered, yields
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P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
10
9
8
7
P/M0
6
5
4
W&V-E
3
W&V-EP
2
W&V-EPD
1
0
0
2
4
6
8
2
10
12
2
wD/M0L x10
Figure 7. Clamped square plate subjected to a central point load—load–displacement curve.
approximately the same ultimate load Pc = 8M0 . For the current example the ultimate load
carried by the structure, calculated through the elasto-plastic investigations is 20% higher than
the one calculated through the application of the elasto-plastic-damage model. We therefore
conclude that neglecting the damage effects can result in unsafe design of the structures.
As expected, we notice that the damage variable only becomes significant when the structure
deforms plastically. This is because the evolution of damage is neglected in the elastic zone in
ductile materials. The current formulation shows a robust performance in this test.
A 4 × 4 mesh is used in this problem for comparison purposes with the reference result.
A finer mesh would not alter the quality of the results significantly. A higher number of plastic
hinges could lead to additional softening, which would however not substantially alter the
estimate of the maximum load carried by the structure. Mesh sensitivity investigations would
be imperative for analysis of postcritical behaviour, which in turn would require accounting for
large strains and strain gradients. Such a study is very important and will be the subject of
future work of the authors. It is however beyond the scope of this paper.
7.2. Spherical dome subjected to a ring of pressure
The problem of a spherical dome with an 18◦ hole at the top, subjected to a ring of pressure
was investigated in the first two authors’ previous paper [4]. It is an important engineering
problem, as well as a discriminating test for accuracy of the finite element representation of
the behaviour of shells. It was shown in Reference [4] that the stress-resultant-based shell
model with the kinematic hardening rule given by Equations (95)–(97) is capable of correctly
predicting the elasto-plastic behaviour of shells, including the Bauschinger effect. In this paper,
we revisit the problem of the spherical dome subjected to a ring of pressure, in order to
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
371
R=10 in
t=0.04 in
E=6.82x107 psi
ν =0.3
σ 0=125 psi
Figure 8. Spherical dome with an 18◦ cut-out; geometry and material properties.
establish the performance of the current formulation in approximating damage due to microvoids. As previously, the structure is loaded into a plastic zone, and subsequently the
pressure is reversed. We examine the elasto-plastic load–displacement curve and compare these
results with that of the curve obtained when the influence of damage taken into account, in
order to test the functioning and accuracy of the proposed yield surface for ductile porous
metals, defined in the stress resultant space. The material parameters n and k3 are the same as
in example 7.1: n = 1.2587, k3 = 0.09. The geometrical and material data are shown in Figure
8 and the resulting load–displacement curves are plotted in Figure 9.
Through the introduction of the porosity function, which characterizes damage into the yield
function we obtain a strong coupling between plasticity and damage. The damage variable is
dependent on the plastic deformation. Therefore, through the application of the robust kinematic
hardening rule, we may model the evolution of damage in the dome that is loaded into the
plastic zone in tension and subsequently the load is reversed, in compression (Figure 9). The
lowered yield point due to the Bauschinger effect is again correctly approximated. The reduction
in stiffness that is caused by damage initiated by the inelastic strains is significant in this case.
It is very important to note that the damage curve (W&V-EPD in Figure 9) as well as the
porosity curve (Figure 10) indicate unloading at the last increment of load before the load
is reversed. This means that considerable porosity has been developed (Figure 10) and the
ultimate load-carrying capacity of the structure has been reached. If the loading force P were
kept at the same level, the porosity
would grow even further, leading to localization and
fracture and ultimately the collapse of the structure. Based on the value of the porosity, the
fracture criterion could be postulated. It cannot, however, be performed consistently using the
current formulation. This is because the strains will reach very high values at the localized
area of deformation. This theory is a large displacement but small strain theory, and it may not
be applied to model localization problems. Furthermore, the gradients of deformation around
the localization area are very high, which can only be approximated by the gradient plasticity
[103] or viscoplasticity theories with damage variables [80–88]. Thus, proposing a fracture
criterion requires further study of the microstructural material characterization. This is beyond
the scope of this work.
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P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
80
60
40
Load
20
0
-1.E-04 -8.E-05 -6.E-05 -4.E-05 -2.E-05 0.E+00 2.E-05 4.E-05 6.E-05
8.E-05 1.E-04
-20
W&V-EP
-40
W&V-EPD
-60
-80
Vertical Displacement
Figure 9. Spherical dome with an 18◦ cut-out—load–displacement curve.
70
60
Load
50
40
30
20
W&V-EPD - porosity
10
0
0
0.01
0.02
0.03
0.04
0.05
0.06
Porosity
Figure 10. Spherical dome with an 18◦ cut-out—porosity as a function of load.
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
373
The plasticity curve (W&V-EP in Figure 9) reaches a plateau at a load about 8% lower
than in the case of the damage analysis. The elasto-plastic analysis is therefore leading to
the overprediction of the ultimate load. It is a very substantial factor from the point of view
of engineering analysis of important structures. Moreover, the theory of plasticity alone does
not provide reliable information about the behaviour of the material or structure after the
ultimate load has been reached. Although the present elasto-plastic-damage model is limited
to rate-independent and small strain problems, it compares much better with the experimental
investigations, showing softening of the material after reaching the ultimate load level.
Figure 10 presents a plot of the porosity as a function of load. The initial porosity in
the virgin material is assumed to be 0 = 0.01. Since the damage variable is a function of the
plastic strains, we only see growth of voids when plasticity occurs. As discussed above at the
ultimate load level of approximately P = 63 lbf/in we observe softening of the material. The
collapse will therefore occur at a load level approximately 8% lower that the load predicted
by the elasto-plastic analysis.
The present approach provides a good approximation of the evolution of damage of the
modelled structure, proving the validity of the original assumptions. The elasto-plastic-damage
model presented here is a very significant advancement over the elasto-plastic formulation.
The computational cost of the performed calculations using the proposed formulation is
much lower than in the case of the shell elements with a layered approach. In order to
substantiate this statement with appropriate data, we compare the number of floating point
operations (FLOPS) per iteration and total CPU time necessary to solve the above problem
by means of 800 elements based on the current elasto-plastic formulation (without damage),
and a layered approach with Gaussian Quadrature method used for numerical integration of
the stiffness matrix. The layered method with 10 layers across the thickness of each element
and reduced-selective integration requires 8.96 × 107 FLOPS per iteration and the total CPU
time of 38.44 s. The current model requires 8.04 × 107 FLOPS per iteration and the total
CPU time of 32.38 s. This means that, based on the current example, the current formulation
is about 16% cheaper than a layered method with implicit stiffness matrix. The savings of
computational effort are even more significant in the case of elasto-plastic-damage model. This
is due to the explicit form of the stiffness matrix, and the application of the single load surface
with the damage variable incorporated. It is noteworthy that a three-dimensional analysis with
solid elements would be even more expensive. In problems with a complicated geometry, the
computational cost of the finite element procedure may be decisive.
8. CONCLUSIONS
The current work presents a mathematically consistent finite element model for the elasto-plastic,
large rotation analysis of thin/thick shells incorporating the influence of damage due to microvoids. An accurate set of thick/thin shell constitutive equations [104], previously developed
by the first two authors is adopted here, along with its finite element implementation [9].
The non-layered yield surface with the stress-resultant-based kinematic hardening rule is used
in this work in order to model the elasto-plastic behaviour of shells, including the Bauschinger
effect. Iliushin’s yield function expressed in terms of stress resultants and couples, modified to
account for the progressive development of the plastic deformation and transverse shear forces
is used. Similar to the case of elastic considerations all the integrals are calculated analytically,
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P. WOELKE, G. Z. VOYIADJIS AND P. PERZYNA
which makes the current formulation extremely efficient as the numerical integration is not
employed at any stage of the computations. The tangent stiffness matrix is obtained explicitly.
Large rotations and displacements that are often associated with inelastic deformation are
described using the Updated Lagrangian method.
The most important and novel feature of this paper is a simple and convenient, yet accurate
description of the damage evolution in plates and shells. Since this work concerns the study
of thick, homogenous isotropic and ductile shells, damage is modelled here as an isotropic,
rate-independent process, caused by the growth of microvoids only. This can be regarded as
a limitation of the current model, since the influence of nucleation due to microcracks is
very important for certain applications. However, the current model is based on the evolution
of porosity defined by Duszek-Perzyna and Perzyna [76], who reported excellent results in
modelling ductile metals. According to Reference [76], the influence of microcracks is very
important when analysing metal matrix composites because of the cracking of the reinforcing
fibres. In the case of homogenous and isotropic shells, the void growth is decisive, and thus it
is the only damage-causing phenomenon described in this work. The evolution of porosity is
introduced into the yield function leading to a strong coupling between plasticity and damage.
The initial porosity is evolving due to the presence of the inelastic strains, which means that
the elastic damage is disregarded in this work.
Only two additional material parameters need to be determined to account for damage here,
as opposed to some higher-order approximations, where sometimes tremendous experimental
data are necessary to calibrate all the required material constants. This would be the case if,
for example, a second-order damage tensor were used. Furthermore, while a more advanced
procedure would be needed to model the elasto-plastic and damage behaviour in anisotropic
materials, the accuracy of the current analysis from the point of view of practical structural
analysis is satisfactory.
The reliability of the presented concepts was evaluated through example problems.
Unfortunately, there is very limited amount of data in references regarding the evolution of
damage in plates and shells. Moreover, it is unrealistic to verify the damage formulation
based on comparisons to other results obtained by approximate methods. The robustness of
this algorithm should be tested against experimental results, particularly for the case of damage
characterization. References providing information about damage in structures that are based on
experimental results are even more difficult to find than the numerical estimates. Nevertheless,
based on the limited references as well as the fact that the results presented here show the
expected pattern of the reduction of the stiffness caused by the evolution of damage, it can be
concluded that the current formulation provides extremely valuable information about damage
in plates and shells.
The results of the examples given in Sections 7.1 and 7.2 proved that neglecting damage
in the analysis of plates and shells leads to the overprediction of the ultimate load carried
by the structure. In the example presented in Section 7.2 both the equilibrium path as well
as the plot of porosity versus load showed softening of the structure at a certain load level.
Softening will be followed by localization, fracture and ultimately collapse of the structure.
These phenomena could not be observed if only the elasto-plastic analysis is performed. It is
necessary to account for damage in the model in order to better approximate the behaviour
of the structure after its ultimate load has been reached. Although addressing the problems
of localization and postulating a fracture criterion require considering large strains, and using
viscoplasticity or gradient plasticity theory with damage as a regularization tool, the formulation
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ELASTO-PLASTIC FINITE ELEMENT ANALYSIS OF SHELLS
375
presented in this paper provides a powerful tool for the comprehensive modelling of shells. The
current elasto-plastic-damage model is a very significant advancement over the elasto-plastic
analysis.
This paper is a continuation of the previous work by the first two authors. The original
objective was to formulate a simple and efficient computational model for the detailed analysis
of plates and shells. Traditionally, shell elements follow a layered approach when non-linear
calculations are performed. This is because most of the yield functions featuring reliable
isotropic and kinematic hardening rules and accounting for the damage effects are expressed
in terms of the stresses. A layered approach requires however discretization of the structure
through the thickness, and calculating a yield function for each layer separately. This is conceptually close to the three-dimensional analysis by means of the solid elements, and the use of
the degenerated shell elements loses its advantage. A non-layered method, on the other hand,
seems to be a natural consequence of the shell element development, as the system of non-linear
equations is expressed in terms of forces and bending moments, and solved without discretization of the shell through the thickness. A reliable yield surface expressed in terms of the stress
resultants would allow for the analysis of shells without ‘through-the-thickness-integration’,
which is simpler and much more efficient. It also uses the shell element concepts and shell
constitutive equations, not only for the linear calculations, but also for the elasto-plastic, damage
and geometrically non-linear analysis. The current work provides a veracious loading surface,
defined in the stress resultant space, featuring both isotropic and kinematic hardening rules, as
well as strong coupling between plasticity and the porosity function, describing damage due
to microvoids. The computations are performed without integration through the thickness, and
since the elastic stiffness matrix is explicit, the tangent elasto-plastic-damage stiffness, is also
given explicitly. The validity of the assumptions is verified through the numerical examples
that proved the reliability of the model.
The nature of the equations derived here allows for further enhancement of this algorithm,
leading to the capabilities of solving viscoplastic and dynamic problems.
ACKNOWLEDGEMENTS
The authors acknowledge the financial support under grant number F33601-01-P-0343 provided by
the Air Force Institute of Technology, WPAFB, Ohio.
The authors would like to thank Professor T. Lodygowski for many fruitful discussions and
suggestions. The help of Dr P. Kattan who offered numerous valuable suggestions is also gratefully acknowledged.
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